ABCF->ab-angle b

Percentage Accurate: 19.7% → 45.4%

Time: 23.8s

Alternatives: 21

Speedup: 5.8×

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {B}^{2} - \left(4 \cdot A\right) \cdot C\\ \frac{-\sqrt{\left(2 \cdot \left(t\_0 \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{t\_0} \end{array} \end{array} \]

FPCore

(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (pow B 2.0) (* (* 4.0 A) C))))
   (/
    (-
     (sqrt
      (*
       (* 2.0 (* t_0 F))
       (- (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0)))))))
    t_0)))

C

double code(double A, double B, double C, double F) {
	double t_0 = pow(B, 2.0) - ((4.0 * A) * C);
	return -sqrt(((2.0 * (t_0 * F)) * ((A + C) - sqrt((pow((A - C), 2.0) + pow(B, 2.0)))))) / t_0;
}

Fortran

real(8) function code(a, b, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: t_0
    t_0 = (b ** 2.0d0) - ((4.0d0 * a) * c)
    code = -sqrt(((2.0d0 * (t_0 * f)) * ((a + c) - sqrt((((a - c) ** 2.0d0) + (b ** 2.0d0)))))) / t_0
end function

Java

public static double code(double A, double B, double C, double F) {
	double t_0 = Math.pow(B, 2.0) - ((4.0 * A) * C);
	return -Math.sqrt(((2.0 * (t_0 * F)) * ((A + C) - Math.sqrt((Math.pow((A - C), 2.0) + Math.pow(B, 2.0)))))) / t_0;
}

Python

def code(A, B, C, F):
	t_0 = math.pow(B, 2.0) - ((4.0 * A) * C)
	return -math.sqrt(((2.0 * (t_0 * F)) * ((A + C) - math.sqrt((math.pow((A - C), 2.0) + math.pow(B, 2.0)))))) / t_0

Julia

function code(A, B, C, F)
	t_0 = Float64((B ^ 2.0) - Float64(Float64(4.0 * A) * C))
	return Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(t_0 * F)) * Float64(Float64(A + C) - sqrt(Float64((Float64(A - C) ^ 2.0) + (B ^ 2.0))))))) / t_0)
end

MATLAB

function tmp = code(A, B, C, F)
	t_0 = (B ^ 2.0) - ((4.0 * A) * C);
	tmp = -sqrt(((2.0 * (t_0 * F)) * ((A + C) - sqrt((((A - C) ^ 2.0) + (B ^ 2.0)))))) / t_0;
end

Wolfram

code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[Power[B, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, N[((-N[Sqrt[N[(N[(2.0 * N[(t$95$0 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision]]

Initial Program: 19.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {B}^{2} - \left(4 \cdot A\right) \cdot C\\ \frac{-\sqrt{\left(2 \cdot \left(t\_0 \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{t\_0} \end{array} \end{array} \]

FPCore

(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (pow B 2.0) (* (* 4.0 A) C))))
   (/
    (-
     (sqrt
      (*
       (* 2.0 (* t_0 F))
       (- (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0)))))))
    t_0)))

C

double code(double A, double B, double C, double F) {
	double t_0 = pow(B, 2.0) - ((4.0 * A) * C);
	return -sqrt(((2.0 * (t_0 * F)) * ((A + C) - sqrt((pow((A - C), 2.0) + pow(B, 2.0)))))) / t_0;
}

Fortran

real(8) function code(a, b, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: t_0
    t_0 = (b ** 2.0d0) - ((4.0d0 * a) * c)
    code = -sqrt(((2.0d0 * (t_0 * f)) * ((a + c) - sqrt((((a - c) ** 2.0d0) + (b ** 2.0d0)))))) / t_0
end function

Java

public static double code(double A, double B, double C, double F) {
	double t_0 = Math.pow(B, 2.0) - ((4.0 * A) * C);
	return -Math.sqrt(((2.0 * (t_0 * F)) * ((A + C) - Math.sqrt((Math.pow((A - C), 2.0) + Math.pow(B, 2.0)))))) / t_0;
}

Python

def code(A, B, C, F):
	t_0 = math.pow(B, 2.0) - ((4.0 * A) * C)
	return -math.sqrt(((2.0 * (t_0 * F)) * ((A + C) - math.sqrt((math.pow((A - C), 2.0) + math.pow(B, 2.0)))))) / t_0

Julia

function code(A, B, C, F)
	t_0 = Float64((B ^ 2.0) - Float64(Float64(4.0 * A) * C))
	return Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(t_0 * F)) * Float64(Float64(A + C) - sqrt(Float64((Float64(A - C) ^ 2.0) + (B ^ 2.0))))))) / t_0)
end

MATLAB

function tmp = code(A, B, C, F)
	t_0 = (B ^ 2.0) - ((4.0 * A) * C);
	tmp = -sqrt(((2.0 * (t_0 * F)) * ((A + C) - sqrt((((A - C) ^ 2.0) + (B ^ 2.0)))))) / t_0;
end

Wolfram

code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[Power[B, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, N[((-N[Sqrt[N[(N[(2.0 * N[(t$95$0 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[B, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision]]

Alternative 1: 45.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := A - \mathsf{hypot}\left(A, B\_m\right)\\ t_1 := \mathsf{hypot}\left(B\_m, A - C\right)\\ t_2 := \left(4 \cdot A\right) \cdot C\\ t_3 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_2\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_2 - {B\_m}^{2}}\\ t_4 := t\_2 - B\_m \cdot B\_m\\ t_5 := C \cdot \left(A \cdot -4\right)\\ t_6 := B\_m \cdot B\_m + t\_5\\ t_7 := B\_m \cdot B\_m + A \cdot A\\ t_8 := \sqrt{\frac{1}{t\_7}}\\ t_9 := A \cdot t\_8 + 1\\ \mathbf{if}\;t\_3 \leq -2 \cdot 10^{-79}:\\ \;\;\;\;\frac{\sqrt{2 \cdot t\_6} \cdot \sqrt{F \cdot \left(A + \left(C - t\_1\right)\right)}}{t\_4}\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\left(F \cdot \left(B\_m \cdot B\_m\right)\right) \cdot t\_0\right) + C \cdot \left(2 \cdot \left(\left(C \cdot F\right) \cdot \left(\left(A \cdot -4\right) \cdot t\_9 + -0.5 \cdot \left(\left(B\_m \cdot B\_m\right) \cdot \left(t\_8 \cdot \left(1 - \frac{A \cdot A}{t\_7}\right)\right)\right)\right) + F \cdot \left(t\_0 \cdot \left(A \cdot -4\right) + \left(B\_m \cdot B\_m\right) \cdot t\_9\right)\right)\right)}}{t\_4}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(A + C, t\_6 \cdot \left(2 \cdot F\right), t\_1 \cdot \left(\left(2 \cdot F\right) \cdot \left(\left(0 - B\_m \cdot B\_m\right) - t\_5\right)\right)\right)}}{t\_4}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(2 \cdot \left(F \cdot \left(C - \mathsf{hypot}\left(B\_m, C\right)\right)\right)\right)}^{0.5}}{0 - B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (- A (hypot A B_m)))
        (t_1 (hypot B_m (- A C)))
        (t_2 (* (* 4.0 A) C))
        (t_3
         (/
          (sqrt
           (*
            (* 2.0 (* (- (pow B_m 2.0) t_2) F))
            (- (+ A C) (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))))))
          (- t_2 (pow B_m 2.0))))
        (t_4 (- t_2 (* B_m B_m)))
        (t_5 (* C (* A -4.0)))
        (t_6 (+ (* B_m B_m) t_5))
        (t_7 (+ (* B_m B_m) (* A A)))
        (t_8 (sqrt (/ 1.0 t_7)))
        (t_9 (+ (* A t_8) 1.0)))
   (if (<= t_3 -2e-79)
     (/ (* (sqrt (* 2.0 t_6)) (sqrt (* F (+ A (- C t_1))))) t_4)
     (if (<= t_3 0.0)
       (/
        (sqrt
         (+
          (* 2.0 (* (* F (* B_m B_m)) t_0))
          (*
           C
           (*
            2.0
            (+
             (*
              (* C F)
              (+
               (* (* A -4.0) t_9)
               (* -0.5 (* (* B_m B_m) (* t_8 (- 1.0 (/ (* A A) t_7)))))))
             (* F (+ (* t_0 (* A -4.0)) (* (* B_m B_m) t_9))))))))
        t_4)
       (if (<= t_3 INFINITY)
         (/
          (sqrt
           (fma
            (+ A C)
            (* t_6 (* 2.0 F))
            (* t_1 (* (* 2.0 F) (- (- 0.0 (* B_m B_m)) t_5)))))
          t_4)
         (/ (pow (* 2.0 (* F (- C (hypot B_m C)))) 0.5) (- 0.0 B_m)))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double t_0 = A - hypot(A, B_m);
	double t_1 = hypot(B_m, (A - C));
	double t_2 = (4.0 * A) * C;
	double t_3 = sqrt(((2.0 * ((pow(B_m, 2.0) - t_2) * F)) * ((A + C) - sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / (t_2 - pow(B_m, 2.0));
	double t_4 = t_2 - (B_m * B_m);
	double t_5 = C * (A * -4.0);
	double t_6 = (B_m * B_m) + t_5;
	double t_7 = (B_m * B_m) + (A * A);
	double t_8 = sqrt((1.0 / t_7));
	double t_9 = (A * t_8) + 1.0;
	double tmp;
	if (t_3 <= -2e-79) {
		tmp = (sqrt((2.0 * t_6)) * sqrt((F * (A + (C - t_1))))) / t_4;
	} else if (t_3 <= 0.0) {
		tmp = sqrt(((2.0 * ((F * (B_m * B_m)) * t_0)) + (C * (2.0 * (((C * F) * (((A * -4.0) * t_9) + (-0.5 * ((B_m * B_m) * (t_8 * (1.0 - ((A * A) / t_7))))))) + (F * ((t_0 * (A * -4.0)) + ((B_m * B_m) * t_9)))))))) / t_4;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt(fma((A + C), (t_6 * (2.0 * F)), (t_1 * ((2.0 * F) * ((0.0 - (B_m * B_m)) - t_5))))) / t_4;
	} else {
		tmp = pow((2.0 * (F * (C - hypot(B_m, C)))), 0.5) / (0.0 - B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	t_0 = Float64(A - hypot(A, B_m))
	t_1 = hypot(B_m, Float64(A - C))
	t_2 = Float64(Float64(4.0 * A) * C)
	t_3 = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_2) * F)) * Float64(Float64(A + C) - sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / Float64(t_2 - (B_m ^ 2.0)))
	t_4 = Float64(t_2 - Float64(B_m * B_m))
	t_5 = Float64(C * Float64(A * -4.0))
	t_6 = Float64(Float64(B_m * B_m) + t_5)
	t_7 = Float64(Float64(B_m * B_m) + Float64(A * A))
	t_8 = sqrt(Float64(1.0 / t_7))
	t_9 = Float64(Float64(A * t_8) + 1.0)
	tmp = 0.0
	if (t_3 <= -2e-79)
		tmp = Float64(Float64(sqrt(Float64(2.0 * t_6)) * sqrt(Float64(F * Float64(A + Float64(C - t_1))))) / t_4);
	elseif (t_3 <= 0.0)
		tmp = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64(F * Float64(B_m * B_m)) * t_0)) + Float64(C * Float64(2.0 * Float64(Float64(Float64(C * F) * Float64(Float64(Float64(A * -4.0) * t_9) + Float64(-0.5 * Float64(Float64(B_m * B_m) * Float64(t_8 * Float64(1.0 - Float64(Float64(A * A) / t_7))))))) + Float64(F * Float64(Float64(t_0 * Float64(A * -4.0)) + Float64(Float64(B_m * B_m) * t_9)))))))) / t_4);
	elseif (t_3 <= Inf)
		tmp = Float64(sqrt(fma(Float64(A + C), Float64(t_6 * Float64(2.0 * F)), Float64(t_1 * Float64(Float64(2.0 * F) * Float64(Float64(0.0 - Float64(B_m * B_m)) - t_5))))) / t_4);
	else
		tmp = Float64((Float64(2.0 * Float64(F * Float64(C - hypot(B_m, C)))) ^ 0.5) / Float64(0.0 - B_m));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(A - N[Sqrt[A ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]}, Block[{t$95$2 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$2), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$2 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$2 - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(N[(B$95$m * B$95$m), $MachinePrecision] + t$95$5), $MachinePrecision]}, Block[{t$95$7 = N[(N[(B$95$m * B$95$m), $MachinePrecision] + N[(A * A), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$8 = N[Sqrt[N[(1.0 / t$95$7), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$9 = N[(N[(A * t$95$8), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[t$95$3, -2e-79], N[(N[(N[Sqrt[N[(2.0 * t$95$6), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(F * N[(A + N[(C - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$4), $MachinePrecision], If[LessEqual[t$95$3, 0.0], N[(N[Sqrt[N[(N[(2.0 * N[(N[(F * N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(C * N[(2.0 * N[(N[(N[(C * F), $MachinePrecision] * N[(N[(N[(A * -4.0), $MachinePrecision] * t$95$9), $MachinePrecision] + N[(-0.5 * N[(N[(B$95$m * B$95$m), $MachinePrecision] * N[(t$95$8 * N[(1.0 - N[(N[(A * A), $MachinePrecision] / t$95$7), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(F * N[(N[(t$95$0 * N[(A * -4.0), $MachinePrecision]), $MachinePrecision] + N[(N[(B$95$m * B$95$m), $MachinePrecision] * t$95$9), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$4), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(N[Sqrt[N[(N[(A + C), $MachinePrecision] * N[(t$95$6 * N[(2.0 * F), $MachinePrecision]), $MachinePrecision] + N[(t$95$1 * N[(N[(2.0 * F), $MachinePrecision] * N[(N[(0.0 - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision] - t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$4), $MachinePrecision], N[(N[Power[N[(2.0 * N[(F * N[(C - N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision] / N[(0.0 - B$95$m), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -2e-79

   1. Initial program 39.6%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Step-by-step derivation

      1. distribute-frac-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\right) \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\color{blue}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right), \color{blue}{\left(\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)\right)}\right) \]

   3. Simplified 43.8%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(B \cdot B + C \cdot \left(A \cdot -4\right)\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
   4. Add Preprocessing

   6. Applied egg-rr 68.1%

      \[\leadsto \frac{\color{blue}{\sqrt{2 \cdot \left(B \cdot B + C \cdot \left(A \cdot -4\right)\right)} \cdot \sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   if -2e-79 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < 0.0

   1. Initial program 22.9%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Step-by-step derivation

      1. distribute-frac-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\right) \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\color{blue}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right), \color{blue}{\left(\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)\right)}\right) \]

   3. Simplified 22.9%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(B \cdot B + C \cdot \left(A \cdot -4\right)\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
   4. Add Preprocessing

   5. Taylor expanded in C around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(2 \cdot \left({B}^{2} \cdot \left(F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)\right)\right) + C \cdot \left(2 \cdot \left(C \cdot \left(F \cdot \left(-4 \cdot \left(A \cdot \left(1 - -1 \cdot \left(A \cdot \sqrt{\frac{1}{{A}^{2} + {B}^{2}}}\right)\right)\right) + \frac{-1}{2} \cdot \left(\left({B}^{2} \cdot \left(1 - \frac{{A}^{2}}{{A}^{2} + {B}^{2}}\right)\right) \cdot \sqrt{\frac{1}{{A}^{2} + {B}^{2}}}\right)\right)\right)\right) + 2 \cdot \left(F \cdot \left(-4 \cdot \left(A \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)\right) + {B}^{2} \cdot \left(1 - -1 \cdot \left(A \cdot \sqrt{\frac{1}{{A}^{2} + {B}^{2}}}\right)\right)\right)\right)\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 45.7%

      \[\leadsto \frac{\sqrt{\color{blue}{2 \cdot \left(\left(\left(B \cdot B\right) \cdot F\right) \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)\right) + C \cdot \left(2 \cdot \left(\left(C \cdot F\right) \cdot \left(\left(-4 \cdot A\right) \cdot \left(1 - \left(-A\right) \cdot \sqrt{\frac{1}{B \cdot B + A \cdot A}}\right) + -0.5 \cdot \left(\left(B \cdot B\right) \cdot \left(\left(1 - \frac{A \cdot A}{B \cdot B + A \cdot A}\right) \cdot \sqrt{\frac{1}{B \cdot B + A \cdot A}}\right)\right)\right) + F \cdot \left(\left(-4 \cdot A\right) \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right) + \left(B \cdot B\right) \cdot \left(1 - \left(-A\right) \cdot \sqrt{\frac{1}{B \cdot B + A \cdot A}}\right)\right)\right)\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   if 0.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < +inf.0

   1. Initial program 39.4%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Step-by-step derivation

      1. distribute-frac-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\right) \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\color{blue}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right), \color{blue}{\left(\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)\right)}\right) \]

   3. Simplified 75.2%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(B \cdot B + C \cdot \left(A \cdot -4\right)\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(\left(B \cdot B + C \cdot \left(A \cdot -4\right)\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \left(\mathsf{neg}\left(\sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(\left(B \cdot B + C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot 2\right)\right) \cdot \left(\left(A + C\right) + \left(\mathsf{neg}\left(\sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(\left(\left(B \cdot B + C \cdot \left(A \cdot -4\right)\right) \cdot F\right) \cdot 2\right) \cdot \left(\left(A + C\right) + \left(\mathsf{neg}\left(\sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(\left(\left(B \cdot B + C \cdot \left(-4 \cdot A\right)\right) \cdot F\right) \cdot 2\right) \cdot \left(\left(A + C\right) + \left(\mathsf{neg}\left(\sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(\left(\left(B \cdot B + C \cdot \left(\left(\mathsf{neg}\left(4\right)\right) \cdot A\right)\right) \cdot F\right) \cdot 2\right) \cdot \left(\left(A + C\right) + \left(\mathsf{neg}\left(\sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. distribute-lft-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(\left(\left(B \cdot B + C \cdot \left(\mathsf{neg}\left(4 \cdot A\right)\right)\right) \cdot F\right) \cdot 2\right) \cdot \left(\left(A + C\right) + \left(\mathsf{neg}\left(\sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      7. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(\left(\left(B \cdot B + \left(\mathsf{neg}\left(C \cdot \left(4 \cdot A\right)\right)\right)\right) \cdot F\right) \cdot 2\right) \cdot \left(\left(A + C\right) + \left(\mathsf{neg}\left(\sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(\left(\left(B \cdot B + \left(\mathsf{neg}\left(\left(4 \cdot A\right) \cdot C\right)\right)\right) \cdot F\right) \cdot 2\right) \cdot \left(\left(A + C\right) + \left(\mathsf{neg}\left(\sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(\left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right) \cdot 2\right) \cdot \left(\left(A + C\right) + \left(\mathsf{neg}\left(\sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      10. pow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right) \cdot 2\right) \cdot \left(\left(A + C\right) + \left(\mathsf{neg}\left(\sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \left(\mathsf{neg}\left(\sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   6. Applied egg-rr 75.3%

      \[\leadsto \frac{\sqrt{\color{blue}{\mathsf{fma}\left(A + C, \left(B \cdot B + C \cdot \left(A \cdot -4\right)\right) \cdot \left(2 \cdot F\right), \left(0 - \mathsf{hypot}\left(B, A - C\right)\right) \cdot \left(\left(B \cdot B + C \cdot \left(A \cdot -4\right)\right) \cdot \left(2 \cdot F\right)\right)\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   if +inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)))

   1. Initial program 0.0%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \color{blue}{\sqrt{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(-1 \cdot \frac{\sqrt{2}}{B}\right), \color{blue}{\left(\sqrt{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \left(\frac{\sqrt{2}}{B}\right)\right), \left(\sqrt{\color{blue}{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\left(\sqrt{2}\right), B\right)\right), \left(\sqrt{F \cdot \color{blue}{\left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \left(\sqrt{F \cdot \left(\color{blue}{C} - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\left(F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\sqrt{B \cdot B + {C}^{2}}\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\sqrt{B \cdot B + C \cdot C}\right)\right)\right)\right)\right) \]

      11. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\mathsf{hypot}\left(B, C\right)\right)\right)\right)\right)\right) \]

      12. hypot-lowering-hypot.f64 23.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \mathsf{hypot.f64}\left(B, C\right)\right)\right)\right)\right) \]

   5. Simplified 23.4%

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(C - \mathsf{hypot}\left(B, C\right)\right)}} \]
   6. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto -1 \cdot \color{blue}{\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)}\right)} \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)}\right) \]

      3. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)}\right)\right) \]

      4. associate-*l/ N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)}}{B}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\left(\sqrt{2} \cdot \sqrt{F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)}\right), B\right)\right) \]

   7. Applied egg-rr 23.5%

      \[\leadsto \color{blue}{-\frac{{\left(2 \cdot \left(F \cdot \left(C - \mathsf{hypot}\left(B, C\right)\right)\right)\right)}^{0.5}}{B}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 46.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -2 \cdot 10^{-79}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(B \cdot B + C \cdot \left(A \cdot -4\right)\right)} \cdot \sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq 0:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\left(F \cdot \left(B \cdot B\right)\right) \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)\right) + C \cdot \left(2 \cdot \left(\left(C \cdot F\right) \cdot \left(\left(A \cdot -4\right) \cdot \left(A \cdot \sqrt{\frac{1}{B \cdot B + A \cdot A}} + 1\right) + -0.5 \cdot \left(\left(B \cdot B\right) \cdot \left(\sqrt{\frac{1}{B \cdot B + A \cdot A}} \cdot \left(1 - \frac{A \cdot A}{B \cdot B + A \cdot A}\right)\right)\right)\right) + F \cdot \left(\left(A - \mathsf{hypot}\left(A, B\right)\right) \cdot \left(A \cdot -4\right) + \left(B \cdot B\right) \cdot \left(A \cdot \sqrt{\frac{1}{B \cdot B + A \cdot A}} + 1\right)\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(A + C, \left(B \cdot B + C \cdot \left(A \cdot -4\right)\right) \cdot \left(2 \cdot F\right), \mathsf{hypot}\left(B, A - C\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(\left(0 - B \cdot B\right) - C \cdot \left(A \cdot -4\right)\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(2 \cdot \left(F \cdot \left(C - \mathsf{hypot}\left(B, C\right)\right)\right)\right)}^{0.5}}{0 - B}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 46.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := C \cdot \left(A \cdot -4\right)\\ t_1 := B\_m \cdot B\_m + t\_0\\ t_2 := \mathsf{hypot}\left(B\_m, A - C\right)\\ t_3 := \left(4 \cdot A\right) \cdot C\\ t_4 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_3\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_3 - {B\_m}^{2}}\\ t_5 := t\_3 - B\_m \cdot B\_m\\ \mathbf{if}\;t\_4 \leq -4 \cdot 10^{-214}:\\ \;\;\;\;\frac{\sqrt{2 \cdot t\_1} \cdot \sqrt{F \cdot \left(A + \left(C - t\_2\right)\right)}}{t\_5}\\ \mathbf{elif}\;t\_4 \leq 0:\\ \;\;\;\;\frac{\sqrt{t\_1 \cdot \left(\frac{F \cdot \left(B\_m \cdot B\_m\right)}{C - A} + 4 \cdot \left(C \cdot F\right)\right)}}{t\_5}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(A + C, t\_1 \cdot \left(2 \cdot F\right), t\_2 \cdot \left(\left(2 \cdot F\right) \cdot \left(\left(0 - B\_m \cdot B\_m\right) - t\_0\right)\right)\right)}}{t\_5}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(2 \cdot \left(F \cdot \left(C - \mathsf{hypot}\left(B\_m, C\right)\right)\right)\right)}^{0.5}}{0 - B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (* C (* A -4.0)))
        (t_1 (+ (* B_m B_m) t_0))
        (t_2 (hypot B_m (- A C)))
        (t_3 (* (* 4.0 A) C))
        (t_4
         (/
          (sqrt
           (*
            (* 2.0 (* (- (pow B_m 2.0) t_3) F))
            (- (+ A C) (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))))))
          (- t_3 (pow B_m 2.0))))
        (t_5 (- t_3 (* B_m B_m))))
   (if (<= t_4 -4e-214)
     (/ (* (sqrt (* 2.0 t_1)) (sqrt (* F (+ A (- C t_2))))) t_5)
     (if (<= t_4 0.0)
       (/ (sqrt (* t_1 (+ (/ (* F (* B_m B_m)) (- C A)) (* 4.0 (* C F))))) t_5)
       (if (<= t_4 INFINITY)
         (/
          (sqrt
           (fma
            (+ A C)
            (* t_1 (* 2.0 F))
            (* t_2 (* (* 2.0 F) (- (- 0.0 (* B_m B_m)) t_0)))))
          t_5)
         (/ (pow (* 2.0 (* F (- C (hypot B_m C)))) 0.5) (- 0.0 B_m)))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double t_0 = C * (A * -4.0);
	double t_1 = (B_m * B_m) + t_0;
	double t_2 = hypot(B_m, (A - C));
	double t_3 = (4.0 * A) * C;
	double t_4 = sqrt(((2.0 * ((pow(B_m, 2.0) - t_3) * F)) * ((A + C) - sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / (t_3 - pow(B_m, 2.0));
	double t_5 = t_3 - (B_m * B_m);
	double tmp;
	if (t_4 <= -4e-214) {
		tmp = (sqrt((2.0 * t_1)) * sqrt((F * (A + (C - t_2))))) / t_5;
	} else if (t_4 <= 0.0) {
		tmp = sqrt((t_1 * (((F * (B_m * B_m)) / (C - A)) + (4.0 * (C * F))))) / t_5;
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = sqrt(fma((A + C), (t_1 * (2.0 * F)), (t_2 * ((2.0 * F) * ((0.0 - (B_m * B_m)) - t_0))))) / t_5;
	} else {
		tmp = pow((2.0 * (F * (C - hypot(B_m, C)))), 0.5) / (0.0 - B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	t_0 = Float64(C * Float64(A * -4.0))
	t_1 = Float64(Float64(B_m * B_m) + t_0)
	t_2 = hypot(B_m, Float64(A - C))
	t_3 = Float64(Float64(4.0 * A) * C)
	t_4 = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_3) * F)) * Float64(Float64(A + C) - sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / Float64(t_3 - (B_m ^ 2.0)))
	t_5 = Float64(t_3 - Float64(B_m * B_m))
	tmp = 0.0
	if (t_4 <= -4e-214)
		tmp = Float64(Float64(sqrt(Float64(2.0 * t_1)) * sqrt(Float64(F * Float64(A + Float64(C - t_2))))) / t_5);
	elseif (t_4 <= 0.0)
		tmp = Float64(sqrt(Float64(t_1 * Float64(Float64(Float64(F * Float64(B_m * B_m)) / Float64(C - A)) + Float64(4.0 * Float64(C * F))))) / t_5);
	elseif (t_4 <= Inf)
		tmp = Float64(sqrt(fma(Float64(A + C), Float64(t_1 * Float64(2.0 * F)), Float64(t_2 * Float64(Float64(2.0 * F) * Float64(Float64(0.0 - Float64(B_m * B_m)) - t_0))))) / t_5);
	else
		tmp = Float64((Float64(2.0 * Float64(F * Float64(C - hypot(B_m, C)))) ^ 0.5) / Float64(0.0 - B_m));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(B$95$m * B$95$m), $MachinePrecision] + t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]}, Block[{t$95$3 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$3), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$3 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$3 - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -4e-214], N[(N[(N[Sqrt[N[(2.0 * t$95$1), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(F * N[(A + N[(C - t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$5), $MachinePrecision], If[LessEqual[t$95$4, 0.0], N[(N[Sqrt[N[(t$95$1 * N[(N[(N[(F * N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision] / N[(C - A), $MachinePrecision]), $MachinePrecision] + N[(4.0 * N[(C * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$5), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[(N[Sqrt[N[(N[(A + C), $MachinePrecision] * N[(t$95$1 * N[(2.0 * F), $MachinePrecision]), $MachinePrecision] + N[(t$95$2 * N[(N[(2.0 * F), $MachinePrecision] * N[(N[(0.0 - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision] - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$5), $MachinePrecision], N[(N[Power[N[(2.0 * N[(F * N[(C - N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision] / N[(0.0 - B$95$m), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -3.99999999999999965e-214

   1. Initial program 45.9%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Step-by-step derivation

      1. distribute-frac-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\right) \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\color{blue}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right), \color{blue}{\left(\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)\right)}\right) \]

   3. Simplified 49.6%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(B \cdot B + C \cdot \left(A \cdot -4\right)\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
   4. Add Preprocessing

   6. Applied egg-rr 71.1%

      \[\leadsto \frac{\color{blue}{\sqrt{2 \cdot \left(B \cdot B + C \cdot \left(A \cdot -4\right)\right)} \cdot \sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   if -3.99999999999999965e-214 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < 0.0

   1. Initial program 3.4%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Step-by-step derivation

      1. distribute-frac-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\right) \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\color{blue}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right), \color{blue}{\left(\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)\right)}\right) \]

   3. Simplified 3.4%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(B \cdot B + C \cdot \left(A \cdot -4\right)\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(B \cdot B + C \cdot \left(A \cdot -4\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(\left(A + C\right) - \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(B \cdot B + C \cdot \left(-4 \cdot A\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(\left(A + C\right) - \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(B \cdot B + C \cdot \left(\left(\mathsf{neg}\left(4\right)\right) \cdot A\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(\left(A + C\right) - \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. distribute-lft-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(B \cdot B + C \cdot \left(\mathsf{neg}\left(4 \cdot A\right)\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(\left(A + C\right) - \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(B \cdot B + \left(\mathsf{neg}\left(C \cdot \left(4 \cdot A\right)\right)\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(\left(A + C\right) - \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(B \cdot B + \left(\mathsf{neg}\left(\left(4 \cdot A\right) \cdot C\right)\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(\left(A + C\right) - \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(\left(A + C\right) - \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      8. pow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(\left(A + C\right) - \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right), \left(\left(2 \cdot F\right) \cdot \left(\left(A + C\right) - \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   6. Applied egg-rr 12.3%

      \[\leadsto \frac{\sqrt{\color{blue}{\left(B \cdot B + C \cdot \left(A \cdot -4\right)\right) \cdot \left(2 \cdot \left(F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   7. Taylor expanded in B around 0

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(C, \mathsf{*.f64}\left(A, -4\right)\right)\right), \color{blue}{\left(-1 \cdot \frac{{B}^{2} \cdot F}{A - C} + 4 \cdot \left(C \cdot F\right)\right)}\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   8. Step-by-step derivation

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(C, \mathsf{*.f64}\left(A, -4\right)\right)\right), \mathsf{+.f64}\left(\left(-1 \cdot \frac{{B}^{2} \cdot F}{A - C}\right), \left(4 \cdot \left(C \cdot F\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(C, \mathsf{*.f64}\left(A, -4\right)\right)\right), \mathsf{+.f64}\left(\left(\mathsf{neg}\left(\frac{{B}^{2} \cdot F}{A - C}\right)\right), \left(4 \cdot \left(C \cdot F\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(C, \mathsf{*.f64}\left(A, -4\right)\right)\right), \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\left(\frac{{B}^{2} \cdot F}{A - C}\right)\right), \left(4 \cdot \left(C \cdot F\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(C, \mathsf{*.f64}\left(A, -4\right)\right)\right), \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(\left({B}^{2} \cdot F\right), \left(A - C\right)\right)\right), \left(4 \cdot \left(C \cdot F\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(C, \mathsf{*.f64}\left(A, -4\right)\right)\right), \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({B}^{2}\right), F\right), \left(A - C\right)\right)\right), \left(4 \cdot \left(C \cdot F\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(C, \mathsf{*.f64}\left(A, -4\right)\right)\right), \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(B \cdot B\right), F\right), \left(A - C\right)\right)\right), \left(4 \cdot \left(C \cdot F\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(C, \mathsf{*.f64}\left(A, -4\right)\right)\right), \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), F\right), \left(A - C\right)\right)\right), \left(4 \cdot \left(C \cdot F\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(C, \mathsf{*.f64}\left(A, -4\right)\right)\right), \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), F\right), \mathsf{\_.f64}\left(A, C\right)\right)\right), \left(4 \cdot \left(C \cdot F\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(C, \mathsf{*.f64}\left(A, -4\right)\right)\right), \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), F\right), \mathsf{\_.f64}\left(A, C\right)\right)\right), \mathsf{*.f64}\left(4, \left(C \cdot F\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      10. *-lowering-*.f64 31.5%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(C, \mathsf{*.f64}\left(A, -4\right)\right)\right), \mathsf{+.f64}\left(\mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), F\right), \mathsf{\_.f64}\left(A, C\right)\right)\right), \mathsf{*.f64}\left(4, \mathsf{*.f64}\left(C, F\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   9. Simplified 31.5%

      \[\leadsto \frac{\sqrt{\left(B \cdot B + C \cdot \left(A \cdot -4\right)\right) \cdot \color{blue}{\left(\left(-\frac{\left(B \cdot B\right) \cdot F}{A - C}\right) + 4 \cdot \left(C \cdot F\right)\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   if 0.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < +inf.0

   1. Initial program 39.4%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Step-by-step derivation

      1. distribute-frac-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\right) \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\color{blue}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right), \color{blue}{\left(\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)\right)}\right) \]

   3. Simplified 75.2%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(B \cdot B + C \cdot \left(A \cdot -4\right)\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(\left(B \cdot B + C \cdot \left(A \cdot -4\right)\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \left(\mathsf{neg}\left(\sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(\left(B \cdot B + C \cdot \left(A \cdot -4\right)\right) \cdot \left(F \cdot 2\right)\right) \cdot \left(\left(A + C\right) + \left(\mathsf{neg}\left(\sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. associate-*r* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(\left(\left(B \cdot B + C \cdot \left(A \cdot -4\right)\right) \cdot F\right) \cdot 2\right) \cdot \left(\left(A + C\right) + \left(\mathsf{neg}\left(\sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(\left(\left(B \cdot B + C \cdot \left(-4 \cdot A\right)\right) \cdot F\right) \cdot 2\right) \cdot \left(\left(A + C\right) + \left(\mathsf{neg}\left(\sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(\left(\left(B \cdot B + C \cdot \left(\left(\mathsf{neg}\left(4\right)\right) \cdot A\right)\right) \cdot F\right) \cdot 2\right) \cdot \left(\left(A + C\right) + \left(\mathsf{neg}\left(\sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. distribute-lft-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(\left(\left(B \cdot B + C \cdot \left(\mathsf{neg}\left(4 \cdot A\right)\right)\right) \cdot F\right) \cdot 2\right) \cdot \left(\left(A + C\right) + \left(\mathsf{neg}\left(\sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      7. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(\left(\left(B \cdot B + \left(\mathsf{neg}\left(C \cdot \left(4 \cdot A\right)\right)\right)\right) \cdot F\right) \cdot 2\right) \cdot \left(\left(A + C\right) + \left(\mathsf{neg}\left(\sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(\left(\left(B \cdot B + \left(\mathsf{neg}\left(\left(4 \cdot A\right) \cdot C\right)\right)\right) \cdot F\right) \cdot 2\right) \cdot \left(\left(A + C\right) + \left(\mathsf{neg}\left(\sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(\left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot F\right) \cdot 2\right) \cdot \left(\left(A + C\right) + \left(\mathsf{neg}\left(\sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      10. pow2 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right) \cdot 2\right) \cdot \left(\left(A + C\right) + \left(\mathsf{neg}\left(\sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \left(\mathsf{neg}\left(\sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   6. Applied egg-rr 75.3%

      \[\leadsto \frac{\sqrt{\color{blue}{\mathsf{fma}\left(A + C, \left(B \cdot B + C \cdot \left(A \cdot -4\right)\right) \cdot \left(2 \cdot F\right), \left(0 - \mathsf{hypot}\left(B, A - C\right)\right) \cdot \left(\left(B \cdot B + C \cdot \left(A \cdot -4\right)\right) \cdot \left(2 \cdot F\right)\right)\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   if +inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)))

   1. Initial program 0.0%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \color{blue}{\sqrt{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(-1 \cdot \frac{\sqrt{2}}{B}\right), \color{blue}{\left(\sqrt{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \left(\frac{\sqrt{2}}{B}\right)\right), \left(\sqrt{\color{blue}{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\left(\sqrt{2}\right), B\right)\right), \left(\sqrt{F \cdot \color{blue}{\left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \left(\sqrt{F \cdot \left(\color{blue}{C} - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\left(F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\sqrt{B \cdot B + {C}^{2}}\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\sqrt{B \cdot B + C \cdot C}\right)\right)\right)\right)\right) \]

      11. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\mathsf{hypot}\left(B, C\right)\right)\right)\right)\right)\right) \]

      12. hypot-lowering-hypot.f64 23.4%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \mathsf{hypot.f64}\left(B, C\right)\right)\right)\right)\right) \]

   5. Simplified 23.4%

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(C - \mathsf{hypot}\left(B, C\right)\right)}} \]
   6. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto -1 \cdot \color{blue}{\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)}\right)} \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)}\right) \]

      3. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)}\right)\right) \]

      4. associate-*l/ N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)}}{B}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\left(\sqrt{2} \cdot \sqrt{F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)}\right), B\right)\right) \]

   7. Applied egg-rr 23.5%

      \[\leadsto \color{blue}{-\frac{{\left(2 \cdot \left(F \cdot \left(C - \mathsf{hypot}\left(B, C\right)\right)\right)\right)}^{0.5}}{B}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 46.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -4 \cdot 10^{-214}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(B \cdot B + C \cdot \left(A \cdot -4\right)\right)} \cdot \sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq 0:\\ \;\;\;\;\frac{\sqrt{\left(B \cdot B + C \cdot \left(A \cdot -4\right)\right) \cdot \left(\frac{F \cdot \left(B \cdot B\right)}{C - A} + 4 \cdot \left(C \cdot F\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(A + C, \left(B \cdot B + C \cdot \left(A \cdot -4\right)\right) \cdot \left(2 \cdot F\right), \mathsf{hypot}\left(B, A - C\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(\left(0 - B \cdot B\right) - C \cdot \left(A \cdot -4\right)\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(2 \cdot \left(F \cdot \left(C - \mathsf{hypot}\left(B, C\right)\right)\right)\right)}^{0.5}}{0 - B}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 39.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 10^{+118}:\\ \;\;\;\;\frac{\sqrt{\left(B\_m \cdot B\_m + C \cdot \left(A \cdot -4\right)\right) \cdot \left(2 \cdot \left(F \cdot \left(A + \left(C - \mathsf{hypot}\left(B\_m, A - C\right)\right)\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - B\_m \cdot B\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(2 \cdot \left(F \cdot \left(C - \mathsf{hypot}\left(B\_m, C\right)\right)\right)\right)}^{0.5}}{0 - B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 1e+118)
   (/
    (sqrt
     (*
      (+ (* B_m B_m) (* C (* A -4.0)))
      (* 2.0 (* F (+ A (- C (hypot B_m (- A C))))))))
    (- (* (* 4.0 A) C) (* B_m B_m)))
   (/ (pow (* 2.0 (* F (- C (hypot B_m C)))) 0.5) (- 0.0 B_m))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (pow(B_m, 2.0) <= 1e+118) {
		tmp = sqrt((((B_m * B_m) + (C * (A * -4.0))) * (2.0 * (F * (A + (C - hypot(B_m, (A - C)))))))) / (((4.0 * A) * C) - (B_m * B_m));
	} else {
		tmp = pow((2.0 * (F * (C - hypot(B_m, C)))), 0.5) / (0.0 - B_m);
	}
	return tmp;
}

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (Math.pow(B_m, 2.0) <= 1e+118) {
		tmp = Math.sqrt((((B_m * B_m) + (C * (A * -4.0))) * (2.0 * (F * (A + (C - Math.hypot(B_m, (A - C)))))))) / (((4.0 * A) * C) - (B_m * B_m));
	} else {
		tmp = Math.pow((2.0 * (F * (C - Math.hypot(B_m, C)))), 0.5) / (0.0 - B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if math.pow(B_m, 2.0) <= 1e+118:
		tmp = math.sqrt((((B_m * B_m) + (C * (A * -4.0))) * (2.0 * (F * (A + (C - math.hypot(B_m, (A - C)))))))) / (((4.0 * A) * C) - (B_m * B_m))
	else:
		tmp = math.pow((2.0 * (F * (C - math.hypot(B_m, C)))), 0.5) / (0.0 - B_m)
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e+118)
		tmp = Float64(sqrt(Float64(Float64(Float64(B_m * B_m) + Float64(C * Float64(A * -4.0))) * Float64(2.0 * Float64(F * Float64(A + Float64(C - hypot(B_m, Float64(A - C)))))))) / Float64(Float64(Float64(4.0 * A) * C) - Float64(B_m * B_m)));
	else
		tmp = Float64((Float64(2.0 * Float64(F * Float64(C - hypot(B_m, C)))) ^ 0.5) / Float64(0.0 - B_m));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if ((B_m ^ 2.0) <= 1e+118)
		tmp = sqrt((((B_m * B_m) + (C * (A * -4.0))) * (2.0 * (F * (A + (C - hypot(B_m, (A - C)))))))) / (((4.0 * A) * C) - (B_m * B_m));
	else
		tmp = ((2.0 * (F * (C - hypot(B_m, C)))) ^ 0.5) / (0.0 - B_m);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e+118], N[(N[Sqrt[N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] + N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(F * N[(A + N[(C - N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision] - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(2.0 * N[(F * N[(C - N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision] / N[(0.0 - B$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 9.99999999999999967e117

   1. Initial program 26.7%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Step-by-step derivation

      1. distribute-frac-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\right) \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\color{blue}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right), \color{blue}{\left(\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)\right)}\right) \]

   3. Simplified 34.9%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(B \cdot B + C \cdot \left(A \cdot -4\right)\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(B \cdot B + C \cdot \left(A \cdot -4\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(\left(A + C\right) - \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(B \cdot B + C \cdot \left(-4 \cdot A\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(\left(A + C\right) - \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(B \cdot B + C \cdot \left(\left(\mathsf{neg}\left(4\right)\right) \cdot A\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(\left(A + C\right) - \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. distribute-lft-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(B \cdot B + C \cdot \left(\mathsf{neg}\left(4 \cdot A\right)\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(\left(A + C\right) - \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(B \cdot B + \left(\mathsf{neg}\left(C \cdot \left(4 \cdot A\right)\right)\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(\left(A + C\right) - \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(B \cdot B + \left(\mathsf{neg}\left(\left(4 \cdot A\right) \cdot C\right)\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(\left(A + C\right) - \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(\left(A + C\right) - \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      8. pow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(\left(A + C\right) - \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right), \left(\left(2 \cdot F\right) \cdot \left(\left(A + C\right) - \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   6. Applied egg-rr 37.9%

      \[\leadsto \frac{\sqrt{\color{blue}{\left(B \cdot B + C \cdot \left(A \cdot -4\right)\right) \cdot \left(2 \cdot \left(F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   if 9.99999999999999967e117 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 9.2%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \color{blue}{\sqrt{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(-1 \cdot \frac{\sqrt{2}}{B}\right), \color{blue}{\left(\sqrt{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \left(\frac{\sqrt{2}}{B}\right)\right), \left(\sqrt{\color{blue}{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\left(\sqrt{2}\right), B\right)\right), \left(\sqrt{F \cdot \color{blue}{\left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \left(\sqrt{F \cdot \left(\color{blue}{C} - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\left(F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\sqrt{B \cdot B + {C}^{2}}\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\sqrt{B \cdot B + C \cdot C}\right)\right)\right)\right)\right) \]

      11. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\mathsf{hypot}\left(B, C\right)\right)\right)\right)\right)\right) \]

      12. hypot-lowering-hypot.f64 32.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \mathsf{hypot.f64}\left(B, C\right)\right)\right)\right)\right) \]

   5. Simplified 32.7%

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(C - \mathsf{hypot}\left(B, C\right)\right)}} \]
   6. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto -1 \cdot \color{blue}{\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)}\right)} \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)}\right) \]

      3. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)}\right)\right) \]

      4. associate-*l/ N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)}}{B}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\left(\sqrt{2} \cdot \sqrt{F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)}\right), B\right)\right) \]

   7. Applied egg-rr 32.8%

      \[\leadsto \color{blue}{-\frac{{\left(2 \cdot \left(F \cdot \left(C - \mathsf{hypot}\left(B, C\right)\right)\right)\right)}^{0.5}}{B}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 36.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{+118}:\\ \;\;\;\;\frac{\sqrt{\left(B \cdot B + C \cdot \left(A \cdot -4\right)\right) \cdot \left(2 \cdot \left(F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(2 \cdot \left(F \cdot \left(C - \mathsf{hypot}\left(B, C\right)\right)\right)\right)}^{0.5}}{0 - B}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 39.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 5.3 \cdot 10^{+59}:\\ \;\;\;\;\frac{\sqrt{\left(\left(B\_m \cdot B\_m + C \cdot \left(A \cdot -4\right)\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B\_m, A - C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B\_m \cdot B\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(2 \cdot \left(F \cdot \left(C - \mathsf{hypot}\left(B\_m, C\right)\right)\right)\right)}^{0.5}}{0 - B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 5.3e+59)
   (/
    (sqrt
     (*
      (* (+ (* B_m B_m) (* C (* A -4.0))) (* 2.0 F))
      (- (+ A C) (hypot B_m (- A C)))))
    (- (* (* 4.0 A) C) (* B_m B_m)))
   (/ (pow (* 2.0 (* F (- C (hypot B_m C)))) 0.5) (- 0.0 B_m))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 5.3e+59) {
		tmp = sqrt(((((B_m * B_m) + (C * (A * -4.0))) * (2.0 * F)) * ((A + C) - hypot(B_m, (A - C))))) / (((4.0 * A) * C) - (B_m * B_m));
	} else {
		tmp = pow((2.0 * (F * (C - hypot(B_m, C)))), 0.5) / (0.0 - B_m);
	}
	return tmp;
}

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 5.3e+59) {
		tmp = Math.sqrt(((((B_m * B_m) + (C * (A * -4.0))) * (2.0 * F)) * ((A + C) - Math.hypot(B_m, (A - C))))) / (((4.0 * A) * C) - (B_m * B_m));
	} else {
		tmp = Math.pow((2.0 * (F * (C - Math.hypot(B_m, C)))), 0.5) / (0.0 - B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 5.3e+59:
		tmp = math.sqrt(((((B_m * B_m) + (C * (A * -4.0))) * (2.0 * F)) * ((A + C) - math.hypot(B_m, (A - C))))) / (((4.0 * A) * C) - (B_m * B_m))
	else:
		tmp = math.pow((2.0 * (F * (C - math.hypot(B_m, C)))), 0.5) / (0.0 - B_m)
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 5.3e+59)
		tmp = Float64(sqrt(Float64(Float64(Float64(Float64(B_m * B_m) + Float64(C * Float64(A * -4.0))) * Float64(2.0 * F)) * Float64(Float64(A + C) - hypot(B_m, Float64(A - C))))) / Float64(Float64(Float64(4.0 * A) * C) - Float64(B_m * B_m)));
	else
		tmp = Float64((Float64(2.0 * Float64(F * Float64(C - hypot(B_m, C)))) ^ 0.5) / Float64(0.0 - B_m));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (B_m <= 5.3e+59)
		tmp = sqrt(((((B_m * B_m) + (C * (A * -4.0))) * (2.0 * F)) * ((A + C) - hypot(B_m, (A - C))))) / (((4.0 * A) * C) - (B_m * B_m));
	else
		tmp = ((2.0 * (F * (C - hypot(B_m, C)))) ^ 0.5) / (0.0 - B_m);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 5.3e+59], N[(N[Sqrt[N[(N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] + N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] - N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision] - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(2.0 * N[(F * N[(C - N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision] / N[(0.0 - B$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 5.2999999999999997e59

   1. Initial program 23.5%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Step-by-step derivation

      1. distribute-frac-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\right) \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\color{blue}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right), \color{blue}{\left(\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)\right)}\right) \]

   3. Simplified 30.0%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(B \cdot B + C \cdot \left(A \cdot -4\right)\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
   4. Add Preprocessing

   if 5.2999999999999997e59 < B

   1. Initial program 6.6%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \color{blue}{\sqrt{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(-1 \cdot \frac{\sqrt{2}}{B}\right), \color{blue}{\left(\sqrt{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \left(\frac{\sqrt{2}}{B}\right)\right), \left(\sqrt{\color{blue}{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\left(\sqrt{2}\right), B\right)\right), \left(\sqrt{F \cdot \color{blue}{\left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \left(\sqrt{F \cdot \left(\color{blue}{C} - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\left(F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\sqrt{B \cdot B + {C}^{2}}\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\sqrt{B \cdot B + C \cdot C}\right)\right)\right)\right)\right) \]

      11. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\mathsf{hypot}\left(B, C\right)\right)\right)\right)\right)\right) \]

      12. hypot-lowering-hypot.f64 60.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \mathsf{hypot.f64}\left(B, C\right)\right)\right)\right)\right) \]

   5. Simplified 60.3%

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(C - \mathsf{hypot}\left(B, C\right)\right)}} \]
   6. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto -1 \cdot \color{blue}{\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)}\right)} \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)}\right) \]

      3. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)}\right)\right) \]

      4. associate-*l/ N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)}}{B}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\left(\sqrt{2} \cdot \sqrt{F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)}\right), B\right)\right) \]

   7. Applied egg-rr 60.7%

      \[\leadsto \color{blue}{-\frac{{\left(2 \cdot \left(F \cdot \left(C - \mathsf{hypot}\left(B, C\right)\right)\right)\right)}^{0.5}}{B}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 35.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 5.3 \cdot 10^{+59}:\\ \;\;\;\;\frac{\sqrt{\left(\left(B \cdot B + C \cdot \left(A \cdot -4\right)\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(2 \cdot \left(F \cdot \left(C - \mathsf{hypot}\left(B, C\right)\right)\right)\right)}^{0.5}}{0 - B}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 34.4% accurate, 2.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C - B\_m \cdot B\_m\\ \mathbf{if}\;B\_m \leq 6.2 \cdot 10^{-117}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(\left(4 \cdot A\right) \cdot \left(B\_m \cdot B\_m + -4 \cdot \left(A \cdot C\right)\right)\right)}}{t\_0}\\ \mathbf{elif}\;B\_m \leq 7.5 \cdot 10^{-45}:\\ \;\;\;\;\frac{\sqrt{\left(B\_m \cdot B\_m + C \cdot \left(A \cdot -4\right)\right) \cdot \left(4 \cdot \left(C \cdot F\right)\right)}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(A - \mathsf{hypot}\left(B\_m, A\right)\right)\right)}}{0 - B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (- (* (* 4.0 A) C) (* B_m B_m))))
   (if (<= B_m 6.2e-117)
     (/ (sqrt (* F (* (* 4.0 A) (+ (* B_m B_m) (* -4.0 (* A C)))))) t_0)
     (if (<= B_m 7.5e-45)
       (/ (sqrt (* (+ (* B_m B_m) (* C (* A -4.0))) (* 4.0 (* C F)))) t_0)
       (/ (sqrt (* 2.0 (* F (- A (hypot B_m A))))) (- 0.0 B_m))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double t_0 = ((4.0 * A) * C) - (B_m * B_m);
	double tmp;
	if (B_m <= 6.2e-117) {
		tmp = sqrt((F * ((4.0 * A) * ((B_m * B_m) + (-4.0 * (A * C)))))) / t_0;
	} else if (B_m <= 7.5e-45) {
		tmp = sqrt((((B_m * B_m) + (C * (A * -4.0))) * (4.0 * (C * F)))) / t_0;
	} else {
		tmp = sqrt((2.0 * (F * (A - hypot(B_m, A))))) / (0.0 - B_m);
	}
	return tmp;
}

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double t_0 = ((4.0 * A) * C) - (B_m * B_m);
	double tmp;
	if (B_m <= 6.2e-117) {
		tmp = Math.sqrt((F * ((4.0 * A) * ((B_m * B_m) + (-4.0 * (A * C)))))) / t_0;
	} else if (B_m <= 7.5e-45) {
		tmp = Math.sqrt((((B_m * B_m) + (C * (A * -4.0))) * (4.0 * (C * F)))) / t_0;
	} else {
		tmp = Math.sqrt((2.0 * (F * (A - Math.hypot(B_m, A))))) / (0.0 - B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	t_0 = ((4.0 * A) * C) - (B_m * B_m)
	tmp = 0
	if B_m <= 6.2e-117:
		tmp = math.sqrt((F * ((4.0 * A) * ((B_m * B_m) + (-4.0 * (A * C)))))) / t_0
	elif B_m <= 7.5e-45:
		tmp = math.sqrt((((B_m * B_m) + (C * (A * -4.0))) * (4.0 * (C * F)))) / t_0
	else:
		tmp = math.sqrt((2.0 * (F * (A - math.hypot(B_m, A))))) / (0.0 - B_m)
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	t_0 = Float64(Float64(Float64(4.0 * A) * C) - Float64(B_m * B_m))
	tmp = 0.0
	if (B_m <= 6.2e-117)
		tmp = Float64(sqrt(Float64(F * Float64(Float64(4.0 * A) * Float64(Float64(B_m * B_m) + Float64(-4.0 * Float64(A * C)))))) / t_0);
	elseif (B_m <= 7.5e-45)
		tmp = Float64(sqrt(Float64(Float64(Float64(B_m * B_m) + Float64(C * Float64(A * -4.0))) * Float64(4.0 * Float64(C * F)))) / t_0);
	else
		tmp = Float64(sqrt(Float64(2.0 * Float64(F * Float64(A - hypot(B_m, A))))) / Float64(0.0 - B_m));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	t_0 = ((4.0 * A) * C) - (B_m * B_m);
	tmp = 0.0;
	if (B_m <= 6.2e-117)
		tmp = sqrt((F * ((4.0 * A) * ((B_m * B_m) + (-4.0 * (A * C)))))) / t_0;
	elseif (B_m <= 7.5e-45)
		tmp = sqrt((((B_m * B_m) + (C * (A * -4.0))) * (4.0 * (C * F)))) / t_0;
	else
		tmp = sqrt((2.0 * (F * (A - hypot(B_m, A))))) / (0.0 - B_m);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision] - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 6.2e-117], N[(N[Sqrt[N[(F * N[(N[(4.0 * A), $MachinePrecision] * N[(N[(B$95$m * B$95$m), $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[B$95$m, 7.5e-45], N[(N[Sqrt[N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] + N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(4.0 * N[(C * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[Sqrt[N[(2.0 * N[(F * N[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(0.0 - B$95$m), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if B < 6.20000000000000022e-117

   1. Initial program 21.5%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Step-by-step derivation

      1. distribute-frac-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\right) \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\color{blue}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right), \color{blue}{\left(\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)\right)}\right) \]

   3. Simplified 27.5%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(B \cdot B + C \cdot \left(A \cdot -4\right)\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(B \cdot B + C \cdot \left(A \cdot -4\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(\left(A + C\right) - \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(B \cdot B + C \cdot \left(-4 \cdot A\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(\left(A + C\right) - \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(B \cdot B + C \cdot \left(\left(\mathsf{neg}\left(4\right)\right) \cdot A\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(\left(A + C\right) - \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. distribute-lft-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(B \cdot B + C \cdot \left(\mathsf{neg}\left(4 \cdot A\right)\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(\left(A + C\right) - \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(B \cdot B + \left(\mathsf{neg}\left(C \cdot \left(4 \cdot A\right)\right)\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(\left(A + C\right) - \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(B \cdot B + \left(\mathsf{neg}\left(\left(4 \cdot A\right) \cdot C\right)\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(\left(A + C\right) - \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(\left(A + C\right) - \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      8. pow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(\left(A + C\right) - \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right), \left(\left(2 \cdot F\right) \cdot \left(\left(A + C\right) - \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   6. Applied egg-rr 29.6%

      \[\leadsto \frac{\sqrt{\color{blue}{\left(B \cdot B + C \cdot \left(A \cdot -4\right)\right) \cdot \left(2 \cdot \left(F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   7. Taylor expanded in A around -inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(C, \mathsf{*.f64}\left(A, -4\right)\right)\right), \color{blue}{\left(4 \cdot \left(A \cdot F\right)\right)}\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   8. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(C, \mathsf{*.f64}\left(A, -4\right)\right)\right), \mathsf{*.f64}\left(4, \left(A \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. *-lowering-*.f64 20.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(C, \mathsf{*.f64}\left(A, -4\right)\right)\right), \mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   9. Simplified 20.7%

      \[\leadsto \frac{\sqrt{\left(B \cdot B + C \cdot \left(A \cdot -4\right)\right) \cdot \color{blue}{\left(4 \cdot \left(A \cdot F\right)\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(B \cdot B + C \cdot \left(A \cdot -4\right)\right) \cdot \left(\left(4 \cdot A\right) \cdot F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       2. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(\left(B \cdot B + C \cdot \left(A \cdot -4\right)\right) \cdot \left(4 \cdot A\right)\right) \cdot F\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(B \cdot B + C \cdot \left(A \cdot -4\right)\right) \cdot \left(4 \cdot A\right)\right), F\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(B \cdot B + C \cdot \left(A \cdot -4\right)\right), \left(4 \cdot A\right)\right), F\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       5. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(B \cdot B\right), \left(C \cdot \left(A \cdot -4\right)\right)\right), \left(4 \cdot A\right)\right), F\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \left(C \cdot \left(A \cdot -4\right)\right)\right), \left(4 \cdot A\right)\right), F\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \left(\left(A \cdot -4\right) \cdot C\right)\right), \left(4 \cdot A\right)\right), F\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \left(\left(-4 \cdot A\right) \cdot C\right)\right), \left(4 \cdot A\right)\right), F\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       9. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \left(-4 \cdot \left(A \cdot C\right)\right)\right), \left(4 \cdot A\right)\right), F\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(-4, \left(A \cdot C\right)\right)\right), \left(4 \cdot A\right)\right), F\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       11. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(-4, \left(C \cdot A\right)\right)\right), \left(4 \cdot A\right)\right), F\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(C, A\right)\right)\right), \left(4 \cdot A\right)\right), F\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       13. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(C, A\right)\right)\right), \left(A \cdot 4\right)\right), F\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       14. *-lowering-*.f64 20.8%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(C, A\right)\right)\right), \mathsf{*.f64}\left(A, 4\right)\right), F\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   11. Applied egg-rr 20.8%

       \[\leadsto \frac{\sqrt{\color{blue}{\left(\left(B \cdot B + -4 \cdot \left(C \cdot A\right)\right) \cdot \left(A \cdot 4\right)\right) \cdot F}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   if 6.20000000000000022e-117 < B < 7.5000000000000006e-45

   1. Initial program 38.3%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Step-by-step derivation

      1. distribute-frac-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\right) \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\color{blue}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right), \color{blue}{\left(\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)\right)}\right) \]

   3. Simplified 38.3%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(B \cdot B + C \cdot \left(A \cdot -4\right)\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(B \cdot B + C \cdot \left(A \cdot -4\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(\left(A + C\right) - \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(B \cdot B + C \cdot \left(-4 \cdot A\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(\left(A + C\right) - \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(B \cdot B + C \cdot \left(\left(\mathsf{neg}\left(4\right)\right) \cdot A\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(\left(A + C\right) - \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. distribute-lft-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(B \cdot B + C \cdot \left(\mathsf{neg}\left(4 \cdot A\right)\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(\left(A + C\right) - \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(B \cdot B + \left(\mathsf{neg}\left(C \cdot \left(4 \cdot A\right)\right)\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(\left(A + C\right) - \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(B \cdot B + \left(\mathsf{neg}\left(\left(4 \cdot A\right) \cdot C\right)\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(\left(A + C\right) - \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(\left(A + C\right) - \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      8. pow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(\left(A + C\right) - \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right), \left(\left(2 \cdot F\right) \cdot \left(\left(A + C\right) - \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   6. Applied egg-rr 46.9%

      \[\leadsto \frac{\sqrt{\color{blue}{\left(B \cdot B + C \cdot \left(A \cdot -4\right)\right) \cdot \left(2 \cdot \left(F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   7. Taylor expanded in C around -inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(C, \mathsf{*.f64}\left(A, -4\right)\right)\right), \color{blue}{\left(4 \cdot \left(C \cdot F\right)\right)}\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   8. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(C, \mathsf{*.f64}\left(A, -4\right)\right)\right), \mathsf{*.f64}\left(4, \left(C \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. *-lowering-*.f64 20.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(C, \mathsf{*.f64}\left(A, -4\right)\right)\right), \mathsf{*.f64}\left(4, \mathsf{*.f64}\left(C, F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   9. Simplified 20.5%

      \[\leadsto \frac{\sqrt{\left(B \cdot B + C \cdot \left(A \cdot -4\right)\right) \cdot \color{blue}{\left(4 \cdot \left(C \cdot F\right)\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   if 7.5000000000000006e-45 < B

   1. Initial program 14.3%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in C around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \color{blue}{\sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(-1 \cdot \frac{\sqrt{2}}{B}\right), \color{blue}{\left(\sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \left(\frac{\sqrt{2}}{B}\right)\right), \left(\sqrt{\color{blue}{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\left(\sqrt{2}\right), B\right)\right), \left(\sqrt{F \cdot \color{blue}{\left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \left(\sqrt{F \cdot \left(\color{blue}{A} - \sqrt{{A}^{2} + {B}^{2}}\right)}\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\left(F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \left(\sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \left(\sqrt{A \cdot A + {B}^{2}}\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \left(\sqrt{A \cdot A + B \cdot B}\right)\right)\right)\right)\right) \]

      11. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \left(\mathsf{hypot}\left(A, B\right)\right)\right)\right)\right)\right) \]

      12. hypot-lowering-hypot.f64 48.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \mathsf{hypot.f64}\left(A, B\right)\right)\right)\right)\right) \]

   5. Simplified 48.6%

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)}} \]
   6. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto -1 \cdot \color{blue}{\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{A \cdot A + B \cdot B}\right)}\right)} \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{A \cdot A + B \cdot B}\right)}\right) \]

      3. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{A \cdot A + B \cdot B}\right)}\right)\right) \]

      4. associate-*l/ N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(A - \sqrt{A \cdot A + B \cdot B}\right)}}{B}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\left(\sqrt{2} \cdot \sqrt{F \cdot \left(A - \sqrt{A \cdot A + B \cdot B}\right)}\right), B\right)\right) \]

   7. Applied egg-rr 48.8%

      \[\leadsto \color{blue}{-\frac{\sqrt{2 \cdot \left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}}{B}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 28.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 6.2 \cdot 10^{-117}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(\left(4 \cdot A\right) \cdot \left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{elif}\;B \leq 7.5 \cdot 10^{-45}:\\ \;\;\;\;\frac{\sqrt{\left(B \cdot B + C \cdot \left(A \cdot -4\right)\right) \cdot \left(4 \cdot \left(C \cdot F\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}}{0 - B}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 33.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 9.2 \cdot 10^{-117}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(\left(4 \cdot A\right) \cdot \left(B\_m \cdot B\_m + -4 \cdot \left(A \cdot C\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - B\_m \cdot B\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(2 \cdot \left(F \cdot \left(C - \mathsf{hypot}\left(B\_m, C\right)\right)\right)\right)}^{0.5}}{0 - B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 9.2e-117)
   (/
    (sqrt (* F (* (* 4.0 A) (+ (* B_m B_m) (* -4.0 (* A C))))))
    (- (* (* 4.0 A) C) (* B_m B_m)))
   (/ (pow (* 2.0 (* F (- C (hypot B_m C)))) 0.5) (- 0.0 B_m))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 9.2e-117) {
		tmp = sqrt((F * ((4.0 * A) * ((B_m * B_m) + (-4.0 * (A * C)))))) / (((4.0 * A) * C) - (B_m * B_m));
	} else {
		tmp = pow((2.0 * (F * (C - hypot(B_m, C)))), 0.5) / (0.0 - B_m);
	}
	return tmp;
}

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 9.2e-117) {
		tmp = Math.sqrt((F * ((4.0 * A) * ((B_m * B_m) + (-4.0 * (A * C)))))) / (((4.0 * A) * C) - (B_m * B_m));
	} else {
		tmp = Math.pow((2.0 * (F * (C - Math.hypot(B_m, C)))), 0.5) / (0.0 - B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 9.2e-117:
		tmp = math.sqrt((F * ((4.0 * A) * ((B_m * B_m) + (-4.0 * (A * C)))))) / (((4.0 * A) * C) - (B_m * B_m))
	else:
		tmp = math.pow((2.0 * (F * (C - math.hypot(B_m, C)))), 0.5) / (0.0 - B_m)
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 9.2e-117)
		tmp = Float64(sqrt(Float64(F * Float64(Float64(4.0 * A) * Float64(Float64(B_m * B_m) + Float64(-4.0 * Float64(A * C)))))) / Float64(Float64(Float64(4.0 * A) * C) - Float64(B_m * B_m)));
	else
		tmp = Float64((Float64(2.0 * Float64(F * Float64(C - hypot(B_m, C)))) ^ 0.5) / Float64(0.0 - B_m));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (B_m <= 9.2e-117)
		tmp = sqrt((F * ((4.0 * A) * ((B_m * B_m) + (-4.0 * (A * C)))))) / (((4.0 * A) * C) - (B_m * B_m));
	else
		tmp = ((2.0 * (F * (C - hypot(B_m, C)))) ^ 0.5) / (0.0 - B_m);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 9.2e-117], N[(N[Sqrt[N[(F * N[(N[(4.0 * A), $MachinePrecision] * N[(N[(B$95$m * B$95$m), $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision] - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(2.0 * N[(F * N[(C - N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision] / N[(0.0 - B$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 9.19999999999999978e-117

   1. Initial program 21.5%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Step-by-step derivation

      1. distribute-frac-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\right) \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\color{blue}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right), \color{blue}{\left(\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)\right)}\right) \]

   3. Simplified 27.5%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(B \cdot B + C \cdot \left(A \cdot -4\right)\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(B \cdot B + C \cdot \left(A \cdot -4\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(\left(A + C\right) - \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(B \cdot B + C \cdot \left(-4 \cdot A\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(\left(A + C\right) - \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(B \cdot B + C \cdot \left(\left(\mathsf{neg}\left(4\right)\right) \cdot A\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(\left(A + C\right) - \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. distribute-lft-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(B \cdot B + C \cdot \left(\mathsf{neg}\left(4 \cdot A\right)\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(\left(A + C\right) - \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(B \cdot B + \left(\mathsf{neg}\left(C \cdot \left(4 \cdot A\right)\right)\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(\left(A + C\right) - \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(B \cdot B + \left(\mathsf{neg}\left(\left(4 \cdot A\right) \cdot C\right)\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(\left(A + C\right) - \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(\left(A + C\right) - \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      8. pow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(\left(A + C\right) - \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right), \left(\left(2 \cdot F\right) \cdot \left(\left(A + C\right) - \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   6. Applied egg-rr 29.6%

      \[\leadsto \frac{\sqrt{\color{blue}{\left(B \cdot B + C \cdot \left(A \cdot -4\right)\right) \cdot \left(2 \cdot \left(F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   7. Taylor expanded in A around -inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(C, \mathsf{*.f64}\left(A, -4\right)\right)\right), \color{blue}{\left(4 \cdot \left(A \cdot F\right)\right)}\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   8. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(C, \mathsf{*.f64}\left(A, -4\right)\right)\right), \mathsf{*.f64}\left(4, \left(A \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. *-lowering-*.f64 20.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(C, \mathsf{*.f64}\left(A, -4\right)\right)\right), \mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   9. Simplified 20.7%

      \[\leadsto \frac{\sqrt{\left(B \cdot B + C \cdot \left(A \cdot -4\right)\right) \cdot \color{blue}{\left(4 \cdot \left(A \cdot F\right)\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(B \cdot B + C \cdot \left(A \cdot -4\right)\right) \cdot \left(\left(4 \cdot A\right) \cdot F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       2. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(\left(B \cdot B + C \cdot \left(A \cdot -4\right)\right) \cdot \left(4 \cdot A\right)\right) \cdot F\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(B \cdot B + C \cdot \left(A \cdot -4\right)\right) \cdot \left(4 \cdot A\right)\right), F\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(B \cdot B + C \cdot \left(A \cdot -4\right)\right), \left(4 \cdot A\right)\right), F\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       5. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(B \cdot B\right), \left(C \cdot \left(A \cdot -4\right)\right)\right), \left(4 \cdot A\right)\right), F\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \left(C \cdot \left(A \cdot -4\right)\right)\right), \left(4 \cdot A\right)\right), F\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \left(\left(A \cdot -4\right) \cdot C\right)\right), \left(4 \cdot A\right)\right), F\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \left(\left(-4 \cdot A\right) \cdot C\right)\right), \left(4 \cdot A\right)\right), F\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       9. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \left(-4 \cdot \left(A \cdot C\right)\right)\right), \left(4 \cdot A\right)\right), F\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(-4, \left(A \cdot C\right)\right)\right), \left(4 \cdot A\right)\right), F\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       11. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(-4, \left(C \cdot A\right)\right)\right), \left(4 \cdot A\right)\right), F\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(C, A\right)\right)\right), \left(4 \cdot A\right)\right), F\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       13. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(C, A\right)\right)\right), \left(A \cdot 4\right)\right), F\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       14. *-lowering-*.f64 20.8%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(C, A\right)\right)\right), \mathsf{*.f64}\left(A, 4\right)\right), F\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   11. Applied egg-rr 20.8%

       \[\leadsto \frac{\sqrt{\color{blue}{\left(\left(B \cdot B + -4 \cdot \left(C \cdot A\right)\right) \cdot \left(A \cdot 4\right)\right) \cdot F}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   if 9.19999999999999978e-117 < B

   1. Initial program 17.5%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \color{blue}{\sqrt{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(-1 \cdot \frac{\sqrt{2}}{B}\right), \color{blue}{\left(\sqrt{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \left(\frac{\sqrt{2}}{B}\right)\right), \left(\sqrt{\color{blue}{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\left(\sqrt{2}\right), B\right)\right), \left(\sqrt{F \cdot \color{blue}{\left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \left(\sqrt{F \cdot \left(\color{blue}{C} - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\left(F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\sqrt{B \cdot B + {C}^{2}}\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\sqrt{B \cdot B + C \cdot C}\right)\right)\right)\right)\right) \]

      11. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\mathsf{hypot}\left(B, C\right)\right)\right)\right)\right)\right) \]

      12. hypot-lowering-hypot.f64 46.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \mathsf{hypot.f64}\left(B, C\right)\right)\right)\right)\right) \]

   5. Simplified 46.3%

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(C - \mathsf{hypot}\left(B, C\right)\right)}} \]
   6. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto -1 \cdot \color{blue}{\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)}\right)} \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)}\right) \]

      3. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)}\right)\right) \]

      4. associate-*l/ N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)}}{B}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\left(\sqrt{2} \cdot \sqrt{F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)}\right), B\right)\right) \]

   7. Applied egg-rr 46.4%

      \[\leadsto \color{blue}{-\frac{{\left(2 \cdot \left(F \cdot \left(C - \mathsf{hypot}\left(B, C\right)\right)\right)\right)}^{0.5}}{B}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 28.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 9.2 \cdot 10^{-117}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(\left(4 \cdot A\right) \cdot \left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(2 \cdot \left(F \cdot \left(C - \mathsf{hypot}\left(B, C\right)\right)\right)\right)}^{0.5}}{0 - B}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 31.8% accurate, 4.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C - B\_m \cdot B\_m\\ \mathbf{if}\;B\_m \leq 6 \cdot 10^{-118}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(\left(4 \cdot A\right) \cdot \left(B\_m \cdot B\_m + -4 \cdot \left(A \cdot C\right)\right)\right)}}{t\_0}\\ \mathbf{elif}\;B\_m \leq 1.75 \cdot 10^{-44}:\\ \;\;\;\;\frac{\sqrt{\left(B\_m \cdot B\_m + C \cdot \left(A \cdot -4\right)\right) \cdot \left(4 \cdot \left(C \cdot F\right)\right)}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{-2 \cdot \left(B\_m \cdot F\right)}}{0 - B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (- (* (* 4.0 A) C) (* B_m B_m))))
   (if (<= B_m 6e-118)
     (/ (sqrt (* F (* (* 4.0 A) (+ (* B_m B_m) (* -4.0 (* A C)))))) t_0)
     (if (<= B_m 1.75e-44)
       (/ (sqrt (* (+ (* B_m B_m) (* C (* A -4.0))) (* 4.0 (* C F)))) t_0)
       (/ (sqrt (* -2.0 (* B_m F))) (- 0.0 B_m))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double t_0 = ((4.0 * A) * C) - (B_m * B_m);
	double tmp;
	if (B_m <= 6e-118) {
		tmp = sqrt((F * ((4.0 * A) * ((B_m * B_m) + (-4.0 * (A * C)))))) / t_0;
	} else if (B_m <= 1.75e-44) {
		tmp = sqrt((((B_m * B_m) + (C * (A * -4.0))) * (4.0 * (C * F)))) / t_0;
	} else {
		tmp = sqrt((-2.0 * (B_m * F))) / (0.0 - B_m);
	}
	return tmp;
}

Fortran

B_m = abs(b)
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((4.0d0 * a) * c) - (b_m * b_m)
    if (b_m <= 6d-118) then
        tmp = sqrt((f * ((4.0d0 * a) * ((b_m * b_m) + ((-4.0d0) * (a * c)))))) / t_0
    else if (b_m <= 1.75d-44) then
        tmp = sqrt((((b_m * b_m) + (c * (a * (-4.0d0)))) * (4.0d0 * (c * f)))) / t_0
    else
        tmp = sqrt(((-2.0d0) * (b_m * f))) / (0.0d0 - b_m)
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double t_0 = ((4.0 * A) * C) - (B_m * B_m);
	double tmp;
	if (B_m <= 6e-118) {
		tmp = Math.sqrt((F * ((4.0 * A) * ((B_m * B_m) + (-4.0 * (A * C)))))) / t_0;
	} else if (B_m <= 1.75e-44) {
		tmp = Math.sqrt((((B_m * B_m) + (C * (A * -4.0))) * (4.0 * (C * F)))) / t_0;
	} else {
		tmp = Math.sqrt((-2.0 * (B_m * F))) / (0.0 - B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	t_0 = ((4.0 * A) * C) - (B_m * B_m)
	tmp = 0
	if B_m <= 6e-118:
		tmp = math.sqrt((F * ((4.0 * A) * ((B_m * B_m) + (-4.0 * (A * C)))))) / t_0
	elif B_m <= 1.75e-44:
		tmp = math.sqrt((((B_m * B_m) + (C * (A * -4.0))) * (4.0 * (C * F)))) / t_0
	else:
		tmp = math.sqrt((-2.0 * (B_m * F))) / (0.0 - B_m)
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	t_0 = Float64(Float64(Float64(4.0 * A) * C) - Float64(B_m * B_m))
	tmp = 0.0
	if (B_m <= 6e-118)
		tmp = Float64(sqrt(Float64(F * Float64(Float64(4.0 * A) * Float64(Float64(B_m * B_m) + Float64(-4.0 * Float64(A * C)))))) / t_0);
	elseif (B_m <= 1.75e-44)
		tmp = Float64(sqrt(Float64(Float64(Float64(B_m * B_m) + Float64(C * Float64(A * -4.0))) * Float64(4.0 * Float64(C * F)))) / t_0);
	else
		tmp = Float64(sqrt(Float64(-2.0 * Float64(B_m * F))) / Float64(0.0 - B_m));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	t_0 = ((4.0 * A) * C) - (B_m * B_m);
	tmp = 0.0;
	if (B_m <= 6e-118)
		tmp = sqrt((F * ((4.0 * A) * ((B_m * B_m) + (-4.0 * (A * C)))))) / t_0;
	elseif (B_m <= 1.75e-44)
		tmp = sqrt((((B_m * B_m) + (C * (A * -4.0))) * (4.0 * (C * F)))) / t_0;
	else
		tmp = sqrt((-2.0 * (B_m * F))) / (0.0 - B_m);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision] - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 6e-118], N[(N[Sqrt[N[(F * N[(N[(4.0 * A), $MachinePrecision] * N[(N[(B$95$m * B$95$m), $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision], If[LessEqual[B$95$m, 1.75e-44], N[(N[Sqrt[N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] + N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(4.0 * N[(C * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[Sqrt[N[(-2.0 * N[(B$95$m * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(0.0 - B$95$m), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if B < 6.00000000000000035e-118

   1. Initial program 21.5%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Step-by-step derivation

      1. distribute-frac-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\right) \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\color{blue}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right), \color{blue}{\left(\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)\right)}\right) \]

   3. Simplified 27.5%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(B \cdot B + C \cdot \left(A \cdot -4\right)\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(B \cdot B + C \cdot \left(A \cdot -4\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(\left(A + C\right) - \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(B \cdot B + C \cdot \left(-4 \cdot A\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(\left(A + C\right) - \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(B \cdot B + C \cdot \left(\left(\mathsf{neg}\left(4\right)\right) \cdot A\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(\left(A + C\right) - \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. distribute-lft-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(B \cdot B + C \cdot \left(\mathsf{neg}\left(4 \cdot A\right)\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(\left(A + C\right) - \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(B \cdot B + \left(\mathsf{neg}\left(C \cdot \left(4 \cdot A\right)\right)\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(\left(A + C\right) - \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(B \cdot B + \left(\mathsf{neg}\left(\left(4 \cdot A\right) \cdot C\right)\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(\left(A + C\right) - \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(\left(A + C\right) - \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      8. pow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(\left(A + C\right) - \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right), \left(\left(2 \cdot F\right) \cdot \left(\left(A + C\right) - \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   6. Applied egg-rr 29.6%

      \[\leadsto \frac{\sqrt{\color{blue}{\left(B \cdot B + C \cdot \left(A \cdot -4\right)\right) \cdot \left(2 \cdot \left(F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   7. Taylor expanded in A around -inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(C, \mathsf{*.f64}\left(A, -4\right)\right)\right), \color{blue}{\left(4 \cdot \left(A \cdot F\right)\right)}\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   8. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(C, \mathsf{*.f64}\left(A, -4\right)\right)\right), \mathsf{*.f64}\left(4, \left(A \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. *-lowering-*.f64 20.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(C, \mathsf{*.f64}\left(A, -4\right)\right)\right), \mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   9. Simplified 20.7%

      \[\leadsto \frac{\sqrt{\left(B \cdot B + C \cdot \left(A \cdot -4\right)\right) \cdot \color{blue}{\left(4 \cdot \left(A \cdot F\right)\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(B \cdot B + C \cdot \left(A \cdot -4\right)\right) \cdot \left(\left(4 \cdot A\right) \cdot F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       2. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(\left(B \cdot B + C \cdot \left(A \cdot -4\right)\right) \cdot \left(4 \cdot A\right)\right) \cdot F\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(B \cdot B + C \cdot \left(A \cdot -4\right)\right) \cdot \left(4 \cdot A\right)\right), F\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(B \cdot B + C \cdot \left(A \cdot -4\right)\right), \left(4 \cdot A\right)\right), F\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       5. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(B \cdot B\right), \left(C \cdot \left(A \cdot -4\right)\right)\right), \left(4 \cdot A\right)\right), F\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \left(C \cdot \left(A \cdot -4\right)\right)\right), \left(4 \cdot A\right)\right), F\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \left(\left(A \cdot -4\right) \cdot C\right)\right), \left(4 \cdot A\right)\right), F\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \left(\left(-4 \cdot A\right) \cdot C\right)\right), \left(4 \cdot A\right)\right), F\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       9. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \left(-4 \cdot \left(A \cdot C\right)\right)\right), \left(4 \cdot A\right)\right), F\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(-4, \left(A \cdot C\right)\right)\right), \left(4 \cdot A\right)\right), F\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       11. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(-4, \left(C \cdot A\right)\right)\right), \left(4 \cdot A\right)\right), F\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(C, A\right)\right)\right), \left(4 \cdot A\right)\right), F\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       13. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(C, A\right)\right)\right), \left(A \cdot 4\right)\right), F\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       14. *-lowering-*.f64 20.8%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(C, A\right)\right)\right), \mathsf{*.f64}\left(A, 4\right)\right), F\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   11. Applied egg-rr 20.8%

       \[\leadsto \frac{\sqrt{\color{blue}{\left(\left(B \cdot B + -4 \cdot \left(C \cdot A\right)\right) \cdot \left(A \cdot 4\right)\right) \cdot F}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   if 6.00000000000000035e-118 < B < 1.7499999999999999e-44

   1. Initial program 38.3%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Step-by-step derivation

      1. distribute-frac-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\right) \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\color{blue}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right), \color{blue}{\left(\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)\right)}\right) \]

   3. Simplified 38.3%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(B \cdot B + C \cdot \left(A \cdot -4\right)\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(B \cdot B + C \cdot \left(A \cdot -4\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(\left(A + C\right) - \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(B \cdot B + C \cdot \left(-4 \cdot A\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(\left(A + C\right) - \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(B \cdot B + C \cdot \left(\left(\mathsf{neg}\left(4\right)\right) \cdot A\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(\left(A + C\right) - \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. distribute-lft-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(B \cdot B + C \cdot \left(\mathsf{neg}\left(4 \cdot A\right)\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(\left(A + C\right) - \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(B \cdot B + \left(\mathsf{neg}\left(C \cdot \left(4 \cdot A\right)\right)\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(\left(A + C\right) - \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(B \cdot B + \left(\mathsf{neg}\left(\left(4 \cdot A\right) \cdot C\right)\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(\left(A + C\right) - \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(\left(A + C\right) - \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      8. pow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(\left(A + C\right) - \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right), \left(\left(2 \cdot F\right) \cdot \left(\left(A + C\right) - \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   6. Applied egg-rr 46.9%

      \[\leadsto \frac{\sqrt{\color{blue}{\left(B \cdot B + C \cdot \left(A \cdot -4\right)\right) \cdot \left(2 \cdot \left(F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   7. Taylor expanded in C around -inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(C, \mathsf{*.f64}\left(A, -4\right)\right)\right), \color{blue}{\left(4 \cdot \left(C \cdot F\right)\right)}\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   8. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(C, \mathsf{*.f64}\left(A, -4\right)\right)\right), \mathsf{*.f64}\left(4, \left(C \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. *-lowering-*.f64 20.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(C, \mathsf{*.f64}\left(A, -4\right)\right)\right), \mathsf{*.f64}\left(4, \mathsf{*.f64}\left(C, F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   9. Simplified 20.5%

      \[\leadsto \frac{\sqrt{\left(B \cdot B + C \cdot \left(A \cdot -4\right)\right) \cdot \color{blue}{\left(4 \cdot \left(C \cdot F\right)\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   if 1.7499999999999999e-44 < B

   1. Initial program 14.3%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in C around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \color{blue}{\sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(-1 \cdot \frac{\sqrt{2}}{B}\right), \color{blue}{\left(\sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \left(\frac{\sqrt{2}}{B}\right)\right), \left(\sqrt{\color{blue}{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\left(\sqrt{2}\right), B\right)\right), \left(\sqrt{F \cdot \color{blue}{\left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \left(\sqrt{F \cdot \left(\color{blue}{A} - \sqrt{{A}^{2} + {B}^{2}}\right)}\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\left(F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \left(\sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \left(\sqrt{A \cdot A + {B}^{2}}\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \left(\sqrt{A \cdot A + B \cdot B}\right)\right)\right)\right)\right) \]

      11. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \left(\mathsf{hypot}\left(A, B\right)\right)\right)\right)\right)\right) \]

      12. hypot-lowering-hypot.f64 48.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \mathsf{hypot.f64}\left(A, B\right)\right)\right)\right)\right) \]

   5. Simplified 48.6%

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)}} \]
   6. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto -1 \cdot \color{blue}{\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{A \cdot A + B \cdot B}\right)}\right)} \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{A \cdot A + B \cdot B}\right)}\right) \]

      3. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{A \cdot A + B \cdot B}\right)}\right)\right) \]

      4. associate-*l/ N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(A - \sqrt{A \cdot A + B \cdot B}\right)}}{B}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\left(\sqrt{2} \cdot \sqrt{F \cdot \left(A - \sqrt{A \cdot A + B \cdot B}\right)}\right), B\right)\right) \]

   7. Applied egg-rr 48.8%

      \[\leadsto \color{blue}{-\frac{\sqrt{2 \cdot \left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}}{B}} \]

   8. Taylor expanded in A around 0

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(-2 \cdot \left(B \cdot F\right)\right)}\right), B\right)\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-2, \left(B \cdot F\right)\right)\right), B\right)\right) \]

      2. *-lowering-*.f64 47.7%

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(B, F\right)\right)\right), B\right)\right) \]

   10. Simplified 47.7%

       \[\leadsto -\frac{\sqrt{\color{blue}{-2 \cdot \left(B \cdot F\right)}}}{B} \]
3. Recombined 3 regimes into one program.

4. Final simplification 28.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 6 \cdot 10^{-118}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(\left(4 \cdot A\right) \cdot \left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{elif}\;B \leq 1.75 \cdot 10^{-44}:\\ \;\;\;\;\frac{\sqrt{\left(B \cdot B + C \cdot \left(A \cdot -4\right)\right) \cdot \left(4 \cdot \left(C \cdot F\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{-2 \cdot \left(B \cdot F\right)}}{0 - B}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 33.1% accurate, 4.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 5 \cdot 10^{-57}:\\ \;\;\;\;\frac{\sqrt{\left(B\_m \cdot B\_m + C \cdot \left(A \cdot -4\right)\right) \cdot \left(4 \cdot \left(A \cdot F\right)\right)}}{\left(4 \cdot A\right) \cdot C - B\_m \cdot B\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{-2 \cdot \left(B\_m \cdot F\right)}}{0 - B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 5e-57)
   (/
    (sqrt (* (+ (* B_m B_m) (* C (* A -4.0))) (* 4.0 (* A F))))
    (- (* (* 4.0 A) C) (* B_m B_m)))
   (/ (sqrt (* -2.0 (* B_m F))) (- 0.0 B_m))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 5e-57) {
		tmp = sqrt((((B_m * B_m) + (C * (A * -4.0))) * (4.0 * (A * F)))) / (((4.0 * A) * C) - (B_m * B_m));
	} else {
		tmp = sqrt((-2.0 * (B_m * F))) / (0.0 - B_m);
	}
	return tmp;
}

Fortran

B_m = abs(b)
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (b_m <= 5d-57) then
        tmp = sqrt((((b_m * b_m) + (c * (a * (-4.0d0)))) * (4.0d0 * (a * f)))) / (((4.0d0 * a) * c) - (b_m * b_m))
    else
        tmp = sqrt(((-2.0d0) * (b_m * f))) / (0.0d0 - b_m)
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 5e-57) {
		tmp = Math.sqrt((((B_m * B_m) + (C * (A * -4.0))) * (4.0 * (A * F)))) / (((4.0 * A) * C) - (B_m * B_m));
	} else {
		tmp = Math.sqrt((-2.0 * (B_m * F))) / (0.0 - B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 5e-57:
		tmp = math.sqrt((((B_m * B_m) + (C * (A * -4.0))) * (4.0 * (A * F)))) / (((4.0 * A) * C) - (B_m * B_m))
	else:
		tmp = math.sqrt((-2.0 * (B_m * F))) / (0.0 - B_m)
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 5e-57)
		tmp = Float64(sqrt(Float64(Float64(Float64(B_m * B_m) + Float64(C * Float64(A * -4.0))) * Float64(4.0 * Float64(A * F)))) / Float64(Float64(Float64(4.0 * A) * C) - Float64(B_m * B_m)));
	else
		tmp = Float64(sqrt(Float64(-2.0 * Float64(B_m * F))) / Float64(0.0 - B_m));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (B_m <= 5e-57)
		tmp = sqrt((((B_m * B_m) + (C * (A * -4.0))) * (4.0 * (A * F)))) / (((4.0 * A) * C) - (B_m * B_m));
	else
		tmp = sqrt((-2.0 * (B_m * F))) / (0.0 - B_m);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 5e-57], N[(N[Sqrt[N[(N[(N[(B$95$m * B$95$m), $MachinePrecision] + N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(4.0 * N[(A * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision] - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(-2.0 * N[(B$95$m * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(0.0 - B$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 5.0000000000000002e-57

   1. Initial program 22.5%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Step-by-step derivation

      1. distribute-frac-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\right) \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\color{blue}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right), \color{blue}{\left(\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)\right)}\right) \]

   3. Simplified 28.2%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(B \cdot B + C \cdot \left(A \cdot -4\right)\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(B \cdot B + C \cdot \left(A \cdot -4\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(\left(A + C\right) - \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(B \cdot B + C \cdot \left(-4 \cdot A\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(\left(A + C\right) - \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(B \cdot B + C \cdot \left(\left(\mathsf{neg}\left(4\right)\right) \cdot A\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(\left(A + C\right) - \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. distribute-lft-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(B \cdot B + C \cdot \left(\mathsf{neg}\left(4 \cdot A\right)\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(\left(A + C\right) - \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(B \cdot B + \left(\mathsf{neg}\left(C \cdot \left(4 \cdot A\right)\right)\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(\left(A + C\right) - \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(B \cdot B + \left(\mathsf{neg}\left(\left(4 \cdot A\right) \cdot C\right)\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(\left(A + C\right) - \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(\left(A + C\right) - \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      8. pow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(\left(A + C\right) - \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right), \left(\left(2 \cdot F\right) \cdot \left(\left(A + C\right) - \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   6. Applied egg-rr 30.6%

      \[\leadsto \frac{\sqrt{\color{blue}{\left(B \cdot B + C \cdot \left(A \cdot -4\right)\right) \cdot \left(2 \cdot \left(F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   7. Taylor expanded in A around -inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(C, \mathsf{*.f64}\left(A, -4\right)\right)\right), \color{blue}{\left(4 \cdot \left(A \cdot F\right)\right)}\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   8. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(C, \mathsf{*.f64}\left(A, -4\right)\right)\right), \mathsf{*.f64}\left(4, \left(A \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. *-lowering-*.f64 20.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(C, \mathsf{*.f64}\left(A, -4\right)\right)\right), \mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   9. Simplified 20.7%

      \[\leadsto \frac{\sqrt{\left(B \cdot B + C \cdot \left(A \cdot -4\right)\right) \cdot \color{blue}{\left(4 \cdot \left(A \cdot F\right)\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   if 5.0000000000000002e-57 < B

   1. Initial program 14.3%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in C around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \color{blue}{\sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(-1 \cdot \frac{\sqrt{2}}{B}\right), \color{blue}{\left(\sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \left(\frac{\sqrt{2}}{B}\right)\right), \left(\sqrt{\color{blue}{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\left(\sqrt{2}\right), B\right)\right), \left(\sqrt{F \cdot \color{blue}{\left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \left(\sqrt{F \cdot \left(\color{blue}{A} - \sqrt{{A}^{2} + {B}^{2}}\right)}\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\left(F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \left(\sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \left(\sqrt{A \cdot A + {B}^{2}}\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \left(\sqrt{A \cdot A + B \cdot B}\right)\right)\right)\right)\right) \]

      11. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \left(\mathsf{hypot}\left(A, B\right)\right)\right)\right)\right)\right) \]

      12. hypot-lowering-hypot.f64 48.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \mathsf{hypot.f64}\left(A, B\right)\right)\right)\right)\right) \]

   5. Simplified 48.6%

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)}} \]
   6. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto -1 \cdot \color{blue}{\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{A \cdot A + B \cdot B}\right)}\right)} \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{A \cdot A + B \cdot B}\right)}\right) \]

      3. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{A \cdot A + B \cdot B}\right)}\right)\right) \]

      4. associate-*l/ N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(A - \sqrt{A \cdot A + B \cdot B}\right)}}{B}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\left(\sqrt{2} \cdot \sqrt{F \cdot \left(A - \sqrt{A \cdot A + B \cdot B}\right)}\right), B\right)\right) \]

   7. Applied egg-rr 48.8%

      \[\leadsto \color{blue}{-\frac{\sqrt{2 \cdot \left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}}{B}} \]

   8. Taylor expanded in A around 0

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(-2 \cdot \left(B \cdot F\right)\right)}\right), B\right)\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-2, \left(B \cdot F\right)\right)\right), B\right)\right) \]

      2. *-lowering-*.f64 47.7%

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(B, F\right)\right)\right), B\right)\right) \]

   10. Simplified 47.7%

       \[\leadsto -\frac{\sqrt{\color{blue}{-2 \cdot \left(B \cdot F\right)}}}{B} \]
3. Recombined 2 regimes into one program.

4. Final simplification 28.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 5 \cdot 10^{-57}:\\ \;\;\;\;\frac{\sqrt{\left(B \cdot B + C \cdot \left(A \cdot -4\right)\right) \cdot \left(4 \cdot \left(A \cdot F\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{-2 \cdot \left(B \cdot F\right)}}{0 - B}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 32.0% accurate, 4.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 5.5 \cdot 10^{-57}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(\left(4 \cdot A\right) \cdot \left(B\_m \cdot B\_m + -4 \cdot \left(A \cdot C\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - B\_m \cdot B\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{-2 \cdot \left(B\_m \cdot F\right)}}{0 - B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 5.5e-57)
   (/
    (sqrt (* F (* (* 4.0 A) (+ (* B_m B_m) (* -4.0 (* A C))))))
    (- (* (* 4.0 A) C) (* B_m B_m)))
   (/ (sqrt (* -2.0 (* B_m F))) (- 0.0 B_m))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 5.5e-57) {
		tmp = sqrt((F * ((4.0 * A) * ((B_m * B_m) + (-4.0 * (A * C)))))) / (((4.0 * A) * C) - (B_m * B_m));
	} else {
		tmp = sqrt((-2.0 * (B_m * F))) / (0.0 - B_m);
	}
	return tmp;
}

Fortran

B_m = abs(b)
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (b_m <= 5.5d-57) then
        tmp = sqrt((f * ((4.0d0 * a) * ((b_m * b_m) + ((-4.0d0) * (a * c)))))) / (((4.0d0 * a) * c) - (b_m * b_m))
    else
        tmp = sqrt(((-2.0d0) * (b_m * f))) / (0.0d0 - b_m)
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 5.5e-57) {
		tmp = Math.sqrt((F * ((4.0 * A) * ((B_m * B_m) + (-4.0 * (A * C)))))) / (((4.0 * A) * C) - (B_m * B_m));
	} else {
		tmp = Math.sqrt((-2.0 * (B_m * F))) / (0.0 - B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 5.5e-57:
		tmp = math.sqrt((F * ((4.0 * A) * ((B_m * B_m) + (-4.0 * (A * C)))))) / (((4.0 * A) * C) - (B_m * B_m))
	else:
		tmp = math.sqrt((-2.0 * (B_m * F))) / (0.0 - B_m)
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 5.5e-57)
		tmp = Float64(sqrt(Float64(F * Float64(Float64(4.0 * A) * Float64(Float64(B_m * B_m) + Float64(-4.0 * Float64(A * C)))))) / Float64(Float64(Float64(4.0 * A) * C) - Float64(B_m * B_m)));
	else
		tmp = Float64(sqrt(Float64(-2.0 * Float64(B_m * F))) / Float64(0.0 - B_m));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (B_m <= 5.5e-57)
		tmp = sqrt((F * ((4.0 * A) * ((B_m * B_m) + (-4.0 * (A * C)))))) / (((4.0 * A) * C) - (B_m * B_m));
	else
		tmp = sqrt((-2.0 * (B_m * F))) / (0.0 - B_m);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 5.5e-57], N[(N[Sqrt[N[(F * N[(N[(4.0 * A), $MachinePrecision] * N[(N[(B$95$m * B$95$m), $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision] - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(-2.0 * N[(B$95$m * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(0.0 - B$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 5.50000000000000011e-57

   1. Initial program 22.5%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Step-by-step derivation

      1. distribute-frac-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\right) \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\color{blue}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right), \color{blue}{\left(\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)\right)}\right) \]

   3. Simplified 28.2%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(B \cdot B + C \cdot \left(A \cdot -4\right)\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(B \cdot B + C \cdot \left(A \cdot -4\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(\left(A + C\right) - \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(B \cdot B + C \cdot \left(-4 \cdot A\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(\left(A + C\right) - \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(B \cdot B + C \cdot \left(\left(\mathsf{neg}\left(4\right)\right) \cdot A\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(\left(A + C\right) - \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. distribute-lft-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(B \cdot B + C \cdot \left(\mathsf{neg}\left(4 \cdot A\right)\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(\left(A + C\right) - \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(B \cdot B + \left(\mathsf{neg}\left(C \cdot \left(4 \cdot A\right)\right)\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(\left(A + C\right) - \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(B \cdot B + \left(\mathsf{neg}\left(\left(4 \cdot A\right) \cdot C\right)\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(\left(A + C\right) - \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(\left(A + C\right) - \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      8. pow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(\left(A + C\right) - \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right), \left(\left(2 \cdot F\right) \cdot \left(\left(A + C\right) - \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   6. Applied egg-rr 30.6%

      \[\leadsto \frac{\sqrt{\color{blue}{\left(B \cdot B + C \cdot \left(A \cdot -4\right)\right) \cdot \left(2 \cdot \left(F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   7. Taylor expanded in A around -inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(C, \mathsf{*.f64}\left(A, -4\right)\right)\right), \color{blue}{\left(4 \cdot \left(A \cdot F\right)\right)}\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   8. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(C, \mathsf{*.f64}\left(A, -4\right)\right)\right), \mathsf{*.f64}\left(4, \left(A \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. *-lowering-*.f64 20.7%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(C, \mathsf{*.f64}\left(A, -4\right)\right)\right), \mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   9. Simplified 20.7%

      \[\leadsto \frac{\sqrt{\left(B \cdot B + C \cdot \left(A \cdot -4\right)\right) \cdot \color{blue}{\left(4 \cdot \left(A \cdot F\right)\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]
   10. Step-by-step derivation

       1. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(B \cdot B + C \cdot \left(A \cdot -4\right)\right) \cdot \left(\left(4 \cdot A\right) \cdot F\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       2. associate-*r* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(\left(B \cdot B + C \cdot \left(A \cdot -4\right)\right) \cdot \left(4 \cdot A\right)\right) \cdot F\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       3. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\left(B \cdot B + C \cdot \left(A \cdot -4\right)\right) \cdot \left(4 \cdot A\right)\right), F\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       4. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\left(B \cdot B + C \cdot \left(A \cdot -4\right)\right), \left(4 \cdot A\right)\right), F\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       5. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\left(B \cdot B\right), \left(C \cdot \left(A \cdot -4\right)\right)\right), \left(4 \cdot A\right)\right), F\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \left(C \cdot \left(A \cdot -4\right)\right)\right), \left(4 \cdot A\right)\right), F\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       7. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \left(\left(A \cdot -4\right) \cdot C\right)\right), \left(4 \cdot A\right)\right), F\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       8. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \left(\left(-4 \cdot A\right) \cdot C\right)\right), \left(4 \cdot A\right)\right), F\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       9. associate-*l* N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \left(-4 \cdot \left(A \cdot C\right)\right)\right), \left(4 \cdot A\right)\right), F\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       10. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(-4, \left(A \cdot C\right)\right)\right), \left(4 \cdot A\right)\right), F\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       11. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(-4, \left(C \cdot A\right)\right)\right), \left(4 \cdot A\right)\right), F\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       12. *-lowering-*.f64 N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(C, A\right)\right)\right), \left(4 \cdot A\right)\right), F\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       13. *-commutative N/A

           \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(C, A\right)\right)\right), \left(A \cdot 4\right)\right), F\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       14. *-lowering-*.f64 20.7%

           \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(C, A\right)\right)\right), \mathsf{*.f64}\left(A, 4\right)\right), F\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   11. Applied egg-rr 20.7%

       \[\leadsto \frac{\sqrt{\color{blue}{\left(\left(B \cdot B + -4 \cdot \left(C \cdot A\right)\right) \cdot \left(A \cdot 4\right)\right) \cdot F}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   if 5.50000000000000011e-57 < B

   1. Initial program 14.3%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in C around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \color{blue}{\sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(-1 \cdot \frac{\sqrt{2}}{B}\right), \color{blue}{\left(\sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \left(\frac{\sqrt{2}}{B}\right)\right), \left(\sqrt{\color{blue}{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\left(\sqrt{2}\right), B\right)\right), \left(\sqrt{F \cdot \color{blue}{\left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \left(\sqrt{F \cdot \left(\color{blue}{A} - \sqrt{{A}^{2} + {B}^{2}}\right)}\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\left(F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \left(\sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \left(\sqrt{A \cdot A + {B}^{2}}\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \left(\sqrt{A \cdot A + B \cdot B}\right)\right)\right)\right)\right) \]

      11. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \left(\mathsf{hypot}\left(A, B\right)\right)\right)\right)\right)\right) \]

      12. hypot-lowering-hypot.f64 48.6%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \mathsf{hypot.f64}\left(A, B\right)\right)\right)\right)\right) \]

   5. Simplified 48.6%

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)}} \]
   6. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto -1 \cdot \color{blue}{\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{A \cdot A + B \cdot B}\right)}\right)} \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{A \cdot A + B \cdot B}\right)}\right) \]

      3. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{A \cdot A + B \cdot B}\right)}\right)\right) \]

      4. associate-*l/ N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(A - \sqrt{A \cdot A + B \cdot B}\right)}}{B}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\left(\sqrt{2} \cdot \sqrt{F \cdot \left(A - \sqrt{A \cdot A + B \cdot B}\right)}\right), B\right)\right) \]

   7. Applied egg-rr 48.8%

      \[\leadsto \color{blue}{-\frac{\sqrt{2 \cdot \left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}}{B}} \]

   8. Taylor expanded in A around 0

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(-2 \cdot \left(B \cdot F\right)\right)}\right), B\right)\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-2, \left(B \cdot F\right)\right)\right), B\right)\right) \]

      2. *-lowering-*.f64 47.7%

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(B, F\right)\right)\right), B\right)\right) \]

   10. Simplified 47.7%

       \[\leadsto -\frac{\sqrt{\color{blue}{-2 \cdot \left(B \cdot F\right)}}}{B} \]
3. Recombined 2 regimes into one program.

4. Final simplification 28.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 5.5 \cdot 10^{-57}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(\left(4 \cdot A\right) \cdot \left(B \cdot B + -4 \cdot \left(A \cdot C\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{-2 \cdot \left(B \cdot F\right)}}{0 - B}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 32.7% accurate, 5.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 10^{-83}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot \left(A \cdot C\right)\right) \cdot \left(4 \cdot \left(A \cdot F\right)\right)}}{\left(4 \cdot A\right) \cdot C - B\_m \cdot B\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{-2 \cdot \left(B\_m \cdot F\right)}}{0 - B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 1e-83)
   (/
    (sqrt (* (* -4.0 (* A C)) (* 4.0 (* A F))))
    (- (* (* 4.0 A) C) (* B_m B_m)))
   (/ (sqrt (* -2.0 (* B_m F))) (- 0.0 B_m))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 1e-83) {
		tmp = sqrt(((-4.0 * (A * C)) * (4.0 * (A * F)))) / (((4.0 * A) * C) - (B_m * B_m));
	} else {
		tmp = sqrt((-2.0 * (B_m * F))) / (0.0 - B_m);
	}
	return tmp;
}

Fortran

B_m = abs(b)
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (b_m <= 1d-83) then
        tmp = sqrt((((-4.0d0) * (a * c)) * (4.0d0 * (a * f)))) / (((4.0d0 * a) * c) - (b_m * b_m))
    else
        tmp = sqrt(((-2.0d0) * (b_m * f))) / (0.0d0 - b_m)
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 1e-83) {
		tmp = Math.sqrt(((-4.0 * (A * C)) * (4.0 * (A * F)))) / (((4.0 * A) * C) - (B_m * B_m));
	} else {
		tmp = Math.sqrt((-2.0 * (B_m * F))) / (0.0 - B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 1e-83:
		tmp = math.sqrt(((-4.0 * (A * C)) * (4.0 * (A * F)))) / (((4.0 * A) * C) - (B_m * B_m))
	else:
		tmp = math.sqrt((-2.0 * (B_m * F))) / (0.0 - B_m)
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 1e-83)
		tmp = Float64(sqrt(Float64(Float64(-4.0 * Float64(A * C)) * Float64(4.0 * Float64(A * F)))) / Float64(Float64(Float64(4.0 * A) * C) - Float64(B_m * B_m)));
	else
		tmp = Float64(sqrt(Float64(-2.0 * Float64(B_m * F))) / Float64(0.0 - B_m));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (B_m <= 1e-83)
		tmp = sqrt(((-4.0 * (A * C)) * (4.0 * (A * F)))) / (((4.0 * A) * C) - (B_m * B_m));
	else
		tmp = sqrt((-2.0 * (B_m * F))) / (0.0 - B_m);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 1e-83], N[(N[Sqrt[N[(N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision] * N[(4.0 * N[(A * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision] - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(-2.0 * N[(B$95$m * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(0.0 - B$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 1e-83

   1. Initial program 21.1%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Step-by-step derivation

      1. distribute-frac-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\right) \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\color{blue}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right), \color{blue}{\left(\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)\right)}\right) \]

   3. Simplified 27.0%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(B \cdot B + C \cdot \left(A \cdot -4\right)\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(B \cdot B + C \cdot \left(A \cdot -4\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(\left(A + C\right) - \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(B \cdot B + C \cdot \left(-4 \cdot A\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(\left(A + C\right) - \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. metadata-eval N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(B \cdot B + C \cdot \left(\left(\mathsf{neg}\left(4\right)\right) \cdot A\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(\left(A + C\right) - \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. distribute-lft-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(B \cdot B + C \cdot \left(\mathsf{neg}\left(4 \cdot A\right)\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(\left(A + C\right) - \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. distribute-rgt-neg-in N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(B \cdot B + \left(\mathsf{neg}\left(C \cdot \left(4 \cdot A\right)\right)\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(\left(A + C\right) - \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(B \cdot B + \left(\mathsf{neg}\left(\left(4 \cdot A\right) \cdot C\right)\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(\left(A + C\right) - \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      7. sub-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left(B \cdot B - \left(4 \cdot A\right) \cdot C\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(\left(A + C\right) - \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      8. pow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot \left(\left(2 \cdot F\right) \cdot \left(\left(A + C\right) - \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right), \left(\left(2 \cdot F\right) \cdot \left(\left(A + C\right) - \sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   6. Applied egg-rr 29.0%

      \[\leadsto \frac{\sqrt{\color{blue}{\left(B \cdot B + C \cdot \left(A \cdot -4\right)\right) \cdot \left(2 \cdot \left(F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   7. Taylor expanded in A around -inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(C, \mathsf{*.f64}\left(A, -4\right)\right)\right), \color{blue}{\left(4 \cdot \left(A \cdot F\right)\right)}\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   8. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(C, \mathsf{*.f64}\left(A, -4\right)\right)\right), \mathsf{*.f64}\left(4, \left(A \cdot F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. *-lowering-*.f64 20.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{*.f64}\left(C, \mathsf{*.f64}\left(A, -4\right)\right)\right), \mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   9. Simplified 20.3%

      \[\leadsto \frac{\sqrt{\left(B \cdot B + C \cdot \left(A \cdot -4\right)\right) \cdot \color{blue}{\left(4 \cdot \left(A \cdot F\right)\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   10. Taylor expanded in B around 0

       \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\color{blue}{\left(-4 \cdot \left(A \cdot C\right)\right)}, \mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   11. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-4, \left(A \cdot C\right)\right), \mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       2. *-commutative N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-4, \left(C \cdot A\right)\right), \mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

       3. *-lowering-*.f64 18.0%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-4, \mathsf{*.f64}\left(C, A\right)\right), \mathsf{*.f64}\left(4, \mathsf{*.f64}\left(A, F\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   12. Simplified 18.0%

       \[\leadsto \frac{\sqrt{\color{blue}{\left(-4 \cdot \left(C \cdot A\right)\right)} \cdot \left(4 \cdot \left(A \cdot F\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   if 1e-83 < B

   1. Initial program 18.3%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in C around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \color{blue}{\sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(-1 \cdot \frac{\sqrt{2}}{B}\right), \color{blue}{\left(\sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \left(\frac{\sqrt{2}}{B}\right)\right), \left(\sqrt{\color{blue}{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\left(\sqrt{2}\right), B\right)\right), \left(\sqrt{F \cdot \color{blue}{\left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \left(\sqrt{F \cdot \left(\color{blue}{A} - \sqrt{{A}^{2} + {B}^{2}}\right)}\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\left(F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \left(\sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \left(\sqrt{A \cdot A + {B}^{2}}\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \left(\sqrt{A \cdot A + B \cdot B}\right)\right)\right)\right)\right) \]

      11. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \left(\mathsf{hypot}\left(A, B\right)\right)\right)\right)\right)\right) \]

      12. hypot-lowering-hypot.f64 47.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \mathsf{hypot.f64}\left(A, B\right)\right)\right)\right)\right) \]

   5. Simplified 47.2%

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)}} \]
   6. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto -1 \cdot \color{blue}{\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{A \cdot A + B \cdot B}\right)}\right)} \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{A \cdot A + B \cdot B}\right)}\right) \]

      3. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{A \cdot A + B \cdot B}\right)}\right)\right) \]

      4. associate-*l/ N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(A - \sqrt{A \cdot A + B \cdot B}\right)}}{B}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\left(\sqrt{2} \cdot \sqrt{F \cdot \left(A - \sqrt{A \cdot A + B \cdot B}\right)}\right), B\right)\right) \]

   7. Applied egg-rr 47.4%

      \[\leadsto \color{blue}{-\frac{\sqrt{2 \cdot \left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}}{B}} \]

   8. Taylor expanded in A around 0

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(-2 \cdot \left(B \cdot F\right)\right)}\right), B\right)\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-2, \left(B \cdot F\right)\right)\right), B\right)\right) \]

      2. *-lowering-*.f64 45.2%

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(B, F\right)\right)\right), B\right)\right) \]

   10. Simplified 45.2%

       \[\leadsto -\frac{\sqrt{\color{blue}{-2 \cdot \left(B \cdot F\right)}}}{B} \]
3. Recombined 2 regimes into one program.

4. Final simplification 26.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 10^{-83}:\\ \;\;\;\;\frac{\sqrt{\left(-4 \cdot \left(A \cdot C\right)\right) \cdot \left(4 \cdot \left(A \cdot F\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{-2 \cdot \left(B \cdot F\right)}}{0 - B}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 32.0% accurate, 5.1× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;F \leq -1.35 \cdot 10^{-307}:\\ \;\;\;\;\frac{\sqrt{-2 \cdot \left(B\_m \cdot F\right)}}{0 - B\_m}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \sqrt{\frac{F \cdot -16 + -4 \cdot \frac{F \cdot \left(B\_m \cdot B\_m\right)}{A \cdot A}}{C}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= F -1.35e-307)
   (/ (sqrt (* -2.0 (* B_m F))) (- 0.0 B_m))
   (*
    0.25
    (sqrt (/ (+ (* F -16.0) (* -4.0 (/ (* F (* B_m B_m)) (* A A)))) C)))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= -1.35e-307) {
		tmp = sqrt((-2.0 * (B_m * F))) / (0.0 - B_m);
	} else {
		tmp = 0.25 * sqrt((((F * -16.0) + (-4.0 * ((F * (B_m * B_m)) / (A * A)))) / C));
	}
	return tmp;
}

Fortran

B_m = abs(b)
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (f <= (-1.35d-307)) then
        tmp = sqrt(((-2.0d0) * (b_m * f))) / (0.0d0 - b_m)
    else
        tmp = 0.25d0 * sqrt((((f * (-16.0d0)) + ((-4.0d0) * ((f * (b_m * b_m)) / (a * a)))) / c))
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= -1.35e-307) {
		tmp = Math.sqrt((-2.0 * (B_m * F))) / (0.0 - B_m);
	} else {
		tmp = 0.25 * Math.sqrt((((F * -16.0) + (-4.0 * ((F * (B_m * B_m)) / (A * A)))) / C));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if F <= -1.35e-307:
		tmp = math.sqrt((-2.0 * (B_m * F))) / (0.0 - B_m)
	else:
		tmp = 0.25 * math.sqrt((((F * -16.0) + (-4.0 * ((F * (B_m * B_m)) / (A * A)))) / C))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (F <= -1.35e-307)
		tmp = Float64(sqrt(Float64(-2.0 * Float64(B_m * F))) / Float64(0.0 - B_m));
	else
		tmp = Float64(0.25 * sqrt(Float64(Float64(Float64(F * -16.0) + Float64(-4.0 * Float64(Float64(F * Float64(B_m * B_m)) / Float64(A * A)))) / C)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (F <= -1.35e-307)
		tmp = sqrt((-2.0 * (B_m * F))) / (0.0 - B_m);
	else
		tmp = 0.25 * sqrt((((F * -16.0) + (-4.0 * ((F * (B_m * B_m)) / (A * A)))) / C));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[F, -1.35e-307], N[(N[Sqrt[N[(-2.0 * N[(B$95$m * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(0.0 - B$95$m), $MachinePrecision]), $MachinePrecision], N[(0.25 * N[Sqrt[N[(N[(N[(F * -16.0), $MachinePrecision] + N[(-4.0 * N[(N[(F * N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision] / N[(A * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / C), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -1.34999999999999993e-307

   1. Initial program 19.3%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in C around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \color{blue}{\sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(-1 \cdot \frac{\sqrt{2}}{B}\right), \color{blue}{\left(\sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \left(\frac{\sqrt{2}}{B}\right)\right), \left(\sqrt{\color{blue}{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\left(\sqrt{2}\right), B\right)\right), \left(\sqrt{F \cdot \color{blue}{\left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \left(\sqrt{F \cdot \left(\color{blue}{A} - \sqrt{{A}^{2} + {B}^{2}}\right)}\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\left(F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \left(\sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \left(\sqrt{A \cdot A + {B}^{2}}\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \left(\sqrt{A \cdot A + B \cdot B}\right)\right)\right)\right)\right) \]

      11. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \left(\mathsf{hypot}\left(A, B\right)\right)\right)\right)\right)\right) \]

      12. hypot-lowering-hypot.f64 18.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \mathsf{hypot.f64}\left(A, B\right)\right)\right)\right)\right) \]

   5. Simplified 18.7%

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)}} \]
   6. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto -1 \cdot \color{blue}{\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{A \cdot A + B \cdot B}\right)}\right)} \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{A \cdot A + B \cdot B}\right)}\right) \]

      3. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{A \cdot A + B \cdot B}\right)}\right)\right) \]

      4. associate-*l/ N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(A - \sqrt{A \cdot A + B \cdot B}\right)}}{B}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\left(\sqrt{2} \cdot \sqrt{F \cdot \left(A - \sqrt{A \cdot A + B \cdot B}\right)}\right), B\right)\right) \]

   7. Applied egg-rr 18.7%

      \[\leadsto \color{blue}{-\frac{\sqrt{2 \cdot \left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}}{B}} \]

   8. Taylor expanded in A around 0

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(-2 \cdot \left(B \cdot F\right)\right)}\right), B\right)\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-2, \left(B \cdot F\right)\right)\right), B\right)\right) \]

      2. *-lowering-*.f64 16.9%

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(B, F\right)\right)\right), B\right)\right) \]

   10. Simplified 16.9%

       \[\leadsto -\frac{\sqrt{\color{blue}{-2 \cdot \left(B \cdot F\right)}}}{B} \]

   if -1.34999999999999993e-307 < F

   1. Initial program 25.9%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Step-by-step derivation

      1. distribute-frac-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\right) \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{\color{blue}{\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)}} \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}\right), \color{blue}{\left(\mathsf{neg}\left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)\right)}\right) \]

   3. Simplified 52.3%

      \[\leadsto \color{blue}{\frac{\sqrt{\left(\left(B \cdot B + C \cdot \left(A \cdot -4\right)\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}} \]
   4. Add Preprocessing

   5. Taylor expanded in A around -inf

      \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left({A}^{2} \cdot \left(-16 \cdot \left(C \cdot F\right) + -1 \cdot \frac{-4 \cdot \left({B}^{2} \cdot F\right) + 4 \cdot \frac{{B}^{2} \cdot \left(C \cdot F\right)}{A}}{A}\right)\right)}\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left({A}^{2}\right), \left(-16 \cdot \left(C \cdot F\right) + -1 \cdot \frac{-4 \cdot \left({B}^{2} \cdot F\right) + 4 \cdot \frac{{B}^{2} \cdot \left(C \cdot F\right)}{A}}{A}\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\color{blue}{\mathsf{*.f64}\left(4, A\right)}, C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(A \cdot A\right), \left(-16 \cdot \left(C \cdot F\right) + -1 \cdot \frac{-4 \cdot \left({B}^{2} \cdot F\right) + 4 \cdot \frac{{B}^{2} \cdot \left(C \cdot F\right)}{A}}{A}\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, A\right), \left(-16 \cdot \left(C \cdot F\right) + -1 \cdot \frac{-4 \cdot \left({B}^{2} \cdot F\right) + 4 \cdot \frac{{B}^{2} \cdot \left(C \cdot F\right)}{A}}{A}\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(\color{blue}{4}, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      4. fma-define N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, A\right), \left(\mathsf{fma}\left(-16, C \cdot F, -1 \cdot \frac{-4 \cdot \left({B}^{2} \cdot F\right) + 4 \cdot \frac{{B}^{2} \cdot \left(C \cdot F\right)}{A}}{A}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, A\right), \left(\mathsf{fma}\left(-16, C \cdot F, \mathsf{neg}\left(\frac{-4 \cdot \left({B}^{2} \cdot F\right) + 4 \cdot \frac{{B}^{2} \cdot \left(C \cdot F\right)}{A}}{A}\right)\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      6. fmm-undef N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, A\right), \left(-16 \cdot \left(C \cdot F\right) - \frac{-4 \cdot \left({B}^{2} \cdot F\right) + 4 \cdot \frac{{B}^{2} \cdot \left(C \cdot F\right)}{A}}{A}\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, A\right), \mathsf{\_.f64}\left(\left(-16 \cdot \left(C \cdot F\right)\right), \left(\frac{-4 \cdot \left({B}^{2} \cdot F\right) + 4 \cdot \frac{{B}^{2} \cdot \left(C \cdot F\right)}{A}}{A}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, \color{blue}{A}\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, A\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(-16, \left(C \cdot F\right)\right), \left(\frac{-4 \cdot \left({B}^{2} \cdot F\right) + 4 \cdot \frac{{B}^{2} \cdot \left(C \cdot F\right)}{A}}{A}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, A\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(C, F\right)\right), \left(\frac{-4 \cdot \left({B}^{2} \cdot F\right) + 4 \cdot \frac{{B}^{2} \cdot \left(C \cdot F\right)}{A}}{A}\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(A, A\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(-16, \mathsf{*.f64}\left(C, F\right)\right), \mathsf{/.f64}\left(\left(-4 \cdot \left({B}^{2} \cdot F\right) + 4 \cdot \frac{{B}^{2} \cdot \left(C \cdot F\right)}{A}\right), A\right)\right)\right)\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(4, A\right), C\right), \mathsf{*.f64}\left(B, B\right)\right)\right) \]

   7. Simplified 23.4%

      \[\leadsto \frac{\sqrt{\color{blue}{\left(A \cdot A\right) \cdot \left(-16 \cdot \left(C \cdot F\right) - \frac{-4 \cdot \left(\left(B \cdot B\right) \cdot F\right) + 4 \cdot \left(\left(B \cdot B\right) \cdot \frac{C \cdot F}{A}\right)}{A}\right)}}}{\left(4 \cdot A\right) \cdot C - B \cdot B} \]

   8. Taylor expanded in C around inf

      \[\leadsto \color{blue}{\frac{1}{4} \cdot \sqrt{\frac{-16 \cdot F - 4 \cdot \frac{{B}^{2} \cdot F}{{A}^{2}}}{C}}} \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \color{blue}{\left(\sqrt{\frac{-16 \cdot F - 4 \cdot \frac{{B}^{2} \cdot F}{{A}^{2}}}{C}}\right)}\right) \]

      2. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{sqrt.f64}\left(\left(\frac{-16 \cdot F - 4 \cdot \frac{{B}^{2} \cdot F}{{A}^{2}}}{C}\right)\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(-16 \cdot F - 4 \cdot \frac{{B}^{2} \cdot F}{{A}^{2}}\right), C\right)\right)\right) \]

      4. cancel-sign-sub-inv N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(-16 \cdot F + \left(\mathsf{neg}\left(4\right)\right) \cdot \frac{{B}^{2} \cdot F}{{A}^{2}}\right), C\right)\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\left(-16 \cdot F + -4 \cdot \frac{{B}^{2} \cdot F}{{A}^{2}}\right), C\right)\right)\right) \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(-16 \cdot F\right), \left(-4 \cdot \frac{{B}^{2} \cdot F}{{A}^{2}}\right)\right), C\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, F\right), \left(-4 \cdot \frac{{B}^{2} \cdot F}{{A}^{2}}\right)\right), C\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, F\right), \mathsf{*.f64}\left(-4, \left(\frac{{B}^{2} \cdot F}{{A}^{2}}\right)\right)\right), C\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, F\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\left({B}^{2} \cdot F\right), \left({A}^{2}\right)\right)\right)\right), C\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, F\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left({B}^{2}\right), F\right), \left({A}^{2}\right)\right)\right)\right), C\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, F\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(B \cdot B\right), F\right), \left({A}^{2}\right)\right)\right)\right), C\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, F\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), F\right), \left({A}^{2}\right)\right)\right)\right), C\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, F\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), F\right), \left(A \cdot A\right)\right)\right)\right), C\right)\right)\right) \]

      14. *-lowering-*.f64 61.3%

          \[\leadsto \mathsf{*.f64}\left(\frac{1}{4}, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(-16, F\right), \mathsf{*.f64}\left(-4, \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), F\right), \mathsf{*.f64}\left(A, A\right)\right)\right)\right), C\right)\right)\right) \]

   10. Simplified 61.3%

       \[\leadsto \color{blue}{0.25 \cdot \sqrt{\frac{-16 \cdot F + -4 \cdot \frac{\left(B \cdot B\right) \cdot F}{A \cdot A}}{C}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 23.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.35 \cdot 10^{-307}:\\ \;\;\;\;\frac{\sqrt{-2 \cdot \left(B \cdot F\right)}}{0 - B}\\ \mathbf{else}:\\ \;\;\;\;0.25 \cdot \sqrt{\frac{F \cdot -16 + -4 \cdot \frac{F \cdot \left(B \cdot B\right)}{A \cdot A}}{C}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 26.6% accurate, 5.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 1.55 \cdot 10^{-83}:\\ \;\;\;\;{\left(0 - \frac{F \cdot \left(B\_m \cdot B\_m\right)}{C}\right)}^{0.5} \cdot \frac{-1}{B\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{-2 \cdot \left(B\_m \cdot F\right)}}{0 - B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 1.55e-83)
   (* (pow (- 0.0 (/ (* F (* B_m B_m)) C)) 0.5) (/ -1.0 B_m))
   (/ (sqrt (* -2.0 (* B_m F))) (- 0.0 B_m))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 1.55e-83) {
		tmp = pow((0.0 - ((F * (B_m * B_m)) / C)), 0.5) * (-1.0 / B_m);
	} else {
		tmp = sqrt((-2.0 * (B_m * F))) / (0.0 - B_m);
	}
	return tmp;
}

Fortran

B_m = abs(b)
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (b_m <= 1.55d-83) then
        tmp = ((0.0d0 - ((f * (b_m * b_m)) / c)) ** 0.5d0) * ((-1.0d0) / b_m)
    else
        tmp = sqrt(((-2.0d0) * (b_m * f))) / (0.0d0 - b_m)
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 1.55e-83) {
		tmp = Math.pow((0.0 - ((F * (B_m * B_m)) / C)), 0.5) * (-1.0 / B_m);
	} else {
		tmp = Math.sqrt((-2.0 * (B_m * F))) / (0.0 - B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 1.55e-83:
		tmp = math.pow((0.0 - ((F * (B_m * B_m)) / C)), 0.5) * (-1.0 / B_m)
	else:
		tmp = math.sqrt((-2.0 * (B_m * F))) / (0.0 - B_m)
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 1.55e-83)
		tmp = Float64((Float64(0.0 - Float64(Float64(F * Float64(B_m * B_m)) / C)) ^ 0.5) * Float64(-1.0 / B_m));
	else
		tmp = Float64(sqrt(Float64(-2.0 * Float64(B_m * F))) / Float64(0.0 - B_m));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (B_m <= 1.55e-83)
		tmp = ((0.0 - ((F * (B_m * B_m)) / C)) ^ 0.5) * (-1.0 / B_m);
	else
		tmp = sqrt((-2.0 * (B_m * F))) / (0.0 - B_m);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 1.55e-83], N[(N[Power[N[(0.0 - N[(N[(F * N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision] * N[(-1.0 / B$95$m), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(-2.0 * N[(B$95$m * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(0.0 - B$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 1.54999999999999996e-83

   1. Initial program 21.0%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \color{blue}{\sqrt{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(-1 \cdot \frac{\sqrt{2}}{B}\right), \color{blue}{\left(\sqrt{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \left(\frac{\sqrt{2}}{B}\right)\right), \left(\sqrt{\color{blue}{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\left(\sqrt{2}\right), B\right)\right), \left(\sqrt{F \cdot \color{blue}{\left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \left(\sqrt{F \cdot \left(\color{blue}{C} - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\left(F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\sqrt{B \cdot B + {C}^{2}}\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\sqrt{B \cdot B + C \cdot C}\right)\right)\right)\right)\right) \]

      11. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\mathsf{hypot}\left(B, C\right)\right)\right)\right)\right)\right) \]

      12. hypot-lowering-hypot.f64 2.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \mathsf{hypot.f64}\left(B, C\right)\right)\right)\right)\right) \]

   5. Simplified 2.7%

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(C - \mathsf{hypot}\left(B, C\right)\right)}} \]
   6. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto -1 \cdot \color{blue}{\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)}\right)} \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)}\right) \]

      3. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)}\right)\right) \]

      4. associate-*l/ N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)}}{B}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\left(\sqrt{2} \cdot \sqrt{F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)}\right), B\right)\right) \]

   7. Applied egg-rr 2.7%

      \[\leadsto \color{blue}{-\frac{{\left(2 \cdot \left(F \cdot \left(C - \mathsf{hypot}\left(B, C\right)\right)\right)\right)}^{0.5}}{B}} \]
   8. Step-by-step derivation

      1. distribute-neg-frac2 N/A

         \[\leadsto \frac{{\left(2 \cdot \left(F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)\right)\right)}^{\frac{1}{2}}}{\color{blue}{\mathsf{neg}\left(B\right)}} \]

      2. div-inv N/A

         \[\leadsto {\left(2 \cdot \left(F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)\right)\right)}^{\frac{1}{2}} \cdot \color{blue}{\frac{1}{\mathsf{neg}\left(B\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(2 \cdot \left(F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)\right)\right)}^{\frac{1}{2}}\right), \color{blue}{\left(\frac{1}{\mathsf{neg}\left(B\right)}\right)}\right) \]

      4. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(2 \cdot \left(F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)\right)\right), \frac{1}{2}\right), \left(\frac{\color{blue}{1}}{\mathsf{neg}\left(B\right)}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\left(F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)\right) \cdot 2\right), \frac{1}{2}\right), \left(\frac{1}{\mathsf{neg}\left(B\right)}\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(F \cdot \left(\left(C - \sqrt{B \cdot B + C \cdot C}\right) \cdot 2\right)\right), \frac{1}{2}\right), \left(\frac{1}{\mathsf{neg}\left(B\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(F, \left(\left(C - \sqrt{B \cdot B + C \cdot C}\right) \cdot 2\right)\right), \frac{1}{2}\right), \left(\frac{1}{\mathsf{neg}\left(B\right)}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(F, \mathsf{*.f64}\left(\left(C - \sqrt{B \cdot B + C \cdot C}\right), 2\right)\right), \frac{1}{2}\right), \left(\frac{1}{\mathsf{neg}\left(B\right)}\right)\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(F, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(C, \left(\sqrt{B \cdot B + C \cdot C}\right)\right), 2\right)\right), \frac{1}{2}\right), \left(\frac{1}{\mathsf{neg}\left(B\right)}\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(F, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(C, \left(\sqrt{C \cdot C + B \cdot B}\right)\right), 2\right)\right), \frac{1}{2}\right), \left(\frac{1}{\mathsf{neg}\left(B\right)}\right)\right) \]

      11. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(F, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(C, \left(\mathsf{hypot}\left(C, B\right)\right)\right), 2\right)\right), \frac{1}{2}\right), \left(\frac{1}{\mathsf{neg}\left(B\right)}\right)\right) \]

      12. hypot-lowering-hypot.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(F, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(C, \mathsf{hypot.f64}\left(C, B\right)\right), 2\right)\right), \frac{1}{2}\right), \left(\frac{1}{\mathsf{neg}\left(B\right)}\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(F, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(C, \mathsf{hypot.f64}\left(C, B\right)\right), 2\right)\right), \frac{1}{2}\right), \left(\frac{\mathsf{neg}\left(-1\right)}{\mathsf{neg}\left(\color{blue}{B}\right)}\right)\right) \]

      14. frac-2neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(F, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(C, \mathsf{hypot.f64}\left(C, B\right)\right), 2\right)\right), \frac{1}{2}\right), \left(\frac{-1}{\color{blue}{B}}\right)\right) \]

      15. /-lowering-/.f64 2.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(F, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(C, \mathsf{hypot.f64}\left(C, B\right)\right), 2\right)\right), \frac{1}{2}\right), \mathsf{/.f64}\left(-1, \color{blue}{B}\right)\right) \]

   9. Applied egg-rr 2.7%

      \[\leadsto \color{blue}{{\left(F \cdot \left(\left(C - \mathsf{hypot}\left(C, B\right)\right) \cdot 2\right)\right)}^{0.5} \cdot \frac{-1}{B}} \]

   10. Taylor expanded in C around inf

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\color{blue}{\left(-1 \cdot \frac{{B}^{2} \cdot F}{C}\right)}, \frac{1}{2}\right), \mathsf{/.f64}\left(-1, B\right)\right) \]
   11. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\mathsf{neg}\left(\frac{{B}^{2} \cdot F}{C}\right)\right), \frac{1}{2}\right), \mathsf{/.f64}\left(-1, B\right)\right) \]

       2. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(0 - \frac{{B}^{2} \cdot F}{C}\right), \frac{1}{2}\right), \mathsf{/.f64}\left(-1, B\right)\right) \]

       3. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(0, \left(\frac{{B}^{2} \cdot F}{C}\right)\right), \frac{1}{2}\right), \mathsf{/.f64}\left(-1, B\right)\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\left({B}^{2} \cdot F\right), C\right)\right), \frac{1}{2}\right), \mathsf{/.f64}\left(-1, B\right)\right) \]

       5. *-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\left(F \cdot {B}^{2}\right), C\right)\right), \frac{1}{2}\right), \mathsf{/.f64}\left(-1, B\right)\right) \]

       6. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \left({B}^{2}\right)\right), C\right)\right), \frac{1}{2}\right), \mathsf{/.f64}\left(-1, B\right)\right) \]

       7. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \left(B \cdot B\right)\right), C\right)\right), \frac{1}{2}\right), \mathsf{/.f64}\left(-1, B\right)\right) \]

       8. *-lowering-*.f64 5.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{*.f64}\left(B, B\right)\right), C\right)\right), \frac{1}{2}\right), \mathsf{/.f64}\left(-1, B\right)\right) \]

   12. Simplified 5.8%

       \[\leadsto {\color{blue}{\left(0 - \frac{F \cdot \left(B \cdot B\right)}{C}\right)}}^{0.5} \cdot \frac{-1}{B} \]

   if 1.54999999999999996e-83 < B

   1. Initial program 18.5%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in C around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \color{blue}{\sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(-1 \cdot \frac{\sqrt{2}}{B}\right), \color{blue}{\left(\sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \left(\frac{\sqrt{2}}{B}\right)\right), \left(\sqrt{\color{blue}{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\left(\sqrt{2}\right), B\right)\right), \left(\sqrt{F \cdot \color{blue}{\left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \left(\sqrt{F \cdot \left(\color{blue}{A} - \sqrt{{A}^{2} + {B}^{2}}\right)}\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\left(F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \left(\sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \left(\sqrt{A \cdot A + {B}^{2}}\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \left(\sqrt{A \cdot A + B \cdot B}\right)\right)\right)\right)\right) \]

      11. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \left(\mathsf{hypot}\left(A, B\right)\right)\right)\right)\right)\right) \]

      12. hypot-lowering-hypot.f64 47.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \mathsf{hypot.f64}\left(A, B\right)\right)\right)\right)\right) \]

   5. Simplified 47.8%

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)}} \]
   6. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto -1 \cdot \color{blue}{\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{A \cdot A + B \cdot B}\right)}\right)} \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{A \cdot A + B \cdot B}\right)}\right) \]

      3. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{A \cdot A + B \cdot B}\right)}\right)\right) \]

      4. associate-*l/ N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(A - \sqrt{A \cdot A + B \cdot B}\right)}}{B}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\left(\sqrt{2} \cdot \sqrt{F \cdot \left(A - \sqrt{A \cdot A + B \cdot B}\right)}\right), B\right)\right) \]

   7. Applied egg-rr 47.9%

      \[\leadsto \color{blue}{-\frac{\sqrt{2 \cdot \left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}}{B}} \]

   8. Taylor expanded in A around 0

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(-2 \cdot \left(B \cdot F\right)\right)}\right), B\right)\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-2, \left(B \cdot F\right)\right)\right), B\right)\right) \]

      2. *-lowering-*.f64 45.7%

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(B, F\right)\right)\right), B\right)\right) \]

   10. Simplified 45.7%

       \[\leadsto -\frac{\sqrt{\color{blue}{-2 \cdot \left(B \cdot F\right)}}}{B} \]
3. Recombined 2 regimes into one program.

4. Final simplification 17.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.55 \cdot 10^{-83}:\\ \;\;\;\;{\left(0 - \frac{F \cdot \left(B \cdot B\right)}{C}\right)}^{0.5} \cdot \frac{-1}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{-2 \cdot \left(B \cdot F\right)}}{0 - B}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 26.5% accurate, 5.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 1.8 \cdot 10^{-83}:\\ \;\;\;\;\frac{-1}{B\_m} \cdot {\left(0 - F \cdot \frac{B\_m \cdot B\_m}{C}\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{-2 \cdot \left(B\_m \cdot F\right)}}{0 - B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 1.8e-83)
   (* (/ -1.0 B_m) (pow (- 0.0 (* F (/ (* B_m B_m) C))) 0.5))
   (/ (sqrt (* -2.0 (* B_m F))) (- 0.0 B_m))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 1.8e-83) {
		tmp = (-1.0 / B_m) * pow((0.0 - (F * ((B_m * B_m) / C))), 0.5);
	} else {
		tmp = sqrt((-2.0 * (B_m * F))) / (0.0 - B_m);
	}
	return tmp;
}

Fortran

B_m = abs(b)
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (b_m <= 1.8d-83) then
        tmp = ((-1.0d0) / b_m) * ((0.0d0 - (f * ((b_m * b_m) / c))) ** 0.5d0)
    else
        tmp = sqrt(((-2.0d0) * (b_m * f))) / (0.0d0 - b_m)
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 1.8e-83) {
		tmp = (-1.0 / B_m) * Math.pow((0.0 - (F * ((B_m * B_m) / C))), 0.5);
	} else {
		tmp = Math.sqrt((-2.0 * (B_m * F))) / (0.0 - B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 1.8e-83:
		tmp = (-1.0 / B_m) * math.pow((0.0 - (F * ((B_m * B_m) / C))), 0.5)
	else:
		tmp = math.sqrt((-2.0 * (B_m * F))) / (0.0 - B_m)
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 1.8e-83)
		tmp = Float64(Float64(-1.0 / B_m) * (Float64(0.0 - Float64(F * Float64(Float64(B_m * B_m) / C))) ^ 0.5));
	else
		tmp = Float64(sqrt(Float64(-2.0 * Float64(B_m * F))) / Float64(0.0 - B_m));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (B_m <= 1.8e-83)
		tmp = (-1.0 / B_m) * ((0.0 - (F * ((B_m * B_m) / C))) ^ 0.5);
	else
		tmp = sqrt((-2.0 * (B_m * F))) / (0.0 - B_m);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 1.8e-83], N[(N[(-1.0 / B$95$m), $MachinePrecision] * N[Power[N[(0.0 - N[(F * N[(N[(B$95$m * B$95$m), $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(-2.0 * N[(B$95$m * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(0.0 - B$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 1.80000000000000006e-83

   1. Initial program 21.0%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \color{blue}{\sqrt{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(-1 \cdot \frac{\sqrt{2}}{B}\right), \color{blue}{\left(\sqrt{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \left(\frac{\sqrt{2}}{B}\right)\right), \left(\sqrt{\color{blue}{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\left(\sqrt{2}\right), B\right)\right), \left(\sqrt{F \cdot \color{blue}{\left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \left(\sqrt{F \cdot \left(\color{blue}{C} - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\left(F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\sqrt{B \cdot B + {C}^{2}}\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\sqrt{B \cdot B + C \cdot C}\right)\right)\right)\right)\right) \]

      11. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\mathsf{hypot}\left(B, C\right)\right)\right)\right)\right)\right) \]

      12. hypot-lowering-hypot.f64 2.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \mathsf{hypot.f64}\left(B, C\right)\right)\right)\right)\right) \]

   5. Simplified 2.7%

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(C - \mathsf{hypot}\left(B, C\right)\right)}} \]
   6. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto -1 \cdot \color{blue}{\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)}\right)} \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)}\right) \]

      3. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)}\right)\right) \]

      4. associate-*l/ N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)}}{B}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\left(\sqrt{2} \cdot \sqrt{F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)}\right), B\right)\right) \]

   7. Applied egg-rr 2.7%

      \[\leadsto \color{blue}{-\frac{{\left(2 \cdot \left(F \cdot \left(C - \mathsf{hypot}\left(B, C\right)\right)\right)\right)}^{0.5}}{B}} \]
   8. Step-by-step derivation

      1. distribute-neg-frac2 N/A

         \[\leadsto \frac{{\left(2 \cdot \left(F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)\right)\right)}^{\frac{1}{2}}}{\color{blue}{\mathsf{neg}\left(B\right)}} \]

      2. div-inv N/A

         \[\leadsto {\left(2 \cdot \left(F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)\right)\right)}^{\frac{1}{2}} \cdot \color{blue}{\frac{1}{\mathsf{neg}\left(B\right)}} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left({\left(2 \cdot \left(F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)\right)\right)}^{\frac{1}{2}}\right), \color{blue}{\left(\frac{1}{\mathsf{neg}\left(B\right)}\right)}\right) \]

      4. pow-lowering-pow.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(2 \cdot \left(F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)\right)\right), \frac{1}{2}\right), \left(\frac{\color{blue}{1}}{\mathsf{neg}\left(B\right)}\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(\left(F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)\right) \cdot 2\right), \frac{1}{2}\right), \left(\frac{1}{\mathsf{neg}\left(B\right)}\right)\right) \]

      6. associate-*l* N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\left(F \cdot \left(\left(C - \sqrt{B \cdot B + C \cdot C}\right) \cdot 2\right)\right), \frac{1}{2}\right), \left(\frac{1}{\mathsf{neg}\left(B\right)}\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(F, \left(\left(C - \sqrt{B \cdot B + C \cdot C}\right) \cdot 2\right)\right), \frac{1}{2}\right), \left(\frac{1}{\mathsf{neg}\left(B\right)}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(F, \mathsf{*.f64}\left(\left(C - \sqrt{B \cdot B + C \cdot C}\right), 2\right)\right), \frac{1}{2}\right), \left(\frac{1}{\mathsf{neg}\left(B\right)}\right)\right) \]

      9. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(F, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(C, \left(\sqrt{B \cdot B + C \cdot C}\right)\right), 2\right)\right), \frac{1}{2}\right), \left(\frac{1}{\mathsf{neg}\left(B\right)}\right)\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(F, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(C, \left(\sqrt{C \cdot C + B \cdot B}\right)\right), 2\right)\right), \frac{1}{2}\right), \left(\frac{1}{\mathsf{neg}\left(B\right)}\right)\right) \]

      11. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(F, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(C, \left(\mathsf{hypot}\left(C, B\right)\right)\right), 2\right)\right), \frac{1}{2}\right), \left(\frac{1}{\mathsf{neg}\left(B\right)}\right)\right) \]

      12. hypot-lowering-hypot.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(F, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(C, \mathsf{hypot.f64}\left(C, B\right)\right), 2\right)\right), \frac{1}{2}\right), \left(\frac{1}{\mathsf{neg}\left(B\right)}\right)\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(F, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(C, \mathsf{hypot.f64}\left(C, B\right)\right), 2\right)\right), \frac{1}{2}\right), \left(\frac{\mathsf{neg}\left(-1\right)}{\mathsf{neg}\left(\color{blue}{B}\right)}\right)\right) \]

      14. frac-2neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(F, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(C, \mathsf{hypot.f64}\left(C, B\right)\right), 2\right)\right), \frac{1}{2}\right), \left(\frac{-1}{\color{blue}{B}}\right)\right) \]

      15. /-lowering-/.f64 2.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(F, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(C, \mathsf{hypot.f64}\left(C, B\right)\right), 2\right)\right), \frac{1}{2}\right), \mathsf{/.f64}\left(-1, \color{blue}{B}\right)\right) \]

   9. Applied egg-rr 2.7%

      \[\leadsto \color{blue}{{\left(F \cdot \left(\left(C - \mathsf{hypot}\left(C, B\right)\right) \cdot 2\right)\right)}^{0.5} \cdot \frac{-1}{B}} \]

   10. Taylor expanded in C around inf

       \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(F, \color{blue}{\left(-1 \cdot \frac{{B}^{2}}{C}\right)}\right), \frac{1}{2}\right), \mathsf{/.f64}\left(-1, B\right)\right) \]
   11. Step-by-step derivation

       1. mul-1-neg N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(F, \left(\mathsf{neg}\left(\frac{{B}^{2}}{C}\right)\right)\right), \frac{1}{2}\right), \mathsf{/.f64}\left(-1, B\right)\right) \]

       2. neg-sub0 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(F, \left(0 - \frac{{B}^{2}}{C}\right)\right), \frac{1}{2}\right), \mathsf{/.f64}\left(-1, B\right)\right) \]

       3. --lowering--.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(0, \left(\frac{{B}^{2}}{C}\right)\right)\right), \frac{1}{2}\right), \mathsf{/.f64}\left(-1, B\right)\right) \]

       4. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\left({B}^{2}\right), C\right)\right)\right), \frac{1}{2}\right), \mathsf{/.f64}\left(-1, B\right)\right) \]

       5. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\left(B \cdot B\right), C\right)\right)\right), \frac{1}{2}\right), \mathsf{/.f64}\left(-1, B\right)\right) \]

       6. *-lowering-*.f64 6.2%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(0, \mathsf{/.f64}\left(\mathsf{*.f64}\left(B, B\right), C\right)\right)\right), \frac{1}{2}\right), \mathsf{/.f64}\left(-1, B\right)\right) \]

   12. Simplified 6.2%

       \[\leadsto {\left(F \cdot \color{blue}{\left(0 - \frac{B \cdot B}{C}\right)}\right)}^{0.5} \cdot \frac{-1}{B} \]

   if 1.80000000000000006e-83 < B

   1. Initial program 18.5%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in C around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \color{blue}{\sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(-1 \cdot \frac{\sqrt{2}}{B}\right), \color{blue}{\left(\sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \left(\frac{\sqrt{2}}{B}\right)\right), \left(\sqrt{\color{blue}{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\left(\sqrt{2}\right), B\right)\right), \left(\sqrt{F \cdot \color{blue}{\left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \left(\sqrt{F \cdot \left(\color{blue}{A} - \sqrt{{A}^{2} + {B}^{2}}\right)}\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\left(F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \left(\sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \left(\sqrt{A \cdot A + {B}^{2}}\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \left(\sqrt{A \cdot A + B \cdot B}\right)\right)\right)\right)\right) \]

      11. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \left(\mathsf{hypot}\left(A, B\right)\right)\right)\right)\right)\right) \]

      12. hypot-lowering-hypot.f64 47.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \mathsf{hypot.f64}\left(A, B\right)\right)\right)\right)\right) \]

   5. Simplified 47.8%

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)}} \]
   6. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto -1 \cdot \color{blue}{\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{A \cdot A + B \cdot B}\right)}\right)} \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{A \cdot A + B \cdot B}\right)}\right) \]

      3. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{A \cdot A + B \cdot B}\right)}\right)\right) \]

      4. associate-*l/ N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(A - \sqrt{A \cdot A + B \cdot B}\right)}}{B}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\left(\sqrt{2} \cdot \sqrt{F \cdot \left(A - \sqrt{A \cdot A + B \cdot B}\right)}\right), B\right)\right) \]

   7. Applied egg-rr 47.9%

      \[\leadsto \color{blue}{-\frac{\sqrt{2 \cdot \left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}}{B}} \]

   8. Taylor expanded in A around 0

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(-2 \cdot \left(B \cdot F\right)\right)}\right), B\right)\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-2, \left(B \cdot F\right)\right)\right), B\right)\right) \]

      2. *-lowering-*.f64 45.7%

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(B, F\right)\right)\right), B\right)\right) \]

   10. Simplified 45.7%

       \[\leadsto -\frac{\sqrt{\color{blue}{-2 \cdot \left(B \cdot F\right)}}}{B} \]
3. Recombined 2 regimes into one program.

4. Final simplification 17.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.8 \cdot 10^{-83}:\\ \;\;\;\;\frac{-1}{B} \cdot {\left(0 - F \cdot \frac{B \cdot B}{C}\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{-2 \cdot \left(B \cdot F\right)}}{0 - B}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 26.4% accurate, 5.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 3.25 \cdot 10^{-83}:\\ \;\;\;\;\frac{{\left(\left(B\_m \cdot B\_m\right) \cdot \left(0 - \frac{F}{C}\right)\right)}^{0.5}}{0 - B\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{-2 \cdot \left(B\_m \cdot F\right)}}{0 - B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 3.25e-83)
   (/ (pow (* (* B_m B_m) (- 0.0 (/ F C))) 0.5) (- 0.0 B_m))
   (/ (sqrt (* -2.0 (* B_m F))) (- 0.0 B_m))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 3.25e-83) {
		tmp = pow(((B_m * B_m) * (0.0 - (F / C))), 0.5) / (0.0 - B_m);
	} else {
		tmp = sqrt((-2.0 * (B_m * F))) / (0.0 - B_m);
	}
	return tmp;
}

Fortran

B_m = abs(b)
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (b_m <= 3.25d-83) then
        tmp = (((b_m * b_m) * (0.0d0 - (f / c))) ** 0.5d0) / (0.0d0 - b_m)
    else
        tmp = sqrt(((-2.0d0) * (b_m * f))) / (0.0d0 - b_m)
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 3.25e-83) {
		tmp = Math.pow(((B_m * B_m) * (0.0 - (F / C))), 0.5) / (0.0 - B_m);
	} else {
		tmp = Math.sqrt((-2.0 * (B_m * F))) / (0.0 - B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 3.25e-83:
		tmp = math.pow(((B_m * B_m) * (0.0 - (F / C))), 0.5) / (0.0 - B_m)
	else:
		tmp = math.sqrt((-2.0 * (B_m * F))) / (0.0 - B_m)
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 3.25e-83)
		tmp = Float64((Float64(Float64(B_m * B_m) * Float64(0.0 - Float64(F / C))) ^ 0.5) / Float64(0.0 - B_m));
	else
		tmp = Float64(sqrt(Float64(-2.0 * Float64(B_m * F))) / Float64(0.0 - B_m));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (B_m <= 3.25e-83)
		tmp = (((B_m * B_m) * (0.0 - (F / C))) ^ 0.5) / (0.0 - B_m);
	else
		tmp = sqrt((-2.0 * (B_m * F))) / (0.0 - B_m);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 3.25e-83], N[(N[Power[N[(N[(B$95$m * B$95$m), $MachinePrecision] * N[(0.0 - N[(F / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision] / N[(0.0 - B$95$m), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(-2.0 * N[(B$95$m * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(0.0 - B$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 3.25e-83

   1. Initial program 21.0%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \color{blue}{\sqrt{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(-1 \cdot \frac{\sqrt{2}}{B}\right), \color{blue}{\left(\sqrt{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \left(\frac{\sqrt{2}}{B}\right)\right), \left(\sqrt{\color{blue}{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\left(\sqrt{2}\right), B\right)\right), \left(\sqrt{F \cdot \color{blue}{\left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \left(\sqrt{F \cdot \left(\color{blue}{C} - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\left(F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\sqrt{B \cdot B + {C}^{2}}\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\sqrt{B \cdot B + C \cdot C}\right)\right)\right)\right)\right) \]

      11. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\mathsf{hypot}\left(B, C\right)\right)\right)\right)\right)\right) \]

      12. hypot-lowering-hypot.f64 2.7%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \mathsf{hypot.f64}\left(B, C\right)\right)\right)\right)\right) \]

   5. Simplified 2.7%

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(C - \mathsf{hypot}\left(B, C\right)\right)}} \]
   6. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto -1 \cdot \color{blue}{\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)}\right)} \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)}\right) \]

      3. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)}\right)\right) \]

      4. associate-*l/ N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)}}{B}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\left(\sqrt{2} \cdot \sqrt{F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)}\right), B\right)\right) \]

   7. Applied egg-rr 2.7%

      \[\leadsto \color{blue}{-\frac{{\left(2 \cdot \left(F \cdot \left(C - \mathsf{hypot}\left(B, C\right)\right)\right)\right)}^{0.5}}{B}} \]

   8. Taylor expanded in C around inf

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\color{blue}{\left(-1 \cdot \frac{{B}^{2} \cdot F}{C}\right)}, \frac{1}{2}\right), B\right)\right) \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(\mathsf{neg}\left(\frac{{B}^{2} \cdot F}{C}\right)\right), \frac{1}{2}\right), B\right)\right) \]

      2. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{neg.f64}\left(\left(\frac{{B}^{2} \cdot F}{C}\right)\right), \frac{1}{2}\right), B\right)\right) \]

      3. associate-/l* N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{neg.f64}\left(\left({B}^{2} \cdot \frac{F}{C}\right)\right), \frac{1}{2}\right), B\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left({B}^{2}\right), \left(\frac{F}{C}\right)\right)\right), \frac{1}{2}\right), B\right)\right) \]

      5. unpow2 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(B \cdot B\right), \left(\frac{F}{C}\right)\right)\right), \frac{1}{2}\right), B\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \left(\frac{F}{C}\right)\right)\right), \frac{1}{2}\right), B\right)\right) \]

      7. /-lowering-/.f64 6.2%

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(B, B\right), \mathsf{/.f64}\left(F, C\right)\right)\right), \frac{1}{2}\right), B\right)\right) \]

   10. Simplified 6.2%

       \[\leadsto -\frac{{\color{blue}{\left(-\left(B \cdot B\right) \cdot \frac{F}{C}\right)}}^{0.5}}{B} \]

   if 3.25e-83 < B

   1. Initial program 18.5%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in C around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \color{blue}{\sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(-1 \cdot \frac{\sqrt{2}}{B}\right), \color{blue}{\left(\sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \left(\frac{\sqrt{2}}{B}\right)\right), \left(\sqrt{\color{blue}{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\left(\sqrt{2}\right), B\right)\right), \left(\sqrt{F \cdot \color{blue}{\left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \left(\sqrt{F \cdot \left(\color{blue}{A} - \sqrt{{A}^{2} + {B}^{2}}\right)}\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\left(F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \left(\sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \left(\sqrt{A \cdot A + {B}^{2}}\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \left(\sqrt{A \cdot A + B \cdot B}\right)\right)\right)\right)\right) \]

      11. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \left(\mathsf{hypot}\left(A, B\right)\right)\right)\right)\right)\right) \]

      12. hypot-lowering-hypot.f64 47.8%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \mathsf{hypot.f64}\left(A, B\right)\right)\right)\right)\right) \]

   5. Simplified 47.8%

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)}} \]
   6. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto -1 \cdot \color{blue}{\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{A \cdot A + B \cdot B}\right)}\right)} \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{A \cdot A + B \cdot B}\right)}\right) \]

      3. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{A \cdot A + B \cdot B}\right)}\right)\right) \]

      4. associate-*l/ N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(A - \sqrt{A \cdot A + B \cdot B}\right)}}{B}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\left(\sqrt{2} \cdot \sqrt{F \cdot \left(A - \sqrt{A \cdot A + B \cdot B}\right)}\right), B\right)\right) \]

   7. Applied egg-rr 47.9%

      \[\leadsto \color{blue}{-\frac{\sqrt{2 \cdot \left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}}{B}} \]

   8. Taylor expanded in A around 0

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(-2 \cdot \left(B \cdot F\right)\right)}\right), B\right)\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-2, \left(B \cdot F\right)\right)\right), B\right)\right) \]

      2. *-lowering-*.f64 45.7%

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(B, F\right)\right)\right), B\right)\right) \]

   10. Simplified 45.7%

       \[\leadsto -\frac{\sqrt{\color{blue}{-2 \cdot \left(B \cdot F\right)}}}{B} \]
3. Recombined 2 regimes into one program.

4. Final simplification 17.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 3.25 \cdot 10^{-83}:\\ \;\;\;\;\frac{{\left(\left(B \cdot B\right) \cdot \left(0 - \frac{F}{C}\right)\right)}^{0.5}}{0 - B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{-2 \cdot \left(B \cdot F\right)}}{0 - B}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 27.1% accurate, 5.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;C \leq -3.3 \cdot 10^{+133}:\\ \;\;\;\;\frac{{\left(4 \cdot \left(C \cdot F\right)\right)}^{0.5}}{0 - B\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{-2 \cdot \left(B\_m \cdot F\right)}}{0 - B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= C -3.3e+133)
   (/ (pow (* 4.0 (* C F)) 0.5) (- 0.0 B_m))
   (/ (sqrt (* -2.0 (* B_m F))) (- 0.0 B_m))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (C <= -3.3e+133) {
		tmp = pow((4.0 * (C * F)), 0.5) / (0.0 - B_m);
	} else {
		tmp = sqrt((-2.0 * (B_m * F))) / (0.0 - B_m);
	}
	return tmp;
}

Fortran

B_m = abs(b)
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (c <= (-3.3d+133)) then
        tmp = ((4.0d0 * (c * f)) ** 0.5d0) / (0.0d0 - b_m)
    else
        tmp = sqrt(((-2.0d0) * (b_m * f))) / (0.0d0 - b_m)
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (C <= -3.3e+133) {
		tmp = Math.pow((4.0 * (C * F)), 0.5) / (0.0 - B_m);
	} else {
		tmp = Math.sqrt((-2.0 * (B_m * F))) / (0.0 - B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if C <= -3.3e+133:
		tmp = math.pow((4.0 * (C * F)), 0.5) / (0.0 - B_m)
	else:
		tmp = math.sqrt((-2.0 * (B_m * F))) / (0.0 - B_m)
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (C <= -3.3e+133)
		tmp = Float64((Float64(4.0 * Float64(C * F)) ^ 0.5) / Float64(0.0 - B_m));
	else
		tmp = Float64(sqrt(Float64(-2.0 * Float64(B_m * F))) / Float64(0.0 - B_m));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (C <= -3.3e+133)
		tmp = ((4.0 * (C * F)) ^ 0.5) / (0.0 - B_m);
	else
		tmp = sqrt((-2.0 * (B_m * F))) / (0.0 - B_m);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[C, -3.3e+133], N[(N[Power[N[(4.0 * N[(C * F), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision] / N[(0.0 - B$95$m), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(-2.0 * N[(B$95$m * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(0.0 - B$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if C < -3.3e133

   1. Initial program 8.2%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \color{blue}{\sqrt{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(-1 \cdot \frac{\sqrt{2}}{B}\right), \color{blue}{\left(\sqrt{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \left(\frac{\sqrt{2}}{B}\right)\right), \left(\sqrt{\color{blue}{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\left(\sqrt{2}\right), B\right)\right), \left(\sqrt{F \cdot \color{blue}{\left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \left(\sqrt{F \cdot \left(\color{blue}{C} - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\left(F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\sqrt{B \cdot B + {C}^{2}}\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\sqrt{B \cdot B + C \cdot C}\right)\right)\right)\right)\right) \]

      11. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\mathsf{hypot}\left(B, C\right)\right)\right)\right)\right)\right) \]

      12. hypot-lowering-hypot.f64 7.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \mathsf{hypot.f64}\left(B, C\right)\right)\right)\right)\right) \]

   5. Simplified 7.3%

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(C - \mathsf{hypot}\left(B, C\right)\right)}} \]
   6. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto -1 \cdot \color{blue}{\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)}\right)} \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)}\right) \]

      3. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)}\right)\right) \]

      4. associate-*l/ N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)}}{B}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\left(\sqrt{2} \cdot \sqrt{F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)}\right), B\right)\right) \]

   7. Applied egg-rr 7.6%

      \[\leadsto \color{blue}{-\frac{{\left(2 \cdot \left(F \cdot \left(C - \mathsf{hypot}\left(B, C\right)\right)\right)\right)}^{0.5}}{B}} \]

   8. Taylor expanded in C around -inf

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\color{blue}{\left(4 \cdot \left(C \cdot F\right)\right)}, \frac{1}{2}\right), B\right)\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(4, \left(C \cdot F\right)\right), \frac{1}{2}\right), B\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(4, \left(F \cdot C\right)\right), \frac{1}{2}\right), B\right)\right) \]

      3. *-lowering-*.f64 7.5%

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(4, \mathsf{*.f64}\left(F, C\right)\right), \frac{1}{2}\right), B\right)\right) \]

   10. Simplified 7.5%

       \[\leadsto -\frac{{\color{blue}{\left(4 \cdot \left(F \cdot C\right)\right)}}^{0.5}}{B} \]

   if -3.3e133 < C

   1. Initial program 22.0%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in C around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \color{blue}{\sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(-1 \cdot \frac{\sqrt{2}}{B}\right), \color{blue}{\left(\sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \left(\frac{\sqrt{2}}{B}\right)\right), \left(\sqrt{\color{blue}{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\left(\sqrt{2}\right), B\right)\right), \left(\sqrt{F \cdot \color{blue}{\left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \left(\sqrt{F \cdot \left(\color{blue}{A} - \sqrt{{A}^{2} + {B}^{2}}\right)}\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\left(F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \left(\sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \left(\sqrt{A \cdot A + {B}^{2}}\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \left(\sqrt{A \cdot A + B \cdot B}\right)\right)\right)\right)\right) \]

      11. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \left(\mathsf{hypot}\left(A, B\right)\right)\right)\right)\right)\right) \]

      12. hypot-lowering-hypot.f64 17.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \mathsf{hypot.f64}\left(A, B\right)\right)\right)\right)\right) \]

   5. Simplified 17.9%

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)}} \]
   6. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto -1 \cdot \color{blue}{\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{A \cdot A + B \cdot B}\right)}\right)} \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{A \cdot A + B \cdot B}\right)}\right) \]

      3. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{A \cdot A + B \cdot B}\right)}\right)\right) \]

      4. associate-*l/ N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(A - \sqrt{A \cdot A + B \cdot B}\right)}}{B}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\left(\sqrt{2} \cdot \sqrt{F \cdot \left(A - \sqrt{A \cdot A + B \cdot B}\right)}\right), B\right)\right) \]

   7. Applied egg-rr 18.0%

      \[\leadsto \color{blue}{-\frac{\sqrt{2 \cdot \left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}}{B}} \]

   8. Taylor expanded in A around 0

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(-2 \cdot \left(B \cdot F\right)\right)}\right), B\right)\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-2, \left(B \cdot F\right)\right)\right), B\right)\right) \]

      2. *-lowering-*.f64 16.6%

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(B, F\right)\right)\right), B\right)\right) \]

   10. Simplified 16.6%

       \[\leadsto -\frac{\sqrt{\color{blue}{-2 \cdot \left(B \cdot F\right)}}}{B} \]
3. Recombined 2 regimes into one program.

4. Final simplification 15.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq -3.3 \cdot 10^{+133}:\\ \;\;\;\;\frac{{\left(4 \cdot \left(C \cdot F\right)\right)}^{0.5}}{0 - B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{-2 \cdot \left(B \cdot F\right)}}{0 - B}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 27.1% accurate, 5.6× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;C \leq -8 \cdot 10^{+134}:\\ \;\;\;\;\frac{2 \cdot \sqrt{C \cdot F}}{0 - B\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{-2 \cdot \left(B\_m \cdot F\right)}}{0 - B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= C -8e+134)
   (/ (* 2.0 (sqrt (* C F))) (- 0.0 B_m))
   (/ (sqrt (* -2.0 (* B_m F))) (- 0.0 B_m))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (C <= -8e+134) {
		tmp = (2.0 * sqrt((C * F))) / (0.0 - B_m);
	} else {
		tmp = sqrt((-2.0 * (B_m * F))) / (0.0 - B_m);
	}
	return tmp;
}

Fortran

B_m = abs(b)
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (c <= (-8d+134)) then
        tmp = (2.0d0 * sqrt((c * f))) / (0.0d0 - b_m)
    else
        tmp = sqrt(((-2.0d0) * (b_m * f))) / (0.0d0 - b_m)
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (C <= -8e+134) {
		tmp = (2.0 * Math.sqrt((C * F))) / (0.0 - B_m);
	} else {
		tmp = Math.sqrt((-2.0 * (B_m * F))) / (0.0 - B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if C <= -8e+134:
		tmp = (2.0 * math.sqrt((C * F))) / (0.0 - B_m)
	else:
		tmp = math.sqrt((-2.0 * (B_m * F))) / (0.0 - B_m)
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (C <= -8e+134)
		tmp = Float64(Float64(2.0 * sqrt(Float64(C * F))) / Float64(0.0 - B_m));
	else
		tmp = Float64(sqrt(Float64(-2.0 * Float64(B_m * F))) / Float64(0.0 - B_m));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (C <= -8e+134)
		tmp = (2.0 * sqrt((C * F))) / (0.0 - B_m);
	else
		tmp = sqrt((-2.0 * (B_m * F))) / (0.0 - B_m);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[C, -8e+134], N[(N[(2.0 * N[Sqrt[N[(C * F), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(0.0 - B$95$m), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(-2.0 * N[(B$95$m * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(0.0 - B$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if C < -7.99999999999999937e134

   1. Initial program 8.2%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in A around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \color{blue}{\sqrt{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(-1 \cdot \frac{\sqrt{2}}{B}\right), \color{blue}{\left(\sqrt{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \left(\frac{\sqrt{2}}{B}\right)\right), \left(\sqrt{\color{blue}{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\left(\sqrt{2}\right), B\right)\right), \left(\sqrt{F \cdot \color{blue}{\left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \left(\sqrt{F \cdot \left(\color{blue}{C} - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\left(F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\sqrt{B \cdot B + {C}^{2}}\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\sqrt{B \cdot B + C \cdot C}\right)\right)\right)\right)\right) \]

      11. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\mathsf{hypot}\left(B, C\right)\right)\right)\right)\right)\right) \]

      12. hypot-lowering-hypot.f64 7.3%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \mathsf{hypot.f64}\left(B, C\right)\right)\right)\right)\right) \]

   5. Simplified 7.3%

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(C - \mathsf{hypot}\left(B, C\right)\right)}} \]
   6. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto -1 \cdot \color{blue}{\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)}\right)} \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)}\right) \]

      3. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)}\right)\right) \]

      4. associate-*l/ N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)}}{B}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\left(\sqrt{2} \cdot \sqrt{F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)}\right), B\right)\right) \]

   7. Applied egg-rr 7.6%

      \[\leadsto \color{blue}{-\frac{{\left(2 \cdot \left(F \cdot \left(C - \mathsf{hypot}\left(B, C\right)\right)\right)\right)}^{0.5}}{B}} \]

   8. Taylor expanded in C around -inf

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\color{blue}{\left(-1 \cdot \left(C \cdot \left(-4 \cdot F + -1 \cdot \frac{{B}^{2} \cdot F}{{C}^{2}}\right)\right)\right)}, \frac{1}{2}\right), B\right)\right) \]
   9. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(\left(-1 \cdot C\right) \cdot \left(-4 \cdot F + -1 \cdot \frac{{B}^{2} \cdot F}{{C}^{2}}\right)\right), \frac{1}{2}\right), B\right)\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\left(-1 \cdot C\right), \left(-4 \cdot F + -1 \cdot \frac{{B}^{2} \cdot F}{{C}^{2}}\right)\right), \frac{1}{2}\right), B\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, C\right), \left(-4 \cdot F + -1 \cdot \frac{{B}^{2} \cdot F}{{C}^{2}}\right)\right), \frac{1}{2}\right), B\right)\right) \]

      4. fma-define N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, C\right), \left(\mathsf{fma}\left(-4, F, -1 \cdot \frac{{B}^{2} \cdot F}{{C}^{2}}\right)\right)\right), \frac{1}{2}\right), B\right)\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, C\right), \left(\mathsf{fma}\left(-4, F, \mathsf{neg}\left(\frac{{B}^{2} \cdot F}{{C}^{2}}\right)\right)\right)\right), \frac{1}{2}\right), B\right)\right) \]

      6. fmm-undef N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, C\right), \left(-4 \cdot F - \frac{{B}^{2} \cdot F}{{C}^{2}}\right)\right), \frac{1}{2}\right), B\right)\right) \]

      7. --lowering--.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, C\right), \mathsf{\_.f64}\left(\left(-4 \cdot F\right), \left(\frac{{B}^{2} \cdot F}{{C}^{2}}\right)\right)\right), \frac{1}{2}\right), B\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, C\right), \mathsf{\_.f64}\left(\left(F \cdot -4\right), \left(\frac{{B}^{2} \cdot F}{{C}^{2}}\right)\right)\right), \frac{1}{2}\right), B\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, C\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, -4\right), \left(\frac{{B}^{2} \cdot F}{{C}^{2}}\right)\right)\right), \frac{1}{2}\right), B\right)\right) \]

      10. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, C\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, -4\right), \mathsf{/.f64}\left(\left({B}^{2} \cdot F\right), \left({C}^{2}\right)\right)\right)\right), \frac{1}{2}\right), B\right)\right) \]

      11. *-commutative N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, C\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, -4\right), \mathsf{/.f64}\left(\left(F \cdot {B}^{2}\right), \left({C}^{2}\right)\right)\right)\right), \frac{1}{2}\right), B\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, C\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, -4\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \left({B}^{2}\right)\right), \left({C}^{2}\right)\right)\right)\right), \frac{1}{2}\right), B\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, C\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, -4\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \left(B \cdot B\right)\right), \left({C}^{2}\right)\right)\right)\right), \frac{1}{2}\right), B\right)\right) \]

      14. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, C\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, -4\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{*.f64}\left(B, B\right)\right), \left({C}^{2}\right)\right)\right)\right), \frac{1}{2}\right), B\right)\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, C\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, -4\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{*.f64}\left(B, B\right)\right), \left(C \cdot C\right)\right)\right)\right), \frac{1}{2}\right), B\right)\right) \]

      16. *-lowering-*.f64 1.5%

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, C\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, -4\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{*.f64}\left(B, B\right)\right), \mathsf{*.f64}\left(C, C\right)\right)\right)\right), \frac{1}{2}\right), B\right)\right) \]

   10. Simplified 1.5%

       \[\leadsto -\frac{{\color{blue}{\left(\left(-1 \cdot C\right) \cdot \left(F \cdot -4 - \frac{F \cdot \left(B \cdot B\right)}{C \cdot C}\right)\right)}}^{0.5}}{B} \]

   11. Taylor expanded in C around inf

       \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\color{blue}{\left(2 \cdot \sqrt{C \cdot F}\right)}, B\right)\right) \]
   12. Step-by-step derivation

       1. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \left(\sqrt{C \cdot F}\right)\right), B\right)\right) \]

       2. sqrt-lowering-sqrt.f64 N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(C \cdot F\right)\right)\right), B\right)\right) \]

       3. *-commutative N/A

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\left(F \cdot C\right)\right)\right), B\right)\right) \]

       4. *-lowering-*.f64 7.2%

          \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{*.f64}\left(2, \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, C\right)\right)\right), B\right)\right) \]

   13. Simplified 7.2%

       \[\leadsto -\frac{\color{blue}{2 \cdot \sqrt{F \cdot C}}}{B} \]

   if -7.99999999999999937e134 < C

   1. Initial program 22.0%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   2. Add Preprocessing

   3. Taylor expanded in C around 0

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}\right)} \]
   4. Step-by-step derivation

      1. associate-*r* N/A

         \[\leadsto \left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \color{blue}{\sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}} \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(-1 \cdot \frac{\sqrt{2}}{B}\right), \color{blue}{\left(\sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}\right)}\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \left(\frac{\sqrt{2}}{B}\right)\right), \left(\sqrt{\color{blue}{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\left(\sqrt{2}\right), B\right)\right), \left(\sqrt{F \cdot \color{blue}{\left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}}\right)\right) \]

      5. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \left(\sqrt{F \cdot \left(\color{blue}{A} - \sqrt{{A}^{2} + {B}^{2}}\right)}\right)\right) \]

      6. sqrt-lowering-sqrt.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\left(F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right) \]

      8. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \left(\sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right)\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \left(\sqrt{A \cdot A + {B}^{2}}\right)\right)\right)\right)\right) \]

      10. unpow2 N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \left(\sqrt{A \cdot A + B \cdot B}\right)\right)\right)\right)\right) \]

      11. hypot-define N/A

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \left(\mathsf{hypot}\left(A, B\right)\right)\right)\right)\right)\right) \]

      12. hypot-lowering-hypot.f64 17.9%

          \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \mathsf{hypot.f64}\left(A, B\right)\right)\right)\right)\right) \]

   5. Simplified 17.9%

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)}} \]
   6. Step-by-step derivation

      1. associate-*l* N/A

         \[\leadsto -1 \cdot \color{blue}{\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{A \cdot A + B \cdot B}\right)}\right)} \]

      2. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{A \cdot A + B \cdot B}\right)}\right) \]

      3. neg-lowering-neg.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{A \cdot A + B \cdot B}\right)}\right)\right) \]

      4. associate-*l/ N/A

         \[\leadsto \mathsf{neg.f64}\left(\left(\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(A - \sqrt{A \cdot A + B \cdot B}\right)}}{B}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\left(\sqrt{2} \cdot \sqrt{F \cdot \left(A - \sqrt{A \cdot A + B \cdot B}\right)}\right), B\right)\right) \]

   7. Applied egg-rr 18.0%

      \[\leadsto \color{blue}{-\frac{\sqrt{2 \cdot \left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}}{B}} \]

   8. Taylor expanded in A around 0

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(-2 \cdot \left(B \cdot F\right)\right)}\right), B\right)\right) \]
   9. Step-by-step derivation

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-2, \left(B \cdot F\right)\right)\right), B\right)\right) \]

      2. *-lowering-*.f64 16.6%

         \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(B, F\right)\right)\right), B\right)\right) \]

   10. Simplified 16.6%

       \[\leadsto -\frac{\sqrt{\color{blue}{-2 \cdot \left(B \cdot F\right)}}}{B} \]
3. Recombined 2 regimes into one program.

4. Final simplification 15.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq -8 \cdot 10^{+134}:\\ \;\;\;\;\frac{2 \cdot \sqrt{C \cdot F}}{0 - B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{-2 \cdot \left(B \cdot F\right)}}{0 - B}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 26.7% accurate, 5.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \frac{\sqrt{-2 \cdot \left(B\_m \cdot F\right)}}{0 - B\_m} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (/ (sqrt (* -2.0 (* B_m F))) (- 0.0 B_m)))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	return sqrt((-2.0 * (B_m * F))) / (0.0 - B_m);
}

Fortran

B_m = abs(b)
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    code = sqrt(((-2.0d0) * (b_m * f))) / (0.0d0 - b_m)
end function

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	return Math.sqrt((-2.0 * (B_m * F))) / (0.0 - B_m);
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	return math.sqrt((-2.0 * (B_m * F))) / (0.0 - B_m)

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	return Float64(sqrt(Float64(-2.0 * Float64(B_m * F))) / Float64(0.0 - B_m))
end

MATLAB

B_m = abs(B);
function tmp = code(A, B_m, C, F)
	tmp = sqrt((-2.0 * (B_m * F))) / (0.0 - B_m);
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := N[(N[Sqrt[N[(-2.0 * N[(B$95$m * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(0.0 - B$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 20.3%

   \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
2. Add Preprocessing

3. Taylor expanded in C around 0

   \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}\right)} \]
4. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \color{blue}{\sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}} \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(-1 \cdot \frac{\sqrt{2}}{B}\right), \color{blue}{\left(\sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}\right)}\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \left(\frac{\sqrt{2}}{B}\right)\right), \left(\sqrt{\color{blue}{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}}\right)\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\left(\sqrt{2}\right), B\right)\right), \left(\sqrt{F \cdot \color{blue}{\left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}}\right)\right) \]

   5. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \left(\sqrt{F \cdot \left(\color{blue}{A} - \sqrt{{A}^{2} + {B}^{2}}\right)}\right)\right) \]

   6. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\left(F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right) \]

   8. --lowering--.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \left(\sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right)\right) \]

   9. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \left(\sqrt{A \cdot A + {B}^{2}}\right)\right)\right)\right)\right) \]

   10. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \left(\sqrt{A \cdot A + B \cdot B}\right)\right)\right)\right)\right) \]

   11. hypot-define N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \left(\mathsf{hypot}\left(A, B\right)\right)\right)\right)\right)\right) \]

   12. hypot-lowering-hypot.f64 16.0%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \mathsf{hypot.f64}\left(A, B\right)\right)\right)\right)\right) \]

5. Simplified 16.0%

   \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)}} \]
6. Step-by-step derivation

   1. associate-*l* N/A

      \[\leadsto -1 \cdot \color{blue}{\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{A \cdot A + B \cdot B}\right)}\right)} \]

   2. mul-1-neg N/A

      \[\leadsto \mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{A \cdot A + B \cdot B}\right)}\right) \]

   3. neg-lowering-neg.f64 N/A

      \[\leadsto \mathsf{neg.f64}\left(\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{A \cdot A + B \cdot B}\right)}\right)\right) \]

   4. associate-*l/ N/A

      \[\leadsto \mathsf{neg.f64}\left(\left(\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(A - \sqrt{A \cdot A + B \cdot B}\right)}}{B}\right)\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\left(\sqrt{2} \cdot \sqrt{F \cdot \left(A - \sqrt{A \cdot A + B \cdot B}\right)}\right), B\right)\right) \]

7. Applied egg-rr 16.1%

   \[\leadsto \color{blue}{-\frac{\sqrt{2 \cdot \left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}}{B}} \]

8. Taylor expanded in A around 0

   \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\color{blue}{\left(-2 \cdot \left(B \cdot F\right)\right)}\right), B\right)\right) \]
9. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-2, \left(B \cdot F\right)\right)\right), B\right)\right) \]

   2. *-lowering-*.f64 15.0%

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(-2, \mathsf{*.f64}\left(B, F\right)\right)\right), B\right)\right) \]

10. Simplified 15.0%

    \[\leadsto -\frac{\sqrt{\color{blue}{-2 \cdot \left(B \cdot F\right)}}}{B} \]

11. Final simplification 15.0%

    \[\leadsto \frac{\sqrt{-2 \cdot \left(B \cdot F\right)}}{0 - B} \]
12. Add Preprocessing

Alternative 18: 5.2% accurate, 5.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \sqrt{A \cdot F} \cdot \frac{-2}{B\_m} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F) :precision binary64 (* (sqrt (* A F)) (/ -2.0 B_m)))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	return sqrt((A * F)) * (-2.0 / B_m);
}

Fortran

B_m = abs(b)
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    code = sqrt((a * f)) * ((-2.0d0) / b_m)
end function

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	return Math.sqrt((A * F)) * (-2.0 / B_m);
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	return math.sqrt((A * F)) * (-2.0 / B_m)

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	return Float64(sqrt(Float64(A * F)) * Float64(-2.0 / B_m))
end

MATLAB

B_m = abs(B);
function tmp = code(A, B_m, C, F)
	tmp = sqrt((A * F)) * (-2.0 / B_m);
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := N[(N[Sqrt[N[(A * F), $MachinePrecision]], $MachinePrecision] * N[(-2.0 / B$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 20.3%

   \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
2. Add Preprocessing

3. Taylor expanded in C around 0

   \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}\right)} \]
4. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \color{blue}{\sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}} \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(-1 \cdot \frac{\sqrt{2}}{B}\right), \color{blue}{\left(\sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}\right)}\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \left(\frac{\sqrt{2}}{B}\right)\right), \left(\sqrt{\color{blue}{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}}\right)\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\left(\sqrt{2}\right), B\right)\right), \left(\sqrt{F \cdot \color{blue}{\left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}}\right)\right) \]

   5. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \left(\sqrt{F \cdot \left(\color{blue}{A} - \sqrt{{A}^{2} + {B}^{2}}\right)}\right)\right) \]

   6. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\left(F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right) \]

   8. --lowering--.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \left(\sqrt{{A}^{2} + {B}^{2}}\right)\right)\right)\right)\right) \]

   9. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \left(\sqrt{A \cdot A + {B}^{2}}\right)\right)\right)\right)\right) \]

   10. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \left(\sqrt{A \cdot A + B \cdot B}\right)\right)\right)\right)\right) \]

   11. hypot-define N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \left(\mathsf{hypot}\left(A, B\right)\right)\right)\right)\right)\right) \]

   12. hypot-lowering-hypot.f64 16.0%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(A, \mathsf{hypot.f64}\left(A, B\right)\right)\right)\right)\right) \]

5. Simplified 16.0%

   \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)}} \]

6. Taylor expanded in A around -inf

   \[\leadsto \color{blue}{\sqrt{A \cdot F} \cdot \frac{{\left(\sqrt{-1}\right)}^{2} \cdot {\left(\sqrt{2}\right)}^{2}}{B}} \]
7. Step-by-step derivation

   1. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(\sqrt{A \cdot F}\right), \color{blue}{\left(\frac{{\left(\sqrt{-1}\right)}^{2} \cdot {\left(\sqrt{2}\right)}^{2}}{B}\right)}\right) \]

   2. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(A \cdot F\right)\right), \left(\frac{\color{blue}{{\left(\sqrt{-1}\right)}^{2} \cdot {\left(\sqrt{2}\right)}^{2}}}{B}\right)\right) \]

   3. *-commutative N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(F \cdot A\right)\right), \left(\frac{\color{blue}{{\left(\sqrt{-1}\right)}^{2}} \cdot {\left(\sqrt{2}\right)}^{2}}{B}\right)\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, A\right)\right), \left(\frac{\color{blue}{{\left(\sqrt{-1}\right)}^{2}} \cdot {\left(\sqrt{2}\right)}^{2}}{B}\right)\right) \]

   5. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, A\right)\right), \left(\frac{\left(\sqrt{-1} \cdot \sqrt{-1}\right) \cdot {\left(\sqrt{2}\right)}^{2}}{B}\right)\right) \]

   6. rem-square-sqrt N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, A\right)\right), \left(\frac{-1 \cdot {\left(\sqrt{2}\right)}^{2}}{B}\right)\right) \]

   7. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, A\right)\right), \left(\frac{-1 \cdot \left(\sqrt{2} \cdot \sqrt{2}\right)}{B}\right)\right) \]

   8. rem-square-sqrt N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, A\right)\right), \left(\frac{-1 \cdot 2}{B}\right)\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, A\right)\right), \left(\frac{-2}{B}\right)\right) \]

   10. /-lowering-/.f64 2.3%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, A\right)\right), \mathsf{/.f64}\left(-2, \color{blue}{B}\right)\right) \]

8. Simplified 2.3%

   \[\leadsto \color{blue}{\sqrt{F \cdot A} \cdot \frac{-2}{B}} \]

9. Final simplification 2.3%

   \[\leadsto \sqrt{A \cdot F} \cdot \frac{-2}{B} \]
10. Add Preprocessing

Alternative 19: 2.6% accurate, 6.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ 0 - \sqrt{\frac{F}{C}} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F) :precision binary64 (- 0.0 (sqrt (/ F C))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	return 0.0 - sqrt((F / C));
}

Fortran

B_m = abs(b)
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    code = 0.0d0 - sqrt((f / c))
end function

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	return 0.0 - Math.sqrt((F / C));
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	return 0.0 - math.sqrt((F / C))

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	return Float64(0.0 - sqrt(Float64(F / C)))
end

MATLAB

B_m = abs(B);
function tmp = code(A, B_m, C, F)
	tmp = 0.0 - sqrt((F / C));
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := N[(0.0 - N[Sqrt[N[(F / C), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 20.3%

   \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
2. Add Preprocessing

3. Taylor expanded in A around 0

   \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]
4. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \color{blue}{\sqrt{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}} \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\left(-1 \cdot \frac{\sqrt{2}}{B}\right), \color{blue}{\left(\sqrt{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)}\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \left(\frac{\sqrt{2}}{B}\right)\right), \left(\sqrt{\color{blue}{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}}\right)\right) \]

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\left(\sqrt{2}\right), B\right)\right), \left(\sqrt{F \cdot \color{blue}{\left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}}\right)\right) \]

   5. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \left(\sqrt{F \cdot \left(\color{blue}{C} - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)\right) \]

   6. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\left(F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right) \]

   7. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right) \]

   8. --lowering--.f64 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\sqrt{{B}^{2} + {C}^{2}}\right)\right)\right)\right)\right) \]

   9. unpow2 N/A

      \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\sqrt{B \cdot B + {C}^{2}}\right)\right)\right)\right)\right) \]

   10. unpow2 N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\sqrt{B \cdot B + C \cdot C}\right)\right)\right)\right)\right) \]

   11. hypot-define N/A

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \left(\mathsf{hypot}\left(B, C\right)\right)\right)\right)\right)\right) \]

   12. hypot-lowering-hypot.f64 16.4%

       \[\leadsto \mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, \mathsf{/.f64}\left(\mathsf{sqrt.f64}\left(2\right), B\right)\right), \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{\_.f64}\left(C, \mathsf{hypot.f64}\left(B, C\right)\right)\right)\right)\right) \]

5. Simplified 16.4%

   \[\leadsto \color{blue}{\left(-1 \cdot \frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(C - \mathsf{hypot}\left(B, C\right)\right)}} \]
6. Step-by-step derivation

   1. associate-*l* N/A

      \[\leadsto -1 \cdot \color{blue}{\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)}\right)} \]

   2. mul-1-neg N/A

      \[\leadsto \mathsf{neg}\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)}\right) \]

   3. neg-lowering-neg.f64 N/A

      \[\leadsto \mathsf{neg.f64}\left(\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)}\right)\right) \]

   4. associate-*l/ N/A

      \[\leadsto \mathsf{neg.f64}\left(\left(\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)}}{B}\right)\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\left(\sqrt{2} \cdot \sqrt{F \cdot \left(C - \sqrt{B \cdot B + C \cdot C}\right)}\right), B\right)\right) \]

7. Applied egg-rr 16.5%

   \[\leadsto \color{blue}{-\frac{{\left(2 \cdot \left(F \cdot \left(C - \mathsf{hypot}\left(B, C\right)\right)\right)\right)}^{0.5}}{B}} \]

8. Taylor expanded in C around -inf

   \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\color{blue}{\left(-1 \cdot \left(C \cdot \left(-4 \cdot F + -1 \cdot \frac{{B}^{2} \cdot F}{{C}^{2}}\right)\right)\right)}, \frac{1}{2}\right), B\right)\right) \]
9. Step-by-step derivation

   1. associate-*r* N/A

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\left(\left(-1 \cdot C\right) \cdot \left(-4 \cdot F + -1 \cdot \frac{{B}^{2} \cdot F}{{C}^{2}}\right)\right), \frac{1}{2}\right), B\right)\right) \]

   2. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\left(-1 \cdot C\right), \left(-4 \cdot F + -1 \cdot \frac{{B}^{2} \cdot F}{{C}^{2}}\right)\right), \frac{1}{2}\right), B\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, C\right), \left(-4 \cdot F + -1 \cdot \frac{{B}^{2} \cdot F}{{C}^{2}}\right)\right), \frac{1}{2}\right), B\right)\right) \]

   4. fma-define N/A

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, C\right), \left(\mathsf{fma}\left(-4, F, -1 \cdot \frac{{B}^{2} \cdot F}{{C}^{2}}\right)\right)\right), \frac{1}{2}\right), B\right)\right) \]

   5. mul-1-neg N/A

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, C\right), \left(\mathsf{fma}\left(-4, F, \mathsf{neg}\left(\frac{{B}^{2} \cdot F}{{C}^{2}}\right)\right)\right)\right), \frac{1}{2}\right), B\right)\right) \]

   6. fmm-undef N/A

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, C\right), \left(-4 \cdot F - \frac{{B}^{2} \cdot F}{{C}^{2}}\right)\right), \frac{1}{2}\right), B\right)\right) \]

   7. --lowering--.f64 N/A

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, C\right), \mathsf{\_.f64}\left(\left(-4 \cdot F\right), \left(\frac{{B}^{2} \cdot F}{{C}^{2}}\right)\right)\right), \frac{1}{2}\right), B\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, C\right), \mathsf{\_.f64}\left(\left(F \cdot -4\right), \left(\frac{{B}^{2} \cdot F}{{C}^{2}}\right)\right)\right), \frac{1}{2}\right), B\right)\right) \]

   9. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, C\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, -4\right), \left(\frac{{B}^{2} \cdot F}{{C}^{2}}\right)\right)\right), \frac{1}{2}\right), B\right)\right) \]

   10. /-lowering-/.f64 N/A

       \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, C\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, -4\right), \mathsf{/.f64}\left(\left({B}^{2} \cdot F\right), \left({C}^{2}\right)\right)\right)\right), \frac{1}{2}\right), B\right)\right) \]

   11. *-commutative N/A

       \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, C\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, -4\right), \mathsf{/.f64}\left(\left(F \cdot {B}^{2}\right), \left({C}^{2}\right)\right)\right)\right), \frac{1}{2}\right), B\right)\right) \]

   12. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, C\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, -4\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \left({B}^{2}\right)\right), \left({C}^{2}\right)\right)\right)\right), \frac{1}{2}\right), B\right)\right) \]

   13. unpow2 N/A

       \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, C\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, -4\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \left(B \cdot B\right)\right), \left({C}^{2}\right)\right)\right)\right), \frac{1}{2}\right), B\right)\right) \]

   14. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, C\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, -4\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{*.f64}\left(B, B\right)\right), \left({C}^{2}\right)\right)\right)\right), \frac{1}{2}\right), B\right)\right) \]

   15. unpow2 N/A

       \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, C\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, -4\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{*.f64}\left(B, B\right)\right), \left(C \cdot C\right)\right)\right)\right), \frac{1}{2}\right), B\right)\right) \]

   16. *-lowering-*.f64 1.6%

       \[\leadsto \mathsf{neg.f64}\left(\mathsf{/.f64}\left(\mathsf{pow.f64}\left(\mathsf{*.f64}\left(\mathsf{*.f64}\left(-1, C\right), \mathsf{\_.f64}\left(\mathsf{*.f64}\left(F, -4\right), \mathsf{/.f64}\left(\mathsf{*.f64}\left(F, \mathsf{*.f64}\left(B, B\right)\right), \mathsf{*.f64}\left(C, C\right)\right)\right)\right), \frac{1}{2}\right), B\right)\right) \]

10. Simplified 1.6%

    \[\leadsto -\frac{{\color{blue}{\left(\left(-1 \cdot C\right) \cdot \left(F \cdot -4 - \frac{F \cdot \left(B \cdot B\right)}{C \cdot C}\right)\right)}}^{0.5}}{B} \]

11. Taylor expanded in B around -inf

    \[\leadsto \color{blue}{\sqrt{\frac{F}{C}} \cdot {\left(\sqrt{-1}\right)}^{2}} \]
12. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto {\left(\sqrt{-1}\right)}^{2} \cdot \color{blue}{\sqrt{\frac{F}{C}}} \]

    2. unpow2 N/A

       \[\leadsto \left(\sqrt{-1} \cdot \sqrt{-1}\right) \cdot \sqrt{\color{blue}{\frac{F}{C}}} \]

    3. rem-square-sqrt N/A

       \[\leadsto -1 \cdot \sqrt{\color{blue}{\frac{F}{C}}} \]

    4. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(-1, \color{blue}{\left(\sqrt{\frac{F}{C}}\right)}\right) \]

    5. sqrt-lowering-sqrt.f64 N/A

       \[\leadsto \mathsf{*.f64}\left(-1, \mathsf{sqrt.f64}\left(\left(\frac{F}{C}\right)\right)\right) \]

    6. /-lowering-/.f64 2.4%

       \[\leadsto \mathsf{*.f64}\left(-1, \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(F, C\right)\right)\right) \]

13. Simplified 2.4%

    \[\leadsto \color{blue}{-1 \cdot \sqrt{\frac{F}{C}}} \]

14. Final simplification 2.4%

    \[\leadsto 0 - \sqrt{\frac{F}{C}} \]
15. Add Preprocessing

Alternative 20: 1.6% accurate, 6.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \sqrt{\frac{2}{\frac{B\_m}{F}}} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F) :precision binary64 (sqrt (/ 2.0 (/ B_m F))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	return sqrt((2.0 / (B_m / F)));
}

Fortran

B_m = abs(b)
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    code = sqrt((2.0d0 / (b_m / f)))
end function

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	return Math.sqrt((2.0 / (B_m / F)));
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	return math.sqrt((2.0 / (B_m / F)))

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	return sqrt(Float64(2.0 / Float64(B_m / F)))
end

MATLAB

B_m = abs(B);
function tmp = code(A, B_m, C, F)
	tmp = sqrt((2.0 / (B_m / F)));
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := N[Sqrt[N[(2.0 / N[(B$95$m / F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 20.3%

   \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
2. Add Preprocessing

3. Taylor expanded in B around -inf

   \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{2}\right)\right)} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \mathsf{neg}\left(\sqrt{\frac{F}{B}} \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{2}\right)\right) \]

   2. neg-lowering-neg.f64 N/A

      \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{\frac{F}{B}} \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{2}\right)\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{\frac{F}{B}}\right), \left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{2}\right)\right)\right) \]

   4. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{F}{B}\right)\right), \left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{2}\right)\right)\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(F, B\right)\right), \left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{2}\right)\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(F, B\right)\right), \left(\left(\sqrt{-1} \cdot \sqrt{-1}\right) \cdot \sqrt{2}\right)\right)\right) \]

   7. rem-square-sqrt N/A

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(F, B\right)\right), \left(-1 \cdot \sqrt{2}\right)\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(F, B\right)\right), \mathsf{*.f64}\left(-1, \left(\sqrt{2}\right)\right)\right)\right) \]

   9. sqrt-lowering-sqrt.f64 2.1%

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(F, B\right)\right), \mathsf{*.f64}\left(-1, \mathsf{sqrt.f64}\left(2\right)\right)\right)\right) \]

5. Simplified 2.1%

   \[\leadsto \color{blue}{-\sqrt{\frac{F}{B}} \cdot \left(-1 \cdot \sqrt{2}\right)} \]
6. Step-by-step derivation

   1. distribute-rgt-neg-in N/A

      \[\leadsto \sqrt{\frac{F}{B}} \cdot \color{blue}{\left(\mathsf{neg}\left(-1 \cdot \sqrt{2}\right)\right)} \]

   2. mul-1-neg N/A

      \[\leadsto \sqrt{\frac{F}{B}} \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\sqrt{2}\right)\right)\right)\right) \]

   3. remove-double-neg N/A

      \[\leadsto \sqrt{\frac{F}{B}} \cdot \sqrt{2} \]

   4. sqrt-unprod N/A

      \[\leadsto \sqrt{\frac{F}{B} \cdot 2} \]

   5. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{F}{B} \cdot 2\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\frac{F}{B}\right), 2\right)\right) \]

   7. /-lowering-/.f64 2.1%

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, B\right), 2\right)\right) \]

7. Applied egg-rr 2.1%

   \[\leadsto \color{blue}{\sqrt{\frac{F}{B} \cdot 2}} \]
8. Step-by-step derivation

   1. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{F}{B} \cdot 2\right)\right) \]

   2. *-commutative N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(2 \cdot \frac{F}{B}\right)\right) \]

   3. clear-num N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(2 \cdot \frac{1}{\frac{B}{F}}\right)\right) \]

   4. un-div-inv N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{2}{\frac{B}{F}}\right)\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \left(\frac{B}{F}\right)\right)\right) \]

   6. /-lowering-/.f64 2.1%

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(2, \mathsf{/.f64}\left(B, F\right)\right)\right) \]

9. Applied egg-rr 2.1%

   \[\leadsto \color{blue}{\sqrt{\frac{2}{\frac{B}{F}}}} \]
10. Add Preprocessing

Alternative 21: 1.6% accurate, 6.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \sqrt{F \cdot \frac{2}{B\_m}} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F) :precision binary64 (sqrt (* F (/ 2.0 B_m))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	return sqrt((F * (2.0 / B_m)));
}

Fortran

B_m = abs(b)
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    code = sqrt((f * (2.0d0 / b_m)))
end function

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	return Math.sqrt((F * (2.0 / B_m)));
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	return math.sqrt((F * (2.0 / B_m)))

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	return sqrt(Float64(F * Float64(2.0 / B_m)))
end

MATLAB

B_m = abs(B);
function tmp = code(A, B_m, C, F)
	tmp = sqrt((F * (2.0 / B_m)));
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := N[Sqrt[N[(F * N[(2.0 / B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 20.3%

   \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
2. Add Preprocessing

3. Taylor expanded in B around -inf

   \[\leadsto \color{blue}{-1 \cdot \left(\sqrt{\frac{F}{B}} \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{2}\right)\right)} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \mathsf{neg}\left(\sqrt{\frac{F}{B}} \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{2}\right)\right) \]

   2. neg-lowering-neg.f64 N/A

      \[\leadsto \mathsf{neg.f64}\left(\left(\sqrt{\frac{F}{B}} \cdot \left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{2}\right)\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\left(\sqrt{\frac{F}{B}}\right), \left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{2}\right)\right)\right) \]

   4. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\left(\frac{F}{B}\right)\right), \left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{2}\right)\right)\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(F, B\right)\right), \left({\left(\sqrt{-1}\right)}^{2} \cdot \sqrt{2}\right)\right)\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(F, B\right)\right), \left(\left(\sqrt{-1} \cdot \sqrt{-1}\right) \cdot \sqrt{2}\right)\right)\right) \]

   7. rem-square-sqrt N/A

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(F, B\right)\right), \left(-1 \cdot \sqrt{2}\right)\right)\right) \]

   8. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(F, B\right)\right), \mathsf{*.f64}\left(-1, \left(\sqrt{2}\right)\right)\right)\right) \]

   9. sqrt-lowering-sqrt.f64 2.1%

      \[\leadsto \mathsf{neg.f64}\left(\mathsf{*.f64}\left(\mathsf{sqrt.f64}\left(\mathsf{/.f64}\left(F, B\right)\right), \mathsf{*.f64}\left(-1, \mathsf{sqrt.f64}\left(2\right)\right)\right)\right) \]

5. Simplified 2.1%

   \[\leadsto \color{blue}{-\sqrt{\frac{F}{B}} \cdot \left(-1 \cdot \sqrt{2}\right)} \]
6. Step-by-step derivation

   1. distribute-rgt-neg-in N/A

      \[\leadsto \sqrt{\frac{F}{B}} \cdot \color{blue}{\left(\mathsf{neg}\left(-1 \cdot \sqrt{2}\right)\right)} \]

   2. mul-1-neg N/A

      \[\leadsto \sqrt{\frac{F}{B}} \cdot \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\sqrt{2}\right)\right)\right)\right) \]

   3. remove-double-neg N/A

      \[\leadsto \sqrt{\frac{F}{B}} \cdot \sqrt{2} \]

   4. sqrt-unprod N/A

      \[\leadsto \sqrt{\frac{F}{B} \cdot 2} \]

   5. sqrt-lowering-sqrt.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{F}{B} \cdot 2\right)\right) \]

   6. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\left(\frac{F}{B}\right), 2\right)\right) \]

   7. /-lowering-/.f64 2.1%

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(F, B\right), 2\right)\right) \]

7. Applied egg-rr 2.1%

   \[\leadsto \color{blue}{\sqrt{\frac{F}{B} \cdot 2}} \]
8. Step-by-step derivation

   1. associate-*l/ N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(\frac{F \cdot 2}{B}\right)\right) \]

   2. associate-/l* N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\left(F \cdot \frac{2}{B}\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \left(\frac{2}{B}\right)\right)\right) \]

   4. /-lowering-/.f64 2.1%

      \[\leadsto \mathsf{sqrt.f64}\left(\mathsf{*.f64}\left(F, \mathsf{/.f64}\left(2, B\right)\right)\right) \]

9. Applied egg-rr 2.1%

   \[\leadsto \sqrt{\color{blue}{F \cdot \frac{2}{B}}} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024150 
(FPCore (A B C F)
  :name "ABCF->ab-angle b"
  :precision binary64
  (/ (- (sqrt (* (* 2.0 (* (- (pow B 2.0) (* (* 4.0 A) C)) F)) (- (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0))))))) (- (pow B 2.0) (* (* 4.0 A) C))))