ABCF->ab-angle a

Percentage Accurate: 19.4% → 60.2%

Time: 35.3s

Alternatives: 12

Speedup: 6.0×

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: 19.4% 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: 60.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := A + \left(C + \mathsf{hypot}\left(B\_m, A - C\right)\right)\\ t_1 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ t_2 := \left(4 \cdot A\right) \cdot C\\ t_3 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_2\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_2 - {B\_m}^{2}}\\ t_4 := {B\_m}^{2} + -4 \cdot \left(A \cdot C\right)\\ \mathbf{if}\;t\_3 \leq -5 \cdot 10^{-158}:\\ \;\;\;\;\sqrt{F \cdot \frac{t\_0}{\mathsf{fma}\left(-4, A \cdot C, {B\_m}^{2}\right)}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{+215}:\\ \;\;\;\;\frac{-1}{\frac{t\_4}{\sqrt{\left(t\_4 \cdot \left(2 \cdot F\right)\right) \cdot \mathsf{fma}\left(-0.5, \frac{{B\_m}^{2}}{A}, 2 \cdot C\right)}}}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(2 \cdot t\_1\right)} \cdot \sqrt{t\_0}}{-t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B\_m} \cdot \left(\sqrt{C + \mathsf{hypot}\left(C, B\_m\right)} \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (+ A (+ C (hypot B_m (- A C)))))
        (t_1 (fma B_m B_m (* A (* C -4.0))))
        (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 (+ (pow B_m 2.0) (* -4.0 (* A C)))))
   (if (<= t_3 -5e-158)
     (* (sqrt (* F (/ t_0 (fma -4.0 (* A C) (pow B_m 2.0))))) (- (sqrt 2.0)))
     (if (<= t_3 2e+215)
       (/
        -1.0
        (/
         t_4
         (sqrt
          (* (* t_4 (* 2.0 F)) (fma -0.5 (/ (pow B_m 2.0) A) (* 2.0 C))))))
       (if (<= t_3 INFINITY)
         (/ (* (sqrt (* F (* 2.0 t_1))) (sqrt t_0)) (- t_1))
         (*
          (/ (sqrt 2.0) B_m)
          (* (sqrt (+ C (hypot C B_m))) (- (sqrt F)))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = A + (C + hypot(B_m, (A - C)));
	double t_1 = fma(B_m, B_m, (A * (C * -4.0)));
	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 = pow(B_m, 2.0) + (-4.0 * (A * C));
	double tmp;
	if (t_3 <= -5e-158) {
		tmp = sqrt((F * (t_0 / fma(-4.0, (A * C), pow(B_m, 2.0))))) * -sqrt(2.0);
	} else if (t_3 <= 2e+215) {
		tmp = -1.0 / (t_4 / sqrt(((t_4 * (2.0 * F)) * fma(-0.5, (pow(B_m, 2.0) / A), (2.0 * C)))));
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = (sqrt((F * (2.0 * t_1))) * sqrt(t_0)) / -t_1;
	} else {
		tmp = (sqrt(2.0) / B_m) * (sqrt((C + hypot(C, B_m))) * -sqrt(F));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(A + Float64(C + hypot(B_m, Float64(A - C))))
	t_1 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	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((B_m ^ 2.0) + Float64(-4.0 * Float64(A * C)))
	tmp = 0.0
	if (t_3 <= -5e-158)
		tmp = Float64(sqrt(Float64(F * Float64(t_0 / fma(-4.0, Float64(A * C), (B_m ^ 2.0))))) * Float64(-sqrt(2.0)));
	elseif (t_3 <= 2e+215)
		tmp = Float64(-1.0 / Float64(t_4 / sqrt(Float64(Float64(t_4 * Float64(2.0 * F)) * fma(-0.5, Float64((B_m ^ 2.0) / A), Float64(2.0 * C))))));
	elseif (t_3 <= Inf)
		tmp = Float64(Float64(sqrt(Float64(F * Float64(2.0 * t_1))) * sqrt(t_0)) / Float64(-t_1));
	else
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(sqrt(Float64(C + hypot(C, B_m))) * Float64(-sqrt(F))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(A + N[(C + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $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[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -5e-158], N[(N[Sqrt[N[(F * N[(t$95$0 / N[(-4.0 * N[(A * C), $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], If[LessEqual[t$95$3, 2e+215], N[(-1.0 / N[(t$95$4 / N[Sqrt[N[(N[(t$95$4 * N[(2.0 * F), $MachinePrecision]), $MachinePrecision] * N[(-0.5 * N[(N[Power[B$95$m, 2.0], $MachinePrecision] / A), $MachinePrecision] + N[(2.0 * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(N[(N[Sqrt[N[(F * N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision] / (-t$95$1)), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[(N[Sqrt[N[(C + N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $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))) < -4.99999999999999972e-158

   1. Initial program 37.7%

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

   3. Taylor expanded in F around 0 44.4%

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

      1. mul-1-neg 44.4%

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

   5. Simplified 68.9%

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

   if -4.99999999999999972e-158 < (/.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))) < 1.99999999999999981e215

   1. Initial program 25.6%

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

   3. Taylor expanded in A around -inf 42.9%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\left(-0.5 \cdot \frac{{B}^{2}}{A} + 2 \cdot C\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   4. Step-by-step derivation

      1. clear-num 42.9%

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

      2. inv-pow 42.9%

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

   5. Applied egg-rr 42.9%

      \[\leadsto \color{blue}{{\left(\frac{{B}^{2} - C \cdot \left(4 \cdot A\right)}{-\sqrt{2 \cdot \left(\left(F \cdot \left({B}^{2} - C \cdot \left(4 \cdot A\right)\right)\right) \cdot \mathsf{fma}\left(C, 2, -0.5 \cdot \frac{{B}^{2}}{A}\right)\right)}}\right)}^{-1}} \]
   6. Step-by-step derivation

      1. unpow-1 42.9%

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

      2. *-commutative 42.9%

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

      3. associate-*r* 42.9%

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

      4. cancel-sign-sub-inv 42.9%

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

      5. metadata-eval 42.9%

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

      6. *-commutative 42.9%

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

      7. associate-*r* 42.9%

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

   7. Simplified 43.0%

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

   if 1.99999999999999981e215 < (/.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 4.3%

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

   3. Simplified 28.4%

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

      1. pow1/2 28.4%

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

      2. associate-*r* 28.4%

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

      3. associate-+r+ 28.4%

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

      4. hypot-undefine 4.3%

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

      5. unpow2 4.3%

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

      6. unpow2 4.3%

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

      7. +-commutative 4.3%

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

      8. unpow-prod-down 4.3%

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

      9. *-commutative 4.3%

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

      10. pow1/2 4.3%

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

   6. Applied egg-rr 75.9%

      \[\leadsto \frac{\color{blue}{{\left(\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot 2\right)}^{0.5} \cdot \sqrt{A + \left(C + \mathsf{hypot}\left(A - C, B\right)\right)}}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
   7. Step-by-step derivation

      1. unpow1/2 75.9%

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

      2. associate-*l* 75.9%

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

      3. hypot-undefine 4.3%

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

      4. unpow2 4.3%

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

      5. unpow2 4.3%

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

      6. +-commutative 4.3%

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

      7. unpow2 4.3%

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

      8. unpow2 4.3%

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

      9. hypot-undefine 75.9%

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

   8. Simplified 75.9%

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

   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 1.8%

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

      1. mul-1-neg 1.8%

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

      2. unpow2 1.8%

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

      3. unpow2 1.8%

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

      4. hypot-define 23.0%

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

   5. Simplified 23.0%

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

      1. pow1/2 23.1%

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

      2. *-commutative 23.1%

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

      3. hypot-undefine 1.9%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot {\left(\left(C + \color{blue}{\sqrt{B \cdot B + C \cdot C}}\right) \cdot F\right)}^{0.5} \]

      4. unpow2 1.9%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot {\left(\left(C + \sqrt{\color{blue}{{B}^{2}} + C \cdot C}\right) \cdot F\right)}^{0.5} \]

      5. unpow2 1.9%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot {\left(\left(C + \sqrt{{B}^{2} + \color{blue}{{C}^{2}}}\right) \cdot F\right)}^{0.5} \]

      6. unpow-prod-down 1.8%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot \color{blue}{\left({\left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}^{0.5} \cdot {F}^{0.5}\right)} \]

   7. Applied egg-rr 33.9%

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

4. Final simplification 48.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -5 \cdot 10^{-158}:\\ \;\;\;\;\sqrt{F \cdot \frac{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)}{\mathsf{fma}\left(-4, A \cdot C, {B}^{2}\right)}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq 2 \cdot 10^{+215}:\\ \;\;\;\;\frac{-1}{\frac{{B}^{2} + -4 \cdot \left(A \cdot C\right)}{\sqrt{\left(\left({B}^{2} + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(2 \cdot F\right)\right) \cdot \mathsf{fma}\left(-0.5, \frac{{B}^{2}}{A}, 2 \cdot C\right)}}}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)} \cdot \sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 58.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ t_1 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 10^{-59}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_0\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{t\_0 - {B\_m}^{2}}\\ \mathbf{elif}\;{B\_m}^{2} \leq 5 \cdot 10^{+196}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(2 \cdot t\_1\right)} \cdot \sqrt{A + \left(C + \mathsf{hypot}\left(B\_m, A - C\right)\right)}}{-t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B\_m} \cdot \left(\sqrt{C + \mathsf{hypot}\left(C, B\_m\right)} \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (* (* 4.0 A) C)) (t_1 (fma B_m B_m (* A (* C -4.0)))))
   (if (<= (pow B_m 2.0) 1e-59)
     (/
      (sqrt (* (* 2.0 (* (- (pow B_m 2.0) t_0) F)) (* 2.0 C)))
      (- t_0 (pow B_m 2.0)))
     (if (<= (pow B_m 2.0) 5e+196)
       (/
        (* (sqrt (* F (* 2.0 t_1))) (sqrt (+ A (+ C (hypot B_m (- A C))))))
        (- t_1))
       (* (/ (sqrt 2.0) B_m) (* (sqrt (+ C (hypot C B_m))) (- (sqrt F))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = (4.0 * A) * C;
	double t_1 = fma(B_m, B_m, (A * (C * -4.0)));
	double tmp;
	if (pow(B_m, 2.0) <= 1e-59) {
		tmp = sqrt(((2.0 * ((pow(B_m, 2.0) - t_0) * F)) * (2.0 * C))) / (t_0 - pow(B_m, 2.0));
	} else if (pow(B_m, 2.0) <= 5e+196) {
		tmp = (sqrt((F * (2.0 * t_1))) * sqrt((A + (C + hypot(B_m, (A - C)))))) / -t_1;
	} else {
		tmp = (sqrt(2.0) / B_m) * (sqrt((C + hypot(C, B_m))) * -sqrt(F));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(Float64(4.0 * A) * C)
	t_1 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e-59)
		tmp = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_0) * F)) * Float64(2.0 * C))) / Float64(t_0 - (B_m ^ 2.0)));
	elseif ((B_m ^ 2.0) <= 5e+196)
		tmp = Float64(Float64(sqrt(Float64(F * Float64(2.0 * t_1))) * sqrt(Float64(A + Float64(C + hypot(B_m, Float64(A - C)))))) / Float64(-t_1));
	else
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(sqrt(Float64(C + hypot(C, B_m))) * Float64(-sqrt(F))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$1 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-59], N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(2.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e+196], N[(N[(N[Sqrt[N[(F * N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(A + N[(C + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / (-t$95$1)), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[(N[Sqrt[N[(C + N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 1e-59

   1. Initial program 21.8%

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

   3. Taylor expanded in A around -inf 30.6%

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

   if 1e-59 < (pow.f64 B #s(literal 2 binary64)) < 4.9999999999999998e196

   1. Initial program 33.8%

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

   3. Simplified 37.0%

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

      1. pow1/2 37.1%

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

      2. associate-*r* 37.1%

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

      3. associate-+r+ 36.2%

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

      4. hypot-undefine 33.8%

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

      5. unpow2 33.8%

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

      6. unpow2 33.8%

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

      7. +-commutative 33.8%

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

      8. unpow-prod-down 42.6%

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

      9. *-commutative 42.6%

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

      10. pow1/2 42.6%

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

   6. Applied egg-rr 54.5%

      \[\leadsto \frac{\color{blue}{{\left(\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot 2\right)}^{0.5} \cdot \sqrt{A + \left(C + \mathsf{hypot}\left(A - C, B\right)\right)}}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
   7. Step-by-step derivation

      1. unpow1/2 54.5%

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

      2. associate-*l* 54.5%

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

      3. hypot-undefine 42.6%

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

      4. unpow2 42.6%

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

      5. unpow2 42.6%

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

      6. +-commutative 42.6%

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

      7. unpow2 42.6%

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

      8. unpow2 42.6%

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

      9. hypot-undefine 54.5%

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

   8. Simplified 54.5%

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

   if 4.9999999999999998e196 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 2.8%

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

   3. Taylor expanded in A around 0 5.2%

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

      1. mul-1-neg 5.2%

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

      2. unpow2 5.2%

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

      3. unpow2 5.2%

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

      4. hypot-define 32.0%

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

   5. Simplified 32.0%

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

      1. pow1/2 32.0%

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

      2. *-commutative 32.0%

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

      3. hypot-undefine 5.2%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot {\left(\left(C + \color{blue}{\sqrt{B \cdot B + C \cdot C}}\right) \cdot F\right)}^{0.5} \]

      4. unpow2 5.2%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot {\left(\left(C + \sqrt{\color{blue}{{B}^{2}} + C \cdot C}\right) \cdot F\right)}^{0.5} \]

      5. unpow2 5.2%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot {\left(\left(C + \sqrt{{B}^{2} + \color{blue}{{C}^{2}}}\right) \cdot F\right)}^{0.5} \]

      6. unpow-prod-down 6.8%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot \color{blue}{\left({\left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}^{0.5} \cdot {F}^{0.5}\right)} \]

   7. Applied egg-rr 49.1%

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

4. Final simplification 41.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{-59}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;{B}^{2} \leq 5 \cdot 10^{+196}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)} \cdot \sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 43.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 5 \cdot 10^{-43}:\\ \;\;\;\;\frac{\sqrt{\left(-8 \cdot \left(A \cdot \left(C \cdot F\right)\right)\right) \cdot \left(A + \left(C + \mathsf{hypot}\left(B\_m, C - A\right)\right)\right)}}{4 \cdot \left(A \cdot C\right) - {B\_m}^{2}}\\ \mathbf{elif}\;{B\_m}^{2} \leq 5 \cdot 10^{+275}:\\ \;\;\;\;-\sqrt{F \cdot \frac{2 \cdot \left(C + \mathsf{hypot}\left(B\_m, C\right)\right)}{{B\_m}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B\_m} \cdot \left(\sqrt{B\_m \cdot \left(\frac{C}{B\_m} + 1\right)} \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 5e-43)
   (/
    (sqrt (* (* -8.0 (* A (* C F))) (+ A (+ C (hypot B_m (- C A))))))
    (- (* 4.0 (* A C)) (pow B_m 2.0)))
   (if (<= (pow B_m 2.0) 5e+275)
     (- (sqrt (* F (/ (* 2.0 (+ C (hypot B_m C))) (pow B_m 2.0)))))
     (*
      (/ (sqrt 2.0) B_m)
      (* (sqrt (* B_m (+ (/ C B_m) 1.0))) (- (sqrt F)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (pow(B_m, 2.0) <= 5e-43) {
		tmp = sqrt(((-8.0 * (A * (C * F))) * (A + (C + hypot(B_m, (C - A)))))) / ((4.0 * (A * C)) - pow(B_m, 2.0));
	} else if (pow(B_m, 2.0) <= 5e+275) {
		tmp = -sqrt((F * ((2.0 * (C + hypot(B_m, C))) / pow(B_m, 2.0))));
	} else {
		tmp = (sqrt(2.0) / B_m) * (sqrt((B_m * ((C / B_m) + 1.0))) * -sqrt(F));
	}
	return tmp;
}

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (Math.pow(B_m, 2.0) <= 5e-43) {
		tmp = Math.sqrt(((-8.0 * (A * (C * F))) * (A + (C + Math.hypot(B_m, (C - A)))))) / ((4.0 * (A * C)) - Math.pow(B_m, 2.0));
	} else if (Math.pow(B_m, 2.0) <= 5e+275) {
		tmp = -Math.sqrt((F * ((2.0 * (C + Math.hypot(B_m, C))) / Math.pow(B_m, 2.0))));
	} else {
		tmp = (Math.sqrt(2.0) / B_m) * (Math.sqrt((B_m * ((C / B_m) + 1.0))) * -Math.sqrt(F));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if math.pow(B_m, 2.0) <= 5e-43:
		tmp = math.sqrt(((-8.0 * (A * (C * F))) * (A + (C + math.hypot(B_m, (C - A)))))) / ((4.0 * (A * C)) - math.pow(B_m, 2.0))
	elif math.pow(B_m, 2.0) <= 5e+275:
		tmp = -math.sqrt((F * ((2.0 * (C + math.hypot(B_m, C))) / math.pow(B_m, 2.0))))
	else:
		tmp = (math.sqrt(2.0) / B_m) * (math.sqrt((B_m * ((C / B_m) + 1.0))) * -math.sqrt(F))
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 5e-43)
		tmp = Float64(sqrt(Float64(Float64(-8.0 * Float64(A * Float64(C * F))) * Float64(A + Float64(C + hypot(B_m, Float64(C - A)))))) / Float64(Float64(4.0 * Float64(A * C)) - (B_m ^ 2.0)));
	elseif ((B_m ^ 2.0) <= 5e+275)
		tmp = Float64(-sqrt(Float64(F * Float64(Float64(2.0 * Float64(C + hypot(B_m, C))) / (B_m ^ 2.0)))));
	else
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(sqrt(Float64(B_m * Float64(Float64(C / B_m) + 1.0))) * Float64(-sqrt(F))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if ((B_m ^ 2.0) <= 5e-43)
		tmp = sqrt(((-8.0 * (A * (C * F))) * (A + (C + hypot(B_m, (C - A)))))) / ((4.0 * (A * C)) - (B_m ^ 2.0));
	elseif ((B_m ^ 2.0) <= 5e+275)
		tmp = -sqrt((F * ((2.0 * (C + hypot(B_m, C))) / (B_m ^ 2.0))));
	else
		tmp = (sqrt(2.0) / B_m) * (sqrt((B_m * ((C / B_m) + 1.0))) * -sqrt(F));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e-43], N[(N[Sqrt[N[(N[(-8.0 * N[(A * N[(C * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(A + N[(C + N[Sqrt[B$95$m ^ 2 + N[(C - A), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision] - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e+275], (-N[Sqrt[N[(F * N[(N[(2.0 * N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[(N[Sqrt[N[(B$95$m * N[(N[(C / B$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 5.00000000000000019e-43

   1. Initial program 21.2%

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

   3. Simplified 29.0%

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

   5. Taylor expanded in A around inf 22.4%

      \[\leadsto \frac{\sqrt{\color{blue}{\left(-8 \cdot \left(A \cdot \left(C \cdot F\right)\right)\right)} \cdot \left(A + \left(C + \mathsf{hypot}\left(B, C - A\right)\right)\right)}}{4 \cdot \left(A \cdot C\right) - {B}^{2}} \]

   if 5.00000000000000019e-43 < (pow.f64 B #s(literal 2 binary64)) < 5.0000000000000003e275

   1. Initial program 28.6%

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

   3. Taylor expanded in A around 0 23.0%

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

      1. mul-1-neg 23.0%

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

      2. unpow2 23.0%

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

      3. unpow2 23.0%

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

      4. hypot-define 25.2%

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

   5. Simplified 25.2%

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

      1. pow1/2 25.2%

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

      2. *-commutative 25.2%

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

      3. hypot-undefine 23.0%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot {\left(\left(C + \color{blue}{\sqrt{B \cdot B + C \cdot C}}\right) \cdot F\right)}^{0.5} \]

      4. unpow2 23.0%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot {\left(\left(C + \sqrt{\color{blue}{{B}^{2}} + C \cdot C}\right) \cdot F\right)}^{0.5} \]

      5. unpow2 23.0%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot {\left(\left(C + \sqrt{{B}^{2} + \color{blue}{{C}^{2}}}\right) \cdot F\right)}^{0.5} \]

      6. unpow-prod-down 25.5%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot \color{blue}{\left({\left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}^{0.5} \cdot {F}^{0.5}\right)} \]

   7. Applied egg-rr 31.0%

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

      1. *-un-lft-identity 31.0%

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

      2. metadata-eval 31.0%

         \[\leadsto -\frac{\color{blue}{\left(3 \cdot 0.3333333333333333\right)} \cdot \sqrt{2}}{B} \cdot \left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \sqrt{F}\right) \]

      3. add-cbrt-cube 20.1%

         \[\leadsto -\frac{\left(3 \cdot 0.3333333333333333\right) \cdot \sqrt{2}}{\color{blue}{\sqrt[3]{\left(B \cdot B\right) \cdot B}}} \cdot \left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \sqrt{F}\right) \]

      4. unpow2 20.1%

         \[\leadsto -\frac{\left(3 \cdot 0.3333333333333333\right) \cdot \sqrt{2}}{\sqrt[3]{\color{blue}{{B}^{2}} \cdot B}} \cdot \left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \sqrt{F}\right) \]

      5. cbrt-prod 30.9%

         \[\leadsto -\frac{\left(3 \cdot 0.3333333333333333\right) \cdot \sqrt{2}}{\color{blue}{\sqrt[3]{{B}^{2}} \cdot \sqrt[3]{B}}} \cdot \left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \sqrt{F}\right) \]

      6. times-frac 30.8%

         \[\leadsto -\color{blue}{\left(\frac{3 \cdot 0.3333333333333333}{\sqrt[3]{{B}^{2}}} \cdot \frac{\sqrt{2}}{\sqrt[3]{B}}\right)} \cdot \left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \sqrt{F}\right) \]

      7. metadata-eval 30.8%

         \[\leadsto -\left(\frac{\color{blue}{1}}{\sqrt[3]{{B}^{2}}} \cdot \frac{\sqrt{2}}{\sqrt[3]{B}}\right) \cdot \left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \sqrt{F}\right) \]

      8. unpow2 30.8%

         \[\leadsto -\left(\frac{1}{\sqrt[3]{\color{blue}{B \cdot B}}} \cdot \frac{\sqrt{2}}{\sqrt[3]{B}}\right) \cdot \left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \sqrt{F}\right) \]

      9. cbrt-prod 30.8%

         \[\leadsto -\left(\frac{1}{\color{blue}{\sqrt[3]{B} \cdot \sqrt[3]{B}}} \cdot \frac{\sqrt{2}}{\sqrt[3]{B}}\right) \cdot \left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \sqrt{F}\right) \]

      10. pow2 30.8%

          \[\leadsto -\left(\frac{1}{\color{blue}{{\left(\sqrt[3]{B}\right)}^{2}}} \cdot \frac{\sqrt{2}}{\sqrt[3]{B}}\right) \cdot \left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \sqrt{F}\right) \]

   9. Applied egg-rr 30.8%

      \[\leadsto -\color{blue}{\left(\frac{1}{{\left(\sqrt[3]{B}\right)}^{2}} \cdot \frac{\sqrt{2}}{\sqrt[3]{B}}\right)} \cdot \left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \sqrt{F}\right) \]
   10. Step-by-step derivation

       1. associate-*l/ 30.9%

          \[\leadsto -\color{blue}{\frac{1 \cdot \frac{\sqrt{2}}{\sqrt[3]{B}}}{{\left(\sqrt[3]{B}\right)}^{2}}} \cdot \left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \sqrt{F}\right) \]

       2. *-lft-identity 30.9%

          \[\leadsto -\frac{\color{blue}{\frac{\sqrt{2}}{\sqrt[3]{B}}}}{{\left(\sqrt[3]{B}\right)}^{2}} \cdot \left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \sqrt{F}\right) \]

   11. Simplified 30.9%

       \[\leadsto -\color{blue}{\frac{\frac{\sqrt{2}}{\sqrt[3]{B}}}{{\left(\sqrt[3]{B}\right)}^{2}}} \cdot \left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \sqrt{F}\right) \]
   12. Step-by-step derivation

       1. add-sqr-sqrt 30.4%

          \[\leadsto -\color{blue}{\sqrt{\frac{\frac{\sqrt{2}}{\sqrt[3]{B}}}{{\left(\sqrt[3]{B}\right)}^{2}} \cdot \left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \sqrt{F}\right)} \cdot \sqrt{\frac{\frac{\sqrt{2}}{\sqrt[3]{B}}}{{\left(\sqrt[3]{B}\right)}^{2}} \cdot \left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \sqrt{F}\right)}} \]

       2. sqrt-unprod 40.6%

          \[\leadsto -\color{blue}{\sqrt{\left(\frac{\frac{\sqrt{2}}{\sqrt[3]{B}}}{{\left(\sqrt[3]{B}\right)}^{2}} \cdot \left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \sqrt{F}\right)\right) \cdot \left(\frac{\frac{\sqrt{2}}{\sqrt[3]{B}}}{{\left(\sqrt[3]{B}\right)}^{2}} \cdot \left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \sqrt{F}\right)\right)}} \]

       3. swap-sqr 33.2%

          \[\leadsto -\sqrt{\color{blue}{\left(\frac{\frac{\sqrt{2}}{\sqrt[3]{B}}}{{\left(\sqrt[3]{B}\right)}^{2}} \cdot \frac{\frac{\sqrt{2}}{\sqrt[3]{B}}}{{\left(\sqrt[3]{B}\right)}^{2}}\right) \cdot \left(\left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \sqrt{F}\right) \cdot \left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \sqrt{F}\right)\right)}} \]

   13. Applied egg-rr 33.7%

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

       1. associate-*r* 41.2%

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

       2. *-commutative 41.2%

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

       3. associate-*l/ 41.2%

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

       4. hypot-undefine 35.7%

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

       5. unpow2 35.7%

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

       6. unpow2 35.7%

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

       7. +-commutative 35.7%

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

       8. unpow2 35.7%

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

       9. unpow2 35.7%

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

       10. hypot-define 41.2%

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

   15. Simplified 41.2%

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

   if 5.0000000000000003e275 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 0.2%

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

   3. Taylor expanded in A around 0 3.2%

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

      1. mul-1-neg 3.2%

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

      2. unpow2 3.2%

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

      3. unpow2 3.2%

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

      4. hypot-define 35.7%

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

   5. Simplified 35.7%

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

      1. pow1/2 35.7%

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

      2. *-commutative 35.7%

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

      3. hypot-undefine 3.2%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot {\left(\left(C + \color{blue}{\sqrt{B \cdot B + C \cdot C}}\right) \cdot F\right)}^{0.5} \]

      4. unpow2 3.2%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot {\left(\left(C + \sqrt{\color{blue}{{B}^{2}} + C \cdot C}\right) \cdot F\right)}^{0.5} \]

      5. unpow2 3.2%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot {\left(\left(C + \sqrt{{B}^{2} + \color{blue}{{C}^{2}}}\right) \cdot F\right)}^{0.5} \]

      6. unpow-prod-down 3.2%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot \color{blue}{\left({\left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}^{0.5} \cdot {F}^{0.5}\right)} \]

   7. Applied egg-rr 52.8%

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

   8. Taylor expanded in B around inf 46.6%

      \[\leadsto -\frac{\sqrt{2}}{B} \cdot \left(\sqrt{\color{blue}{B \cdot \left(1 + \frac{C}{B}\right)}} \cdot \sqrt{F}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 33.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 5 \cdot 10^{-43}:\\ \;\;\;\;\frac{\sqrt{\left(-8 \cdot \left(A \cdot \left(C \cdot F\right)\right)\right) \cdot \left(A + \left(C + \mathsf{hypot}\left(B, C - A\right)\right)\right)}}{4 \cdot \left(A \cdot C\right) - {B}^{2}}\\ \mathbf{elif}\;{B}^{2} \leq 5 \cdot 10^{+275}:\\ \;\;\;\;-\sqrt{F \cdot \frac{2 \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)}{{B}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{B \cdot \left(\frac{C}{B} + 1\right)} \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 54.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ \mathbf{if}\;{B\_m}^{2} \leq 5 \cdot 10^{-44}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_0\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{t\_0 - {B\_m}^{2}}\\ \mathbf{elif}\;{B\_m}^{2} \leq 5 \cdot 10^{+275}:\\ \;\;\;\;-\sqrt{F \cdot \frac{2 \cdot \left(C + \mathsf{hypot}\left(B\_m, C\right)\right)}{{B\_m}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B\_m} \cdot \left(\sqrt{B\_m \cdot \left(\frac{C}{B\_m} + 1\right)} \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (* (* 4.0 A) C)))
   (if (<= (pow B_m 2.0) 5e-44)
     (/
      (sqrt (* (* 2.0 (* (- (pow B_m 2.0) t_0) F)) (* 2.0 C)))
      (- t_0 (pow B_m 2.0)))
     (if (<= (pow B_m 2.0) 5e+275)
       (- (sqrt (* F (/ (* 2.0 (+ C (hypot B_m C))) (pow B_m 2.0)))))
       (*
        (/ (sqrt 2.0) B_m)
        (* (sqrt (* B_m (+ (/ C B_m) 1.0))) (- (sqrt F))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = (4.0 * A) * C;
	double tmp;
	if (pow(B_m, 2.0) <= 5e-44) {
		tmp = sqrt(((2.0 * ((pow(B_m, 2.0) - t_0) * F)) * (2.0 * C))) / (t_0 - pow(B_m, 2.0));
	} else if (pow(B_m, 2.0) <= 5e+275) {
		tmp = -sqrt((F * ((2.0 * (C + hypot(B_m, C))) / pow(B_m, 2.0))));
	} else {
		tmp = (sqrt(2.0) / B_m) * (sqrt((B_m * ((C / B_m) + 1.0))) * -sqrt(F));
	}
	return tmp;
}

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double t_0 = (4.0 * A) * C;
	double tmp;
	if (Math.pow(B_m, 2.0) <= 5e-44) {
		tmp = Math.sqrt(((2.0 * ((Math.pow(B_m, 2.0) - t_0) * F)) * (2.0 * C))) / (t_0 - Math.pow(B_m, 2.0));
	} else if (Math.pow(B_m, 2.0) <= 5e+275) {
		tmp = -Math.sqrt((F * ((2.0 * (C + Math.hypot(B_m, C))) / Math.pow(B_m, 2.0))));
	} else {
		tmp = (Math.sqrt(2.0) / B_m) * (Math.sqrt((B_m * ((C / B_m) + 1.0))) * -Math.sqrt(F));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	t_0 = (4.0 * A) * C
	tmp = 0
	if math.pow(B_m, 2.0) <= 5e-44:
		tmp = math.sqrt(((2.0 * ((math.pow(B_m, 2.0) - t_0) * F)) * (2.0 * C))) / (t_0 - math.pow(B_m, 2.0))
	elif math.pow(B_m, 2.0) <= 5e+275:
		tmp = -math.sqrt((F * ((2.0 * (C + math.hypot(B_m, C))) / math.pow(B_m, 2.0))))
	else:
		tmp = (math.sqrt(2.0) / B_m) * (math.sqrt((B_m * ((C / B_m) + 1.0))) * -math.sqrt(F))
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(Float64(4.0 * A) * C)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 5e-44)
		tmp = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_0) * F)) * Float64(2.0 * C))) / Float64(t_0 - (B_m ^ 2.0)));
	elseif ((B_m ^ 2.0) <= 5e+275)
		tmp = Float64(-sqrt(Float64(F * Float64(Float64(2.0 * Float64(C + hypot(B_m, C))) / (B_m ^ 2.0)))));
	else
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(sqrt(Float64(B_m * Float64(Float64(C / B_m) + 1.0))) * Float64(-sqrt(F))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	t_0 = (4.0 * A) * C;
	tmp = 0.0;
	if ((B_m ^ 2.0) <= 5e-44)
		tmp = sqrt(((2.0 * (((B_m ^ 2.0) - t_0) * F)) * (2.0 * C))) / (t_0 - (B_m ^ 2.0));
	elseif ((B_m ^ 2.0) <= 5e+275)
		tmp = -sqrt((F * ((2.0 * (C + hypot(B_m, C))) / (B_m ^ 2.0))));
	else
		tmp = (sqrt(2.0) / B_m) * (sqrt((B_m * ((C / B_m) + 1.0))) * -sqrt(F));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e-44], N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(2.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e+275], (-N[Sqrt[N[(F * N[(N[(2.0 * N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[(N[Sqrt[N[(B$95$m * N[(N[(C / B$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 5.00000000000000039e-44

   1. Initial program 21.3%

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

   3. Taylor expanded in A around -inf 29.9%

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

   if 5.00000000000000039e-44 < (pow.f64 B #s(literal 2 binary64)) < 5.0000000000000003e275

   1. Initial program 28.2%

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

   3. Taylor expanded in A around 0 22.6%

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

      1. mul-1-neg 22.6%

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

      2. unpow2 22.6%

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

      3. unpow2 22.6%

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

      4. hypot-define 24.8%

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

   5. Simplified 24.8%

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

      1. pow1/2 24.8%

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

      2. *-commutative 24.8%

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

      3. hypot-undefine 22.7%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot {\left(\left(C + \color{blue}{\sqrt{B \cdot B + C \cdot C}}\right) \cdot F\right)}^{0.5} \]

      4. unpow2 22.7%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot {\left(\left(C + \sqrt{\color{blue}{{B}^{2}} + C \cdot C}\right) \cdot F\right)}^{0.5} \]

      5. unpow2 22.7%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot {\left(\left(C + \sqrt{{B}^{2} + \color{blue}{{C}^{2}}}\right) \cdot F\right)}^{0.5} \]

      6. unpow-prod-down 25.1%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot \color{blue}{\left({\left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}^{0.5} \cdot {F}^{0.5}\right)} \]

   7. Applied egg-rr 30.6%

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

      1. *-un-lft-identity 30.6%

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

      2. metadata-eval 30.6%

         \[\leadsto -\frac{\color{blue}{\left(3 \cdot 0.3333333333333333\right)} \cdot \sqrt{2}}{B} \cdot \left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \sqrt{F}\right) \]

      3. add-cbrt-cube 19.8%

         \[\leadsto -\frac{\left(3 \cdot 0.3333333333333333\right) \cdot \sqrt{2}}{\color{blue}{\sqrt[3]{\left(B \cdot B\right) \cdot B}}} \cdot \left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \sqrt{F}\right) \]

      4. unpow2 19.8%

         \[\leadsto -\frac{\left(3 \cdot 0.3333333333333333\right) \cdot \sqrt{2}}{\sqrt[3]{\color{blue}{{B}^{2}} \cdot B}} \cdot \left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \sqrt{F}\right) \]

      5. cbrt-prod 30.4%

         \[\leadsto -\frac{\left(3 \cdot 0.3333333333333333\right) \cdot \sqrt{2}}{\color{blue}{\sqrt[3]{{B}^{2}} \cdot \sqrt[3]{B}}} \cdot \left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \sqrt{F}\right) \]

      6. times-frac 30.3%

         \[\leadsto -\color{blue}{\left(\frac{3 \cdot 0.3333333333333333}{\sqrt[3]{{B}^{2}}} \cdot \frac{\sqrt{2}}{\sqrt[3]{B}}\right)} \cdot \left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \sqrt{F}\right) \]

      7. metadata-eval 30.3%

         \[\leadsto -\left(\frac{\color{blue}{1}}{\sqrt[3]{{B}^{2}}} \cdot \frac{\sqrt{2}}{\sqrt[3]{B}}\right) \cdot \left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \sqrt{F}\right) \]

      8. unpow2 30.3%

         \[\leadsto -\left(\frac{1}{\sqrt[3]{\color{blue}{B \cdot B}}} \cdot \frac{\sqrt{2}}{\sqrt[3]{B}}\right) \cdot \left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \sqrt{F}\right) \]

      9. cbrt-prod 30.4%

         \[\leadsto -\left(\frac{1}{\color{blue}{\sqrt[3]{B} \cdot \sqrt[3]{B}}} \cdot \frac{\sqrt{2}}{\sqrt[3]{B}}\right) \cdot \left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \sqrt{F}\right) \]

      10. pow2 30.4%

          \[\leadsto -\left(\frac{1}{\color{blue}{{\left(\sqrt[3]{B}\right)}^{2}}} \cdot \frac{\sqrt{2}}{\sqrt[3]{B}}\right) \cdot \left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \sqrt{F}\right) \]

   9. Applied egg-rr 30.4%

      \[\leadsto -\color{blue}{\left(\frac{1}{{\left(\sqrt[3]{B}\right)}^{2}} \cdot \frac{\sqrt{2}}{\sqrt[3]{B}}\right)} \cdot \left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \sqrt{F}\right) \]
   10. Step-by-step derivation

       1. associate-*l/ 30.4%

          \[\leadsto -\color{blue}{\frac{1 \cdot \frac{\sqrt{2}}{\sqrt[3]{B}}}{{\left(\sqrt[3]{B}\right)}^{2}}} \cdot \left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \sqrt{F}\right) \]

       2. *-lft-identity 30.4%

          \[\leadsto -\frac{\color{blue}{\frac{\sqrt{2}}{\sqrt[3]{B}}}}{{\left(\sqrt[3]{B}\right)}^{2}} \cdot \left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \sqrt{F}\right) \]

   11. Simplified 30.4%

       \[\leadsto -\color{blue}{\frac{\frac{\sqrt{2}}{\sqrt[3]{B}}}{{\left(\sqrt[3]{B}\right)}^{2}}} \cdot \left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \sqrt{F}\right) \]
   12. Step-by-step derivation

       1. add-sqr-sqrt 30.0%

          \[\leadsto -\color{blue}{\sqrt{\frac{\frac{\sqrt{2}}{\sqrt[3]{B}}}{{\left(\sqrt[3]{B}\right)}^{2}} \cdot \left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \sqrt{F}\right)} \cdot \sqrt{\frac{\frac{\sqrt{2}}{\sqrt[3]{B}}}{{\left(\sqrt[3]{B}\right)}^{2}} \cdot \left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \sqrt{F}\right)}} \]

       2. sqrt-unprod 40.0%

          \[\leadsto -\color{blue}{\sqrt{\left(\frac{\frac{\sqrt{2}}{\sqrt[3]{B}}}{{\left(\sqrt[3]{B}\right)}^{2}} \cdot \left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \sqrt{F}\right)\right) \cdot \left(\frac{\frac{\sqrt{2}}{\sqrt[3]{B}}}{{\left(\sqrt[3]{B}\right)}^{2}} \cdot \left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \sqrt{F}\right)\right)}} \]

       3. swap-sqr 32.7%

          \[\leadsto -\sqrt{\color{blue}{\left(\frac{\frac{\sqrt{2}}{\sqrt[3]{B}}}{{\left(\sqrt[3]{B}\right)}^{2}} \cdot \frac{\frac{\sqrt{2}}{\sqrt[3]{B}}}{{\left(\sqrt[3]{B}\right)}^{2}}\right) \cdot \left(\left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \sqrt{F}\right) \cdot \left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \sqrt{F}\right)\right)}} \]

   13. Applied egg-rr 33.2%

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

       1. associate-*r* 40.5%

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

       2. *-commutative 40.5%

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

       3. associate-*l/ 40.5%

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

       4. hypot-undefine 35.1%

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

       5. unpow2 35.1%

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

       6. unpow2 35.1%

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

       7. +-commutative 35.1%

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

       8. unpow2 35.1%

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

       9. unpow2 35.1%

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

       10. hypot-define 40.5%

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

   15. Simplified 40.5%

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

   if 5.0000000000000003e275 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 0.2%

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

   3. Taylor expanded in A around 0 3.2%

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

      1. mul-1-neg 3.2%

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

      2. unpow2 3.2%

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

      3. unpow2 3.2%

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

      4. hypot-define 35.7%

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

   5. Simplified 35.7%

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

      1. pow1/2 35.7%

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

      2. *-commutative 35.7%

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

      3. hypot-undefine 3.2%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot {\left(\left(C + \color{blue}{\sqrt{B \cdot B + C \cdot C}}\right) \cdot F\right)}^{0.5} \]

      4. unpow2 3.2%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot {\left(\left(C + \sqrt{\color{blue}{{B}^{2}} + C \cdot C}\right) \cdot F\right)}^{0.5} \]

      5. unpow2 3.2%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot {\left(\left(C + \sqrt{{B}^{2} + \color{blue}{{C}^{2}}}\right) \cdot F\right)}^{0.5} \]

      6. unpow-prod-down 3.2%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot \color{blue}{\left({\left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}^{0.5} \cdot {F}^{0.5}\right)} \]

   7. Applied egg-rr 52.8%

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

   8. Taylor expanded in B around inf 46.6%

      \[\leadsto -\frac{\sqrt{2}}{B} \cdot \left(\sqrt{\color{blue}{B \cdot \left(1 + \frac{C}{B}\right)}} \cdot \sqrt{F}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 37.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 5 \cdot 10^{-44}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;{B}^{2} \leq 5 \cdot 10^{+275}:\\ \;\;\;\;-\sqrt{F \cdot \frac{2 \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)}{{B}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{B \cdot \left(\frac{C}{B} + 1\right)} \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 57.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ \mathbf{if}\;{B\_m}^{2} \leq 5 \cdot 10^{-44}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_0\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{t\_0 - {B\_m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B\_m} \cdot \left(\sqrt{C + \mathsf{hypot}\left(C, B\_m\right)} \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (* (* 4.0 A) C)))
   (if (<= (pow B_m 2.0) 5e-44)
     (/
      (sqrt (* (* 2.0 (* (- (pow B_m 2.0) t_0) F)) (* 2.0 C)))
      (- t_0 (pow B_m 2.0)))
     (* (/ (sqrt 2.0) B_m) (* (sqrt (+ C (hypot C B_m))) (- (sqrt F)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = (4.0 * A) * C;
	double tmp;
	if (pow(B_m, 2.0) <= 5e-44) {
		tmp = sqrt(((2.0 * ((pow(B_m, 2.0) - t_0) * F)) * (2.0 * C))) / (t_0 - pow(B_m, 2.0));
	} else {
		tmp = (sqrt(2.0) / B_m) * (sqrt((C + hypot(C, B_m))) * -sqrt(F));
	}
	return tmp;
}

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double t_0 = (4.0 * A) * C;
	double tmp;
	if (Math.pow(B_m, 2.0) <= 5e-44) {
		tmp = Math.sqrt(((2.0 * ((Math.pow(B_m, 2.0) - t_0) * F)) * (2.0 * C))) / (t_0 - Math.pow(B_m, 2.0));
	} else {
		tmp = (Math.sqrt(2.0) / B_m) * (Math.sqrt((C + Math.hypot(C, B_m))) * -Math.sqrt(F));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	t_0 = (4.0 * A) * C
	tmp = 0
	if math.pow(B_m, 2.0) <= 5e-44:
		tmp = math.sqrt(((2.0 * ((math.pow(B_m, 2.0) - t_0) * F)) * (2.0 * C))) / (t_0 - math.pow(B_m, 2.0))
	else:
		tmp = (math.sqrt(2.0) / B_m) * (math.sqrt((C + math.hypot(C, B_m))) * -math.sqrt(F))
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(Float64(4.0 * A) * C)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 5e-44)
		tmp = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_0) * F)) * Float64(2.0 * C))) / Float64(t_0 - (B_m ^ 2.0)));
	else
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(sqrt(Float64(C + hypot(C, B_m))) * Float64(-sqrt(F))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	t_0 = (4.0 * A) * C;
	tmp = 0.0;
	if ((B_m ^ 2.0) <= 5e-44)
		tmp = sqrt(((2.0 * (((B_m ^ 2.0) - t_0) * F)) * (2.0 * C))) / (t_0 - (B_m ^ 2.0));
	else
		tmp = (sqrt(2.0) / B_m) * (sqrt((C + hypot(C, B_m))) * -sqrt(F));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e-44], N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(2.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[(N[Sqrt[N[(C + N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 5.00000000000000039e-44

   1. Initial program 21.3%

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

   3. Taylor expanded in A around -inf 29.9%

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

   if 5.00000000000000039e-44 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 12.6%

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

   3. Taylor expanded in A around 0 11.8%

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

      1. mul-1-neg 11.8%

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

      2. unpow2 11.8%

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

      3. unpow2 11.8%

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

      4. hypot-define 30.9%

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

   5. Simplified 30.9%

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

      1. pow1/2 30.9%

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

      2. *-commutative 30.9%

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

      3. hypot-undefine 11.8%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot {\left(\left(C + \color{blue}{\sqrt{B \cdot B + C \cdot C}}\right) \cdot F\right)}^{0.5} \]

      4. unpow2 11.8%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot {\left(\left(C + \sqrt{\color{blue}{{B}^{2}} + C \cdot C}\right) \cdot F\right)}^{0.5} \]

      5. unpow2 11.8%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot {\left(\left(C + \sqrt{{B}^{2} + \color{blue}{{C}^{2}}}\right) \cdot F\right)}^{0.5} \]

      6. unpow-prod-down 12.9%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot \color{blue}{\left({\left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}^{0.5} \cdot {F}^{0.5}\right)} \]

   7. Applied egg-rr 42.9%

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

4. Final simplification 36.6%

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

Alternative 6: 37.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;F \leq 1.2 \cdot 10^{+144}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot F\right) \cdot \left(C + \mathsf{hypot}\left(C, B\_m\right)\right)}}{-B\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B\_m} \cdot \left(\sqrt{B\_m \cdot \left(\frac{C}{B\_m} + 1\right)} \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= F 1.2e+144)
   (/ (sqrt (* (* 2.0 F) (+ C (hypot C B_m)))) (- B_m))
   (* (/ (sqrt 2.0) B_m) (* (sqrt (* B_m (+ (/ C B_m) 1.0))) (- (sqrt F))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= 1.2e+144) {
		tmp = sqrt(((2.0 * F) * (C + hypot(C, B_m)))) / -B_m;
	} else {
		tmp = (sqrt(2.0) / B_m) * (sqrt((B_m * ((C / B_m) + 1.0))) * -sqrt(F));
	}
	return tmp;
}

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= 1.2e+144) {
		tmp = Math.sqrt(((2.0 * F) * (C + Math.hypot(C, B_m)))) / -B_m;
	} else {
		tmp = (Math.sqrt(2.0) / B_m) * (Math.sqrt((B_m * ((C / B_m) + 1.0))) * -Math.sqrt(F));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if F <= 1.2e+144:
		tmp = math.sqrt(((2.0 * F) * (C + math.hypot(C, B_m)))) / -B_m
	else:
		tmp = (math.sqrt(2.0) / B_m) * (math.sqrt((B_m * ((C / B_m) + 1.0))) * -math.sqrt(F))
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (F <= 1.2e+144)
		tmp = Float64(sqrt(Float64(Float64(2.0 * F) * Float64(C + hypot(C, B_m)))) / Float64(-B_m));
	else
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(sqrt(Float64(B_m * Float64(Float64(C / B_m) + 1.0))) * Float64(-sqrt(F))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (F <= 1.2e+144)
		tmp = sqrt(((2.0 * F) * (C + hypot(C, B_m)))) / -B_m;
	else
		tmp = (sqrt(2.0) / B_m) * (sqrt((B_m * ((C / B_m) + 1.0))) * -sqrt(F));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[F, 1.2e+144], N[(N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] * N[(C + N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[(N[Sqrt[N[(B$95$m * N[(N[(C / B$95$m), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < 1.2e144

   1. Initial program 20.6%

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

   3. Taylor expanded in A around 0 9.2%

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

      1. mul-1-neg 9.2%

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

      2. unpow2 9.2%

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

      3. unpow2 9.2%

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

      4. hypot-define 22.7%

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

   5. Simplified 22.7%

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

      1. associate-*l/ 22.7%

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

      2. sqrt-unprod 22.8%

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

      3. hypot-undefine 9.3%

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

      4. unpow2 9.3%

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

      5. unpow2 9.3%

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

      6. +-commutative 9.3%

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

      7. unpow2 9.3%

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

      8. unpow2 9.3%

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

      9. hypot-define 22.8%

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

   7. Applied egg-rr 22.8%

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

      1. associate-*r* 22.8%

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

      2. *-commutative 22.8%

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

   9. Simplified 22.8%

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

   if 1.2e144 < F

   1. Initial program 5.5%

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

   3. Taylor expanded in A around 0 6.8%

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

      1. mul-1-neg 6.8%

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

      2. unpow2 6.8%

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

      3. unpow2 6.8%

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

      4. hypot-define 7.1%

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

   5. Simplified 7.1%

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

      1. pow1/2 7.1%

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

      2. *-commutative 7.1%

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

      3. hypot-undefine 6.8%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot {\left(\left(C + \color{blue}{\sqrt{B \cdot B + C \cdot C}}\right) \cdot F\right)}^{0.5} \]

      4. unpow2 6.8%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot {\left(\left(C + \sqrt{\color{blue}{{B}^{2}} + C \cdot C}\right) \cdot F\right)}^{0.5} \]

      5. unpow2 6.8%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot {\left(\left(C + \sqrt{{B}^{2} + \color{blue}{{C}^{2}}}\right) \cdot F\right)}^{0.5} \]

      6. unpow-prod-down 9.1%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot \color{blue}{\left({\left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}^{0.5} \cdot {F}^{0.5}\right)} \]

   7. Applied egg-rr 30.6%

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

   8. Taylor expanded in B around inf 25.8%

      \[\leadsto -\frac{\sqrt{2}}{B} \cdot \left(\sqrt{\color{blue}{B \cdot \left(1 + \frac{C}{B}\right)}} \cdot \sqrt{F}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 23.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 1.2 \cdot 10^{+144}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot F\right) \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)}}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{B \cdot \left(\frac{C}{B} + 1\right)} \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 37.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;F \leq 1.2 \cdot 10^{+144}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot F\right) \cdot \left(C + \mathsf{hypot}\left(C, B\_m\right)\right)}}{-B\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B\_m} \cdot \left(\sqrt{B\_m + C} \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= F 1.2e+144)
   (/ (sqrt (* (* 2.0 F) (+ C (hypot C B_m)))) (- B_m))
   (* (/ (sqrt 2.0) B_m) (* (sqrt (+ B_m C)) (- (sqrt F))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= 1.2e+144) {
		tmp = sqrt(((2.0 * F) * (C + hypot(C, B_m)))) / -B_m;
	} else {
		tmp = (sqrt(2.0) / B_m) * (sqrt((B_m + C)) * -sqrt(F));
	}
	return tmp;
}

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= 1.2e+144) {
		tmp = Math.sqrt(((2.0 * F) * (C + Math.hypot(C, B_m)))) / -B_m;
	} else {
		tmp = (Math.sqrt(2.0) / B_m) * (Math.sqrt((B_m + C)) * -Math.sqrt(F));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if F <= 1.2e+144:
		tmp = math.sqrt(((2.0 * F) * (C + math.hypot(C, B_m)))) / -B_m
	else:
		tmp = (math.sqrt(2.0) / B_m) * (math.sqrt((B_m + C)) * -math.sqrt(F))
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (F <= 1.2e+144)
		tmp = Float64(sqrt(Float64(Float64(2.0 * F) * Float64(C + hypot(C, B_m)))) / Float64(-B_m));
	else
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(sqrt(Float64(B_m + C)) * Float64(-sqrt(F))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (F <= 1.2e+144)
		tmp = sqrt(((2.0 * F) * (C + hypot(C, B_m)))) / -B_m;
	else
		tmp = (sqrt(2.0) / B_m) * (sqrt((B_m + C)) * -sqrt(F));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[F, 1.2e+144], N[(N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] * N[(C + N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[(N[Sqrt[N[(B$95$m + C), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < 1.2e144

   1. Initial program 20.6%

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

   3. Taylor expanded in A around 0 9.2%

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

      1. mul-1-neg 9.2%

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

      2. unpow2 9.2%

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

      3. unpow2 9.2%

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

      4. hypot-define 22.7%

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

   5. Simplified 22.7%

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

      1. associate-*l/ 22.7%

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

      2. sqrt-unprod 22.8%

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

      3. hypot-undefine 9.3%

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

      4. unpow2 9.3%

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

      5. unpow2 9.3%

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

      6. +-commutative 9.3%

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

      7. unpow2 9.3%

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

      8. unpow2 9.3%

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

      9. hypot-define 22.8%

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

   7. Applied egg-rr 22.8%

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

      1. associate-*r* 22.8%

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

      2. *-commutative 22.8%

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

   9. Simplified 22.8%

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

   if 1.2e144 < F

   1. Initial program 5.5%

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

   3. Taylor expanded in A around 0 6.8%

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

      1. mul-1-neg 6.8%

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

      2. unpow2 6.8%

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

      3. unpow2 6.8%

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

      4. hypot-define 7.1%

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

   5. Simplified 7.1%

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

      1. pow1/2 7.1%

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

      2. *-commutative 7.1%

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

      3. hypot-undefine 6.8%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot {\left(\left(C + \color{blue}{\sqrt{B \cdot B + C \cdot C}}\right) \cdot F\right)}^{0.5} \]

      4. unpow2 6.8%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot {\left(\left(C + \sqrt{\color{blue}{{B}^{2}} + C \cdot C}\right) \cdot F\right)}^{0.5} \]

      5. unpow2 6.8%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot {\left(\left(C + \sqrt{{B}^{2} + \color{blue}{{C}^{2}}}\right) \cdot F\right)}^{0.5} \]

      6. unpow-prod-down 9.1%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot \color{blue}{\left({\left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}^{0.5} \cdot {F}^{0.5}\right)} \]

   7. Applied egg-rr 30.6%

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

   8. Taylor expanded in C around 0 25.8%

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

4. Final simplification 23.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 1.2 \cdot 10^{+144}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot F\right) \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)}}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{B + C} \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 37.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;F \leq 1.2 \cdot 10^{+144}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot F\right) \cdot \left(C + \mathsf{hypot}\left(C, B\_m\right)\right)}}{-B\_m}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F} \cdot \left(-\sqrt{\frac{2}{B\_m}}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= F 1.2e+144)
   (/ (sqrt (* (* 2.0 F) (+ C (hypot C B_m)))) (- B_m))
   (* (sqrt F) (- (sqrt (/ 2.0 B_m))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= 1.2e+144) {
		tmp = sqrt(((2.0 * F) * (C + hypot(C, B_m)))) / -B_m;
	} else {
		tmp = sqrt(F) * -sqrt((2.0 / B_m));
	}
	return tmp;
}

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= 1.2e+144) {
		tmp = Math.sqrt(((2.0 * F) * (C + Math.hypot(C, B_m)))) / -B_m;
	} else {
		tmp = Math.sqrt(F) * -Math.sqrt((2.0 / B_m));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if F <= 1.2e+144:
		tmp = math.sqrt(((2.0 * F) * (C + math.hypot(C, B_m)))) / -B_m
	else:
		tmp = math.sqrt(F) * -math.sqrt((2.0 / B_m))
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (F <= 1.2e+144)
		tmp = Float64(sqrt(Float64(Float64(2.0 * F) * Float64(C + hypot(C, B_m)))) / Float64(-B_m));
	else
		tmp = Float64(sqrt(F) * Float64(-sqrt(Float64(2.0 / B_m))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (F <= 1.2e+144)
		tmp = sqrt(((2.0 * F) * (C + hypot(C, B_m)))) / -B_m;
	else
		tmp = sqrt(F) * -sqrt((2.0 / B_m));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[F, 1.2e+144], N[(N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] * N[(C + N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision], N[(N[Sqrt[F], $MachinePrecision] * (-N[Sqrt[N[(2.0 / B$95$m), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < 1.2e144

   1. Initial program 20.6%

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

   3. Taylor expanded in A around 0 9.2%

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

      1. mul-1-neg 9.2%

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

      2. unpow2 9.2%

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

      3. unpow2 9.2%

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

      4. hypot-define 22.7%

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

   5. Simplified 22.7%

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

      1. associate-*l/ 22.7%

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

      2. sqrt-unprod 22.8%

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

      3. hypot-undefine 9.3%

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

      4. unpow2 9.3%

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

      5. unpow2 9.3%

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

      6. +-commutative 9.3%

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

      7. unpow2 9.3%

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

      8. unpow2 9.3%

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

      9. hypot-define 22.8%

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

   7. Applied egg-rr 22.8%

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

      1. associate-*r* 22.8%

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

      2. *-commutative 22.8%

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

   9. Simplified 22.8%

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

   if 1.2e144 < F

   1. Initial program 5.5%

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

   3. Taylor expanded in B around inf 25.2%

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

      1. mul-1-neg 25.2%

         \[\leadsto \color{blue}{-\sqrt{\frac{F}{B}} \cdot \sqrt{2}} \]

   5. Simplified 25.2%

      \[\leadsto \color{blue}{-\sqrt{\frac{F}{B}} \cdot \sqrt{2}} \]
   6. Step-by-step derivation

      1. sqrt-unprod 25.2%

         \[\leadsto -\color{blue}{\sqrt{\frac{F}{B} \cdot 2}} \]

   7. Applied egg-rr 25.2%

      \[\leadsto -\color{blue}{\sqrt{\frac{F}{B} \cdot 2}} \]
   8. Step-by-step derivation

      1. *-commutative 25.2%

         \[\leadsto -\sqrt{\color{blue}{2 \cdot \frac{F}{B}}} \]

      2. clear-num 25.2%

         \[\leadsto -\sqrt{2 \cdot \color{blue}{\frac{1}{\frac{B}{F}}}} \]

      3. un-div-inv 25.2%

         \[\leadsto -\sqrt{\color{blue}{\frac{2}{\frac{B}{F}}}} \]

   9. Applied egg-rr 25.2%

      \[\leadsto -\sqrt{\color{blue}{\frac{2}{\frac{B}{F}}}} \]
   10. Step-by-step derivation

       1. associate-/r/ 25.2%

          \[\leadsto -\sqrt{\color{blue}{\frac{2}{B} \cdot F}} \]

   11. Simplified 25.2%

       \[\leadsto -\sqrt{\color{blue}{\frac{2}{B} \cdot F}} \]
   12. Step-by-step derivation

       1. sqrt-prod 25.2%

          \[\leadsto -\color{blue}{\sqrt{\frac{2}{B}} \cdot \sqrt{F}} \]

   13. Applied egg-rr 25.2%

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

4. Final simplification 23.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 1.2 \cdot 10^{+144}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot F\right) \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)}}{-B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F} \cdot \left(-\sqrt{\frac{2}{B}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 36.1% accurate, 3.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;C \leq 7.8 \cdot 10^{+244}:\\ \;\;\;\;\sqrt{F} \cdot \left(-\sqrt{\frac{2}{B\_m}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{C \cdot F}}{B\_m} \cdot \left(-2\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= C 7.8e+244)
   (* (sqrt F) (- (sqrt (/ 2.0 B_m))))
   (* (/ (sqrt (* C F)) B_m) (- 2.0))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (C <= 7.8e+244) {
		tmp = sqrt(F) * -sqrt((2.0 / B_m));
	} else {
		tmp = (sqrt((C * F)) / B_m) * -2.0;
	}
	return tmp;
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
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 <= 7.8d+244) then
        tmp = sqrt(f) * -sqrt((2.0d0 / b_m))
    else
        tmp = (sqrt((c * f)) / b_m) * -2.0d0
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (C <= 7.8e+244) {
		tmp = Math.sqrt(F) * -Math.sqrt((2.0 / B_m));
	} else {
		tmp = (Math.sqrt((C * F)) / B_m) * -2.0;
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if C <= 7.8e+244:
		tmp = math.sqrt(F) * -math.sqrt((2.0 / B_m))
	else:
		tmp = (math.sqrt((C * F)) / B_m) * -2.0
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (C <= 7.8e+244)
		tmp = Float64(sqrt(F) * Float64(-sqrt(Float64(2.0 / B_m))));
	else
		tmp = Float64(Float64(sqrt(Float64(C * F)) / B_m) * Float64(-2.0));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (C <= 7.8e+244)
		tmp = sqrt(F) * -sqrt((2.0 / B_m));
	else
		tmp = (sqrt((C * F)) / B_m) * -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[C, 7.8e+244], N[(N[Sqrt[F], $MachinePrecision] * (-N[Sqrt[N[(2.0 / B$95$m), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(N[(N[Sqrt[N[(C * F), $MachinePrecision]], $MachinePrecision] / B$95$m), $MachinePrecision] * (-2.0)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if C < 7.8000000000000001e244

   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 B around inf 17.0%

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

      1. mul-1-neg 17.0%

         \[\leadsto \color{blue}{-\sqrt{\frac{F}{B}} \cdot \sqrt{2}} \]

   5. Simplified 17.0%

      \[\leadsto \color{blue}{-\sqrt{\frac{F}{B}} \cdot \sqrt{2}} \]
   6. Step-by-step derivation

      1. sqrt-unprod 17.1%

         \[\leadsto -\color{blue}{\sqrt{\frac{F}{B} \cdot 2}} \]

   7. Applied egg-rr 17.1%

      \[\leadsto -\color{blue}{\sqrt{\frac{F}{B} \cdot 2}} \]
   8. Step-by-step derivation

      1. *-commutative 17.1%

         \[\leadsto -\sqrt{\color{blue}{2 \cdot \frac{F}{B}}} \]

      2. clear-num 16.0%

         \[\leadsto -\sqrt{2 \cdot \color{blue}{\frac{1}{\frac{B}{F}}}} \]

      3. un-div-inv 16.0%

         \[\leadsto -\sqrt{\color{blue}{\frac{2}{\frac{B}{F}}}} \]

   9. Applied egg-rr 16.0%

      \[\leadsto -\sqrt{\color{blue}{\frac{2}{\frac{B}{F}}}} \]
   10. Step-by-step derivation

       1. associate-/r/ 17.2%

          \[\leadsto -\sqrt{\color{blue}{\frac{2}{B} \cdot F}} \]

   11. Simplified 17.2%

       \[\leadsto -\sqrt{\color{blue}{\frac{2}{B} \cdot F}} \]
   12. Step-by-step derivation

       1. sqrt-prod 22.2%

          \[\leadsto -\color{blue}{\sqrt{\frac{2}{B}} \cdot \sqrt{F}} \]

   13. Applied egg-rr 22.2%

       \[\leadsto -\color{blue}{\sqrt{\frac{2}{B}} \cdot \sqrt{F}} \]

   if 7.8000000000000001e244 < C

   1. Initial program 1.4%

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

   3. Taylor expanded in A around 0 1.3%

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

      1. mul-1-neg 1.3%

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

      2. unpow2 1.3%

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

      3. unpow2 1.3%

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

      4. hypot-define 25.7%

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

   5. Simplified 25.7%

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

      1. pow1/2 26.3%

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

      2. *-commutative 26.3%

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

      3. hypot-undefine 2.4%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot {\left(\left(C + \color{blue}{\sqrt{B \cdot B + C \cdot C}}\right) \cdot F\right)}^{0.5} \]

      4. unpow2 2.4%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot {\left(\left(C + \sqrt{\color{blue}{{B}^{2}} + C \cdot C}\right) \cdot F\right)}^{0.5} \]

      5. unpow2 2.4%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot {\left(\left(C + \sqrt{{B}^{2} + \color{blue}{{C}^{2}}}\right) \cdot F\right)}^{0.5} \]

      6. unpow-prod-down 1.3%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot \color{blue}{\left({\left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}^{0.5} \cdot {F}^{0.5}\right)} \]

   7. Applied egg-rr 33.6%

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

      1. *-un-lft-identity 33.6%

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

      2. metadata-eval 33.6%

         \[\leadsto -\frac{\color{blue}{\left(3 \cdot 0.3333333333333333\right)} \cdot \sqrt{2}}{B} \cdot \left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \sqrt{F}\right) \]

      3. add-cbrt-cube 1.8%

         \[\leadsto -\frac{\left(3 \cdot 0.3333333333333333\right) \cdot \sqrt{2}}{\color{blue}{\sqrt[3]{\left(B \cdot B\right) \cdot B}}} \cdot \left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \sqrt{F}\right) \]

      4. unpow2 1.8%

         \[\leadsto -\frac{\left(3 \cdot 0.3333333333333333\right) \cdot \sqrt{2}}{\sqrt[3]{\color{blue}{{B}^{2}} \cdot B}} \cdot \left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \sqrt{F}\right) \]

      5. cbrt-prod 9.7%

         \[\leadsto -\frac{\left(3 \cdot 0.3333333333333333\right) \cdot \sqrt{2}}{\color{blue}{\sqrt[3]{{B}^{2}} \cdot \sqrt[3]{B}}} \cdot \left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \sqrt{F}\right) \]

      6. times-frac 9.7%

         \[\leadsto -\color{blue}{\left(\frac{3 \cdot 0.3333333333333333}{\sqrt[3]{{B}^{2}}} \cdot \frac{\sqrt{2}}{\sqrt[3]{B}}\right)} \cdot \left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \sqrt{F}\right) \]

      7. metadata-eval 9.7%

         \[\leadsto -\left(\frac{\color{blue}{1}}{\sqrt[3]{{B}^{2}}} \cdot \frac{\sqrt{2}}{\sqrt[3]{B}}\right) \cdot \left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \sqrt{F}\right) \]

      8. unpow2 9.7%

         \[\leadsto -\left(\frac{1}{\sqrt[3]{\color{blue}{B \cdot B}}} \cdot \frac{\sqrt{2}}{\sqrt[3]{B}}\right) \cdot \left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \sqrt{F}\right) \]

      9. cbrt-prod 33.5%

         \[\leadsto -\left(\frac{1}{\color{blue}{\sqrt[3]{B} \cdot \sqrt[3]{B}}} \cdot \frac{\sqrt{2}}{\sqrt[3]{B}}\right) \cdot \left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \sqrt{F}\right) \]

      10. pow2 33.5%

          \[\leadsto -\left(\frac{1}{\color{blue}{{\left(\sqrt[3]{B}\right)}^{2}}} \cdot \frac{\sqrt{2}}{\sqrt[3]{B}}\right) \cdot \left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \sqrt{F}\right) \]

   9. Applied egg-rr 33.5%

      \[\leadsto -\color{blue}{\left(\frac{1}{{\left(\sqrt[3]{B}\right)}^{2}} \cdot \frac{\sqrt{2}}{\sqrt[3]{B}}\right)} \cdot \left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \sqrt{F}\right) \]
   10. Step-by-step derivation

       1. associate-*l/ 33.6%

          \[\leadsto -\color{blue}{\frac{1 \cdot \frac{\sqrt{2}}{\sqrt[3]{B}}}{{\left(\sqrt[3]{B}\right)}^{2}}} \cdot \left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \sqrt{F}\right) \]

       2. *-lft-identity 33.6%

          \[\leadsto -\frac{\color{blue}{\frac{\sqrt{2}}{\sqrt[3]{B}}}}{{\left(\sqrt[3]{B}\right)}^{2}} \cdot \left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \sqrt{F}\right) \]

   11. Simplified 33.6%

       \[\leadsto -\color{blue}{\frac{\frac{\sqrt{2}}{\sqrt[3]{B}}}{{\left(\sqrt[3]{B}\right)}^{2}}} \cdot \left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \sqrt{F}\right) \]

   12. Taylor expanded in B around 0 23.8%

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

       1. associate-*l/ 23.8%

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

       2. unpow2 23.8%

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

       3. rem-square-sqrt 23.8%

          \[\leadsto -\frac{\color{blue}{2} \cdot \sqrt{C \cdot F}}{B} \]

       4. associate-/l* 23.8%

          \[\leadsto -\color{blue}{2 \cdot \frac{\sqrt{C \cdot F}}{B}} \]

   14. Simplified 23.8%

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

4. Final simplification 22.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq 7.8 \cdot 10^{+244}:\\ \;\;\;\;\sqrt{F} \cdot \left(-\sqrt{\frac{2}{B}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{C \cdot F}}{B} \cdot \left(-2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 28.2% accurate, 5.6× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;C \leq 8 \cdot 10^{+174}:\\ \;\;\;\;-{\left(\frac{2 \cdot F}{B\_m}\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{C \cdot F}}{B\_m} \cdot \left(-2\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= C 8e+174)
   (- (pow (/ (* 2.0 F) B_m) 0.5))
   (* (/ (sqrt (* C F)) B_m) (- 2.0))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (C <= 8e+174) {
		tmp = -pow(((2.0 * F) / B_m), 0.5);
	} else {
		tmp = (sqrt((C * F)) / B_m) * -2.0;
	}
	return tmp;
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
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+174) then
        tmp = -(((2.0d0 * f) / b_m) ** 0.5d0)
    else
        tmp = (sqrt((c * f)) / b_m) * -2.0d0
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (C <= 8e+174) {
		tmp = -Math.pow(((2.0 * F) / B_m), 0.5);
	} else {
		tmp = (Math.sqrt((C * F)) / B_m) * -2.0;
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if C <= 8e+174:
		tmp = -math.pow(((2.0 * F) / B_m), 0.5)
	else:
		tmp = (math.sqrt((C * F)) / B_m) * -2.0
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (C <= 8e+174)
		tmp = Float64(-(Float64(Float64(2.0 * F) / B_m) ^ 0.5));
	else
		tmp = Float64(Float64(sqrt(Float64(C * F)) / B_m) * Float64(-2.0));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (C <= 8e+174)
		tmp = -(((2.0 * F) / B_m) ^ 0.5);
	else
		tmp = (sqrt((C * F)) / B_m) * -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[C, 8e+174], (-N[Power[N[(N[(2.0 * F), $MachinePrecision] / B$95$m), $MachinePrecision], 0.5], $MachinePrecision]), N[(N[(N[Sqrt[N[(C * F), $MachinePrecision]], $MachinePrecision] / B$95$m), $MachinePrecision] * (-2.0)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if C < 8.00000000000000055e174

   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 B around inf 17.9%

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

      1. mul-1-neg 17.9%

         \[\leadsto \color{blue}{-\sqrt{\frac{F}{B}} \cdot \sqrt{2}} \]

   5. Simplified 17.9%

      \[\leadsto \color{blue}{-\sqrt{\frac{F}{B}} \cdot \sqrt{2}} \]
   6. Step-by-step derivation

      1. sqrt-unprod 18.0%

         \[\leadsto -\color{blue}{\sqrt{\frac{F}{B} \cdot 2}} \]

   7. Applied egg-rr 18.0%

      \[\leadsto -\color{blue}{\sqrt{\frac{F}{B} \cdot 2}} \]
   8. Step-by-step derivation

      1. *-commutative 18.0%

         \[\leadsto -\sqrt{\color{blue}{2 \cdot \frac{F}{B}}} \]

      2. clear-num 16.9%

         \[\leadsto -\sqrt{2 \cdot \color{blue}{\frac{1}{\frac{B}{F}}}} \]

      3. un-div-inv 16.9%

         \[\leadsto -\sqrt{\color{blue}{\frac{2}{\frac{B}{F}}}} \]

   9. Applied egg-rr 16.9%

      \[\leadsto -\sqrt{\color{blue}{\frac{2}{\frac{B}{F}}}} \]
   10. Step-by-step derivation

       1. associate-/r/ 18.1%

          \[\leadsto -\sqrt{\color{blue}{\frac{2}{B} \cdot F}} \]

   11. Simplified 18.1%

       \[\leadsto -\sqrt{\color{blue}{\frac{2}{B} \cdot F}} \]
   12. Step-by-step derivation

       1. pow1/2 18.2%

          \[\leadsto -\color{blue}{{\left(\frac{2}{B} \cdot F\right)}^{0.5}} \]

       2. associate-*l/ 18.2%

          \[\leadsto -{\color{blue}{\left(\frac{2 \cdot F}{B}\right)}}^{0.5} \]

   13. Applied egg-rr 18.2%

       \[\leadsto -\color{blue}{{\left(\frac{2 \cdot F}{B}\right)}^{0.5}} \]

   if 8.00000000000000055e174 < C

   1. Initial program 1.8%

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

   3. Taylor expanded in A around 0 1.4%

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

      1. mul-1-neg 1.4%

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

      2. unpow2 1.4%

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

      3. unpow2 1.4%

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

      4. hypot-define 19.5%

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

   5. Simplified 19.5%

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

      1. pow1/2 19.9%

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

      2. *-commutative 19.9%

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

      3. hypot-undefine 2.3%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot {\left(\left(C + \color{blue}{\sqrt{B \cdot B + C \cdot C}}\right) \cdot F\right)}^{0.5} \]

      4. unpow2 2.3%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot {\left(\left(C + \sqrt{\color{blue}{{B}^{2}} + C \cdot C}\right) \cdot F\right)}^{0.5} \]

      5. unpow2 2.3%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot {\left(\left(C + \sqrt{{B}^{2} + \color{blue}{{C}^{2}}}\right) \cdot F\right)}^{0.5} \]

      6. unpow-prod-down 1.4%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot \color{blue}{\left({\left(C + \sqrt{{B}^{2} + {C}^{2}}\right)}^{0.5} \cdot {F}^{0.5}\right)} \]

   7. Applied egg-rr 30.1%

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

      1. *-un-lft-identity 30.1%

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

      2. metadata-eval 30.1%

         \[\leadsto -\frac{\color{blue}{\left(3 \cdot 0.3333333333333333\right)} \cdot \sqrt{2}}{B} \cdot \left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \sqrt{F}\right) \]

      3. add-cbrt-cube 1.9%

         \[\leadsto -\frac{\left(3 \cdot 0.3333333333333333\right) \cdot \sqrt{2}}{\color{blue}{\sqrt[3]{\left(B \cdot B\right) \cdot B}}} \cdot \left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \sqrt{F}\right) \]

      4. unpow2 1.9%

         \[\leadsto -\frac{\left(3 \cdot 0.3333333333333333\right) \cdot \sqrt{2}}{\sqrt[3]{\color{blue}{{B}^{2}} \cdot B}} \cdot \left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \sqrt{F}\right) \]

      5. cbrt-prod 12.5%

         \[\leadsto -\frac{\left(3 \cdot 0.3333333333333333\right) \cdot \sqrt{2}}{\color{blue}{\sqrt[3]{{B}^{2}} \cdot \sqrt[3]{B}}} \cdot \left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \sqrt{F}\right) \]

      6. times-frac 12.5%

         \[\leadsto -\color{blue}{\left(\frac{3 \cdot 0.3333333333333333}{\sqrt[3]{{B}^{2}}} \cdot \frac{\sqrt{2}}{\sqrt[3]{B}}\right)} \cdot \left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \sqrt{F}\right) \]

      7. metadata-eval 12.5%

         \[\leadsto -\left(\frac{\color{blue}{1}}{\sqrt[3]{{B}^{2}}} \cdot \frac{\sqrt{2}}{\sqrt[3]{B}}\right) \cdot \left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \sqrt{F}\right) \]

      8. unpow2 12.5%

         \[\leadsto -\left(\frac{1}{\sqrt[3]{\color{blue}{B \cdot B}}} \cdot \frac{\sqrt{2}}{\sqrt[3]{B}}\right) \cdot \left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \sqrt{F}\right) \]

      9. cbrt-prod 29.9%

         \[\leadsto -\left(\frac{1}{\color{blue}{\sqrt[3]{B} \cdot \sqrt[3]{B}}} \cdot \frac{\sqrt{2}}{\sqrt[3]{B}}\right) \cdot \left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \sqrt{F}\right) \]

      10. pow2 29.9%

          \[\leadsto -\left(\frac{1}{\color{blue}{{\left(\sqrt[3]{B}\right)}^{2}}} \cdot \frac{\sqrt{2}}{\sqrt[3]{B}}\right) \cdot \left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \sqrt{F}\right) \]

   9. Applied egg-rr 29.9%

      \[\leadsto -\color{blue}{\left(\frac{1}{{\left(\sqrt[3]{B}\right)}^{2}} \cdot \frac{\sqrt{2}}{\sqrt[3]{B}}\right)} \cdot \left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \sqrt{F}\right) \]
   10. Step-by-step derivation

       1. associate-*l/ 30.0%

          \[\leadsto -\color{blue}{\frac{1 \cdot \frac{\sqrt{2}}{\sqrt[3]{B}}}{{\left(\sqrt[3]{B}\right)}^{2}}} \cdot \left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \sqrt{F}\right) \]

       2. *-lft-identity 30.0%

          \[\leadsto -\frac{\color{blue}{\frac{\sqrt{2}}{\sqrt[3]{B}}}}{{\left(\sqrt[3]{B}\right)}^{2}} \cdot \left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \sqrt{F}\right) \]

   11. Simplified 30.0%

       \[\leadsto -\color{blue}{\frac{\frac{\sqrt{2}}{\sqrt[3]{B}}}{{\left(\sqrt[3]{B}\right)}^{2}}} \cdot \left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \sqrt{F}\right) \]

   12. Taylor expanded in B around 0 15.3%

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

       1. associate-*l/ 15.3%

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

       2. unpow2 15.3%

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

       3. rem-square-sqrt 15.3%

          \[\leadsto -\frac{\color{blue}{2} \cdot \sqrt{C \cdot F}}{B} \]

       4. associate-/l* 15.3%

          \[\leadsto -\color{blue}{2 \cdot \frac{\sqrt{C \cdot F}}{B}} \]

   14. Simplified 15.3%

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

4. Final simplification 17.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq 8 \cdot 10^{+174}:\\ \;\;\;\;-{\left(\frac{2 \cdot F}{B}\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{C \cdot F}}{B} \cdot \left(-2\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 26.8% accurate, 5.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ -{\left(\frac{2 \cdot F}{B\_m}\right)}^{0.5} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F) :precision binary64 (- (pow (/ (* 2.0 F) B_m) 0.5)))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	return -pow(((2.0 * F) / B_m), 0.5);
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
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 = -(((2.0d0 * f) / b_m) ** 0.5d0)
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	return -Math.pow(((2.0 * F) / B_m), 0.5);
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	return -math.pow(((2.0 * F) / B_m), 0.5)

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	return Float64(-(Float64(Float64(2.0 * F) / B_m) ^ 0.5))
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp = code(A, B_m, C, F)
	tmp = -(((2.0 * F) / B_m) ^ 0.5);
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := (-N[Power[N[(N[(2.0 * F), $MachinePrecision] / B$95$m), $MachinePrecision], 0.5], $MachinePrecision])

Derivation

1. Initial program 16.8%

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

3. Taylor expanded in B around inf 16.4%

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

   1. mul-1-neg 16.4%

      \[\leadsto \color{blue}{-\sqrt{\frac{F}{B}} \cdot \sqrt{2}} \]

5. Simplified 16.4%

   \[\leadsto \color{blue}{-\sqrt{\frac{F}{B}} \cdot \sqrt{2}} \]
6. Step-by-step derivation

   1. sqrt-unprod 16.5%

      \[\leadsto -\color{blue}{\sqrt{\frac{F}{B} \cdot 2}} \]

7. Applied egg-rr 16.5%

   \[\leadsto -\color{blue}{\sqrt{\frac{F}{B} \cdot 2}} \]
8. Step-by-step derivation

   1. *-commutative 16.5%

      \[\leadsto -\sqrt{\color{blue}{2 \cdot \frac{F}{B}}} \]

   2. clear-num 15.4%

      \[\leadsto -\sqrt{2 \cdot \color{blue}{\frac{1}{\frac{B}{F}}}} \]

   3. un-div-inv 15.4%

      \[\leadsto -\sqrt{\color{blue}{\frac{2}{\frac{B}{F}}}} \]

9. Applied egg-rr 15.4%

   \[\leadsto -\sqrt{\color{blue}{\frac{2}{\frac{B}{F}}}} \]
10. Step-by-step derivation

    1. associate-/r/ 16.6%

       \[\leadsto -\sqrt{\color{blue}{\frac{2}{B} \cdot F}} \]

11. Simplified 16.6%

    \[\leadsto -\sqrt{\color{blue}{\frac{2}{B} \cdot F}} \]
12. Step-by-step derivation

    1. pow1/2 16.7%

       \[\leadsto -\color{blue}{{\left(\frac{2}{B} \cdot F\right)}^{0.5}} \]

    2. associate-*l/ 16.7%

       \[\leadsto -{\color{blue}{\left(\frac{2 \cdot F}{B}\right)}}^{0.5} \]

13. Applied egg-rr 16.7%

    \[\leadsto -\color{blue}{{\left(\frac{2 \cdot F}{B}\right)}^{0.5}} \]

14. Final simplification 16.7%

    \[\leadsto -{\left(\frac{2 \cdot F}{B}\right)}^{0.5} \]
15. Add Preprocessing

Alternative 12: 26.8% accurate, 6.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ -\sqrt{F \cdot \frac{2}{B\_m}} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F) :precision binary64 (- (sqrt (* F (/ 2.0 B_m)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	return -sqrt((F * (2.0 / B_m)));
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
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);
assert A < B_m && B_m < C && C < F;
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)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	return -math.sqrt((F * (2.0 / B_m)))

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	return Float64(-sqrt(Float64(F * Float64(2.0 / B_m))))
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp = code(A, B_m, C, F)
	tmp = -sqrt((F * (2.0 / B_m)));
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := (-N[Sqrt[N[(F * N[(2.0 / B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])

Derivation

1. Initial program 16.8%

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

3. Taylor expanded in B around inf 16.4%

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

   1. mul-1-neg 16.4%

      \[\leadsto \color{blue}{-\sqrt{\frac{F}{B}} \cdot \sqrt{2}} \]

5. Simplified 16.4%

   \[\leadsto \color{blue}{-\sqrt{\frac{F}{B}} \cdot \sqrt{2}} \]
6. Step-by-step derivation

   1. sqrt-unprod 16.5%

      \[\leadsto -\color{blue}{\sqrt{\frac{F}{B} \cdot 2}} \]

7. Applied egg-rr 16.5%

   \[\leadsto -\color{blue}{\sqrt{\frac{F}{B} \cdot 2}} \]
8. Step-by-step derivation

   1. *-commutative 16.5%

      \[\leadsto -\sqrt{\color{blue}{2 \cdot \frac{F}{B}}} \]

   2. clear-num 15.4%

      \[\leadsto -\sqrt{2 \cdot \color{blue}{\frac{1}{\frac{B}{F}}}} \]

   3. un-div-inv 15.4%

      \[\leadsto -\sqrt{\color{blue}{\frac{2}{\frac{B}{F}}}} \]

9. Applied egg-rr 15.4%

   \[\leadsto -\sqrt{\color{blue}{\frac{2}{\frac{B}{F}}}} \]
10. Step-by-step derivation

    1. associate-/r/ 16.6%

       \[\leadsto -\sqrt{\color{blue}{\frac{2}{B} \cdot F}} \]

11. Simplified 16.6%

    \[\leadsto -\sqrt{\color{blue}{\frac{2}{B} \cdot F}} \]

12. Final simplification 16.6%

    \[\leadsto -\sqrt{F \cdot \frac{2}{B}} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024100 
(FPCore (A B C F)
  :name "ABCF->ab-angle a"
  :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))))