ABCF->ab-angle a

Percentage Accurate: 18.5% → 59.6%

Time: 31.6s

Alternatives: 14

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: 18.5% 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: 59.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := 2 \cdot t\_0\\ t_2 := -t\_0\\ t_3 := \left(4 \cdot A\right) \cdot C\\ t_4 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_3\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_3 - {B\_m}^{2}}\\ \mathbf{if}\;t\_4 \leq -4 \cdot 10^{-184}:\\ \;\;\;\;\frac{\left(\sqrt{t\_1} \cdot \sqrt{F}\right) \cdot \sqrt{A + \left(C + \mathsf{hypot}\left(B\_m, A - C\right)\right)}}{t\_2}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;\frac{\sqrt{F \cdot t\_1} \cdot \sqrt{-0.5 \cdot \frac{{B\_m}^{2}}{A} + 2 \cdot C}}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B\_m} \cdot \left(\sqrt{C + \mathsf{hypot}\left(B\_m, C\right)} \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \end{array} \]

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 (fma B_m B_m (* A (* C -4.0))))
        (t_1 (* 2.0 t_0))
        (t_2 (- t_0))
        (t_3 (* (* 4.0 A) C))
        (t_4
         (/
          (sqrt
           (*
            (* 2.0 (* (- (pow B_m 2.0) t_3) F))
            (+ (+ A C) (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))))))
          (- t_3 (pow B_m 2.0)))))
   (if (<= t_4 -4e-184)
     (/ (* (* (sqrt t_1) (sqrt F)) (sqrt (+ A (+ C (hypot B_m (- A C)))))) t_2)
     (if (<= t_4 INFINITY)
       (/
        (* (sqrt (* F t_1)) (sqrt (+ (* -0.5 (/ (pow B_m 2.0) A)) (* 2.0 C))))
        t_2)
       (* (/ (sqrt 2.0) B_m) (* (sqrt (+ C (hypot 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 t_0 = fma(B_m, B_m, (A * (C * -4.0)));
	double t_1 = 2.0 * t_0;
	double t_2 = -t_0;
	double t_3 = (4.0 * A) * C;
	double t_4 = sqrt(((2.0 * ((pow(B_m, 2.0) - t_3) * F)) * ((A + C) + sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / (t_3 - pow(B_m, 2.0));
	double tmp;
	if (t_4 <= -4e-184) {
		tmp = ((sqrt(t_1) * sqrt(F)) * sqrt((A + (C + hypot(B_m, (A - C)))))) / t_2;
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = (sqrt((F * t_1)) * sqrt(((-0.5 * (pow(B_m, 2.0) / A)) + (2.0 * C)))) / t_2;
	} else {
		tmp = (sqrt(2.0) / B_m) * (sqrt((C + hypot(B_m, C))) * -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 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	t_1 = Float64(2.0 * t_0)
	t_2 = Float64(-t_0)
	t_3 = Float64(Float64(4.0 * A) * C)
	t_4 = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_3) * F)) * Float64(Float64(A + C) + sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / Float64(t_3 - (B_m ^ 2.0)))
	tmp = 0.0
	if (t_4 <= -4e-184)
		tmp = Float64(Float64(Float64(sqrt(t_1) * sqrt(F)) * sqrt(Float64(A + Float64(C + hypot(B_m, Float64(A - C)))))) / t_2);
	elseif (t_4 <= Inf)
		tmp = Float64(Float64(sqrt(Float64(F * t_1)) * sqrt(Float64(Float64(-0.5 * Float64((B_m ^ 2.0) / A)) + Float64(2.0 * C)))) / t_2);
	else
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(sqrt(Float64(C + hypot(B_m, C))) * 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[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 * t$95$0), $MachinePrecision]}, Block[{t$95$2 = (-t$95$0)}, Block[{t$95$3 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$3), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$3 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -4e-184], N[(N[(N[(N[Sqrt[t$95$1], $MachinePrecision] * N[Sqrt[F], $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$2), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[(N[(N[Sqrt[N[(F * t$95$1), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(N[(-0.5 * N[(N[Power[B$95$m, 2.0], $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision] + N[(2.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[(N[Sqrt[N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 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))) < -4.0000000000000002e-184

   1. Initial program 43.3%

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

   3. Simplified 51.0%

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

      1. pow1/2 51.0%

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

      2. associate-*r* 51.0%

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

      3. associate-+r+ 50.3%

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

      4. hypot-undefine 43.3%

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

      5. unpow2 43.3%

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

      6. unpow2 43.3%

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

      7. +-commutative 43.3%

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

      8. unpow-prod-down 50.9%

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

      9. *-commutative 50.9%

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

      10. pow1/2 50.9%

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

   6. Applied egg-rr 64.9%

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

      1. unpow1/2 64.9%

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

      2. associate-*l* 64.9%

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

      3. hypot-undefine 50.9%

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

      4. unpow2 50.9%

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

      5. unpow2 50.9%

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

      6. +-commutative 50.9%

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

      7. unpow2 50.9%

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

      8. unpow2 50.9%

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

      9. hypot-undefine 64.9%

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

   8. Simplified 64.9%

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

      1. pow1/2 64.9%

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

      2. *-commutative 64.9%

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

      3. unpow-prod-down 76.9%

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

      4. pow1/2 76.9%

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

      5. *-commutative 76.9%

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

      6. pow1/2 76.9%

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

   10. Applied egg-rr 76.9%

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

   if -4.0000000000000002e-184 < (/.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 13.5%

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

   3. Simplified 23.1%

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

      1. pow1/2 23.1%

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

      2. associate-*r* 23.1%

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

      3. associate-+r+ 20.9%

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

      4. hypot-undefine 13.5%

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

      5. unpow2 13.5%

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

      6. unpow2 13.5%

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

      7. +-commutative 13.5%

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

      8. unpow-prod-down 13.6%

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

      9. *-commutative 13.6%

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

      10. pow1/2 13.6%

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

   6. Applied egg-rr 29.9%

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

      1. unpow1/2 29.9%

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

      2. associate-*l* 29.9%

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

      3. hypot-undefine 15.2%

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

      4. unpow2 15.2%

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

      5. unpow2 15.2%

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

      6. +-commutative 15.2%

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

      7. unpow2 15.2%

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

      8. unpow2 15.2%

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

      9. hypot-undefine 29.9%

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

   8. Simplified 29.9%

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

   9. Taylor expanded in A around -inf 29.8%

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

   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 20.4%

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

   5. Simplified 20.4%

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

      1. pow1/2 20.4%

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

      2. *-commutative 20.4%

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

      3. unpow-prod-down 29.8%

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

      4. pow1/2 29.8%

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

      5. pow1/2 29.8%

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

   7. Applied egg-rr 29.8%

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

4. Final simplification 46.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -4 \cdot 10^{-184}:\\ \;\;\;\;\frac{\left(\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \cdot \sqrt{F}\right) \cdot \sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)} \cdot \sqrt{-0.5 \cdot \frac{{B}^{2}}{A} + 2 \cdot C}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{C + \mathsf{hypot}\left(B, C\right)} \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 57.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := \sqrt{C + \mathsf{hypot}\left(B\_m, C\right)}\\ \mathbf{if}\;{B\_m}^{2} \leq 5 \cdot 10^{+159}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(2 \cdot t\_0\right)} \cdot \left(-t\_1\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B\_m} \cdot \left(t\_1 \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \end{array} \]

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 (fma B_m B_m (* A (* C -4.0)))) (t_1 (sqrt (+ C (hypot B_m C)))))
   (if (<= (pow B_m 2.0) 5e+159)
     (/ (* (sqrt (* F (* 2.0 t_0))) (- t_1)) t_0)
     (* (/ (sqrt 2.0) B_m) (* t_1 (- (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 = fma(B_m, B_m, (A * (C * -4.0)));
	double t_1 = sqrt((C + hypot(B_m, C)));
	double tmp;
	if (pow(B_m, 2.0) <= 5e+159) {
		tmp = (sqrt((F * (2.0 * t_0))) * -t_1) / t_0;
	} else {
		tmp = (sqrt(2.0) / B_m) * (t_1 * -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 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	t_1 = sqrt(Float64(C + hypot(B_m, C)))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 5e+159)
		tmp = Float64(Float64(sqrt(Float64(F * Float64(2.0 * t_0))) * Float64(-t_1)) / t_0);
	else
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(t_1 * 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[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e+159], N[(N[(N[Sqrt[N[(F * N[(2.0 * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-t$95$1)), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[(t$95$1 * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 5.00000000000000003e159

   1. Initial program 24.2%

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

   3. Simplified 31.8%

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

      1. pow1/2 31.8%

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

      2. associate-*r* 31.8%

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

      3. associate-+r+ 30.4%

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

      4. hypot-undefine 24.3%

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

      5. unpow2 24.3%

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

      6. unpow2 24.3%

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

      7. +-commutative 24.3%

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

      8. unpow-prod-down 27.5%

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

      9. *-commutative 27.5%

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

      10. pow1/2 27.5%

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

   6. Applied egg-rr 40.8%

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

      1. unpow1/2 40.8%

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

      2. associate-*l* 40.8%

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

      3. hypot-undefine 28.1%

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

      4. unpow2 28.1%

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

      5. unpow2 28.1%

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

      6. +-commutative 28.1%

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

      7. unpow2 28.1%

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

      8. unpow2 28.1%

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

      9. hypot-undefine 40.8%

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

   8. Simplified 40.8%

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

   9. Taylor expanded in A around 0 22.4%

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

       1. unpow2 22.4%

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

       2. unpow2 22.4%

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

       3. hypot-undefine 29.4%

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

   11. Simplified 29.4%

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

   if 5.00000000000000003e159 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 8.6%

      \[\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]
   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 29.1%

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

   5. Simplified 29.1%

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

      1. pow1/2 29.1%

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

      2. *-commutative 29.1%

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

      3. unpow-prod-down 41.0%

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

      4. pow1/2 41.0%

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

      5. pow1/2 41.0%

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

   7. Applied egg-rr 41.0%

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

4. Final simplification 34.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 5 \cdot 10^{+159}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)} \cdot \left(-\sqrt{C + \mathsf{hypot}\left(B, C\right)}\right)}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{C + \mathsf{hypot}\left(B, C\right)} \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 42.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 10^{-312}:\\ \;\;\;\;\frac{\sqrt{\left(-8 \cdot \left(A \cdot \left(C \cdot F\right)\right)\right) \cdot \left(A + \left(C + \mathsf{hypot}\left(B\_m, C - A\right)\right)\right)}}{4 \cdot \left(A \cdot C\right) - {B\_m}^{2}}\\ \mathbf{elif}\;{B\_m}^{2} \leq 5 \cdot 10^{+234}:\\ \;\;\;\;-\sqrt{\left(2 \cdot F\right) \cdot \frac{C + \mathsf{hypot}\left(B\_m, C\right)}{{B\_m}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B\_m} \cdot \left(\sqrt{F} \cdot \left(-\sqrt{B\_m + C}\right)\right)\\ \end{array} \end{array} \]

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) 1e-312)
   (/
    (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+234)
     (- (sqrt (* (* 2.0 F) (/ (+ C (hypot B_m C)) (pow B_m 2.0)))))
     (* (/ (sqrt 2.0) B_m) (* (sqrt F) (- (sqrt (+ B_m C))))))))

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) <= 1e-312) {
		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+234) {
		tmp = -sqrt(((2.0 * F) * ((C + hypot(B_m, C)) / pow(B_m, 2.0))));
	} else {
		tmp = (sqrt(2.0) / B_m) * (sqrt(F) * -sqrt((B_m + C)));
	}
	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) <= 1e-312) {
		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+234) {
		tmp = -Math.sqrt(((2.0 * F) * ((C + Math.hypot(B_m, C)) / Math.pow(B_m, 2.0))));
	} else {
		tmp = (Math.sqrt(2.0) / B_m) * (Math.sqrt(F) * -Math.sqrt((B_m + C)));
	}
	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) <= 1e-312:
		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+234:
		tmp = -math.sqrt(((2.0 * F) * ((C + math.hypot(B_m, C)) / math.pow(B_m, 2.0))))
	else:
		tmp = (math.sqrt(2.0) / B_m) * (math.sqrt(F) * -math.sqrt((B_m + C)))
	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) <= 1e-312)
		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+234)
		tmp = Float64(-sqrt(Float64(Float64(2.0 * F) * Float64(Float64(C + hypot(B_m, C)) / (B_m ^ 2.0)))));
	else
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(sqrt(F) * Float64(-sqrt(Float64(B_m + C)))));
	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) <= 1e-312)
		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+234)
		tmp = -sqrt(((2.0 * F) * ((C + hypot(B_m, C)) / (B_m ^ 2.0))));
	else
		tmp = (sqrt(2.0) / B_m) * (sqrt(F) * -sqrt((B_m + C)));
	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], 1e-312], 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+234], (-N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] * N[(N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $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[F], $MachinePrecision] * (-N[Sqrt[N[(B$95$m + C), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 9.9999999999847e-313

   1. Initial program 16.5%

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

   3. Simplified 25.0%

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

   5. Taylor expanded in A around inf 19.1%

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

   if 9.9999999999847e-313 < (pow.f64 B #s(literal 2 binary64)) < 5.0000000000000003e234

   1. Initial program 30.8%

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

   3. Taylor expanded in A around 0 15.6%

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

      1. mul-1-neg 15.6%

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

      2. unpow2 15.6%

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

      3. unpow2 15.6%

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

      4. hypot-define 16.0%

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

   5. Simplified 16.0%

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

      1. pow1/2 16.0%

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

      2. *-commutative 16.0%

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

      3. unpow-prod-down 20.7%

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

      4. pow1/2 20.7%

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

      5. pow1/2 20.7%

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

   7. Applied egg-rr 20.7%

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

      1. add-sqr-sqrt 20.0%

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

      2. sqrt-unprod 37.2%

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

      3. associate-*l/ 37.2%

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

      4. sqrt-unprod 29.8%

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

      5. *-commutative 29.8%

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

      6. sqrt-prod 29.8%

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

      7. associate-*l/ 29.8%

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

      8. sqrt-unprod 29.8%

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

      9. *-commutative 29.8%

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

      10. sqrt-prod 29.8%

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

   9. Applied egg-rr 29.9%

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

       1. associate-/l* 37.5%

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

       2. *-commutative 37.5%

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

   11. Simplified 37.5%

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

   if 5.0000000000000003e234 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 5.6%

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

   3. Taylor expanded in A around 0 8.1%

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

      1. mul-1-neg 8.1%

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

      2. unpow2 8.1%

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

      3. unpow2 8.1%

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

      4. hypot-define 30.8%

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

   5. Simplified 30.8%

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

      1. pow1/2 30.8%

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

      2. *-commutative 30.8%

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

      3. unpow-prod-down 43.5%

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

      4. pow1/2 43.5%

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

      5. pow1/2 43.5%

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

   7. Applied egg-rr 43.5%

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

   8. Taylor expanded in C around 0 38.9%

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

4. Final simplification 33.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{-312}:\\ \;\;\;\;\frac{\sqrt{\left(-8 \cdot \left(A \cdot \left(C \cdot F\right)\right)\right) \cdot \left(A + \left(C + \mathsf{hypot}\left(B, C - A\right)\right)\right)}}{4 \cdot \left(A \cdot C\right) - {B}^{2}}\\ \mathbf{elif}\;{B}^{2} \leq 5 \cdot 10^{+234}:\\ \;\;\;\;-\sqrt{\left(2 \cdot F\right) \cdot \frac{C + \mathsf{hypot}\left(B, C\right)}{{B}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{F} \cdot \left(-\sqrt{B + C}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 56.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ \mathbf{if}\;{B\_m}^{2} \leq 10^{-50}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_0\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{t\_0 - {B\_m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B\_m} \cdot \left(\sqrt{C + \mathsf{hypot}\left(B\_m, C\right)} \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \end{array} \]

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) 1e-50)
     (/
      (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 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 t_0 = (4.0 * A) * C;
	double tmp;
	if (pow(B_m, 2.0) <= 1e-50) {
		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(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 t_0 = (4.0 * A) * C;
	double tmp;
	if (Math.pow(B_m, 2.0) <= 1e-50) {
		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(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):
	t_0 = (4.0 * A) * C
	tmp = 0
	if math.pow(B_m, 2.0) <= 1e-50:
		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(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)
	t_0 = Float64(Float64(4.0 * A) * C)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e-50)
		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(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)
	t_0 = (4.0 * A) * C;
	tmp = 0.0;
	if ((B_m ^ 2.0) <= 1e-50)
		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(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_] := Block[{t$95$0 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-50], 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[B$95$m ^ 2 + C ^ 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)) < 1.00000000000000001e-50

   1. Initial program 19.3%

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

   3. Taylor expanded in A around -inf 18.7%

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

   if 1.00000000000000001e-50 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 16.4%

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

   3. Taylor expanded in A around 0 12.7%

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

      1. mul-1-neg 12.7%

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

      2. unpow2 12.7%

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

      3. unpow2 12.7%

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

      4. hypot-define 27.5%

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

   5. Simplified 27.5%

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

      1. pow1/2 27.5%

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

      2. *-commutative 27.5%

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

      3. unpow-prod-down 37.4%

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

      4. pow1/2 37.4%

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

      5. pow1/2 37.4%

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

   7. Applied egg-rr 37.4%

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

4. Final simplification 29.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{-50}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{C + \mathsf{hypot}\left(B, C\right)} \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 51.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ \mathbf{if}\;{B\_m}^{2} \leq 10^{-50}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_0\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{t\_0 - {B\_m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B\_m} \cdot \left(\sqrt{F} \cdot \left(-\sqrt{B\_m + C}\right)\right)\\ \end{array} \end{array} \]

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) 1e-50)
     (/
      (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 F) (- (sqrt (+ B_m C))))))))

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) <= 1e-50) {
		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(F) * -sqrt((B_m + C)));
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = (4.0d0 * a) * c
    if ((b_m ** 2.0d0) <= 1d-50) then
        tmp = sqrt(((2.0d0 * (((b_m ** 2.0d0) - t_0) * f)) * (2.0d0 * c))) / (t_0 - (b_m ** 2.0d0))
    else
        tmp = (sqrt(2.0d0) / b_m) * (sqrt(f) * -sqrt((b_m + c)))
    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 t_0 = (4.0 * A) * C;
	double tmp;
	if (Math.pow(B_m, 2.0) <= 1e-50) {
		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(F) * -Math.sqrt((B_m + C)));
	}
	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) <= 1e-50:
		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(F) * -math.sqrt((B_m + C)))
	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) <= 1e-50)
		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(F) * Float64(-sqrt(Float64(B_m + C)))));
	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) <= 1e-50)
		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(F) * -sqrt((B_m + C)));
	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], 1e-50], 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[F], $MachinePrecision] * (-N[Sqrt[N[(B$95$m + C), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 1.00000000000000001e-50

   1. Initial program 19.3%

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

   3. Taylor expanded in A around -inf 18.7%

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

   if 1.00000000000000001e-50 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 16.4%

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

   3. Taylor expanded in A around 0 12.7%

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

      1. mul-1-neg 12.7%

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

      2. unpow2 12.7%

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

      3. unpow2 12.7%

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

      4. hypot-define 27.5%

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

   5. Simplified 27.5%

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

      1. pow1/2 27.5%

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

      2. *-commutative 27.5%

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

      3. unpow-prod-down 37.4%

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

      4. pow1/2 37.4%

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

      5. pow1/2 37.4%

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

   7. Applied egg-rr 37.4%

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

   8. Taylor expanded in C around 0 32.4%

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

4. Final simplification 26.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{-50}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{F} \cdot \left(-\sqrt{B + C}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 36.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;C \leq 5.8 \cdot 10^{+177}:\\ \;\;\;\;\frac{\sqrt{2}}{B\_m} \cdot \left(\sqrt{F} \cdot \left(-\sqrt{B\_m + C}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{F} \cdot \sqrt{2 \cdot C}\right) \cdot \frac{-\sqrt{2}}{B\_m}\\ \end{array} \end{array} \]

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 5.8e+177)
   (* (/ (sqrt 2.0) B_m) (* (sqrt F) (- (sqrt (+ B_m C)))))
   (* (* (sqrt F) (sqrt (* 2.0 C))) (/ (- (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 (C <= 5.8e+177) {
		tmp = (sqrt(2.0) / B_m) * (sqrt(F) * -sqrt((B_m + C)));
	} else {
		tmp = (sqrt(F) * sqrt((2.0 * C))) * (-sqrt(2.0) / B_m);
	}
	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 <= 5.8d+177) then
        tmp = (sqrt(2.0d0) / b_m) * (sqrt(f) * -sqrt((b_m + c)))
    else
        tmp = (sqrt(f) * sqrt((2.0d0 * c))) * (-sqrt(2.0d0) / b_m)
    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 <= 5.8e+177) {
		tmp = (Math.sqrt(2.0) / B_m) * (Math.sqrt(F) * -Math.sqrt((B_m + C)));
	} else {
		tmp = (Math.sqrt(F) * Math.sqrt((2.0 * C))) * (-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 C <= 5.8e+177:
		tmp = (math.sqrt(2.0) / B_m) * (math.sqrt(F) * -math.sqrt((B_m + C)))
	else:
		tmp = (math.sqrt(F) * math.sqrt((2.0 * C))) * (-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 (C <= 5.8e+177)
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(sqrt(F) * Float64(-sqrt(Float64(B_m + C)))));
	else
		tmp = Float64(Float64(sqrt(F) * sqrt(Float64(2.0 * C))) * Float64(Float64(-sqrt(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 (C <= 5.8e+177)
		tmp = (sqrt(2.0) / B_m) * (sqrt(F) * -sqrt((B_m + C)));
	else
		tmp = (sqrt(F) * sqrt((2.0 * C))) * (-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[C, 5.8e+177], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[(N[Sqrt[F], $MachinePrecision] * (-N[Sqrt[N[(B$95$m + C), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(2.0 * C), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[((-N[Sqrt[2.0], $MachinePrecision]) / B$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if C < 5.80000000000000027e177

   1. Initial program 19.7%

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

   3. Taylor expanded in A around 0 10.2%

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

      1. mul-1-neg 10.2%

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

      2. unpow2 10.2%

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

      3. unpow2 10.2%

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

      4. hypot-define 19.9%

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

   5. Simplified 19.9%

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

      1. pow1/2 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. unpow-prod-down 26.6%

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

      4. pow1/2 26.6%

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

      5. pow1/2 26.6%

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

   7. Applied egg-rr 26.6%

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

   8. Taylor expanded in C around 0 22.3%

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

   if 5.80000000000000027e177 < 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 1.0%

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

      1. mul-1-neg 1.0%

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

      2. unpow2 1.0%

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

      3. unpow2 1.0%

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

      4. hypot-define 1.5%

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

   5. Simplified 1.5%

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

      1. pow1/2 1.5%

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

      2. *-commutative 1.5%

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

      3. unpow-prod-down 8.4%

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

      4. pow1/2 8.4%

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

      5. pow1/2 8.4%

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

   7. Applied egg-rr 8.4%

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

   8. Taylor expanded in C around inf 5.7%

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

4. Final simplification 20.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq 5.8 \cdot 10^{+177}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{F} \cdot \left(-\sqrt{B + C}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{F} \cdot \sqrt{2 \cdot C}\right) \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 36.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;C \leq 6 \cdot 10^{+172}:\\ \;\;\;\;\sqrt{\frac{1}{B\_m}} \cdot \left(-\sqrt{2 \cdot F}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{F} \cdot \sqrt{2 \cdot C}\right) \cdot \frac{-\sqrt{2}}{B\_m}\\ \end{array} \end{array} \]

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 6e+172)
   (* (sqrt (/ 1.0 B_m)) (- (sqrt (* 2.0 F))))
   (* (* (sqrt F) (sqrt (* 2.0 C))) (/ (- (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 (C <= 6e+172) {
		tmp = sqrt((1.0 / B_m)) * -sqrt((2.0 * F));
	} else {
		tmp = (sqrt(F) * sqrt((2.0 * C))) * (-sqrt(2.0) / B_m);
	}
	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 <= 6d+172) then
        tmp = sqrt((1.0d0 / b_m)) * -sqrt((2.0d0 * f))
    else
        tmp = (sqrt(f) * sqrt((2.0d0 * c))) * (-sqrt(2.0d0) / b_m)
    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 <= 6e+172) {
		tmp = Math.sqrt((1.0 / B_m)) * -Math.sqrt((2.0 * F));
	} else {
		tmp = (Math.sqrt(F) * Math.sqrt((2.0 * C))) * (-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 C <= 6e+172:
		tmp = math.sqrt((1.0 / B_m)) * -math.sqrt((2.0 * F))
	else:
		tmp = (math.sqrt(F) * math.sqrt((2.0 * C))) * (-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 (C <= 6e+172)
		tmp = Float64(sqrt(Float64(1.0 / B_m)) * Float64(-sqrt(Float64(2.0 * F))));
	else
		tmp = Float64(Float64(sqrt(F) * sqrt(Float64(2.0 * C))) * Float64(Float64(-sqrt(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 (C <= 6e+172)
		tmp = sqrt((1.0 / B_m)) * -sqrt((2.0 * F));
	else
		tmp = (sqrt(F) * sqrt((2.0 * C))) * (-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[C, 6e+172], N[(N[Sqrt[N[(1.0 / B$95$m), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(2.0 * C), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[((-N[Sqrt[2.0], $MachinePrecision]) / B$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if C < 5.9999999999999998e172

   1. Initial program 20.0%

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

   3. Taylor expanded in B around inf 17.6%

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

      1. mul-1-neg 17.6%

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

   5. Simplified 17.6%

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

      1. pow1 17.6%

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

      2. sqrt-unprod 17.6%

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

   7. Applied egg-rr 17.6%

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

      1. unpow1 17.6%

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

      2. associate-*l/ 17.7%

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

   9. Simplified 17.7%

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

       1. pow1/2 17.8%

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

       2. div-inv 17.8%

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

       3. unpow-prod-down 22.8%

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

       4. pow1/2 22.8%

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

   11. Applied egg-rr 22.8%

       \[\leadsto -\color{blue}{\sqrt{F \cdot 2} \cdot {\left(\frac{1}{B}\right)}^{0.5}} \]
   12. Step-by-step derivation

       1. unpow1/2 22.8%

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

       2. *-commutative 22.8%

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

       3. *-commutative 22.8%

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

   13. Simplified 22.8%

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

   if 5.9999999999999998e172 < C

   1. Initial program 1.7%

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

   3. Taylor expanded in A around 0 1.1%

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

      1. mul-1-neg 1.1%

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

      2. unpow2 1.1%

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

      3. unpow2 1.1%

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

      4. hypot-define 1.6%

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

   5. Simplified 1.6%

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

      1. pow1/2 1.6%

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

      2. *-commutative 1.6%

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

      3. unpow-prod-down 10.8%

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

      4. pow1/2 10.8%

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

      5. pow1/2 10.8%

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

   7. Applied egg-rr 10.8%

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

   8. Taylor expanded in C around inf 8.3%

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

4. Final simplification 21.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq 6 \cdot 10^{+172}:\\ \;\;\;\;\sqrt{\frac{1}{B}} \cdot \left(-\sqrt{2 \cdot F}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{F} \cdot \sqrt{2 \cdot C}\right) \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 38.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;F \leq 2.3 \cdot 10^{+51}:\\ \;\;\;\;\frac{\sqrt{\left(C + \mathsf{hypot}\left(B\_m, C\right)\right) \cdot \left(2 \cdot F\right)}}{-B\_m}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{B\_m}} \cdot \left(-\sqrt{2 \cdot F}\right)\\ \end{array} \end{array} \]

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 2.3e+51)
   (/ (sqrt (* (+ C (hypot B_m C)) (* 2.0 F))) (- B_m))
   (* (sqrt (/ 1.0 B_m)) (- (sqrt (* 2.0 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 <= 2.3e+51) {
		tmp = sqrt(((C + hypot(B_m, C)) * (2.0 * F))) / -B_m;
	} else {
		tmp = sqrt((1.0 / B_m)) * -sqrt((2.0 * 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 <= 2.3e+51) {
		tmp = Math.sqrt(((C + Math.hypot(B_m, C)) * (2.0 * F))) / -B_m;
	} else {
		tmp = Math.sqrt((1.0 / B_m)) * -Math.sqrt((2.0 * 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 <= 2.3e+51:
		tmp = math.sqrt(((C + math.hypot(B_m, C)) * (2.0 * F))) / -B_m
	else:
		tmp = math.sqrt((1.0 / B_m)) * -math.sqrt((2.0 * 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 <= 2.3e+51)
		tmp = Float64(sqrt(Float64(Float64(C + hypot(B_m, C)) * Float64(2.0 * F))) / Float64(-B_m));
	else
		tmp = Float64(sqrt(Float64(1.0 / B_m)) * Float64(-sqrt(Float64(2.0 * 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 <= 2.3e+51)
		tmp = sqrt(((C + hypot(B_m, C)) * (2.0 * F))) / -B_m;
	else
		tmp = sqrt((1.0 / B_m)) * -sqrt((2.0 * 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, 2.3e+51], N[(N[Sqrt[N[(N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision] * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision], N[(N[Sqrt[N[(1.0 / B$95$m), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < 2.30000000000000005e51

   1. Initial program 20.7%

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

   3. Taylor expanded in A around 0 9.3%

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

      1. mul-1-neg 9.3%

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

      2. unpow2 9.3%

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

      3. unpow2 9.3%

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

      4. hypot-define 21.6%

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

   5. Simplified 21.6%

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

      1. associate-*l/ 21.5%

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

      2. sqrt-unprod 21.6%

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

   7. Applied egg-rr 21.6%

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

      1. associate-*r* 21.6%

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

   9. Simplified 21.6%

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

   if 2.30000000000000005e51 < F

   1. Initial program 11.6%

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

   3. Taylor expanded in B around inf 22.1%

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

      1. mul-1-neg 22.1%

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

   5. Simplified 22.1%

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

      1. pow1 22.1%

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

      2. sqrt-unprod 22.2%

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

   7. Applied egg-rr 22.2%

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

      1. unpow1 22.2%

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

      2. associate-*l/ 22.2%

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

   9. Simplified 22.2%

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

       1. pow1/2 22.7%

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

       2. div-inv 22.6%

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

       3. unpow-prod-down 22.4%

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

       4. pow1/2 22.4%

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

   11. Applied egg-rr 22.4%

       \[\leadsto -\color{blue}{\sqrt{F \cdot 2} \cdot {\left(\frac{1}{B}\right)}^{0.5}} \]
   12. Step-by-step derivation

       1. unpow1/2 22.4%

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

       2. *-commutative 22.4%

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

       3. *-commutative 22.4%

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

   13. Simplified 22.4%

       \[\leadsto -\color{blue}{\sqrt{\frac{1}{B}} \cdot \sqrt{2 \cdot F}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 21.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 2.3 \cdot 10^{+51}:\\ \;\;\;\;\frac{\sqrt{\left(C + \mathsf{hypot}\left(B, C\right)\right) \cdot \left(2 \cdot F\right)}}{-B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{1}{B}} \cdot \left(-\sqrt{2 \cdot F}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 34.6% accurate, 3.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \sqrt{\frac{1}{B\_m}} \cdot \left(-\sqrt{2 \cdot F}\right) \end{array} \]

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 (/ 1.0 B_m)) (- (sqrt (* 2.0 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) {
	return sqrt((1.0 / B_m)) * -sqrt((2.0 * F));
}

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((1.0d0 / b_m)) * -sqrt((2.0d0 * f))
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((1.0 / B_m)) * -Math.sqrt((2.0 * F));
}

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((1.0 / B_m)) * -math.sqrt((2.0 * F))

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(1.0 / B_m)) * Float64(-sqrt(Float64(2.0 * F))))
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((1.0 / B_m)) * -sqrt((2.0 * F));
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[(N[Sqrt[N[(1.0 / B$95$m), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]

Derivation

1. Initial program 17.6%

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

3. Taylor expanded in B around inf 15.8%

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

   1. mul-1-neg 15.8%

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

5. Simplified 15.8%

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

   1. pow1 15.8%

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

   2. sqrt-unprod 15.9%

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

7. Applied egg-rr 15.9%

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

   1. unpow1 15.9%

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

   2. associate-*l/ 15.9%

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

9. Simplified 15.9%

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

    1. pow1/2 16.0%

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

    2. div-inv 16.0%

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

    3. unpow-prod-down 20.3%

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

    4. pow1/2 20.3%

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

11. Applied egg-rr 20.3%

    \[\leadsto -\color{blue}{\sqrt{F \cdot 2} \cdot {\left(\frac{1}{B}\right)}^{0.5}} \]
12. Step-by-step derivation

    1. unpow1/2 20.3%

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

    2. *-commutative 20.3%

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

    3. *-commutative 20.3%

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

13. Simplified 20.3%

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

14. Final simplification 20.3%

    \[\leadsto \sqrt{\frac{1}{B}} \cdot \left(-\sqrt{2 \cdot F}\right) \]
15. Add Preprocessing

Alternative 10: 34.6% accurate, 3.1× speedup

Math

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

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 (/ 2.0 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) {
	return sqrt((2.0 / B_m)) * -sqrt(F);
}

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((2.0d0 / b_m)) * -sqrt(f)
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((2.0 / B_m)) * -Math.sqrt(F);
}

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((2.0 / B_m)) * -math.sqrt(F)

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(2.0 / B_m)) * Float64(-sqrt(F)))
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((2.0 / B_m)) * -sqrt(F);
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[(N[Sqrt[N[(2.0 / B$95$m), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision]

Derivation

1. Initial program 17.6%

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

3. Taylor expanded in B around inf 15.8%

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

   1. mul-1-neg 15.8%

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

5. Simplified 15.8%

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

   1. pow1 15.8%

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

   2. sqrt-unprod 15.9%

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

7. Applied egg-rr 15.9%

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

   1. unpow1 15.9%

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

   2. associate-*l/ 15.9%

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

9. Simplified 15.9%

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

    1. pow1/2 16.0%

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

    2. associate-/l* 16.0%

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

    3. unpow-prod-down 20.2%

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

    4. pow1/2 20.2%

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

11. Applied egg-rr 20.2%

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

    1. unpow1/2 20.2%

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

13. Simplified 20.2%

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

14. Final simplification 20.2%

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

Alternative 11: 26.3% accurate, 3.1× speedup

Math

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

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 (fabs (* 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(fabs((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(abs((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(Math.abs((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(math.fabs((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(abs(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(abs((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[Abs[N[(F * N[(2.0 / B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision])

Derivation

1. Initial program 17.6%

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

3. Taylor expanded in B around inf 15.8%

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

   1. mul-1-neg 15.8%

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

5. Simplified 15.8%

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

   1. pow1 15.8%

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

   2. sqrt-unprod 15.9%

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

7. Applied egg-rr 15.9%

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

   1. unpow1 15.9%

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

   2. associate-*l/ 15.9%

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

9. Simplified 15.9%

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

    1. *-un-lft-identity 15.9%

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

    2. associate-/l* 15.9%

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

11. Applied egg-rr 15.9%

    \[\leadsto -\color{blue}{1 \cdot \sqrt{F \cdot \frac{2}{B}}} \]
12. Step-by-step derivation

    1. *-lft-identity 15.9%

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

13. Simplified 15.9%

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

    1. add-sqr-sqrt 15.9%

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

    2. pow1/2 15.9%

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

    3. pow1/2 16.0%

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

    4. pow-prod-down 16.3%

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

    5. pow2 16.3%

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

    6. associate-*r/ 16.4%

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

15. Applied egg-rr 16.4%

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

    1. unpow1/2 16.4%

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

    2. unpow2 16.4%

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

    3. rem-sqrt-square 29.6%

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

    4. associate-/l* 29.6%

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

17. Simplified 29.6%

    \[\leadsto -\sqrt{\color{blue}{\left|F \cdot \frac{2}{B}\right|}} \]
18. Add Preprocessing

Alternative 12: 26.2% 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 17.6%

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

3. Taylor expanded in B around inf 15.8%

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

   1. mul-1-neg 15.8%

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

5. Simplified 15.8%

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

   1. pow1 15.8%

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

   2. sqrt-unprod 15.9%

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

7. Applied egg-rr 15.9%

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

   1. unpow1 15.9%

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

   2. associate-*l/ 15.9%

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

9. Simplified 15.9%

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

    1. *-un-lft-identity 15.9%

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

    2. associate-/l* 15.9%

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

11. Applied egg-rr 15.9%

    \[\leadsto -\color{blue}{1 \cdot \sqrt{F \cdot \frac{2}{B}}} \]
12. Step-by-step derivation

    1. *-lft-identity 15.9%

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

13. Simplified 15.9%

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

    1. pow1/2 16.0%

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

    2. associate-*r/ 16.0%

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

15. Applied egg-rr 16.0%

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

16. Final simplification 16.0%

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

Alternative 13: 26.1% accurate, 6.0× speedup

Math

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

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 (/ (* 2.0 F) 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(((2.0 * F) / 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(((2.0d0 * f) / 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(((2.0 * F) / 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(((2.0 * F) / 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(Float64(2.0 * F) / 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(((2.0 * F) / 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[(N[(2.0 * F), $MachinePrecision] / B$95$m), $MachinePrecision]], $MachinePrecision])

Derivation

1. Initial program 17.6%

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

3. Taylor expanded in B around inf 15.8%

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

   1. mul-1-neg 15.8%

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

5. Simplified 15.8%

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

   1. pow1 15.8%

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

   2. sqrt-unprod 15.9%

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

7. Applied egg-rr 15.9%

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

   1. unpow1 15.9%

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

   2. associate-*l/ 15.9%

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

9. Simplified 15.9%

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

10. Final simplification 15.9%

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

Alternative 14: 26.2% 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 17.6%

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

3. Taylor expanded in B around inf 15.8%

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

   1. mul-1-neg 15.8%

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

5. Simplified 15.8%

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

   1. pow1 15.8%

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

   2. sqrt-unprod 15.9%

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

7. Applied egg-rr 15.9%

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

   1. unpow1 15.9%

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

   2. associate-*l/ 15.9%

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

9. Simplified 15.9%

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

    1. *-un-lft-identity 15.9%

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

    2. associate-/l* 15.9%

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

11. Applied egg-rr 15.9%

    \[\leadsto -\color{blue}{1 \cdot \sqrt{F \cdot \frac{2}{B}}} \]
12. Step-by-step derivation

    1. *-lft-identity 15.9%

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

13. Simplified 15.9%

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

Reproduce

herbie shell --seed 2024089 
(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))))