ABCF->ab-angle a

Percentage Accurate: 19.0% → 51.6%

Time: 30.6s

Alternatives: 16

Speedup: 6.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 19.0% 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: 51.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := \mathsf{hypot}\left(A - C, B_m\right)\\ t_1 := \mathsf{fma}\left(B_m, B_m, A \cdot \left(C \cdot -4\right)\right)\\ t_2 := {B_m}^{2} - \left(4 \cdot A\right) \cdot C\\ t_3 := 2 \cdot \left(t_2 \cdot F\right)\\ t_4 := \frac{-\sqrt{t_3 \cdot \left(\left(A + C\right) + \sqrt{{B_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t_2}\\ \mathbf{if}\;t_4 \leq -2 \cdot 10^{-167}:\\ \;\;\;\;\frac{\sqrt{A + \left(C + t_0\right)} \cdot \left(-\sqrt{2 \cdot \left(F \cdot t_1\right)}\right)}{t_2}\\ \mathbf{elif}\;t_4 \leq 0:\\ \;\;\;\;\frac{-\sqrt{t_3 \cdot \left(-0.5 \cdot \frac{{B_m}^{2}}{A} + 2 \cdot C\right)}}{t_2}\\ \mathbf{elif}\;t_4 \leq \infty:\\ \;\;\;\;\sqrt{\left(\left(A + C\right) + t_0\right) \cdot \left(t_1 \cdot \left(2 \cdot F\right)\right)} \cdot \frac{-1}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{A + \mathsf{hypot}\left(B_m, A\right)} \cdot \sqrt{F}\right) \cdot \left(-\frac{\sqrt{2}}{B_m}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (hypot (- A C) B_m))
        (t_1 (fma B_m B_m (* A (* C -4.0))))
        (t_2 (- (pow B_m 2.0) (* (* 4.0 A) C)))
        (t_3 (* 2.0 (* t_2 F)))
        (t_4
         (/
          (-
           (sqrt
            (* t_3 (+ (+ A C) (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0)))))))
          t_2)))
   (if (<= t_4 -2e-167)
     (/ (* (sqrt (+ A (+ C t_0))) (- (sqrt (* 2.0 (* F t_1))))) t_2)
     (if (<= t_4 0.0)
       (/ (- (sqrt (* t_3 (+ (* -0.5 (/ (pow B_m 2.0) A)) (* 2.0 C))))) t_2)
       (if (<= t_4 INFINITY)
         (* (sqrt (* (+ (+ A C) t_0) (* t_1 (* 2.0 F)))) (/ -1.0 t_1))
         (*
          (* (sqrt (+ A (hypot B_m A))) (sqrt F))
          (- (/ (sqrt 2.0) B_m))))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double t_0 = hypot((A - C), B_m);
	double t_1 = fma(B_m, B_m, (A * (C * -4.0)));
	double t_2 = pow(B_m, 2.0) - ((4.0 * A) * C);
	double t_3 = 2.0 * (t_2 * F);
	double t_4 = -sqrt((t_3 * ((A + C) + sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / t_2;
	double tmp;
	if (t_4 <= -2e-167) {
		tmp = (sqrt((A + (C + t_0))) * -sqrt((2.0 * (F * t_1)))) / t_2;
	} else if (t_4 <= 0.0) {
		tmp = -sqrt((t_3 * ((-0.5 * (pow(B_m, 2.0) / A)) + (2.0 * C)))) / t_2;
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = sqrt((((A + C) + t_0) * (t_1 * (2.0 * F)))) * (-1.0 / t_1);
	} else {
		tmp = (sqrt((A + hypot(B_m, A))) * sqrt(F)) * -(sqrt(2.0) / B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	t_0 = hypot(Float64(A - C), B_m)
	t_1 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	t_2 = Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_3 = Float64(2.0 * Float64(t_2 * F))
	t_4 = Float64(Float64(-sqrt(Float64(t_3 * Float64(Float64(A + C) + sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0))))))) / t_2)
	tmp = 0.0
	if (t_4 <= -2e-167)
		tmp = Float64(Float64(sqrt(Float64(A + Float64(C + t_0))) * Float64(-sqrt(Float64(2.0 * Float64(F * t_1))))) / t_2);
	elseif (t_4 <= 0.0)
		tmp = Float64(Float64(-sqrt(Float64(t_3 * Float64(Float64(-0.5 * Float64((B_m ^ 2.0) / A)) + Float64(2.0 * C))))) / t_2);
	elseif (t_4 <= Inf)
		tmp = Float64(sqrt(Float64(Float64(Float64(A + C) + t_0) * Float64(t_1 * Float64(2.0 * F)))) * Float64(-1.0 / t_1));
	else
		tmp = Float64(Float64(sqrt(Float64(A + hypot(B_m, A))) * sqrt(F)) * Float64(-Float64(sqrt(2.0) / B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision]}, Block[{t$95$1 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(2.0 * N[(t$95$2 * F), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[((-N[Sqrt[N[(t$95$3 * 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]) / t$95$2), $MachinePrecision]}, If[LessEqual[t$95$4, -2e-167], N[(N[(N[Sqrt[N[(A + N[(C + t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(2.0 * N[(F * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[t$95$4, 0.0], N[((-N[Sqrt[N[(t$95$3 * 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], If[LessEqual[t$95$4, Infinity], N[(N[Sqrt[N[(N[(N[(A + C), $MachinePrecision] + t$95$0), $MachinePrecision] * N[(t$95$1 * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(A + N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] * (-N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision])), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))) < -2e-167

   1. Initial program 41.1%

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

      1. sqrt-prod 45.0%

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

      2. associate-*r* 45.0%

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

      3. associate-*l* 45.0%

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

      4. associate-+l+ 45.0%

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

      5. unpow2 45.0%

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

      6. unpow2 45.0%

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

      7. hypot-def 65.3%

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

   3. Applied egg-rr 65.3%

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

      1. *-commutative 65.3%

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

      2. associate-*l* 65.3%

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

      3. *-commutative 65.3%

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

      4. unpow2 65.3%

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

      5. fma-neg 65.3%

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

      6. distribute-lft-neg-in 65.3%

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

      7. metadata-eval 65.3%

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

      8. *-commutative 65.3%

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

      9. associate-*l* 65.3%

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

   5. Simplified 65.3%

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

   if -2e-167 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))) < -0.0

   1. Initial program 3.5%

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

   2. Taylor expanded in A around -inf 42.7%

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

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

   1. Initial program 52.2%

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

   3. Simplified 60.8%

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

      1. pow1/2 60.8%

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

      2. pow-to-exp 57.3%

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

   5. Applied egg-rr 57.3%

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

   7. Applied egg-rr 60.8%

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

   if +inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 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. Taylor expanded in C around 0 1.9%

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

      1. mul-1-neg 1.9%

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

      2. *-commutative 1.9%

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

      3. distribute-rgt-neg-in 1.9%

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

      4. +-commutative 1.9%

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

      5. unpow2 1.9%

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

      6. unpow2 1.9%

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

      7. hypot-def 23.2%

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

   4. Simplified 23.2%

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

      1. pow1/2 23.3%

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

      2. *-commutative 23.3%

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

      3. unpow-prod-down 32.0%

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

      4. pow1/2 32.0%

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

      5. pow1/2 32.0%

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

   6. Applied egg-rr 32.0%

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

4. Final simplification 48.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \leq -2 \cdot 10^{-167}:\\ \;\;\;\;\frac{\sqrt{A + \left(C + \mathsf{hypot}\left(A - C, B\right)\right)} \cdot \left(-\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)}\right)}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\\ \mathbf{elif}\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \leq 0:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(-0.5 \cdot \frac{{B}^{2}}{A} + 2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\\ \mathbf{elif}\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \leq \infty:\\ \;\;\;\;\sqrt{\left(\left(A + C\right) + \mathsf{hypot}\left(A - C, B\right)\right) \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(2 \cdot F\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{A + \mathsf{hypot}\left(B, A\right)} \cdot \sqrt{F}\right) \cdot \left(-\frac{\sqrt{2}}{B}\right)\\ \end{array} \]

Alternative 2: 51.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B_m, B_m, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;{B_m}^{2} \leq 2 \cdot 10^{+93}:\\ \;\;\;\;\frac{\sqrt{\left(A + C\right) + \mathsf{hypot}\left(A - C, B_m\right)} \cdot \left(-\sqrt{t_0 \cdot \left(2 \cdot F\right)}\right)}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B_m} \cdot \left(\sqrt{F} \cdot \left(-\sqrt{C + \mathsf{hypot}\left(B_m, C\right)}\right)\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* A (* C -4.0)))))
   (if (<= (pow B_m 2.0) 2e+93)
     (/
      (* (sqrt (+ (+ A C) (hypot (- A C) B_m))) (- (sqrt (* t_0 (* 2.0 F)))))
      t_0)
     (* (/ (sqrt 2.0) B_m) (* (sqrt F) (- (sqrt (+ C (hypot B_m C)))))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (A * (C * -4.0)));
	double tmp;
	if (pow(B_m, 2.0) <= 2e+93) {
		tmp = (sqrt(((A + C) + hypot((A - C), B_m))) * -sqrt((t_0 * (2.0 * F)))) / t_0;
	} else {
		tmp = (sqrt(2.0) / B_m) * (sqrt(F) * -sqrt((C + hypot(B_m, C))));
	}
	return tmp;
}

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 2e+93)
		tmp = Float64(Float64(sqrt(Float64(Float64(A + C) + hypot(Float64(A - C), B_m))) * Float64(-sqrt(Float64(t_0 * Float64(2.0 * F))))) / t_0);
	else
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(sqrt(F) * Float64(-sqrt(Float64(C + hypot(B_m, C))))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
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]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+93], N[(N[(N[Sqrt[N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(t$95$0 * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[(N[Sqrt[F], $MachinePrecision] * (-N[Sqrt[N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B 2) < 2.00000000000000009e93

   1. Initial program 27.5%

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

   3. Simplified 34.7%

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

      1. flip-+ 14.8%

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

      2. hypot-udef 14.8%

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

      3. unpow2 14.8%

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

      4. unpow2 14.8%

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

      5. hypot-udef 14.8%

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

      6. unpow2 14.8%

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

      7. unpow2 14.8%

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

      8. add-sqr-sqrt 14.8%

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

      9. +-commutative 14.8%

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

      10. div-sub 14.8%

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

   5. Applied egg-rr 14.8%

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

      1. div-sub 14.8%

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

   7. Simplified 14.8%

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

      1. pow1/2 14.9%

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

      2. *-commutative 14.9%

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

      3. unpow2 14.9%

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

      4. unpow2 14.9%

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

      5. flip-+ 34.7%

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

      6. unpow-prod-down 39.8%

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

      7. pow1/2 39.8%

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

      8. associate-+r+ 39.0%

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

      9. pow1/2 39.0%

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

   9. Applied egg-rr 39.0%

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

   if 2.00000000000000009e93 < (pow.f64 B 2)

   1. Initial program 11.0%

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

   2. Taylor expanded in A around 0 10.9%

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

      1. mul-1-neg 10.9%

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

      2. *-commutative 10.9%

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

      3. distribute-rgt-neg-in 10.9%

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

      4. unpow2 10.9%

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

      5. unpow2 10.9%

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

      6. hypot-def 32.1%

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

   4. Simplified 32.1%

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

      1. pow1/2 32.1%

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

      2. *-commutative 32.1%

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

      3. unpow-prod-down 40.3%

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

      4. pow1/2 40.3%

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

      5. pow1/2 40.3%

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

   6. Applied egg-rr 40.3%

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

4. Final simplification 39.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 2 \cdot 10^{+93}:\\ \;\;\;\;\frac{\sqrt{\left(A + C\right) + \mathsf{hypot}\left(A - C, B\right)} \cdot \left(-\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(2 \cdot F\right)}\right)}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{F} \cdot \left(-\sqrt{C + \mathsf{hypot}\left(B, C\right)}\right)\right)\\ \end{array} \]

Alternative 3: 48.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B_m, B_m, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;{B_m}^{2} \leq 40:\\ \;\;\;\;\frac{-\sqrt{\left(t_0 \cdot \left(2 \cdot F\right)\right) \cdot \left(A + \left(C + \mathsf{hypot}\left(B_m, A - C\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B_m} \cdot \left(\sqrt{F} \cdot \left(-\sqrt{C + \mathsf{hypot}\left(B_m, C\right)}\right)\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* A (* C -4.0)))))
   (if (<= (pow B_m 2.0) 40.0)
     (/ (- (sqrt (* (* t_0 (* 2.0 F)) (+ A (+ C (hypot B_m (- A C))))))) t_0)
     (* (/ (sqrt 2.0) B_m) (* (sqrt F) (- (sqrt (+ C (hypot B_m C)))))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (A * (C * -4.0)));
	double tmp;
	if (pow(B_m, 2.0) <= 40.0) {
		tmp = -sqrt(((t_0 * (2.0 * F)) * (A + (C + hypot(B_m, (A - C)))))) / t_0;
	} else {
		tmp = (sqrt(2.0) / B_m) * (sqrt(F) * -sqrt((C + hypot(B_m, C))));
	}
	return tmp;
}

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 40.0)
		tmp = Float64(Float64(-sqrt(Float64(Float64(t_0 * Float64(2.0 * F)) * Float64(A + Float64(C + hypot(B_m, Float64(A - C))))))) / t_0);
	else
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(sqrt(F) * Float64(-sqrt(Float64(C + hypot(B_m, C))))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
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]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 40.0], N[((-N[Sqrt[N[(N[(t$95$0 * N[(2.0 * F), $MachinePrecision]), $MachinePrecision] * N[(A + N[(C + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[(N[Sqrt[F], $MachinePrecision] * (-N[Sqrt[N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B 2) < 40

   1. Initial program 25.7%

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

   3. Simplified 34.1%

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

   if 40 < (pow.f64 B 2)

   1. Initial program 15.7%

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

   2. Taylor expanded in A around 0 11.9%

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

      1. mul-1-neg 11.9%

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

      2. *-commutative 11.9%

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

      3. distribute-rgt-neg-in 11.9%

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

      4. unpow2 11.9%

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

      5. unpow2 11.9%

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

      6. hypot-def 29.5%

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

   4. Simplified 29.5%

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

      1. pow1/2 29.5%

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

      2. *-commutative 29.5%

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

      3. unpow-prod-down 37.0%

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

      4. pow1/2 37.0%

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

      5. pow1/2 37.0%

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

   6. Applied egg-rr 37.0%

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

4. Final simplification 35.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 40:\\ \;\;\;\;\frac{-\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{F} \cdot \left(-\sqrt{C + \mathsf{hypot}\left(B, C\right)}\right)\right)\\ \end{array} \]

Alternative 4: 43.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := {B_m}^{2} - \left(4 \cdot A\right) \cdot C\\ \mathbf{if}\;{B_m}^{2} \leq 10^{-12}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(t_0 \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B_m} \cdot \left(\sqrt{F} \cdot \left(-\sqrt{C + \mathsf{hypot}\left(B_m, C\right)}\right)\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (- (pow B_m 2.0) (* (* 4.0 A) C))))
   (if (<= (pow B_m 2.0) 1e-12)
     (/ (- (sqrt (* (* 2.0 (* t_0 F)) (* 2.0 C)))) t_0)
     (* (/ (sqrt 2.0) B_m) (* (sqrt F) (- (sqrt (+ C (hypot B_m C)))))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double t_0 = pow(B_m, 2.0) - ((4.0 * A) * C);
	double tmp;
	if (pow(B_m, 2.0) <= 1e-12) {
		tmp = -sqrt(((2.0 * (t_0 * F)) * (2.0 * C))) / t_0;
	} else {
		tmp = (sqrt(2.0) / B_m) * (sqrt(F) * -sqrt((C + hypot(B_m, C))));
	}
	return tmp;
}

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double t_0 = Math.pow(B_m, 2.0) - ((4.0 * A) * C);
	double tmp;
	if (Math.pow(B_m, 2.0) <= 1e-12) {
		tmp = -Math.sqrt(((2.0 * (t_0 * F)) * (2.0 * C))) / t_0;
	} else {
		tmp = (Math.sqrt(2.0) / B_m) * (Math.sqrt(F) * -Math.sqrt((C + Math.hypot(B_m, C))));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	t_0 = math.pow(B_m, 2.0) - ((4.0 * A) * C)
	tmp = 0
	if math.pow(B_m, 2.0) <= 1e-12:
		tmp = -math.sqrt(((2.0 * (t_0 * F)) * (2.0 * C))) / t_0
	else:
		tmp = (math.sqrt(2.0) / B_m) * (math.sqrt(F) * -math.sqrt((C + math.hypot(B_m, C))))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	t_0 = Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e-12)
		tmp = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(t_0 * F)) * Float64(2.0 * C)))) / t_0);
	else
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(sqrt(F) * Float64(-sqrt(Float64(C + hypot(B_m, C))))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	t_0 = (B_m ^ 2.0) - ((4.0 * A) * C);
	tmp = 0.0;
	if ((B_m ^ 2.0) <= 1e-12)
		tmp = -sqrt(((2.0 * (t_0 * F)) * (2.0 * C))) / t_0;
	else
		tmp = (sqrt(2.0) / B_m) * (sqrt(F) * -sqrt((C + hypot(B_m, C))));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-12], N[((-N[Sqrt[N[(N[(2.0 * N[(t$95$0 * F), $MachinePrecision]), $MachinePrecision] * N[(2.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[(N[Sqrt[F], $MachinePrecision] * (-N[Sqrt[N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B 2) < 9.9999999999999998e-13

   1. Initial program 25.2%

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

   2. Taylor expanded in A around -inf 32.2%

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

   if 9.9999999999999998e-13 < (pow.f64 B 2)

   1. Initial program 16.5%

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

   2. Taylor expanded in A around 0 12.1%

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

      1. mul-1-neg 12.1%

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

      2. *-commutative 12.1%

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

      3. distribute-rgt-neg-in 12.1%

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

      4. unpow2 12.1%

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

      5. unpow2 12.1%

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

      6. hypot-def 28.9%

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

   4. Simplified 28.9%

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

      1. pow1/2 29.0%

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

      2. *-commutative 29.0%

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

      3. unpow-prod-down 36.1%

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

      4. pow1/2 36.1%

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

      5. pow1/2 36.1%

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

   6. Applied egg-rr 36.1%

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

4. Final simplification 34.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{-12}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{F} \cdot \left(-\sqrt{C + \mathsf{hypot}\left(B, C\right)}\right)\right)\\ \end{array} \]

Alternative 5: 41.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := -\frac{\sqrt{2}}{B_m}\\ t_1 := {B_m}^{2} - \left(4 \cdot A\right) \cdot C\\ \mathbf{if}\;B_m \leq 1.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(t_1 \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{t_1}\\ \mathbf{elif}\;B_m \leq 1.26 \cdot 10^{+174}:\\ \;\;\;\;t_0 \cdot \sqrt{F \cdot \left(C + \mathsf{hypot}\left(B_m, C\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{A + \mathsf{hypot}\left(B_m, A\right)} \cdot \sqrt{F}\right) \cdot t_0\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (- (/ (sqrt 2.0) B_m))) (t_1 (- (pow B_m 2.0) (* (* 4.0 A) C))))
   (if (<= B_m 1.2e-6)
     (/ (- (sqrt (* (* 2.0 (* t_1 F)) (* 2.0 C)))) t_1)
     (if (<= B_m 1.26e+174)
       (* t_0 (sqrt (* F (+ C (hypot B_m C)))))
       (* (* (sqrt (+ A (hypot B_m A))) (sqrt F)) t_0)))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double t_0 = -(sqrt(2.0) / B_m);
	double t_1 = pow(B_m, 2.0) - ((4.0 * A) * C);
	double tmp;
	if (B_m <= 1.2e-6) {
		tmp = -sqrt(((2.0 * (t_1 * F)) * (2.0 * C))) / t_1;
	} else if (B_m <= 1.26e+174) {
		tmp = t_0 * sqrt((F * (C + hypot(B_m, C))));
	} else {
		tmp = (sqrt((A + hypot(B_m, A))) * sqrt(F)) * t_0;
	}
	return tmp;
}

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double t_0 = -(Math.sqrt(2.0) / B_m);
	double t_1 = Math.pow(B_m, 2.0) - ((4.0 * A) * C);
	double tmp;
	if (B_m <= 1.2e-6) {
		tmp = -Math.sqrt(((2.0 * (t_1 * F)) * (2.0 * C))) / t_1;
	} else if (B_m <= 1.26e+174) {
		tmp = t_0 * Math.sqrt((F * (C + Math.hypot(B_m, C))));
	} else {
		tmp = (Math.sqrt((A + Math.hypot(B_m, A))) * Math.sqrt(F)) * t_0;
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	t_0 = -(math.sqrt(2.0) / B_m)
	t_1 = math.pow(B_m, 2.0) - ((4.0 * A) * C)
	tmp = 0
	if B_m <= 1.2e-6:
		tmp = -math.sqrt(((2.0 * (t_1 * F)) * (2.0 * C))) / t_1
	elif B_m <= 1.26e+174:
		tmp = t_0 * math.sqrt((F * (C + math.hypot(B_m, C))))
	else:
		tmp = (math.sqrt((A + math.hypot(B_m, A))) * math.sqrt(F)) * t_0
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	t_0 = Float64(-Float64(sqrt(2.0) / B_m))
	t_1 = Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))
	tmp = 0.0
	if (B_m <= 1.2e-6)
		tmp = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(t_1 * F)) * Float64(2.0 * C)))) / t_1);
	elseif (B_m <= 1.26e+174)
		tmp = Float64(t_0 * sqrt(Float64(F * Float64(C + hypot(B_m, C)))));
	else
		tmp = Float64(Float64(sqrt(Float64(A + hypot(B_m, A))) * sqrt(F)) * t_0);
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	t_0 = -(sqrt(2.0) / B_m);
	t_1 = (B_m ^ 2.0) - ((4.0 * A) * C);
	tmp = 0.0;
	if (B_m <= 1.2e-6)
		tmp = -sqrt(((2.0 * (t_1 * F)) * (2.0 * C))) / t_1;
	elseif (B_m <= 1.26e+174)
		tmp = t_0 * sqrt((F * (C + hypot(B_m, C))));
	else
		tmp = (sqrt((A + hypot(B_m, A))) * sqrt(F)) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = (-N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision])}, Block[{t$95$1 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 1.2e-6], N[((-N[Sqrt[N[(N[(2.0 * N[(t$95$1 * F), $MachinePrecision]), $MachinePrecision] * N[(2.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision], If[LessEqual[B$95$m, 1.26e+174], N[(t$95$0 * N[Sqrt[N[(F * N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(A + N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if B < 1.1999999999999999e-6

   1. Initial program 22.0%

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

   2. Taylor expanded in A around -inf 22.4%

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

   if 1.1999999999999999e-6 < B < 1.26000000000000006e174

   1. Initial program 32.2%

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

   2. Taylor expanded in A around 0 40.7%

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

      1. mul-1-neg 40.7%

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

      2. *-commutative 40.7%

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

      3. distribute-rgt-neg-in 40.7%

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

      4. unpow2 40.7%

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

      5. unpow2 40.7%

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

      6. hypot-def 49.1%

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

   4. Simplified 49.1%

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

   if 1.26000000000000006e174 < B

   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. Taylor expanded in C around 0 2.3%

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

      1. mul-1-neg 2.3%

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

      2. *-commutative 2.3%

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

      3. distribute-rgt-neg-in 2.3%

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

      4. +-commutative 2.3%

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

      5. unpow2 2.3%

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

      6. unpow2 2.3%

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

      7. hypot-def 63.3%

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

   4. Simplified 63.3%

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

      1. pow1/2 63.3%

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

      2. *-commutative 63.3%

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

      3. unpow-prod-down 89.6%

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

      4. pow1/2 89.6%

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

      5. pow1/2 89.6%

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

   6. Applied egg-rr 89.6%

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

4. Final simplification 34.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.2 \cdot 10^{-6}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\\ \mathbf{elif}\;B \leq 1.26 \cdot 10^{+174}:\\ \;\;\;\;\left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{A + \mathsf{hypot}\left(B, A\right)} \cdot \sqrt{F}\right) \cdot \left(-\frac{\sqrt{2}}{B}\right)\\ \end{array} \]

Alternative 6: 40.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := \frac{\sqrt{2}}{B_m}\\ t_1 := {B_m}^{2} - \left(4 \cdot A\right) \cdot C\\ \mathbf{if}\;B_m \leq 8.9 \cdot 10^{-7}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(t_1 \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{t_1}\\ \mathbf{elif}\;B_m \leq 1.25 \cdot 10^{+174}:\\ \;\;\;\;\left(-t_0\right) \cdot \sqrt{F \cdot \left(C + \mathsf{hypot}\left(B_m, C\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;-t_0 \cdot \left(\sqrt{F} \cdot \sqrt{B_m + C}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (/ (sqrt 2.0) B_m)) (t_1 (- (pow B_m 2.0) (* (* 4.0 A) C))))
   (if (<= B_m 8.9e-7)
     (/ (- (sqrt (* (* 2.0 (* t_1 F)) (* 2.0 C)))) t_1)
     (if (<= B_m 1.25e+174)
       (* (- t_0) (sqrt (* F (+ C (hypot B_m C)))))
       (- (* t_0 (* (sqrt F) (sqrt (+ B_m C)))))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double t_0 = sqrt(2.0) / B_m;
	double t_1 = pow(B_m, 2.0) - ((4.0 * A) * C);
	double tmp;
	if (B_m <= 8.9e-7) {
		tmp = -sqrt(((2.0 * (t_1 * F)) * (2.0 * C))) / t_1;
	} else if (B_m <= 1.25e+174) {
		tmp = -t_0 * sqrt((F * (C + hypot(B_m, C))));
	} else {
		tmp = -(t_0 * (sqrt(F) * sqrt((B_m + C))));
	}
	return tmp;
}

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double t_0 = Math.sqrt(2.0) / B_m;
	double t_1 = Math.pow(B_m, 2.0) - ((4.0 * A) * C);
	double tmp;
	if (B_m <= 8.9e-7) {
		tmp = -Math.sqrt(((2.0 * (t_1 * F)) * (2.0 * C))) / t_1;
	} else if (B_m <= 1.25e+174) {
		tmp = -t_0 * Math.sqrt((F * (C + Math.hypot(B_m, C))));
	} else {
		tmp = -(t_0 * (Math.sqrt(F) * Math.sqrt((B_m + C))));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	t_0 = math.sqrt(2.0) / B_m
	t_1 = math.pow(B_m, 2.0) - ((4.0 * A) * C)
	tmp = 0
	if B_m <= 8.9e-7:
		tmp = -math.sqrt(((2.0 * (t_1 * F)) * (2.0 * C))) / t_1
	elif B_m <= 1.25e+174:
		tmp = -t_0 * math.sqrt((F * (C + math.hypot(B_m, C))))
	else:
		tmp = -(t_0 * (math.sqrt(F) * math.sqrt((B_m + C))))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	t_0 = Float64(sqrt(2.0) / B_m)
	t_1 = Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C))
	tmp = 0.0
	if (B_m <= 8.9e-7)
		tmp = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(t_1 * F)) * Float64(2.0 * C)))) / t_1);
	elseif (B_m <= 1.25e+174)
		tmp = Float64(Float64(-t_0) * sqrt(Float64(F * Float64(C + hypot(B_m, C)))));
	else
		tmp = Float64(-Float64(t_0 * Float64(sqrt(F) * sqrt(Float64(B_m + C)))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	t_0 = sqrt(2.0) / B_m;
	t_1 = (B_m ^ 2.0) - ((4.0 * A) * C);
	tmp = 0.0;
	if (B_m <= 8.9e-7)
		tmp = -sqrt(((2.0 * (t_1 * F)) * (2.0 * C))) / t_1;
	elseif (B_m <= 1.25e+174)
		tmp = -t_0 * sqrt((F * (C + hypot(B_m, C))));
	else
		tmp = -(t_0 * (sqrt(F) * sqrt((B_m + C))));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 8.9e-7], N[((-N[Sqrt[N[(N[(2.0 * N[(t$95$1 * F), $MachinePrecision]), $MachinePrecision] * N[(2.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision], If[LessEqual[B$95$m, 1.25e+174], N[((-t$95$0) * N[Sqrt[N[(F * N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], (-N[(t$95$0 * 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 B < 8.899999999999999e-7

   1. Initial program 22.0%

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

   2. Taylor expanded in A around -inf 22.4%

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

   if 8.899999999999999e-7 < B < 1.2499999999999999e174

   1. Initial program 32.2%

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

   2. Taylor expanded in A around 0 40.7%

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

      1. mul-1-neg 40.7%

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

      2. *-commutative 40.7%

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

      3. distribute-rgt-neg-in 40.7%

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

      4. unpow2 40.7%

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

      5. unpow2 40.7%

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

      6. hypot-def 49.1%

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

   4. Simplified 49.1%

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

   if 1.2499999999999999e174 < B

   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. Taylor expanded in A around 0 2.3%

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

      1. mul-1-neg 2.3%

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

      2. *-commutative 2.3%

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

      3. distribute-rgt-neg-in 2.3%

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

      4. unpow2 2.3%

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

      5. unpow2 2.3%

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

      6. hypot-def 59.8%

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

   4. Simplified 59.8%

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

      1. pow1/2 59.8%

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

      2. *-commutative 59.8%

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

      3. unpow-prod-down 83.1%

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

      4. pow1/2 83.1%

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

      5. pow1/2 83.1%

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

   6. Applied egg-rr 83.1%

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

   7. Taylor expanded in C around 0 82.8%

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

4. Final simplification 34.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 8.9 \cdot 10^{-7}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\\ \mathbf{elif}\;B \leq 1.25 \cdot 10^{+174}:\\ \;\;\;\;\left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{2}}{B} \cdot \left(\sqrt{F} \cdot \sqrt{B + C}\right)\\ \end{array} \]

Alternative 7: 37.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := \frac{\sqrt{2}}{B_m}\\ t_1 := C + \mathsf{hypot}\left(B_m, C\right)\\ \mathbf{if}\;B_m \leq 9 \cdot 10^{-61}:\\ \;\;\;\;\frac{-\sqrt{t_1 \cdot \left(2 \cdot \left(-4 \cdot \left(A \cdot \left(C \cdot F\right)\right)\right)\right)}}{{B_m}^{2} - \left(4 \cdot A\right) \cdot C}\\ \mathbf{elif}\;B_m \leq 1.3 \cdot 10^{+174}:\\ \;\;\;\;\left(-t_0\right) \cdot \sqrt{F \cdot t_1}\\ \mathbf{else}:\\ \;\;\;\;-t_0 \cdot \left(\sqrt{F} \cdot \sqrt{B_m + C}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (/ (sqrt 2.0) B_m)) (t_1 (+ C (hypot B_m C))))
   (if (<= B_m 9e-61)
     (/
      (- (sqrt (* t_1 (* 2.0 (* -4.0 (* A (* C F)))))))
      (- (pow B_m 2.0) (* (* 4.0 A) C)))
     (if (<= B_m 1.3e+174)
       (* (- t_0) (sqrt (* F t_1)))
       (- (* t_0 (* (sqrt F) (sqrt (+ B_m C)))))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double t_0 = sqrt(2.0) / B_m;
	double t_1 = C + hypot(B_m, C);
	double tmp;
	if (B_m <= 9e-61) {
		tmp = -sqrt((t_1 * (2.0 * (-4.0 * (A * (C * F)))))) / (pow(B_m, 2.0) - ((4.0 * A) * C));
	} else if (B_m <= 1.3e+174) {
		tmp = -t_0 * sqrt((F * t_1));
	} else {
		tmp = -(t_0 * (sqrt(F) * sqrt((B_m + C))));
	}
	return tmp;
}

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double t_0 = Math.sqrt(2.0) / B_m;
	double t_1 = C + Math.hypot(B_m, C);
	double tmp;
	if (B_m <= 9e-61) {
		tmp = -Math.sqrt((t_1 * (2.0 * (-4.0 * (A * (C * F)))))) / (Math.pow(B_m, 2.0) - ((4.0 * A) * C));
	} else if (B_m <= 1.3e+174) {
		tmp = -t_0 * Math.sqrt((F * t_1));
	} else {
		tmp = -(t_0 * (Math.sqrt(F) * Math.sqrt((B_m + C))));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	t_0 = math.sqrt(2.0) / B_m
	t_1 = C + math.hypot(B_m, C)
	tmp = 0
	if B_m <= 9e-61:
		tmp = -math.sqrt((t_1 * (2.0 * (-4.0 * (A * (C * F)))))) / (math.pow(B_m, 2.0) - ((4.0 * A) * C))
	elif B_m <= 1.3e+174:
		tmp = -t_0 * math.sqrt((F * t_1))
	else:
		tmp = -(t_0 * (math.sqrt(F) * math.sqrt((B_m + C))))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	t_0 = Float64(sqrt(2.0) / B_m)
	t_1 = Float64(C + hypot(B_m, C))
	tmp = 0.0
	if (B_m <= 9e-61)
		tmp = Float64(Float64(-sqrt(Float64(t_1 * Float64(2.0 * Float64(-4.0 * Float64(A * Float64(C * F))))))) / Float64((B_m ^ 2.0) - Float64(Float64(4.0 * A) * C)));
	elseif (B_m <= 1.3e+174)
		tmp = Float64(Float64(-t_0) * sqrt(Float64(F * t_1)));
	else
		tmp = Float64(-Float64(t_0 * Float64(sqrt(F) * sqrt(Float64(B_m + C)))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	t_0 = sqrt(2.0) / B_m;
	t_1 = C + hypot(B_m, C);
	tmp = 0.0;
	if (B_m <= 9e-61)
		tmp = -sqrt((t_1 * (2.0 * (-4.0 * (A * (C * F)))))) / ((B_m ^ 2.0) - ((4.0 * A) * C));
	elseif (B_m <= 1.3e+174)
		tmp = -t_0 * sqrt((F * t_1));
	else
		tmp = -(t_0 * (sqrt(F) * sqrt((B_m + C))));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision]}, Block[{t$95$1 = N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 9e-61], N[((-N[Sqrt[N[(t$95$1 * N[(2.0 * N[(-4.0 * N[(A * N[(C * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 1.3e+174], N[((-t$95$0) * N[Sqrt[N[(F * t$95$1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], (-N[(t$95$0 * 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 B < 9e-61

   1. Initial program 22.3%

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

   2. Taylor expanded in A around 0 21.2%

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

      1. unpow2 21.2%

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

      2. unpow2 21.2%

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

      3. hypot-def 23.6%

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

   4. Simplified 23.6%

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

   5. Taylor expanded in B around 0 14.5%

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

   if 9e-61 < B < 1.2999999999999999e174

   1. Initial program 29.6%

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

   2. Taylor expanded in A around 0 39.1%

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

      1. mul-1-neg 39.1%

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

      2. *-commutative 39.1%

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

      3. distribute-rgt-neg-in 39.1%

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

      4. unpow2 39.1%

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

      5. unpow2 39.1%

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

      6. hypot-def 46.3%

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

   4. Simplified 46.3%

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

   if 1.2999999999999999e174 < B

   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. Taylor expanded in A around 0 2.3%

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

      1. mul-1-neg 2.3%

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

      2. *-commutative 2.3%

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

      3. distribute-rgt-neg-in 2.3%

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

      4. unpow2 2.3%

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

      5. unpow2 2.3%

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

      6. hypot-def 59.8%

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

   4. Simplified 59.8%

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

      1. pow1/2 59.8%

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

      2. *-commutative 59.8%

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

      3. unpow-prod-down 83.1%

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

      4. pow1/2 83.1%

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

      5. pow1/2 83.1%

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

   6. Applied egg-rr 83.1%

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

   7. Taylor expanded in C around 0 82.8%

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

4. Final simplification 28.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 9 \cdot 10^{-61}:\\ \;\;\;\;\frac{-\sqrt{\left(C + \mathsf{hypot}\left(B, C\right)\right) \cdot \left(2 \cdot \left(-4 \cdot \left(A \cdot \left(C \cdot F\right)\right)\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\\ \mathbf{elif}\;B \leq 1.3 \cdot 10^{+174}:\\ \;\;\;\;\left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{2}}{B} \cdot \left(\sqrt{F} \cdot \sqrt{B + C}\right)\\ \end{array} \]

Alternative 8: 35.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := \frac{\sqrt{2}}{B_m}\\ \mathbf{if}\;A \leq -1.52 \cdot 10^{+98}:\\ \;\;\;\;\left(-t_0\right) \cdot \sqrt{-0.5 \cdot \frac{{B_m}^{2} \cdot F}{A}}\\ \mathbf{elif}\;A \leq 2.85 \cdot 10^{+153}:\\ \;\;\;\;-t_0 \cdot \left(\sqrt{F} \cdot \sqrt{B_m + C}\right)\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{\left(2 \cdot F\right) \cdot \left(A + \mathsf{hypot}\left(B_m, A\right)\right)}}{B_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (/ (sqrt 2.0) B_m)))
   (if (<= A -1.52e+98)
     (* (- t_0) (sqrt (* -0.5 (/ (* (pow B_m 2.0) F) A))))
     (if (<= A 2.85e+153)
       (- (* t_0 (* (sqrt F) (sqrt (+ B_m C)))))
       (- (/ (sqrt (* (* 2.0 F) (+ A (hypot B_m A)))) B_m))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double t_0 = sqrt(2.0) / B_m;
	double tmp;
	if (A <= -1.52e+98) {
		tmp = -t_0 * sqrt((-0.5 * ((pow(B_m, 2.0) * F) / A)));
	} else if (A <= 2.85e+153) {
		tmp = -(t_0 * (sqrt(F) * sqrt((B_m + C))));
	} else {
		tmp = -(sqrt(((2.0 * F) * (A + hypot(B_m, A)))) / B_m);
	}
	return tmp;
}

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double t_0 = Math.sqrt(2.0) / B_m;
	double tmp;
	if (A <= -1.52e+98) {
		tmp = -t_0 * Math.sqrt((-0.5 * ((Math.pow(B_m, 2.0) * F) / A)));
	} else if (A <= 2.85e+153) {
		tmp = -(t_0 * (Math.sqrt(F) * Math.sqrt((B_m + C))));
	} else {
		tmp = -(Math.sqrt(((2.0 * F) * (A + Math.hypot(B_m, A)))) / B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	t_0 = math.sqrt(2.0) / B_m
	tmp = 0
	if A <= -1.52e+98:
		tmp = -t_0 * math.sqrt((-0.5 * ((math.pow(B_m, 2.0) * F) / A)))
	elif A <= 2.85e+153:
		tmp = -(t_0 * (math.sqrt(F) * math.sqrt((B_m + C))))
	else:
		tmp = -(math.sqrt(((2.0 * F) * (A + math.hypot(B_m, A)))) / B_m)
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	t_0 = Float64(sqrt(2.0) / B_m)
	tmp = 0.0
	if (A <= -1.52e+98)
		tmp = Float64(Float64(-t_0) * sqrt(Float64(-0.5 * Float64(Float64((B_m ^ 2.0) * F) / A))));
	elseif (A <= 2.85e+153)
		tmp = Float64(-Float64(t_0 * Float64(sqrt(F) * sqrt(Float64(B_m + C)))));
	else
		tmp = Float64(-Float64(sqrt(Float64(Float64(2.0 * F) * Float64(A + hypot(B_m, A)))) / B_m));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	t_0 = sqrt(2.0) / B_m;
	tmp = 0.0;
	if (A <= -1.52e+98)
		tmp = -t_0 * sqrt((-0.5 * (((B_m ^ 2.0) * F) / A)));
	elseif (A <= 2.85e+153)
		tmp = -(t_0 * (sqrt(F) * sqrt((B_m + C))));
	else
		tmp = -(sqrt(((2.0 * F) * (A + hypot(B_m, A)))) / B_m);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision]}, If[LessEqual[A, -1.52e+98], N[((-t$95$0) * N[Sqrt[N[(-0.5 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] * F), $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[A, 2.85e+153], (-N[(t$95$0 * N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(B$95$m + C), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), (-N[(N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] * N[(A + N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / B$95$m), $MachinePrecision])]]]

Derivation

1. Split input into 3 regimes

2. if A < -1.52000000000000014e98

   1. Initial program 1.6%

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

   2. Taylor expanded in C around 0 2.2%

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

      1. mul-1-neg 2.2%

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

      2. *-commutative 2.2%

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

      3. distribute-rgt-neg-in 2.2%

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

      4. +-commutative 2.2%

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

      5. unpow2 2.2%

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

      6. unpow2 2.2%

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

      7. hypot-def 6.8%

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

   4. Simplified 6.8%

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

   5. Taylor expanded in A around -inf 20.5%

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

   if -1.52000000000000014e98 < A < 2.84999999999999993e153

   1. Initial program 28.6%

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

   2. Taylor expanded in A around 0 11.8%

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

      1. mul-1-neg 11.8%

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

      2. *-commutative 11.8%

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

      3. distribute-rgt-neg-in 11.8%

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

      4. unpow2 11.8%

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

      5. unpow2 11.8%

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

      6. hypot-def 23.2%

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

   4. Simplified 23.2%

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

      1. pow1/2 23.3%

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

      2. *-commutative 23.3%

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

      3. unpow-prod-down 28.2%

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

      4. pow1/2 28.2%

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

      5. pow1/2 28.2%

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

   6. Applied egg-rr 28.2%

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

   7. Taylor expanded in C around 0 23.2%

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

   if 2.84999999999999993e153 < A

   1. Initial program 4.5%

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

   2. Taylor expanded in C around 0 1.0%

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

      1. mul-1-neg 1.0%

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

      2. *-commutative 1.0%

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

      3. distribute-rgt-neg-in 1.0%

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

      4. +-commutative 1.0%

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

      5. unpow2 1.0%

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

      6. unpow2 1.0%

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

      7. hypot-def 13.2%

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

   4. Simplified 13.2%

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

      1. distribute-lft-in 13.1%

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

   6. Applied egg-rr 13.1%

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

      1. distribute-rgt-neg-out 13.1%

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

      2. add-sqr-sqrt 12.3%

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

      3. sqrt-unprod 8.3%

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

      4. sqr-neg 8.3%

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

      5. sqrt-unprod 13.2%

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

      6. add-sqr-sqrt 13.7%

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

      7. distribute-lft-in 13.7%

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

      8. *-commutative 13.7%

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

      9. distribute-frac-neg 13.7%

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

      10. associate-*l/ 13.8%

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

   8. Applied egg-rr 13.2%

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

      1. *-commutative 13.2%

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

      2. *-commutative 13.2%

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

      3. associate-*l* 13.2%

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

   10. Simplified 13.2%

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

4. Final simplification 21.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -1.52 \cdot 10^{+98}:\\ \;\;\;\;\left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{-0.5 \cdot \frac{{B}^{2} \cdot F}{A}}\\ \mathbf{elif}\;A \leq 2.85 \cdot 10^{+153}:\\ \;\;\;\;-\frac{\sqrt{2}}{B} \cdot \left(\sqrt{F} \cdot \sqrt{B + C}\right)\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{\left(2 \cdot F\right) \cdot \left(A + \mathsf{hypot}\left(B, A\right)\right)}}{B}\\ \end{array} \]

Alternative 9: 35.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := \frac{\sqrt{2}}{B_m}\\ \mathbf{if}\;A \leq -2.85 \cdot 10^{+99}:\\ \;\;\;\;t_0 \cdot \left(-\sqrt{F \cdot \left(-0.5 \cdot \frac{{B_m}^{2}}{A}\right)}\right)\\ \mathbf{elif}\;A \leq 8 \cdot 10^{+150}:\\ \;\;\;\;-t_0 \cdot \left(\sqrt{F} \cdot \sqrt{B_m + C}\right)\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{\left(2 \cdot F\right) \cdot \left(A + \mathsf{hypot}\left(B_m, A\right)\right)}}{B_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (/ (sqrt 2.0) B_m)))
   (if (<= A -2.85e+99)
     (* t_0 (- (sqrt (* F (* -0.5 (/ (pow B_m 2.0) A))))))
     (if (<= A 8e+150)
       (- (* t_0 (* (sqrt F) (sqrt (+ B_m C)))))
       (- (/ (sqrt (* (* 2.0 F) (+ A (hypot B_m A)))) B_m))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double t_0 = sqrt(2.0) / B_m;
	double tmp;
	if (A <= -2.85e+99) {
		tmp = t_0 * -sqrt((F * (-0.5 * (pow(B_m, 2.0) / A))));
	} else if (A <= 8e+150) {
		tmp = -(t_0 * (sqrt(F) * sqrt((B_m + C))));
	} else {
		tmp = -(sqrt(((2.0 * F) * (A + hypot(B_m, A)))) / B_m);
	}
	return tmp;
}

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double t_0 = Math.sqrt(2.0) / B_m;
	double tmp;
	if (A <= -2.85e+99) {
		tmp = t_0 * -Math.sqrt((F * (-0.5 * (Math.pow(B_m, 2.0) / A))));
	} else if (A <= 8e+150) {
		tmp = -(t_0 * (Math.sqrt(F) * Math.sqrt((B_m + C))));
	} else {
		tmp = -(Math.sqrt(((2.0 * F) * (A + Math.hypot(B_m, A)))) / B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	t_0 = math.sqrt(2.0) / B_m
	tmp = 0
	if A <= -2.85e+99:
		tmp = t_0 * -math.sqrt((F * (-0.5 * (math.pow(B_m, 2.0) / A))))
	elif A <= 8e+150:
		tmp = -(t_0 * (math.sqrt(F) * math.sqrt((B_m + C))))
	else:
		tmp = -(math.sqrt(((2.0 * F) * (A + math.hypot(B_m, A)))) / B_m)
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	t_0 = Float64(sqrt(2.0) / B_m)
	tmp = 0.0
	if (A <= -2.85e+99)
		tmp = Float64(t_0 * Float64(-sqrt(Float64(F * Float64(-0.5 * Float64((B_m ^ 2.0) / A))))));
	elseif (A <= 8e+150)
		tmp = Float64(-Float64(t_0 * Float64(sqrt(F) * sqrt(Float64(B_m + C)))));
	else
		tmp = Float64(-Float64(sqrt(Float64(Float64(2.0 * F) * Float64(A + hypot(B_m, A)))) / B_m));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	t_0 = sqrt(2.0) / B_m;
	tmp = 0.0;
	if (A <= -2.85e+99)
		tmp = t_0 * -sqrt((F * (-0.5 * ((B_m ^ 2.0) / A))));
	elseif (A <= 8e+150)
		tmp = -(t_0 * (sqrt(F) * sqrt((B_m + C))));
	else
		tmp = -(sqrt(((2.0 * F) * (A + hypot(B_m, A)))) / B_m);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision]}, If[LessEqual[A, -2.85e+99], N[(t$95$0 * (-N[Sqrt[N[(F * N[(-0.5 * N[(N[Power[B$95$m, 2.0], $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], If[LessEqual[A, 8e+150], (-N[(t$95$0 * N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(B$95$m + C), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), (-N[(N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] * N[(A + N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / B$95$m), $MachinePrecision])]]]

Derivation

1. Split input into 3 regimes

2. if A < -2.85000000000000002e99

   1. Initial program 1.6%

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

   2. Taylor expanded in C around 0 2.2%

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

      1. mul-1-neg 2.2%

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

      2. *-commutative 2.2%

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

      3. distribute-rgt-neg-in 2.2%

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

      4. +-commutative 2.2%

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

      5. unpow2 2.2%

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

      6. unpow2 2.2%

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

      7. hypot-def 6.8%

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

   4. Simplified 6.8%

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

   5. Taylor expanded in A around -inf 22.6%

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

   if -2.85000000000000002e99 < A < 7.99999999999999985e150

   1. Initial program 28.6%

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

   2. Taylor expanded in A around 0 11.8%

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

      1. mul-1-neg 11.8%

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

      2. *-commutative 11.8%

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

      3. distribute-rgt-neg-in 11.8%

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

      4. unpow2 11.8%

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

      5. unpow2 11.8%

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

      6. hypot-def 23.2%

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

   4. Simplified 23.2%

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

      1. pow1/2 23.3%

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

      2. *-commutative 23.3%

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

      3. unpow-prod-down 28.2%

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

      4. pow1/2 28.2%

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

      5. pow1/2 28.2%

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

   6. Applied egg-rr 28.2%

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

   7. Taylor expanded in C around 0 23.2%

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

   if 7.99999999999999985e150 < A

   1. Initial program 4.5%

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

   2. Taylor expanded in C around 0 1.0%

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

      1. mul-1-neg 1.0%

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

      2. *-commutative 1.0%

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

      3. distribute-rgt-neg-in 1.0%

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

      4. +-commutative 1.0%

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

      5. unpow2 1.0%

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

      6. unpow2 1.0%

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

      7. hypot-def 13.2%

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

   4. Simplified 13.2%

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

      1. distribute-lft-in 13.1%

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

   6. Applied egg-rr 13.1%

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

      1. distribute-rgt-neg-out 13.1%

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

      2. add-sqr-sqrt 12.3%

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

      3. sqrt-unprod 8.3%

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

      4. sqr-neg 8.3%

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

      5. sqrt-unprod 13.2%

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

      6. add-sqr-sqrt 13.7%

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

      7. distribute-lft-in 13.7%

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

      8. *-commutative 13.7%

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

      9. distribute-frac-neg 13.7%

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

      10. associate-*l/ 13.8%

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

   8. Applied egg-rr 13.2%

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

      1. *-commutative 13.2%

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

      2. *-commutative 13.2%

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

      3. associate-*l* 13.2%

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

   10. Simplified 13.2%

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

4. Final simplification 21.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -2.85 \cdot 10^{+99}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(-0.5 \cdot \frac{{B}^{2}}{A}\right)}\right)\\ \mathbf{elif}\;A \leq 8 \cdot 10^{+150}:\\ \;\;\;\;-\frac{\sqrt{2}}{B} \cdot \left(\sqrt{F} \cdot \sqrt{B + C}\right)\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{\left(2 \cdot F\right) \cdot \left(A + \mathsf{hypot}\left(B, A\right)\right)}}{B}\\ \end{array} \]

Alternative 10: 35.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := \frac{\sqrt{2}}{B_m}\\ \mathbf{if}\;F \leq 3.4 \cdot 10^{+116}:\\ \;\;\;\;\left(-t_0\right) \cdot \sqrt{F \cdot \left(C + \mathsf{hypot}\left(B_m, C\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;-t_0 \cdot \left(\sqrt{F} \cdot \sqrt{B_m + C}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (/ (sqrt 2.0) B_m)))
   (if (<= F 3.4e+116)
     (* (- t_0) (sqrt (* F (+ C (hypot B_m C)))))
     (- (* t_0 (* (sqrt F) (sqrt (+ B_m C))))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double t_0 = sqrt(2.0) / B_m;
	double tmp;
	if (F <= 3.4e+116) {
		tmp = -t_0 * sqrt((F * (C + hypot(B_m, C))));
	} else {
		tmp = -(t_0 * (sqrt(F) * sqrt((B_m + C))));
	}
	return tmp;
}

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double t_0 = Math.sqrt(2.0) / B_m;
	double tmp;
	if (F <= 3.4e+116) {
		tmp = -t_0 * Math.sqrt((F * (C + Math.hypot(B_m, C))));
	} else {
		tmp = -(t_0 * (Math.sqrt(F) * Math.sqrt((B_m + C))));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	t_0 = math.sqrt(2.0) / B_m
	tmp = 0
	if F <= 3.4e+116:
		tmp = -t_0 * math.sqrt((F * (C + math.hypot(B_m, C))))
	else:
		tmp = -(t_0 * (math.sqrt(F) * math.sqrt((B_m + C))))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	t_0 = Float64(sqrt(2.0) / B_m)
	tmp = 0.0
	if (F <= 3.4e+116)
		tmp = Float64(Float64(-t_0) * sqrt(Float64(F * Float64(C + hypot(B_m, C)))));
	else
		tmp = Float64(-Float64(t_0 * Float64(sqrt(F) * sqrt(Float64(B_m + C)))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	t_0 = sqrt(2.0) / B_m;
	tmp = 0.0;
	if (F <= 3.4e+116)
		tmp = -t_0 * sqrt((F * (C + hypot(B_m, C))));
	else
		tmp = -(t_0 * (sqrt(F) * sqrt((B_m + C))));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision]}, If[LessEqual[F, 3.4e+116], N[((-t$95$0) * N[Sqrt[N[(F * N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], (-N[(t$95$0 * 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 F < 3.40000000000000023e116

   1. Initial program 22.7%

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

   2. Taylor expanded in A around 0 9.5%

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

      1. mul-1-neg 9.5%

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

      2. *-commutative 9.5%

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

      3. distribute-rgt-neg-in 9.5%

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

      4. unpow2 9.5%

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

      5. unpow2 9.5%

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

      6. hypot-def 22.0%

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

   4. Simplified 22.0%

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

   if 3.40000000000000023e116 < F

   1. Initial program 15.2%

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

   2. Taylor expanded in A around 0 7.4%

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

      1. mul-1-neg 7.4%

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

      2. *-commutative 7.4%

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

      3. distribute-rgt-neg-in 7.4%

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

      4. unpow2 7.4%

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

      5. unpow2 7.4%

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

      6. hypot-def 7.6%

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

   4. Simplified 7.6%

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

      1. pow1/2 7.6%

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

      2. *-commutative 7.6%

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

      3. unpow-prod-down 21.3%

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

      4. pow1/2 21.3%

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

      5. pow1/2 21.3%

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

   6. Applied egg-rr 21.3%

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

   7. Taylor expanded in C around 0 17.5%

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

4. Final simplification 20.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 3.4 \cdot 10^{+116}:\\ \;\;\;\;\left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{F \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\sqrt{2}}{B} \cdot \left(\sqrt{F} \cdot \sqrt{B + C}\right)\\ \end{array} \]

Alternative 11: 33.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := \frac{\sqrt{2}}{B_m}\\ \mathbf{if}\;C \leq 3000000000:\\ \;\;\;\;t_0 \cdot \left(\sqrt{F} \cdot \left(-\sqrt{B_m}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t_0 \cdot \left(\sqrt{F} \cdot \left(-\sqrt{2 \cdot C}\right)\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (/ (sqrt 2.0) B_m)))
   (if (<= C 3000000000.0)
     (* t_0 (* (sqrt F) (- (sqrt B_m))))
     (* t_0 (* (sqrt F) (- (sqrt (* 2.0 C))))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double t_0 = sqrt(2.0) / B_m;
	double tmp;
	if (C <= 3000000000.0) {
		tmp = t_0 * (sqrt(F) * -sqrt(B_m));
	} else {
		tmp = t_0 * (sqrt(F) * -sqrt((2.0 * C)));
	}
	return tmp;
}

Fortran

B_m = abs(B)
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sqrt(2.0d0) / b_m
    if (c <= 3000000000.0d0) then
        tmp = t_0 * (sqrt(f) * -sqrt(b_m))
    else
        tmp = t_0 * (sqrt(f) * -sqrt((2.0d0 * c)))
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double t_0 = Math.sqrt(2.0) / B_m;
	double tmp;
	if (C <= 3000000000.0) {
		tmp = t_0 * (Math.sqrt(F) * -Math.sqrt(B_m));
	} else {
		tmp = t_0 * (Math.sqrt(F) * -Math.sqrt((2.0 * C)));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	t_0 = math.sqrt(2.0) / B_m
	tmp = 0
	if C <= 3000000000.0:
		tmp = t_0 * (math.sqrt(F) * -math.sqrt(B_m))
	else:
		tmp = t_0 * (math.sqrt(F) * -math.sqrt((2.0 * C)))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	t_0 = Float64(sqrt(2.0) / B_m)
	tmp = 0.0
	if (C <= 3000000000.0)
		tmp = Float64(t_0 * Float64(sqrt(F) * Float64(-sqrt(B_m))));
	else
		tmp = Float64(t_0 * Float64(sqrt(F) * Float64(-sqrt(Float64(2.0 * C)))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	t_0 = sqrt(2.0) / B_m;
	tmp = 0.0;
	if (C <= 3000000000.0)
		tmp = t_0 * (sqrt(F) * -sqrt(B_m));
	else
		tmp = t_0 * (sqrt(F) * -sqrt((2.0 * C)));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision]}, If[LessEqual[C, 3000000000.0], N[(t$95$0 * N[(N[Sqrt[F], $MachinePrecision] * (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], N[(t$95$0 * N[(N[Sqrt[F], $MachinePrecision] * (-N[Sqrt[N[(2.0 * C), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if C < 3e9

   1. Initial program 18.6%

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

   2. Taylor expanded in A around 0 9.7%

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

      1. mul-1-neg 9.7%

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

      2. *-commutative 9.7%

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

      3. distribute-rgt-neg-in 9.7%

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

      4. unpow2 9.7%

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

      5. unpow2 9.7%

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

      6. hypot-def 18.7%

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

   4. Simplified 18.7%

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

      1. pow1/2 18.7%

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

      2. *-commutative 18.7%

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

      3. unpow-prod-down 21.9%

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

      4. pow1/2 21.9%

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

      5. pow1/2 21.9%

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

   6. Applied egg-rr 21.9%

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

   7. Taylor expanded in C around 0 20.2%

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

   if 3e9 < C

   1. Initial program 25.7%

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

   2. Taylor expanded in A around 0 7.0%

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

      1. mul-1-neg 7.0%

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

      2. *-commutative 7.0%

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

      3. distribute-rgt-neg-in 7.0%

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

      4. unpow2 7.0%

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

      5. unpow2 7.0%

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

      6. hypot-def 16.4%

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

   4. Simplified 16.4%

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

      1. pow1/2 16.8%

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

      2. *-commutative 16.8%

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

      3. unpow-prod-down 21.6%

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

      4. pow1/2 21.6%

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

      5. pow1/2 21.6%

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

   6. Applied egg-rr 21.6%

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

   7. Taylor expanded in C around inf 13.3%

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

4. Final simplification 18.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq 3000000000:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{F} \cdot \left(-\sqrt{B}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{F} \cdot \left(-\sqrt{2 \cdot C}\right)\right)\\ \end{array} \]

Alternative 12: 36.0% accurate, 2.1× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;F \leq 3 \cdot 10^{+110}:\\ \;\;\;\;-\frac{\sqrt{\left(2 \cdot F\right) \cdot \left(A + \mathsf{hypot}\left(B_m, A\right)\right)}}{B_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F}}{\sqrt{B_m}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= F 3e+110)
   (- (/ (sqrt (* (* 2.0 F) (+ A (hypot B_m A)))) B_m))
   (* (/ (sqrt F) (sqrt B_m)) (- (sqrt 2.0)))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= 3e+110) {
		tmp = -(sqrt(((2.0 * F) * (A + hypot(B_m, A)))) / B_m);
	} else {
		tmp = (sqrt(F) / sqrt(B_m)) * -sqrt(2.0);
	}
	return tmp;
}

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= 3e+110) {
		tmp = -(Math.sqrt(((2.0 * F) * (A + Math.hypot(B_m, A)))) / B_m);
	} else {
		tmp = (Math.sqrt(F) / Math.sqrt(B_m)) * -Math.sqrt(2.0);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if F <= 3e+110:
		tmp = -(math.sqrt(((2.0 * F) * (A + math.hypot(B_m, A)))) / B_m)
	else:
		tmp = (math.sqrt(F) / math.sqrt(B_m)) * -math.sqrt(2.0)
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (F <= 3e+110)
		tmp = Float64(-Float64(sqrt(Float64(Float64(2.0 * F) * Float64(A + hypot(B_m, A)))) / B_m));
	else
		tmp = Float64(Float64(sqrt(F) / sqrt(B_m)) * Float64(-sqrt(2.0)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (F <= 3e+110)
		tmp = -(sqrt(((2.0 * F) * (A + hypot(B_m, A)))) / B_m);
	else
		tmp = (sqrt(F) / sqrt(B_m)) * -sqrt(2.0);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[F, 3e+110], (-N[(N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] * N[(A + N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / B$95$m), $MachinePrecision]), N[(N[(N[Sqrt[F], $MachinePrecision] / N[Sqrt[B$95$m], $MachinePrecision]), $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < 3.00000000000000007e110

   1. Initial program 22.5%

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

   2. Taylor expanded in C around 0 8.0%

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

      1. mul-1-neg 8.0%

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

      2. *-commutative 8.0%

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

      3. distribute-rgt-neg-in 8.0%

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

      4. +-commutative 8.0%

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

      5. unpow2 8.0%

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

      6. unpow2 8.0%

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

      7. hypot-def 20.7%

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

   4. Simplified 20.7%

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

      1. distribute-lft-in 20.6%

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

   6. Applied egg-rr 20.6%

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

      1. distribute-rgt-neg-out 20.6%

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

      2. add-sqr-sqrt 19.3%

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

      3. sqrt-unprod 16.8%

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

      4. sqr-neg 16.8%

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

      5. sqrt-unprod 13.6%

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

      6. add-sqr-sqrt 15.2%

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

      7. distribute-lft-in 15.3%

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

      8. *-commutative 15.3%

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

      9. distribute-frac-neg 15.3%

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

      10. associate-*l/ 15.3%

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

   8. Applied egg-rr 20.7%

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

      1. *-commutative 20.7%

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

      2. *-commutative 20.7%

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

      3. associate-*l* 20.7%

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

   10. Simplified 20.7%

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

   if 3.00000000000000007e110 < F

   1. Initial program 16.0%

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

   2. Taylor expanded in A around 0 7.2%

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

      1. mul-1-neg 7.2%

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

      2. *-commutative 7.2%

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

      3. distribute-rgt-neg-in 7.2%

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

      4. unpow2 7.2%

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

      5. unpow2 7.2%

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

      6. hypot-def 7.5%

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

   4. Simplified 7.5%

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

   5. Taylor expanded in C around 0 16.5%

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

      1. mul-1-neg 16.5%

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

   7. Simplified 16.5%

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

      1. sqrt-div 16.6%

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

   9. Applied egg-rr 16.6%

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

4. Final simplification 19.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 3 \cdot 10^{+110}:\\ \;\;\;\;-\frac{\sqrt{\left(2 \cdot F\right) \cdot \left(A + \mathsf{hypot}\left(B, A\right)\right)}}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{F}}{\sqrt{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]

Alternative 13: 35.5% accurate, 3.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;F \leq 4.6 \cdot 10^{+111}:\\ \;\;\;\;-\frac{\sqrt{\left(2 \cdot F\right) \cdot \left(A + \mathsf{hypot}\left(B_m, A\right)\right)}}{B_m}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{2 \cdot F}{B_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= F 4.6e+111)
   (- (/ (sqrt (* (* 2.0 F) (+ A (hypot B_m A)))) B_m))
   (- (sqrt (/ (* 2.0 F) B_m)))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= 4.6e+111) {
		tmp = -(sqrt(((2.0 * F) * (A + hypot(B_m, A)))) / B_m);
	} else {
		tmp = -sqrt(((2.0 * F) / B_m));
	}
	return tmp;
}

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= 4.6e+111) {
		tmp = -(Math.sqrt(((2.0 * F) * (A + Math.hypot(B_m, A)))) / B_m);
	} else {
		tmp = -Math.sqrt(((2.0 * F) / B_m));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if F <= 4.6e+111:
		tmp = -(math.sqrt(((2.0 * F) * (A + math.hypot(B_m, A)))) / B_m)
	else:
		tmp = -math.sqrt(((2.0 * F) / B_m))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (F <= 4.6e+111)
		tmp = Float64(-Float64(sqrt(Float64(Float64(2.0 * F) * Float64(A + hypot(B_m, A)))) / B_m));
	else
		tmp = Float64(-sqrt(Float64(Float64(2.0 * F) / B_m)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (F <= 4.6e+111)
		tmp = -(sqrt(((2.0 * F) * (A + hypot(B_m, A)))) / B_m);
	else
		tmp = -sqrt(((2.0 * F) / B_m));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[F, 4.6e+111], (-N[(N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] * N[(A + N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / B$95$m), $MachinePrecision]), (-N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] / B$95$m), $MachinePrecision]], $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if F < 4.60000000000000004e111

   1. Initial program 22.5%

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

   2. Taylor expanded in C around 0 8.0%

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

      1. mul-1-neg 8.0%

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

      2. *-commutative 8.0%

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

      3. distribute-rgt-neg-in 8.0%

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

      4. +-commutative 8.0%

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

      5. unpow2 8.0%

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

      6. unpow2 8.0%

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

      7. hypot-def 20.7%

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

   4. Simplified 20.7%

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

      1. distribute-lft-in 20.6%

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

   6. Applied egg-rr 20.6%

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

      1. distribute-rgt-neg-out 20.6%

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

      2. add-sqr-sqrt 19.3%

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

      3. sqrt-unprod 16.8%

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

      4. sqr-neg 16.8%

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

      5. sqrt-unprod 13.6%

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

      6. add-sqr-sqrt 15.2%

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

      7. distribute-lft-in 15.3%

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

      8. *-commutative 15.3%

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

      9. distribute-frac-neg 15.3%

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

      10. associate-*l/ 15.3%

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

   8. Applied egg-rr 20.7%

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

      1. *-commutative 20.7%

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

      2. *-commutative 20.7%

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

      3. associate-*l* 20.7%

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

   10. Simplified 20.7%

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

   if 4.60000000000000004e111 < F

   1. Initial program 16.0%

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

   2. Taylor expanded in A around 0 7.2%

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

      1. mul-1-neg 7.2%

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

      2. *-commutative 7.2%

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

      3. distribute-rgt-neg-in 7.2%

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

      4. unpow2 7.2%

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

      5. unpow2 7.2%

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

      6. hypot-def 7.5%

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

   4. Simplified 7.5%

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

   5. Taylor expanded in C around 0 16.5%

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

      1. mul-1-neg 16.5%

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

   7. Simplified 16.5%

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

      1. expm1-log1p-u 16.2%

         \[\leadsto -\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)\right)} \]

      2. expm1-udef 9.6%

         \[\leadsto -\color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} - 1\right)} \]

      3. sqrt-unprod 9.6%

         \[\leadsto -\left(e^{\mathsf{log1p}\left(\color{blue}{\sqrt{\frac{F}{B} \cdot 2}}\right)} - 1\right) \]

   9. Applied egg-rr 9.6%

      \[\leadsto -\color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{F}{B} \cdot 2}\right)} - 1\right)} \]
   10. Step-by-step derivation

       1. expm1-def 16.3%

          \[\leadsto -\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{F}{B} \cdot 2}\right)\right)} \]

       2. expm1-log1p 16.7%

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

       3. *-commutative 16.7%

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

       4. associate-*r/ 16.7%

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

   11. Simplified 16.7%

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

4. Final simplification 19.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 4.6 \cdot 10^{+111}:\\ \;\;\;\;-\frac{\sqrt{\left(2 \cdot F\right) \cdot \left(A + \mathsf{hypot}\left(B, A\right)\right)}}{B}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{2 \cdot F}{B}}\\ \end{array} \]

Alternative 14: 34.2% accurate, 3.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;F \leq 4.5 \cdot 10^{-38}:\\ \;\;\;\;\left(-\frac{\sqrt{2}}{B_m}\right) \cdot \sqrt{B_m \cdot F}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{2 \cdot F}{B_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= F 4.5e-38)
   (* (- (/ (sqrt 2.0) B_m)) (sqrt (* B_m F)))
   (- (sqrt (/ (* 2.0 F) B_m)))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= 4.5e-38) {
		tmp = -(sqrt(2.0) / B_m) * sqrt((B_m * F));
	} else {
		tmp = -sqrt(((2.0 * F) / B_m));
	}
	return tmp;
}

Fortran

B_m = abs(B)
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (f <= 4.5d-38) then
        tmp = -(sqrt(2.0d0) / b_m) * sqrt((b_m * f))
    else
        tmp = -sqrt(((2.0d0 * f) / b_m))
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= 4.5e-38) {
		tmp = -(Math.sqrt(2.0) / B_m) * Math.sqrt((B_m * F));
	} else {
		tmp = -Math.sqrt(((2.0 * F) / B_m));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if F <= 4.5e-38:
		tmp = -(math.sqrt(2.0) / B_m) * math.sqrt((B_m * F))
	else:
		tmp = -math.sqrt(((2.0 * F) / B_m))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (F <= 4.5e-38)
		tmp = Float64(Float64(-Float64(sqrt(2.0) / B_m)) * sqrt(Float64(B_m * F)));
	else
		tmp = Float64(-sqrt(Float64(Float64(2.0 * F) / B_m)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (F <= 4.5e-38)
		tmp = -(sqrt(2.0) / B_m) * sqrt((B_m * F));
	else
		tmp = -sqrt(((2.0 * F) / B_m));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[F, 4.5e-38], N[((-N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision]) * N[Sqrt[N[(B$95$m * F), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], (-N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] / B$95$m), $MachinePrecision]], $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if F < 4.50000000000000009e-38

   1. Initial program 22.4%

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

   2. Taylor expanded in A around 0 9.5%

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

      1. mul-1-neg 9.5%

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

      2. *-commutative 9.5%

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

      3. distribute-rgt-neg-in 9.5%

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

      4. unpow2 9.5%

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

      5. unpow2 9.5%

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

      6. hypot-def 22.3%

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

   4. Simplified 22.3%

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

   5. Taylor expanded in C around 0 19.9%

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

   if 4.50000000000000009e-38 < F

   1. Initial program 18.8%

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

   2. Taylor expanded in A around 0 8.3%

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

      1. mul-1-neg 8.3%

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

      2. *-commutative 8.3%

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

      3. distribute-rgt-neg-in 8.3%

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

      4. unpow2 8.3%

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

      5. unpow2 8.3%

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

      6. hypot-def 13.5%

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

   4. Simplified 13.5%

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

   5. Taylor expanded in C around 0 16.3%

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

      1. mul-1-neg 16.3%

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

   7. Simplified 16.3%

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

      1. expm1-log1p-u 16.1%

         \[\leadsto -\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)\right)} \]

      2. expm1-udef 7.3%

         \[\leadsto -\color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} - 1\right)} \]

      3. sqrt-unprod 7.3%

         \[\leadsto -\left(e^{\mathsf{log1p}\left(\color{blue}{\sqrt{\frac{F}{B} \cdot 2}}\right)} - 1\right) \]

   9. Applied egg-rr 7.3%

      \[\leadsto -\color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{F}{B} \cdot 2}\right)} - 1\right)} \]
   10. Step-by-step derivation

       1. expm1-def 16.2%

          \[\leadsto -\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{F}{B} \cdot 2}\right)\right)} \]

       2. expm1-log1p 16.4%

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

       3. *-commutative 16.4%

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

       4. associate-*r/ 16.4%

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

   11. Simplified 16.4%

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

4. Final simplification 18.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 4.5 \cdot 10^{-38}:\\ \;\;\;\;\left(-\frac{\sqrt{2}}{B}\right) \cdot \sqrt{B \cdot F}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{2 \cdot F}{B}}\\ \end{array} \]

Alternative 15: 26.0% accurate, 5.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;C \leq 150000:\\ \;\;\;\;-\sqrt{\frac{2 \cdot F}{B_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{B_m} \cdot \left(-\sqrt{C \cdot F}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= C 150000.0)
   (- (sqrt (/ (* 2.0 F) B_m)))
   (* (/ 2.0 B_m) (- (sqrt (* C F))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (C <= 150000.0) {
		tmp = -sqrt(((2.0 * F) / B_m));
	} else {
		tmp = (2.0 / B_m) * -sqrt((C * F));
	}
	return tmp;
}

Fortran

B_m = abs(B)
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (c <= 150000.0d0) then
        tmp = -sqrt(((2.0d0 * f) / b_m))
    else
        tmp = (2.0d0 / b_m) * -sqrt((c * f))
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (C <= 150000.0) {
		tmp = -Math.sqrt(((2.0 * F) / B_m));
	} else {
		tmp = (2.0 / B_m) * -Math.sqrt((C * F));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if C <= 150000.0:
		tmp = -math.sqrt(((2.0 * F) / B_m))
	else:
		tmp = (2.0 / B_m) * -math.sqrt((C * F))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (C <= 150000.0)
		tmp = Float64(-sqrt(Float64(Float64(2.0 * F) / B_m)));
	else
		tmp = Float64(Float64(2.0 / B_m) * Float64(-sqrt(Float64(C * F))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (C <= 150000.0)
		tmp = -sqrt(((2.0 * F) / B_m));
	else
		tmp = (2.0 / B_m) * -sqrt((C * F));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[C, 150000.0], (-N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] / B$95$m), $MachinePrecision]], $MachinePrecision]), N[(N[(2.0 / B$95$m), $MachinePrecision] * (-N[Sqrt[N[(C * F), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if C < 1.5e5

   1. Initial program 18.7%

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

   2. Taylor expanded in A around 0 9.7%

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

      1. mul-1-neg 9.7%

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

      2. *-commutative 9.7%

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

      3. distribute-rgt-neg-in 9.7%

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

      4. unpow2 9.7%

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

      5. unpow2 9.7%

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

      6. hypot-def 18.8%

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

   4. Simplified 18.8%

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

   5. Taylor expanded in C around 0 14.4%

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

      1. mul-1-neg 14.4%

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

   7. Simplified 14.4%

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

      1. expm1-log1p-u 14.2%

         \[\leadsto -\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)\right)} \]

      2. expm1-udef 6.4%

         \[\leadsto -\color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} - 1\right)} \]

      3. sqrt-unprod 6.4%

         \[\leadsto -\left(e^{\mathsf{log1p}\left(\color{blue}{\sqrt{\frac{F}{B} \cdot 2}}\right)} - 1\right) \]

   9. Applied egg-rr 6.4%

      \[\leadsto -\color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{F}{B} \cdot 2}\right)} - 1\right)} \]
   10. Step-by-step derivation

       1. expm1-def 14.2%

          \[\leadsto -\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{F}{B} \cdot 2}\right)\right)} \]

       2. expm1-log1p 14.4%

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

       3. *-commutative 14.4%

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

       4. associate-*r/ 14.4%

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

   11. Simplified 14.4%

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

   if 1.5e5 < C

   1. Initial program 25.3%

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

   2. Taylor expanded in A around 0 6.9%

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

      1. mul-1-neg 6.9%

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

      2. *-commutative 6.9%

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

      3. distribute-rgt-neg-in 6.9%

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

      4. unpow2 6.9%

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

      5. unpow2 6.9%

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

      6. hypot-def 16.3%

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

   4. Simplified 16.3%

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

      1. pow1/2 16.6%

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

      2. *-commutative 16.6%

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

      3. unpow-prod-down 21.4%

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

      4. pow1/2 21.4%

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

      5. pow1/2 21.4%

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

   6. Applied egg-rr 21.4%

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

   7. Taylor expanded in B around 0 10.4%

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

      1. mul-1-neg 10.4%

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

      2. distribute-rgt-neg-in 10.4%

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

      3. unpow2 10.4%

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

      4. rem-square-sqrt 10.5%

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

   9. Simplified 10.5%

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

4. Final simplification 13.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq 150000:\\ \;\;\;\;-\sqrt{\frac{2 \cdot F}{B}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{B} \cdot \left(-\sqrt{C \cdot F}\right)\\ \end{array} \]

Alternative 16: 27.4% accurate, 6.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ -\sqrt{\frac{2 \cdot F}{B_m}} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F) :precision binary64 (- (sqrt (/ (* 2.0 F) B_m))))

C

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

Fortran

B_m = abs(B)
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    code = -sqrt(((2.0d0 * f) / b_m))
end function

Java

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

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	return -math.sqrt(((2.0 * F) / B_m))

Julia

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

MATLAB

B_m = abs(B);
function tmp = code(A, B_m, C, F)
	tmp = -sqrt(((2.0 * F) / B_m));
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := (-N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] / B$95$m), $MachinePrecision]], $MachinePrecision])

Derivation

1. Initial program 20.6%

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

2. Taylor expanded in A around 0 8.9%

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

   1. mul-1-neg 8.9%

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

   2. *-commutative 8.9%

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

   3. distribute-rgt-neg-in 8.9%

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

   4. unpow2 8.9%

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

   5. unpow2 8.9%

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

   6. hypot-def 18.0%

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

4. Simplified 18.0%

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

5. Taylor expanded in C around 0 12.5%

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

   1. mul-1-neg 12.5%

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

7. Simplified 12.5%

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

   1. expm1-log1p-u 12.4%

      \[\leadsto -\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)\right)} \]

   2. expm1-udef 5.5%

      \[\leadsto -\color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{F}{B}} \cdot \sqrt{2}\right)} - 1\right)} \]

   3. sqrt-unprod 5.5%

      \[\leadsto -\left(e^{\mathsf{log1p}\left(\color{blue}{\sqrt{\frac{F}{B} \cdot 2}}\right)} - 1\right) \]

9. Applied egg-rr 5.5%

   \[\leadsto -\color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\frac{F}{B} \cdot 2}\right)} - 1\right)} \]
10. Step-by-step derivation

    1. expm1-def 12.4%

       \[\leadsto -\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{F}{B} \cdot 2}\right)\right)} \]

    2. expm1-log1p 12.5%

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

    3. *-commutative 12.5%

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

    4. associate-*r/ 12.5%

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

11. Simplified 12.5%

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

12. Final simplification 12.5%

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

Reproduce

herbie shell --seed 2023322 
(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))))