ABCF->ab-angle a

Percentage Accurate: 18.8% → 52.6%

Time: 33.5s

Alternatives: 12

Speedup: 3.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 18.8% 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: 52.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;{B_m}^{2} \leq 5 \cdot 10^{+131}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\mathsf{fma}\left(B_m, B_m, C \cdot \left(A \cdot -4\right)\right) \cdot F\right)} \cdot \left(-\sqrt{A + \left(C + \mathsf{hypot}\left(B_m, A - C\right)\right)}\right)}{\mathsf{fma}\left(B_m, B_m, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(B_m, C\right)} \cdot \sqrt{F}\right) \cdot \frac{-\sqrt{2}}{B_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 5e+131)
   (/
    (*
     (sqrt (* 2.0 (* (fma B_m B_m (* C (* A -4.0))) F)))
     (- (sqrt (+ A (+ C (hypot B_m (- A C)))))))
    (fma B_m B_m (* A (* C -4.0))))
   (* (* (sqrt (+ C (hypot B_m C))) (sqrt F)) (/ (- (sqrt 2.0)) B_m))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (pow(B_m, 2.0) <= 5e+131) {
		tmp = (sqrt((2.0 * (fma(B_m, B_m, (C * (A * -4.0))) * F))) * -sqrt((A + (C + hypot(B_m, (A - C)))))) / fma(B_m, B_m, (A * (C * -4.0)));
	} else {
		tmp = (sqrt((C + hypot(B_m, C))) * sqrt(F)) * (-sqrt(2.0) / B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 5e+131)
		tmp = Float64(Float64(sqrt(Float64(2.0 * Float64(fma(B_m, B_m, Float64(C * Float64(A * -4.0))) * F))) * Float64(-sqrt(Float64(A + Float64(C + hypot(B_m, Float64(A - C))))))) / fma(B_m, B_m, Float64(A * Float64(C * -4.0))));
	else
		tmp = Float64(Float64(sqrt(Float64(C + hypot(B_m, C))) * 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_] := If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e+131], N[(N[(N[Sqrt[N[(2.0 * N[(N[(B$95$m * B$95$m + N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(A + N[(C + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 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 2 regimes

2. if (pow.f64 B 2) < 4.99999999999999995e131

   1. Initial program 23.6%

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

      1. neg-sub0 23.6%

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

      2. div-sub 23.6%

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

      3. associate-*l* 23.6%

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

   4. Applied egg-rr 30.1%

      \[\leadsto \color{blue}{\frac{0}{{B}^{2} - 4 \cdot \left(A \cdot C\right)} - \frac{\sqrt{\left(\left(2 \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right) \cdot F\right) \cdot \left(A + \left(C + \mathsf{hypot}\left(A - C, B\right)\right)\right)}}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}} \]
   5. Step-by-step derivation

      1. div0 30.1%

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

      2. neg-sub0 30.1%

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

      3. distribute-neg-frac 30.1%

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

   6. Simplified 30.1%

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

      1. pow1/2 30.1%

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

      2. associate-*r* 30.1%

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

      3. unpow-prod-down 42.2%

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

      4. associate-*r* 42.2%

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

      5. *-commutative 42.2%

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

      6. associate-*l* 41.6%

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

      7. pow1/2 41.6%

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

      8. associate-+r+ 40.4%

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

      9. +-commutative 40.4%

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

      10. +-commutative 40.4%

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

   8. Applied egg-rr 40.4%

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

      1. unpow1/2 40.4%

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

      2. associate-*l* 40.4%

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

      3. +-commutative 40.4%

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

      4. +-commutative 40.4%

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

      5. hypot-def 27.4%

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

      6. unpow2 27.4%

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

      7. unpow2 27.4%

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

      8. +-commutative 27.4%

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

      9. associate-+r+ 27.8%

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

      10. unpow2 27.8%

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

      11. unpow2 27.8%

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

      12. hypot-def 41.6%

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

   10. Simplified 41.6%

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

   if 4.99999999999999995e131 < (pow.f64 B 2)

   1. Initial program 6.8%

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

   3. Taylor expanded in A around 0 10.6%

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

      1. mul-1-neg 10.6%

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

      2. *-commutative 10.6%

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

      3. distribute-rgt-neg-in 10.6%

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

      4. unpow2 10.6%

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

      5. unpow2 10.6%

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

      6. hypot-def 23.7%

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

   5. Simplified 23.7%

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

      1. pow1/2 23.7%

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

      2. *-commutative 23.7%

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

      3. unpow-prod-down 40.1%

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

      4. pow1/2 40.1%

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

      5. pow1/2 40.1%

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

   7. Applied egg-rr 40.1%

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

4. Final simplification 40.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 5 \cdot 10^{+131}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot F\right)} \cdot \left(-\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)}\right)}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(B, C\right)} \cdot \sqrt{F}\right) \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 43.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;{B_m}^{2} \leq 10^{-98}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(2 \cdot \left(F \cdot \left({B_m}^{2} + -4 \cdot \left(C \cdot A\right)\right)\right)\right)} \cdot \left(-\sqrt{C}\right)}{{B_m}^{2} - C \cdot \left(A \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(B_m, C\right)} \cdot \sqrt{F}\right) \cdot \frac{-\sqrt{2}}{B_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 1e-98)
   (/
    (*
     (sqrt (* 2.0 (* 2.0 (* F (+ (pow B_m 2.0) (* -4.0 (* C A)))))))
     (- (sqrt C)))
    (- (pow B_m 2.0) (* C (* A 4.0))))
   (* (* (sqrt (+ C (hypot B_m C))) (sqrt F)) (/ (- (sqrt 2.0)) B_m))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (pow(B_m, 2.0) <= 1e-98) {
		tmp = (sqrt((2.0 * (2.0 * (F * (pow(B_m, 2.0) + (-4.0 * (C * A))))))) * -sqrt(C)) / (pow(B_m, 2.0) - (C * (A * 4.0)));
	} else {
		tmp = (sqrt((C + hypot(B_m, C))) * sqrt(F)) * (-sqrt(2.0) / B_m);
	}
	return tmp;
}

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (Math.pow(B_m, 2.0) <= 1e-98) {
		tmp = (Math.sqrt((2.0 * (2.0 * (F * (Math.pow(B_m, 2.0) + (-4.0 * (C * A))))))) * -Math.sqrt(C)) / (Math.pow(B_m, 2.0) - (C * (A * 4.0)));
	} else {
		tmp = (Math.sqrt((C + Math.hypot(B_m, C))) * Math.sqrt(F)) * (-Math.sqrt(2.0) / B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if math.pow(B_m, 2.0) <= 1e-98:
		tmp = (math.sqrt((2.0 * (2.0 * (F * (math.pow(B_m, 2.0) + (-4.0 * (C * A))))))) * -math.sqrt(C)) / (math.pow(B_m, 2.0) - (C * (A * 4.0)))
	else:
		tmp = (math.sqrt((C + math.hypot(B_m, C))) * math.sqrt(F)) * (-math.sqrt(2.0) / B_m)
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e-98)
		tmp = Float64(Float64(sqrt(Float64(2.0 * Float64(2.0 * Float64(F * Float64((B_m ^ 2.0) + Float64(-4.0 * Float64(C * A))))))) * Float64(-sqrt(C))) / Float64((B_m ^ 2.0) - Float64(C * Float64(A * 4.0))));
	else
		tmp = Float64(Float64(sqrt(Float64(C + hypot(B_m, C))) * sqrt(F)) * Float64(Float64(-sqrt(2.0)) / B_m));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if ((B_m ^ 2.0) <= 1e-98)
		tmp = (sqrt((2.0 * (2.0 * (F * ((B_m ^ 2.0) + (-4.0 * (C * A))))))) * -sqrt(C)) / ((B_m ^ 2.0) - (C * (A * 4.0)));
	else
		tmp = (sqrt((C + hypot(B_m, C))) * sqrt(F)) * (-sqrt(2.0) / B_m);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-98], N[(N[(N[Sqrt[N[(2.0 * N[(2.0 * N[(F * N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[(-4.0 * N[(C * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[C], $MachinePrecision])), $MachinePrecision] / N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(C * N[(A * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 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 2 regimes

2. if (pow.f64 B 2) < 9.99999999999999939e-99

   1. Initial program 14.0%

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

   3. Taylor expanded in A around -inf 28.3%

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

      1. pow1/2 28.4%

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

      2. associate-*r* 28.4%

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

      3. unpow-prod-down 29.1%

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

      4. *-commutative 29.1%

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

      5. associate-*l* 29.2%

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

      6. *-commutative 29.2%

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

      7. pow1/2 29.2%

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

   5. Applied egg-rr 29.2%

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

      1. unpow1/2 29.2%

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

      2. associate-*l* 29.2%

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

      3. cancel-sign-sub-inv 29.2%

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

      4. metadata-eval 29.2%

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

   7. Simplified 29.2%

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

   if 9.99999999999999939e-99 < (pow.f64 B 2)

   1. Initial program 17.5%

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

   3. Taylor expanded in A around 0 15.9%

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

      1. mul-1-neg 15.9%

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

      2. *-commutative 15.9%

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

      3. distribute-rgt-neg-in 15.9%

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

      4. unpow2 15.9%

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

      5. unpow2 15.9%

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

      6. hypot-def 25.6%

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

   5. Simplified 25.6%

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

      1. pow1/2 25.6%

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

      2. *-commutative 25.6%

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

      3. unpow-prod-down 37.6%

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

      4. pow1/2 37.6%

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

      5. pow1/2 37.6%

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

   7. Applied egg-rr 37.6%

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

4. Final simplification 34.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{-98}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(2 \cdot \left(F \cdot \left({B}^{2} + -4 \cdot \left(C \cdot A\right)\right)\right)\right)} \cdot \left(-\sqrt{C}\right)}{{B}^{2} - C \cdot \left(A \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(B, C\right)} \cdot \sqrt{F}\right) \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 43.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := {B_m}^{2} + -4 \cdot \left(C \cdot A\right)\\ \mathbf{if}\;{B_m}^{2} \leq 10^{-98}:\\ \;\;\;\;\frac{1}{\frac{t_0}{-\sqrt{\left(t_0 \cdot \left(2 \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(B_m, C\right)} \cdot \sqrt{F}\right) \cdot \frac{-\sqrt{2}}{B_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (+ (pow B_m 2.0) (* -4.0 (* C A)))))
   (if (<= (pow B_m 2.0) 1e-98)
     (/ 1.0 (/ t_0 (- (sqrt (* (* t_0 (* 2.0 F)) (* 2.0 C))))))
     (* (* (sqrt (+ C (hypot B_m C))) (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 = pow(B_m, 2.0) + (-4.0 * (C * A));
	double tmp;
	if (pow(B_m, 2.0) <= 1e-98) {
		tmp = 1.0 / (t_0 / -sqrt(((t_0 * (2.0 * F)) * (2.0 * C))));
	} else {
		tmp = (sqrt((C + hypot(B_m, C))) * sqrt(F)) * (-sqrt(2.0) / B_m);
	}
	return tmp;
}

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double t_0 = Math.pow(B_m, 2.0) + (-4.0 * (C * A));
	double tmp;
	if (Math.pow(B_m, 2.0) <= 1e-98) {
		tmp = 1.0 / (t_0 / -Math.sqrt(((t_0 * (2.0 * F)) * (2.0 * C))));
	} else {
		tmp = (Math.sqrt((C + Math.hypot(B_m, C))) * Math.sqrt(F)) * (-Math.sqrt(2.0) / B_m);
	}
	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 * (C * A))
	tmp = 0
	if math.pow(B_m, 2.0) <= 1e-98:
		tmp = 1.0 / (t_0 / -math.sqrt(((t_0 * (2.0 * F)) * (2.0 * C))))
	else:
		tmp = (math.sqrt((C + math.hypot(B_m, C))) * math.sqrt(F)) * (-math.sqrt(2.0) / B_m)
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	t_0 = Float64((B_m ^ 2.0) + Float64(-4.0 * Float64(C * A)))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e-98)
		tmp = Float64(1.0 / Float64(t_0 / Float64(-sqrt(Float64(Float64(t_0 * Float64(2.0 * F)) * Float64(2.0 * C))))));
	else
		tmp = Float64(Float64(sqrt(Float64(C + hypot(B_m, C))) * sqrt(F)) * Float64(Float64(-sqrt(2.0)) / B_m));
	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 * (C * A));
	tmp = 0.0;
	if ((B_m ^ 2.0) <= 1e-98)
		tmp = 1.0 / (t_0 / -sqrt(((t_0 * (2.0 * F)) * (2.0 * C))));
	else
		tmp = (sqrt((C + hypot(B_m, C))) * sqrt(F)) * (-sqrt(2.0) / B_m);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[(-4.0 * N[(C * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-98], N[(1.0 / N[(t$95$0 / (-N[Sqrt[N[(N[(t$95$0 * N[(2.0 * F), $MachinePrecision]), $MachinePrecision] * N[(2.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 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 2 regimes

2. if (pow.f64 B 2) < 9.99999999999999939e-99

   1. Initial program 14.0%

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

   3. Taylor expanded in A around -inf 28.3%

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

      1. clear-num 28.3%

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

      2. inv-pow 28.3%

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

   5. Applied egg-rr 28.3%

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

      1. unpow-1 28.3%

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

      2. cancel-sign-sub-inv 28.3%

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

      3. metadata-eval 28.3%

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

      4. associate-*r* 28.4%

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

      5. associate-*r* 28.4%

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

      6. cancel-sign-sub-inv 28.4%

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

      7. metadata-eval 28.4%

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

   7. Simplified 28.4%

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

   if 9.99999999999999939e-99 < (pow.f64 B 2)

   1. Initial program 17.5%

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

   3. Taylor expanded in A around 0 15.9%

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

      1. mul-1-neg 15.9%

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

      2. *-commutative 15.9%

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

      3. distribute-rgt-neg-in 15.9%

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

      4. unpow2 15.9%

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

      5. unpow2 15.9%

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

      6. hypot-def 25.6%

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

   5. Simplified 25.6%

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

      1. pow1/2 25.6%

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

      2. *-commutative 25.6%

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

      3. unpow-prod-down 37.6%

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

      4. pow1/2 37.6%

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

      5. pow1/2 37.6%

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

   7. Applied egg-rr 37.6%

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

4. Final simplification 34.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{-98}:\\ \;\;\;\;\frac{1}{\frac{{B}^{2} + -4 \cdot \left(C \cdot A\right)}{-\sqrt{\left(\left({B}^{2} + -4 \cdot \left(C \cdot A\right)\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(B, C\right)} \cdot \sqrt{F}\right) \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 40.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := {B_m}^{2} + -4 \cdot \left(C \cdot A\right)\\ \mathbf{if}\;B_m \leq 7.5 \cdot 10^{-49}:\\ \;\;\;\;\frac{1}{\frac{t_0}{-\sqrt{\left(t_0 \cdot \left(2 \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B_m} \cdot \left(\sqrt{B_m + C} \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (+ (pow B_m 2.0) (* -4.0 (* C A)))))
   (if (<= B_m 7.5e-49)
     (/ 1.0 (/ t_0 (- (sqrt (* (* t_0 (* 2.0 F)) (* 2.0 C))))))
     (* (/ (sqrt 2.0) B_m) (* (sqrt (+ B_m C)) (- (sqrt F)))))))

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 * (C * A));
	double tmp;
	if (B_m <= 7.5e-49) {
		tmp = 1.0 / (t_0 / -sqrt(((t_0 * (2.0 * F)) * (2.0 * C))));
	} else {
		tmp = (sqrt(2.0) / B_m) * (sqrt((B_m + C)) * -sqrt(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) :: t_0
    real(8) :: tmp
    t_0 = (b_m ** 2.0d0) + ((-4.0d0) * (c * a))
    if (b_m <= 7.5d-49) then
        tmp = 1.0d0 / (t_0 / -sqrt(((t_0 * (2.0d0 * f)) * (2.0d0 * c))))
    else
        tmp = (sqrt(2.0d0) / b_m) * (sqrt((b_m + c)) * -sqrt(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 t_0 = Math.pow(B_m, 2.0) + (-4.0 * (C * A));
	double tmp;
	if (B_m <= 7.5e-49) {
		tmp = 1.0 / (t_0 / -Math.sqrt(((t_0 * (2.0 * F)) * (2.0 * C))));
	} else {
		tmp = (Math.sqrt(2.0) / B_m) * (Math.sqrt((B_m + C)) * -Math.sqrt(F));
	}
	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 * (C * A))
	tmp = 0
	if B_m <= 7.5e-49:
		tmp = 1.0 / (t_0 / -math.sqrt(((t_0 * (2.0 * F)) * (2.0 * C))))
	else:
		tmp = (math.sqrt(2.0) / B_m) * (math.sqrt((B_m + C)) * -math.sqrt(F))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	t_0 = Float64((B_m ^ 2.0) + Float64(-4.0 * Float64(C * A)))
	tmp = 0.0
	if (B_m <= 7.5e-49)
		tmp = Float64(1.0 / Float64(t_0 / Float64(-sqrt(Float64(Float64(t_0 * Float64(2.0 * F)) * Float64(2.0 * C))))));
	else
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(sqrt(Float64(B_m + C)) * Float64(-sqrt(F))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	t_0 = (B_m ^ 2.0) + (-4.0 * (C * A));
	tmp = 0.0;
	if (B_m <= 7.5e-49)
		tmp = 1.0 / (t_0 / -sqrt(((t_0 * (2.0 * F)) * (2.0 * C))));
	else
		tmp = (sqrt(2.0) / B_m) * (sqrt((B_m + C)) * -sqrt(F));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[(-4.0 * N[(C * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 7.5e-49], N[(1.0 / N[(t$95$0 / (-N[Sqrt[N[(N[(t$95$0 * N[(2.0 * F), $MachinePrecision]), $MachinePrecision] * N[(2.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[(N[Sqrt[N[(B$95$m + C), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if B < 7.4999999999999998e-49

   1. Initial program 12.1%

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

   3. Taylor expanded in A around -inf 18.8%

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

      1. clear-num 18.8%

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

      2. inv-pow 18.8%

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

   5. Applied egg-rr 18.8%

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

      1. unpow-1 18.8%

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

      2. cancel-sign-sub-inv 18.8%

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

      3. metadata-eval 18.8%

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

      4. associate-*r* 18.8%

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

      5. associate-*r* 18.8%

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

      6. cancel-sign-sub-inv 18.8%

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

      7. metadata-eval 18.8%

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

   7. Simplified 18.8%

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

   if 7.4999999999999998e-49 < B

   1. Initial program 24.1%

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

   3. Taylor expanded in A around 0 28.2%

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

      1. mul-1-neg 28.2%

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

      2. *-commutative 28.2%

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

      3. distribute-rgt-neg-in 28.2%

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

      4. unpow2 28.2%

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

      5. unpow2 28.2%

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

      6. hypot-def 45.0%

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

   5. Simplified 45.0%

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

      1. pow1/2 45.0%

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

      2. *-commutative 45.0%

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

      3. unpow-prod-down 66.9%

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

      4. pow1/2 66.9%

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

      5. pow1/2 66.9%

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

   7. Applied egg-rr 66.9%

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

   8. Taylor expanded in C around 0 60.4%

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

4. Final simplification 32.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 7.5 \cdot 10^{-49}:\\ \;\;\;\;\frac{1}{\frac{{B}^{2} + -4 \cdot \left(C \cdot A\right)}{-\sqrt{\left(\left({B}^{2} + -4 \cdot \left(C \cdot A\right)\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{B + C} \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 40.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := {B_m}^{2} - C \cdot \left(A \cdot 4\right)\\ \mathbf{if}\;B_m \leq 1.9 \cdot 10^{-49}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot t_0\right)\right) \cdot \left(2 \cdot C\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B_m} \cdot \left(\sqrt{B_m + C} \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (- (pow B_m 2.0) (* C (* A 4.0)))))
   (if (<= B_m 1.9e-49)
     (/ (- (sqrt (* (* 2.0 (* F t_0)) (* 2.0 C)))) t_0)
     (* (/ (sqrt 2.0) B_m) (* (sqrt (+ B_m C)) (- (sqrt F)))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double t_0 = pow(B_m, 2.0) - (C * (A * 4.0));
	double tmp;
	if (B_m <= 1.9e-49) {
		tmp = -sqrt(((2.0 * (F * t_0)) * (2.0 * C))) / t_0;
	} else {
		tmp = (sqrt(2.0) / B_m) * (sqrt((B_m + C)) * -sqrt(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) :: t_0
    real(8) :: tmp
    t_0 = (b_m ** 2.0d0) - (c * (a * 4.0d0))
    if (b_m <= 1.9d-49) then
        tmp = -sqrt(((2.0d0 * (f * t_0)) * (2.0d0 * c))) / t_0
    else
        tmp = (sqrt(2.0d0) / b_m) * (sqrt((b_m + c)) * -sqrt(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 t_0 = Math.pow(B_m, 2.0) - (C * (A * 4.0));
	double tmp;
	if (B_m <= 1.9e-49) {
		tmp = -Math.sqrt(((2.0 * (F * t_0)) * (2.0 * C))) / t_0;
	} else {
		tmp = (Math.sqrt(2.0) / B_m) * (Math.sqrt((B_m + C)) * -Math.sqrt(F));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	t_0 = math.pow(B_m, 2.0) - (C * (A * 4.0))
	tmp = 0
	if B_m <= 1.9e-49:
		tmp = -math.sqrt(((2.0 * (F * t_0)) * (2.0 * C))) / t_0
	else:
		tmp = (math.sqrt(2.0) / B_m) * (math.sqrt((B_m + C)) * -math.sqrt(F))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	t_0 = Float64((B_m ^ 2.0) - Float64(C * Float64(A * 4.0)))
	tmp = 0.0
	if (B_m <= 1.9e-49)
		tmp = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(F * t_0)) * Float64(2.0 * C)))) / t_0);
	else
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(sqrt(Float64(B_m + C)) * Float64(-sqrt(F))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	t_0 = (B_m ^ 2.0) - (C * (A * 4.0));
	tmp = 0.0;
	if (B_m <= 1.9e-49)
		tmp = -sqrt(((2.0 * (F * t_0)) * (2.0 * C))) / t_0;
	else
		tmp = (sqrt(2.0) / B_m) * (sqrt((B_m + C)) * -sqrt(F));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(C * N[(A * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 1.9e-49], N[((-N[Sqrt[N[(N[(2.0 * N[(F * t$95$0), $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[N[(B$95$m + C), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if B < 1.8999999999999999e-49

   1. Initial program 12.1%

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

   3. Taylor expanded in A around -inf 18.8%

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

   if 1.8999999999999999e-49 < B

   1. Initial program 24.1%

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

   3. Taylor expanded in A around 0 28.2%

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

      1. mul-1-neg 28.2%

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

      2. *-commutative 28.2%

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

      3. distribute-rgt-neg-in 28.2%

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

      4. unpow2 28.2%

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

      5. unpow2 28.2%

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

      6. hypot-def 45.0%

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

   5. Simplified 45.0%

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

      1. pow1/2 45.0%

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

      2. *-commutative 45.0%

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

      3. unpow-prod-down 66.9%

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

      4. pow1/2 66.9%

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

      5. pow1/2 66.9%

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

   7. Applied egg-rr 66.9%

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

   8. Taylor expanded in C around 0 60.4%

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

4. Final simplification 32.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.9 \cdot 10^{-49}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right)\right) \cdot \left(2 \cdot C\right)}}{{B}^{2} - C \cdot \left(A \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{B + C} \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 40.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;B_m \leq 4.8 \cdot 10^{-49}:\\ \;\;\;\;\frac{-\sqrt{4 \cdot \left(C \cdot \left(F \cdot \left({B_m}^{2} - 4 \cdot \left(C \cdot A\right)\right)\right)\right)}}{{B_m}^{2} - C \cdot \left(A \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B_m} \cdot \left(\sqrt{B_m + C} \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 4.8e-49)
   (/
    (- (sqrt (* 4.0 (* C (* F (- (pow B_m 2.0) (* 4.0 (* C A))))))))
    (- (pow B_m 2.0) (* C (* A 4.0))))
   (* (/ (sqrt 2.0) B_m) (* (sqrt (+ B_m C)) (- (sqrt F))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 4.8e-49) {
		tmp = -sqrt((4.0 * (C * (F * (pow(B_m, 2.0) - (4.0 * (C * A))))))) / (pow(B_m, 2.0) - (C * (A * 4.0)));
	} else {
		tmp = (sqrt(2.0) / B_m) * (sqrt((B_m + C)) * -sqrt(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 (b_m <= 4.8d-49) then
        tmp = -sqrt((4.0d0 * (c * (f * ((b_m ** 2.0d0) - (4.0d0 * (c * a))))))) / ((b_m ** 2.0d0) - (c * (a * 4.0d0)))
    else
        tmp = (sqrt(2.0d0) / b_m) * (sqrt((b_m + c)) * -sqrt(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 (B_m <= 4.8e-49) {
		tmp = -Math.sqrt((4.0 * (C * (F * (Math.pow(B_m, 2.0) - (4.0 * (C * A))))))) / (Math.pow(B_m, 2.0) - (C * (A * 4.0)));
	} else {
		tmp = (Math.sqrt(2.0) / B_m) * (Math.sqrt((B_m + C)) * -Math.sqrt(F));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 4.8e-49:
		tmp = -math.sqrt((4.0 * (C * (F * (math.pow(B_m, 2.0) - (4.0 * (C * A))))))) / (math.pow(B_m, 2.0) - (C * (A * 4.0)))
	else:
		tmp = (math.sqrt(2.0) / B_m) * (math.sqrt((B_m + C)) * -math.sqrt(F))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 4.8e-49)
		tmp = Float64(Float64(-sqrt(Float64(4.0 * Float64(C * Float64(F * Float64((B_m ^ 2.0) - Float64(4.0 * Float64(C * A)))))))) / Float64((B_m ^ 2.0) - Float64(C * Float64(A * 4.0))));
	else
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(sqrt(Float64(B_m + C)) * Float64(-sqrt(F))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (B_m <= 4.8e-49)
		tmp = -sqrt((4.0 * (C * (F * ((B_m ^ 2.0) - (4.0 * (C * A))))))) / ((B_m ^ 2.0) - (C * (A * 4.0)));
	else
		tmp = (sqrt(2.0) / B_m) * (sqrt((B_m + C)) * -sqrt(F));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 4.8e-49], N[((-N[Sqrt[N[(4.0 * N[(C * N[(F * N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(4.0 * N[(C * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(C * N[(A * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[(N[Sqrt[N[(B$95$m + C), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 4.79999999999999985e-49

   1. Initial program 12.1%

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

   3. Taylor expanded in A around -inf 18.8%

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

   4. Taylor expanded in F around 0 18.7%

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

   if 4.79999999999999985e-49 < B

   1. Initial program 24.1%

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

   3. Taylor expanded in A around 0 28.2%

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

      1. mul-1-neg 28.2%

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

      2. *-commutative 28.2%

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

      3. distribute-rgt-neg-in 28.2%

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

      4. unpow2 28.2%

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

      5. unpow2 28.2%

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

      6. hypot-def 45.0%

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

   5. Simplified 45.0%

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

      1. pow1/2 45.0%

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

      2. *-commutative 45.0%

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

      3. unpow-prod-down 66.9%

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

      4. pow1/2 66.9%

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

      5. pow1/2 66.9%

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

   7. Applied egg-rr 66.9%

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

   8. Taylor expanded in C around 0 60.4%

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

4. Final simplification 32.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 4.8 \cdot 10^{-49}:\\ \;\;\;\;\frac{-\sqrt{4 \cdot \left(C \cdot \left(F \cdot \left({B}^{2} - 4 \cdot \left(C \cdot A\right)\right)\right)\right)}}{{B}^{2} - C \cdot \left(A \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{B + C} \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 39.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;B_m \leq 1.05 \cdot 10^{-49}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot C\right) \cdot \left(2 \cdot \left(-4 \cdot \left(A \cdot \left(C \cdot F\right)\right)\right)\right)}}{{B_m}^{2} - C \cdot \left(A \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B_m} \cdot \left(\sqrt{B_m + C} \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 1.05e-49)
   (/
    (- (sqrt (* (* 2.0 C) (* 2.0 (* -4.0 (* A (* C F)))))))
    (- (pow B_m 2.0) (* C (* A 4.0))))
   (* (/ (sqrt 2.0) B_m) (* (sqrt (+ B_m C)) (- (sqrt F))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 1.05e-49) {
		tmp = -sqrt(((2.0 * C) * (2.0 * (-4.0 * (A * (C * F)))))) / (pow(B_m, 2.0) - (C * (A * 4.0)));
	} else {
		tmp = (sqrt(2.0) / B_m) * (sqrt((B_m + C)) * -sqrt(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 (b_m <= 1.05d-49) then
        tmp = -sqrt(((2.0d0 * c) * (2.0d0 * ((-4.0d0) * (a * (c * f)))))) / ((b_m ** 2.0d0) - (c * (a * 4.0d0)))
    else
        tmp = (sqrt(2.0d0) / b_m) * (sqrt((b_m + c)) * -sqrt(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 (B_m <= 1.05e-49) {
		tmp = -Math.sqrt(((2.0 * C) * (2.0 * (-4.0 * (A * (C * F)))))) / (Math.pow(B_m, 2.0) - (C * (A * 4.0)));
	} else {
		tmp = (Math.sqrt(2.0) / B_m) * (Math.sqrt((B_m + C)) * -Math.sqrt(F));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 1.05e-49:
		tmp = -math.sqrt(((2.0 * C) * (2.0 * (-4.0 * (A * (C * F)))))) / (math.pow(B_m, 2.0) - (C * (A * 4.0)))
	else:
		tmp = (math.sqrt(2.0) / B_m) * (math.sqrt((B_m + C)) * -math.sqrt(F))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 1.05e-49)
		tmp = Float64(Float64(-sqrt(Float64(Float64(2.0 * C) * Float64(2.0 * Float64(-4.0 * Float64(A * Float64(C * F))))))) / Float64((B_m ^ 2.0) - Float64(C * Float64(A * 4.0))));
	else
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(sqrt(Float64(B_m + C)) * Float64(-sqrt(F))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (B_m <= 1.05e-49)
		tmp = -sqrt(((2.0 * C) * (2.0 * (-4.0 * (A * (C * F)))))) / ((B_m ^ 2.0) - (C * (A * 4.0)));
	else
		tmp = (sqrt(2.0) / B_m) * (sqrt((B_m + C)) * -sqrt(F));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 1.05e-49], N[((-N[Sqrt[N[(N[(2.0 * C), $MachinePrecision] * 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[(C * N[(A * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[(N[Sqrt[N[(B$95$m + C), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 1.0499999999999999e-49

   1. Initial program 12.1%

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

   3. Taylor expanded in A around -inf 18.8%

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

   4. Taylor expanded in B around 0 16.8%

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

   if 1.0499999999999999e-49 < B

   1. Initial program 24.1%

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

   3. Taylor expanded in A around 0 28.2%

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

      1. mul-1-neg 28.2%

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

      2. *-commutative 28.2%

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

      3. distribute-rgt-neg-in 28.2%

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

      4. unpow2 28.2%

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

      5. unpow2 28.2%

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

      6. hypot-def 45.0%

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

   5. Simplified 45.0%

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

      1. pow1/2 45.0%

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

      2. *-commutative 45.0%

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

      3. unpow-prod-down 66.9%

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

      4. pow1/2 66.9%

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

      5. pow1/2 66.9%

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

   7. Applied egg-rr 66.9%

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

   8. Taylor expanded in C around 0 60.4%

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

4. Final simplification 31.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.05 \cdot 10^{-49}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot C\right) \cdot \left(2 \cdot \left(-4 \cdot \left(A \cdot \left(C \cdot F\right)\right)\right)\right)}}{{B}^{2} - C \cdot \left(A \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{B + C} \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 39.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;B_m \leq 7.8 \cdot 10^{-50}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot C\right) \cdot \left(2 \cdot \left(-4 \cdot \left(A \cdot \left(C \cdot F\right)\right)\right)\right)}}{{B_m}^{2} - C \cdot \left(A \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B_m} \cdot \left(\sqrt{F} \cdot \sqrt{B_m}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 7.8e-50)
   (/
    (- (sqrt (* (* 2.0 C) (* 2.0 (* -4.0 (* A (* C F)))))))
    (- (pow B_m 2.0) (* C (* A 4.0))))
   (* (/ (- (sqrt 2.0)) B_m) (* (sqrt F) (sqrt B_m)))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 7.8e-50) {
		tmp = -sqrt(((2.0 * C) * (2.0 * (-4.0 * (A * (C * F)))))) / (pow(B_m, 2.0) - (C * (A * 4.0)));
	} else {
		tmp = (-sqrt(2.0) / B_m) * (sqrt(F) * sqrt(B_m));
	}
	return tmp;
}

Fortran

B_m = abs(B)
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (b_m <= 7.8d-50) then
        tmp = -sqrt(((2.0d0 * c) * (2.0d0 * ((-4.0d0) * (a * (c * f)))))) / ((b_m ** 2.0d0) - (c * (a * 4.0d0)))
    else
        tmp = (-sqrt(2.0d0) / b_m) * (sqrt(f) * sqrt(b_m))
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 7.8e-50) {
		tmp = -Math.sqrt(((2.0 * C) * (2.0 * (-4.0 * (A * (C * F)))))) / (Math.pow(B_m, 2.0) - (C * (A * 4.0)));
	} else {
		tmp = (-Math.sqrt(2.0) / B_m) * (Math.sqrt(F) * Math.sqrt(B_m));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 7.8e-50:
		tmp = -math.sqrt(((2.0 * C) * (2.0 * (-4.0 * (A * (C * F)))))) / (math.pow(B_m, 2.0) - (C * (A * 4.0)))
	else:
		tmp = (-math.sqrt(2.0) / B_m) * (math.sqrt(F) * math.sqrt(B_m))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 7.8e-50)
		tmp = Float64(Float64(-sqrt(Float64(Float64(2.0 * C) * Float64(2.0 * Float64(-4.0 * Float64(A * Float64(C * F))))))) / Float64((B_m ^ 2.0) - Float64(C * Float64(A * 4.0))));
	else
		tmp = Float64(Float64(Float64(-sqrt(2.0)) / B_m) * Float64(sqrt(F) * sqrt(B_m)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (B_m <= 7.8e-50)
		tmp = -sqrt(((2.0 * C) * (2.0 * (-4.0 * (A * (C * F)))))) / ((B_m ^ 2.0) - (C * (A * 4.0)));
	else
		tmp = (-sqrt(2.0) / B_m) * (sqrt(F) * sqrt(B_m));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 7.8e-50], N[((-N[Sqrt[N[(N[(2.0 * C), $MachinePrecision] * 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[(C * N[(A * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-N[Sqrt[2.0], $MachinePrecision]) / B$95$m), $MachinePrecision] * N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[B$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 7.80000000000000042e-50

   1. Initial program 12.1%

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

   3. Taylor expanded in A around -inf 18.8%

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

   4. Taylor expanded in B around 0 16.8%

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

   if 7.80000000000000042e-50 < B

   1. Initial program 24.1%

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

   3. Taylor expanded in A around 0 28.2%

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

      1. mul-1-neg 28.2%

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

      2. *-commutative 28.2%

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

      3. distribute-rgt-neg-in 28.2%

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

      4. unpow2 28.2%

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

      5. unpow2 28.2%

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

      6. hypot-def 45.0%

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

   5. Simplified 45.0%

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

      1. pow1/2 45.0%

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

      2. *-commutative 45.0%

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

      3. unpow-prod-down 66.9%

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

      4. pow1/2 66.9%

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

      5. pow1/2 66.9%

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

   7. Applied egg-rr 66.9%

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

   8. Taylor expanded in C around 0 60.0%

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

4. Final simplification 31.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 7.8 \cdot 10^{-50}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot C\right) \cdot \left(2 \cdot \left(-4 \cdot \left(A \cdot \left(C \cdot F\right)\right)\right)\right)}}{{B}^{2} - C \cdot \left(A \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \left(\sqrt{F} \cdot \sqrt{B}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 38.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;F \leq -3.2 \cdot 10^{-283}:\\ \;\;\;\;\sqrt{\frac{-F}{C}}\\ \mathbf{elif}\;F \leq 2.6 \cdot 10^{-69}:\\ \;\;\;\;\frac{\sqrt{2}}{B_m} \cdot \left(-\sqrt{B_m \cdot F}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B_m}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= F -3.2e-283)
   (sqrt (/ (- F) C))
   (if (<= F 2.6e-69)
     (* (/ (sqrt 2.0) B_m) (- (sqrt (* B_m F))))
     (* (sqrt (/ F 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 <= -3.2e-283) {
		tmp = sqrt((-F / C));
	} else if (F <= 2.6e-69) {
		tmp = (sqrt(2.0) / B_m) * -sqrt((B_m * F));
	} else {
		tmp = sqrt((F / B_m)) * -sqrt(2.0);
	}
	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 <= (-3.2d-283)) then
        tmp = sqrt((-f / c))
    else if (f <= 2.6d-69) then
        tmp = (sqrt(2.0d0) / b_m) * -sqrt((b_m * f))
    else
        tmp = sqrt((f / b_m)) * -sqrt(2.0d0)
    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 <= -3.2e-283) {
		tmp = Math.sqrt((-F / C));
	} else if (F <= 2.6e-69) {
		tmp = (Math.sqrt(2.0) / B_m) * -Math.sqrt((B_m * F));
	} else {
		tmp = Math.sqrt((F / 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 <= -3.2e-283:
		tmp = math.sqrt((-F / C))
	elif F <= 2.6e-69:
		tmp = (math.sqrt(2.0) / B_m) * -math.sqrt((B_m * F))
	else:
		tmp = math.sqrt((F / 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 <= -3.2e-283)
		tmp = sqrt(Float64(Float64(-F) / C));
	elseif (F <= 2.6e-69)
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(-sqrt(Float64(B_m * F))));
	else
		tmp = Float64(sqrt(Float64(F / 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 <= -3.2e-283)
		tmp = sqrt((-F / C));
	elseif (F <= 2.6e-69)
		tmp = (sqrt(2.0) / B_m) * -sqrt((B_m * F));
	else
		tmp = sqrt((F / 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, -3.2e-283], N[Sqrt[N[((-F) / C), $MachinePrecision]], $MachinePrecision], If[LessEqual[F, 2.6e-69], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * (-N[Sqrt[N[(B$95$m * F), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[N[(F / B$95$m), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -3.20000000000000012e-283

   1. Initial program 13.1%

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

      1. add-sqr-sqrt 13.1%

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

      2. sqrt-unprod 9.8%

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

      3. frac-times 8.5%

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

   4. Applied egg-rr 19.0%

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

      1. associate-/l* 19.5%

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

      2. associate-*l* 19.5%

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

      3. *-commutative 19.5%

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

      4. unpow2 19.5%

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

      5. fma-neg 19.5%

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

      6. distribute-lft-neg-in 19.5%

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

      7. metadata-eval 19.5%

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

      8. *-commutative 19.5%

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

      9. *-commutative 19.5%

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

   6. Simplified 19.5%

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

   7. Taylor expanded in B around 0 55.7%

      \[\leadsto \sqrt{\color{blue}{-1 \cdot \frac{F}{C}}} \]
   8. Step-by-step derivation

      1. mul-1-neg 55.7%

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

   9. Simplified 55.7%

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

   if -3.20000000000000012e-283 < F < 2.6000000000000002e-69

   1. Initial program 19.8%

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

   3. Taylor expanded in A around 0 15.4%

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

      1. mul-1-neg 15.4%

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

      2. *-commutative 15.4%

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

      3. distribute-rgt-neg-in 15.4%

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

      4. unpow2 15.4%

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

      5. unpow2 15.4%

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

      6. hypot-def 25.9%

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

   5. Simplified 25.9%

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

   6. Taylor expanded in C around 0 23.1%

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

   if 2.6000000000000002e-69 < F

   1. Initial program 14.5%

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

   3. Taylor expanded in A around 0 11.1%

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

      1. mul-1-neg 11.1%

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

      2. *-commutative 11.1%

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

      3. distribute-rgt-neg-in 11.1%

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

      4. unpow2 11.1%

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

      5. unpow2 11.1%

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

      6. hypot-def 15.8%

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

   5. Simplified 15.8%

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

   6. Taylor expanded in C around 0 23.6%

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

      1. mul-1-neg 23.6%

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

   8. Simplified 23.6%

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

4. Final simplification 26.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3.2 \cdot 10^{-283}:\\ \;\;\;\;\sqrt{\frac{-F}{C}}\\ \mathbf{elif}\;F \leq 2.6 \cdot 10^{-69}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{B \cdot F}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 31.8% accurate, 3.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;F \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{-F}{C}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B_m}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= F -2e-310) (sqrt (/ (- F) C)) (* (sqrt (/ F 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 <= -2e-310) {
		tmp = sqrt((-F / C));
	} else {
		tmp = sqrt((F / B_m)) * -sqrt(2.0);
	}
	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 <= (-2d-310)) then
        tmp = sqrt((-f / c))
    else
        tmp = sqrt((f / b_m)) * -sqrt(2.0d0)
    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 <= -2e-310) {
		tmp = Math.sqrt((-F / C));
	} else {
		tmp = Math.sqrt((F / 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 <= -2e-310:
		tmp = math.sqrt((-F / C))
	else:
		tmp = math.sqrt((F / 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 <= -2e-310)
		tmp = sqrt(Float64(Float64(-F) / C));
	else
		tmp = Float64(sqrt(Float64(F / 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 <= -2e-310)
		tmp = sqrt((-F / C));
	else
		tmp = sqrt((F / 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, -2e-310], N[Sqrt[N[((-F) / C), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[N[(F / B$95$m), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -1.999999999999994e-310

   1. Initial program 15.6%

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

      1. add-sqr-sqrt 15.6%

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

      2. sqrt-unprod 12.2%

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

      3. frac-times 11.0%

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

   4. Applied egg-rr 20.8%

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

      1. associate-/l* 21.3%

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

      2. associate-*l* 21.3%

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

      3. *-commutative 21.3%

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

      4. unpow2 21.3%

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

      5. fma-neg 21.3%

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

      6. distribute-lft-neg-in 21.3%

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

      7. metadata-eval 21.3%

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

      8. *-commutative 21.3%

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

      9. *-commutative 21.3%

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

   6. Simplified 21.3%

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

   7. Taylor expanded in B around 0 52.3%

      \[\leadsto \sqrt{\color{blue}{-1 \cdot \frac{F}{C}}} \]
   8. Step-by-step derivation

      1. mul-1-neg 52.3%

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

   9. Simplified 52.3%

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

   if -1.999999999999994e-310 < F

   1. Initial program 16.2%

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

   3. Taylor expanded in A around 0 12.8%

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

      1. mul-1-neg 12.8%

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

      2. *-commutative 12.8%

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

      3. distribute-rgt-neg-in 12.8%

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

      4. unpow2 12.8%

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

      5. unpow2 12.8%

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

      6. hypot-def 19.8%

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

   5. Simplified 19.8%

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

   6. Taylor expanded in C around 0 21.0%

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

      1. mul-1-neg 21.0%

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

   8. Simplified 21.0%

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

4. Final simplification 24.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\sqrt{\frac{-F}{C}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 14.4% accurate, 5.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;C \leq 6 \cdot 10^{+74}:\\ \;\;\;\;\sqrt{\frac{-F}{C}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-F}{A}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= C 6e+74) (sqrt (/ (- F) C)) (sqrt (/ (- F) A))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (C <= 6e+74) {
		tmp = sqrt((-F / C));
	} else {
		tmp = sqrt((-F / A));
	}
	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 <= 6d+74) then
        tmp = sqrt((-f / c))
    else
        tmp = sqrt((-f / a))
    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 <= 6e+74) {
		tmp = Math.sqrt((-F / C));
	} else {
		tmp = Math.sqrt((-F / A));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if C <= 6e+74:
		tmp = math.sqrt((-F / C))
	else:
		tmp = math.sqrt((-F / A))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (C <= 6e+74)
		tmp = sqrt(Float64(Float64(-F) / C));
	else
		tmp = sqrt(Float64(Float64(-F) / A));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (C <= 6e+74)
		tmp = sqrt((-F / C));
	else
		tmp = sqrt((-F / A));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[C, 6e+74], N[Sqrt[N[((-F) / C), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[((-F) / A), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if C < 6e74

   1. Initial program 17.1%

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

      1. add-sqr-sqrt 1.2%

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

      2. sqrt-unprod 1.3%

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

      3. frac-times 0.9%

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

   4. Applied egg-rr 2.2%

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

      1. associate-/l* 2.4%

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

      2. associate-*l* 2.4%

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

      3. *-commutative 2.4%

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

      4. unpow2 2.4%

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

      5. fma-neg 2.4%

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

      6. distribute-lft-neg-in 2.4%

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

      7. metadata-eval 2.4%

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

      8. *-commutative 2.4%

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

      9. *-commutative 2.4%

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

   6. Simplified 2.2%

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

   7. Taylor expanded in B around 0 13.8%

      \[\leadsto \sqrt{\color{blue}{-1 \cdot \frac{F}{C}}} \]
   8. Step-by-step derivation

      1. mul-1-neg 13.8%

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

   9. Simplified 13.8%

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

   if 6e74 < C

   1. Initial program 12.5%

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

      1. add-sqr-sqrt 6.0%

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

      2. sqrt-unprod 6.0%

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

      3. frac-times 5.7%

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

   4. Applied egg-rr 9.5%

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

      1. associate-/l* 9.6%

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

      2. associate-*l* 9.6%

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

      3. *-commutative 9.6%

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

      4. unpow2 9.6%

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

      5. fma-neg 9.6%

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

      6. distribute-lft-neg-in 9.6%

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

      7. metadata-eval 9.6%

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

      8. *-commutative 9.6%

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

      9. *-commutative 9.6%

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

   6. Simplified 9.6%

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

   7. Taylor expanded in C around inf 19.8%

      \[\leadsto \sqrt{\color{blue}{-1 \cdot \frac{F}{A}}} \]
   8. Step-by-step derivation

      1. mul-1-neg 19.8%

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

   9. Simplified 19.8%

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

4. Final simplification 15.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq 6 \cdot 10^{+74}:\\ \;\;\;\;\sqrt{\frac{-F}{C}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{-F}{A}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 11.3% accurate, 6.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 16.1%

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

   1. add-sqr-sqrt 2.2%

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

   2. sqrt-unprod 2.3%

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

   3. frac-times 1.9%

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

4. Applied egg-rr 3.7%

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

   1. associate-/l* 3.9%

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

   2. associate-*l* 3.9%

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

   3. *-commutative 3.9%

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

   4. unpow2 3.9%

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

   5. fma-neg 3.9%

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

   6. distribute-lft-neg-in 3.9%

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

   7. metadata-eval 3.9%

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

   8. *-commutative 3.9%

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

   9. *-commutative 3.9%

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

6. Simplified 3.7%

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

7. Taylor expanded in C around inf 8.7%

   \[\leadsto \sqrt{\color{blue}{-1 \cdot \frac{F}{A}}} \]
8. Step-by-step derivation

   1. mul-1-neg 8.7%

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

9. Simplified 8.7%

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

10. Final simplification 8.7%

    \[\leadsto \sqrt{\frac{-F}{A}} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024017 
(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))))