ABCF->ab-angle a

Percentage Accurate: 18.6% → 52.2%

Time: 37.2s

Alternatives: 20

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: 18.6% 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: 52.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 5 \cdot 10^{+69}:\\ \;\;\;\;\frac{\sqrt{A + \left(C + \mathsf{hypot}\left(A - C, B\_m\right)\right)} \cdot \left(-\sqrt{t\_0 \cdot \left(2 \cdot F\right)}\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B\_m} \cdot \left(\sqrt{C + \mathsf{hypot}\left(C, B\_m\right)} \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* A (* C -4.0)))))
   (if (<= (pow B_m 2.0) 5e+69)
     (/
      (* (sqrt (+ A (+ C (hypot (- A C) B_m)))) (- (sqrt (* t_0 (* 2.0 F)))))
      t_0)
     (* (/ (sqrt 2.0) B_m) (* (sqrt (+ C (hypot C B_m))) (- (sqrt F)))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (A * (C * -4.0)));
	double tmp;
	if (pow(B_m, 2.0) <= 5e+69) {
		tmp = (sqrt((A + (C + hypot((A - C), B_m)))) * -sqrt((t_0 * (2.0 * F)))) / t_0;
	} else {
		tmp = (sqrt(2.0) / B_m) * (sqrt((C + hypot(C, B_m))) * -sqrt(F));
	}
	return tmp;
}

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 5e+69)
		tmp = Float64(Float64(sqrt(Float64(A + Float64(C + hypot(Float64(A - C), B_m)))) * Float64(-sqrt(Float64(t_0 * Float64(2.0 * F))))) / t_0);
	else
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(sqrt(Float64(C + hypot(C, B_m))) * Float64(-sqrt(F))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e+69], N[(N[(N[Sqrt[N[(A + N[(C + N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(t$95$0 * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[(N[Sqrt[N[(C + N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B 2) < 5.00000000000000036e69

   1. Initial program 23.0%

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

   3. Simplified 32.9%

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

      1. add-exp-log 31.5%

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

      2. hypot-udef 23.3%

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

      3. unpow2 23.3%

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

      4. unpow2 23.3%

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

      5. +-commutative 23.3%

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

      6. unpow2 23.3%

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

      7. unpow2 23.3%

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

      8. hypot-def 31.5%

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

   6. Applied egg-rr 31.5%

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

      1. pow1/2 31.5%

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

      2. *-commutative 31.5%

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

      3. unpow-prod-down 37.7%

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

      4. pow1/2 37.7%

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

      5. rem-exp-log 39.7%

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

      6. pow1/2 39.7%

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

   8. Applied egg-rr 39.7%

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

   if 5.00000000000000036e69 < (pow.f64 B 2)

   1. Initial program 11.7%

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

   3. Taylor expanded in A around 0 9.2%

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

      1. mul-1-neg 9.2%

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

      2. distribute-rgt-neg-in 9.2%

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

      3. unpow2 9.2%

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

      4. unpow2 9.2%

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

      5. hypot-def 25.1%

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

   5. Simplified 25.1%

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

      1. pow1/2 25.1%

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

      2. *-commutative 25.1%

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

      3. unpow-prod-down 36.6%

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

      4. pow1/2 36.6%

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

      5. hypot-udef 10.9%

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

      6. unpow2 10.9%

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

      7. unpow2 10.9%

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

      8. +-commutative 10.9%

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

      9. unpow2 10.9%

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

      10. unpow2 10.9%

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

      11. hypot-def 36.6%

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

      12. pow1/2 36.6%

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

   7. Applied egg-rr 36.6%

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

4. Final simplification 38.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 5 \cdot 10^{+69}:\\ \;\;\;\;\frac{\sqrt{A + \left(C + \mathsf{hypot}\left(A - C, B\right)\right)} \cdot \left(-\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(2 \cdot F\right)}\right)}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 48.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 5 \cdot 10^{+69}:\\ \;\;\;\;\frac{-\sqrt{\left(t\_0 \cdot \left(2 \cdot F\right)\right) \cdot \left(A + \left(C + \mathsf{hypot}\left(B\_m, A - C\right)\right)\right)}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B\_m} \cdot \left(\sqrt{C + \mathsf{hypot}\left(C, B\_m\right)} \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* A (* C -4.0)))))
   (if (<= (pow B_m 2.0) 5e+69)
     (/ (- (sqrt (* (* t_0 (* 2.0 F)) (+ A (+ C (hypot B_m (- A C))))))) t_0)
     (* (/ (sqrt 2.0) B_m) (* (sqrt (+ C (hypot C B_m))) (- (sqrt F)))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (A * (C * -4.0)));
	double tmp;
	if (pow(B_m, 2.0) <= 5e+69) {
		tmp = -sqrt(((t_0 * (2.0 * F)) * (A + (C + hypot(B_m, (A - C)))))) / t_0;
	} else {
		tmp = (sqrt(2.0) / B_m) * (sqrt((C + hypot(C, B_m))) * -sqrt(F));
	}
	return tmp;
}

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 5e+69)
		tmp = Float64(Float64(-sqrt(Float64(Float64(t_0 * Float64(2.0 * F)) * Float64(A + Float64(C + hypot(B_m, Float64(A - C))))))) / t_0);
	else
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(sqrt(Float64(C + hypot(C, B_m))) * Float64(-sqrt(F))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e+69], N[((-N[Sqrt[N[(N[(t$95$0 * N[(2.0 * F), $MachinePrecision]), $MachinePrecision] * N[(A + N[(C + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[(N[Sqrt[N[(C + N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B 2) < 5.00000000000000036e69

   1. Initial program 23.0%

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

   3. Simplified 32.9%

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

   if 5.00000000000000036e69 < (pow.f64 B 2)

   1. Initial program 11.7%

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

   3. Taylor expanded in A around 0 9.2%

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

      1. mul-1-neg 9.2%

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

      2. distribute-rgt-neg-in 9.2%

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

      3. unpow2 9.2%

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

      4. unpow2 9.2%

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

      5. hypot-def 25.1%

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

   5. Simplified 25.1%

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

      1. pow1/2 25.1%

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

      2. *-commutative 25.1%

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

      3. unpow-prod-down 36.6%

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

      4. pow1/2 36.6%

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

      5. hypot-udef 10.9%

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

      6. unpow2 10.9%

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

      7. unpow2 10.9%

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

      8. +-commutative 10.9%

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

      9. unpow2 10.9%

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

      10. unpow2 10.9%

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

      11. hypot-def 36.6%

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

      12. pow1/2 36.6%

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

   7. Applied egg-rr 36.6%

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

4. Final simplification 34.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 5 \cdot 10^{+69}:\\ \;\;\;\;\frac{-\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 40.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := {B\_m}^{2} - C \cdot \left(A \cdot 4\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 5 \cdot 10^{-82}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot t\_0\right)\right) \cdot \left(2 \cdot C\right)}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{C + \mathsf{hypot}\left(C, B\_m\right)} \cdot \left(\frac{\sqrt{2}}{B\_m} \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \end{array} \]

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 (<= (pow B_m 2.0) 5e-82)
     (/ (- (sqrt (* (* 2.0 (* F t_0)) (* 2.0 C)))) t_0)
     (* (sqrt (+ C (hypot C B_m))) (* (/ (sqrt 2.0) B_m) (- (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 (pow(B_m, 2.0) <= 5e-82) {
		tmp = -sqrt(((2.0 * (F * t_0)) * (2.0 * C))) / t_0;
	} else {
		tmp = sqrt((C + hypot(C, B_m))) * ((sqrt(2.0) / B_m) * -sqrt(F));
	}
	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) - (C * (A * 4.0));
	double tmp;
	if (Math.pow(B_m, 2.0) <= 5e-82) {
		tmp = -Math.sqrt(((2.0 * (F * t_0)) * (2.0 * C))) / t_0;
	} else {
		tmp = Math.sqrt((C + Math.hypot(C, B_m))) * ((Math.sqrt(2.0) / B_m) * -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 math.pow(B_m, 2.0) <= 5e-82:
		tmp = -math.sqrt(((2.0 * (F * t_0)) * (2.0 * C))) / t_0
	else:
		tmp = math.sqrt((C + math.hypot(C, B_m))) * ((math.sqrt(2.0) / B_m) * -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 ^ 2.0) <= 5e-82)
		tmp = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(F * t_0)) * Float64(2.0 * C)))) / t_0);
	else
		tmp = Float64(sqrt(Float64(C + hypot(C, B_m))) * Float64(Float64(sqrt(2.0) / B_m) * 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 ^ 2.0) <= 5e-82)
		tmp = -sqrt(((2.0 * (F * t_0)) * (2.0 * C))) / t_0;
	else
		tmp = sqrt((C + hypot(C, B_m))) * ((sqrt(2.0) / B_m) * -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[N[Power[B$95$m, 2.0], $MachinePrecision], 5e-82], 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[Sqrt[N[(C + N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B 2) < 4.9999999999999998e-82

   1. Initial program 17.4%

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

   3. Taylor expanded in A around -inf 28.9%

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

   if 4.9999999999999998e-82 < (pow.f64 B 2)

   1. Initial program 18.7%

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

   3. Taylor expanded in A around 0 9.1%

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

      1. mul-1-neg 9.1%

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

   5. Simplified 9.1%

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

      1. add-cbrt-cube 7.1%

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

      2. pow3 7.1%

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

   7. Applied egg-rr 12.4%

      \[\leadsto -\color{blue}{\sqrt[3]{{\left(\frac{{\left(2 \cdot \left(F \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)\right)\right)}^{0.5}}{B}\right)}^{3}}} \]
   8. Step-by-step derivation

      1. rem-cbrt-cube 22.3%

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

      2. div-inv 22.3%

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

      3. unpow1/2 22.3%

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

      4. *-commutative 22.3%

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

      5. sqrt-prod 22.2%

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

      6. sqrt-prod 31.1%

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

      7. *-commutative 31.1%

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

      8. add-sqr-sqrt 31.1%

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

      9. sqrt-prod 22.2%

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

      10. sqr-neg 22.2%

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

      11. sqrt-unprod 0.6%

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

      12. add-sqr-sqrt 34.4%

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

      13. associate-*r* 34.3%

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

   9. Applied egg-rr 28.4%

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

4. Final simplification 28.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 5 \cdot 10^{-82}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right)\right) \cdot \left(2 \cdot C\right)}}{{B}^{2} - C \cdot \left(A \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \left(\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 45.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := {B\_m}^{2} - C \cdot \left(A \cdot 4\right)\\ t_1 := \mathsf{fma}\left(B\_m, B\_m, -4 \cdot \left(A \cdot C\right)\right)\\ t_2 := 2 \cdot \left(F \cdot t\_0\right)\\ \mathbf{if}\;B\_m \leq 3.85 \cdot 10^{-41}:\\ \;\;\;\;\frac{-\sqrt{t\_2 \cdot \left(2 \cdot C\right)}}{t\_0}\\ \mathbf{elif}\;B\_m \leq 5.2 \cdot 10^{-7}:\\ \;\;\;\;\frac{-\sqrt{t\_2 \cdot \left(A + \mathsf{hypot}\left(B\_m, A\right)\right)}}{t\_0}\\ \mathbf{elif}\;B\_m \leq 1:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot t\_1\right) \cdot \left(F \cdot \left(\mathsf{hypot}\left(A - C, B\_m\right) + \left(A + C\right)\right)\right)}}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B\_m} \cdot \left(\sqrt{C + \mathsf{hypot}\left(C, B\_m\right)} \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (- (pow B_m 2.0) (* C (* A 4.0))))
        (t_1 (fma B_m B_m (* -4.0 (* A C))))
        (t_2 (* 2.0 (* F t_0))))
   (if (<= B_m 3.85e-41)
     (/ (- (sqrt (* t_2 (* 2.0 C)))) t_0)
     (if (<= B_m 5.2e-7)
       (/ (- (sqrt (* t_2 (+ A (hypot B_m A))))) t_0)
       (if (<= B_m 1.0)
         (/
          (- (sqrt (* (* 2.0 t_1) (* F (+ (hypot (- A C) B_m) (+ A C))))))
          t_1)
         (*
          (/ (sqrt 2.0) B_m)
          (* (sqrt (+ C (hypot C B_m))) (- (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 t_1 = fma(B_m, B_m, (-4.0 * (A * C)));
	double t_2 = 2.0 * (F * t_0);
	double tmp;
	if (B_m <= 3.85e-41) {
		tmp = -sqrt((t_2 * (2.0 * C))) / t_0;
	} else if (B_m <= 5.2e-7) {
		tmp = -sqrt((t_2 * (A + hypot(B_m, A)))) / t_0;
	} else if (B_m <= 1.0) {
		tmp = -sqrt(((2.0 * t_1) * (F * (hypot((A - C), B_m) + (A + C))))) / t_1;
	} else {
		tmp = (sqrt(2.0) / B_m) * (sqrt((C + hypot(C, B_m))) * -sqrt(F));
	}
	return tmp;
}

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	t_0 = Float64((B_m ^ 2.0) - Float64(C * Float64(A * 4.0)))
	t_1 = fma(B_m, B_m, Float64(-4.0 * Float64(A * C)))
	t_2 = Float64(2.0 * Float64(F * t_0))
	tmp = 0.0
	if (B_m <= 3.85e-41)
		tmp = Float64(Float64(-sqrt(Float64(t_2 * Float64(2.0 * C)))) / t_0);
	elseif (B_m <= 5.2e-7)
		tmp = Float64(Float64(-sqrt(Float64(t_2 * Float64(A + hypot(B_m, A))))) / t_0);
	elseif (B_m <= 1.0)
		tmp = Float64(Float64(-sqrt(Float64(Float64(2.0 * t_1) * Float64(F * Float64(hypot(Float64(A - C), B_m) + Float64(A + C)))))) / t_1);
	else
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(sqrt(Float64(C + hypot(C, B_m))) * Float64(-sqrt(F))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
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]}, Block[{t$95$1 = N[(B$95$m * B$95$m + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 3.85e-41], N[((-N[Sqrt[N[(t$95$2 * N[(2.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], If[LessEqual[B$95$m, 5.2e-7], N[((-N[Sqrt[N[(t$95$2 * N[(A + N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], If[LessEqual[B$95$m, 1.0], N[((-N[Sqrt[N[(N[(2.0 * t$95$1), $MachinePrecision] * N[(F * N[(N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision] + N[(A + C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[(N[Sqrt[N[(C + N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if B < 3.8499999999999999e-41

   1. Initial program 19.1%

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

   3. Taylor expanded in A around -inf 19.2%

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

   if 3.8499999999999999e-41 < B < 5.19999999999999998e-7

   1. Initial program 35.9%

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

   3. Taylor expanded in C around 0 37.2%

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

      1. +-commutative 37.2%

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

      2. unpow2 37.2%

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

      3. unpow2 37.2%

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

      4. hypot-def 68.7%

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

   5. Simplified 68.7%

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

   if 5.19999999999999998e-7 < B < 1

   1. Initial program 100.0%

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

      1. neg-sub0 100.0%

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

      2. div-sub 100.0%

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

      3. associate-*l* 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. div0 100.0%

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

      2. neg-sub0 100.0%

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

      3. distribute-neg-frac 100.0%

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

   6. Simplified 100.0%

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

   if 1 < B

   1. Initial program 12.6%

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

   3. Taylor expanded in A around 0 17.3%

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

      1. mul-1-neg 17.3%

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

      2. distribute-rgt-neg-in 17.3%

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

      3. unpow2 17.3%

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

      4. unpow2 17.3%

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

      5. hypot-def 43.5%

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

   5. Simplified 43.5%

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

      1. pow1/2 43.5%

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

      2. *-commutative 43.5%

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

      3. unpow-prod-down 61.9%

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

      4. pow1/2 61.9%

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

      5. hypot-udef 20.1%

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

      6. unpow2 20.1%

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

      7. unpow2 20.1%

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

      8. +-commutative 20.1%

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

      9. unpow2 20.1%

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

      10. unpow2 20.1%

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

      11. hypot-def 61.9%

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

      12. pow1/2 61.9%

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

   7. Applied egg-rr 61.9%

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

4. Final simplification 32.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 3.85 \cdot 10^{-41}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right)\right) \cdot \left(2 \cdot C\right)}}{{B}^{2} - C \cdot \left(A \cdot 4\right)}\\ \mathbf{elif}\;B \leq 5.2 \cdot 10^{-7}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right)\right) \cdot \left(A + \mathsf{hypot}\left(B, A\right)\right)}}{{B}^{2} - C \cdot \left(A \cdot 4\right)}\\ \mathbf{elif}\;B \leq 1:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)\right) \cdot \left(F \cdot \left(\mathsf{hypot}\left(A - C, B\right) + \left(A + C\right)\right)\right)}}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 45.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := {B\_m}^{2} - C \cdot \left(A \cdot 4\right)\\ t_1 := 2 \cdot \left(F \cdot t\_0\right)\\ \mathbf{if}\;B\_m \leq 2.6 \cdot 10^{-41}:\\ \;\;\;\;\frac{-\sqrt{t\_1 \cdot \left(2 \cdot C\right)}}{t\_0}\\ \mathbf{elif}\;B\_m \leq 0.96:\\ \;\;\;\;\frac{-\sqrt{t\_1 \cdot \left(A + \mathsf{hypot}\left(B\_m, A\right)\right)}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B\_m} \cdot \left(\sqrt{C + \mathsf{hypot}\left(C, B\_m\right)} \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (- (pow B_m 2.0) (* C (* A 4.0)))) (t_1 (* 2.0 (* F t_0))))
   (if (<= B_m 2.6e-41)
     (/ (- (sqrt (* t_1 (* 2.0 C)))) t_0)
     (if (<= B_m 0.96)
       (/ (- (sqrt (* t_1 (+ A (hypot B_m A))))) t_0)
       (* (/ (sqrt 2.0) B_m) (* (sqrt (+ C (hypot C B_m))) (- (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 t_1 = 2.0 * (F * t_0);
	double tmp;
	if (B_m <= 2.6e-41) {
		tmp = -sqrt((t_1 * (2.0 * C))) / t_0;
	} else if (B_m <= 0.96) {
		tmp = -sqrt((t_1 * (A + hypot(B_m, A)))) / t_0;
	} else {
		tmp = (sqrt(2.0) / B_m) * (sqrt((C + hypot(C, B_m))) * -sqrt(F));
	}
	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) - (C * (A * 4.0));
	double t_1 = 2.0 * (F * t_0);
	double tmp;
	if (B_m <= 2.6e-41) {
		tmp = -Math.sqrt((t_1 * (2.0 * C))) / t_0;
	} else if (B_m <= 0.96) {
		tmp = -Math.sqrt((t_1 * (A + Math.hypot(B_m, A)))) / t_0;
	} else {
		tmp = (Math.sqrt(2.0) / B_m) * (Math.sqrt((C + Math.hypot(C, B_m))) * -Math.sqrt(F));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	t_0 = math.pow(B_m, 2.0) - (C * (A * 4.0))
	t_1 = 2.0 * (F * t_0)
	tmp = 0
	if B_m <= 2.6e-41:
		tmp = -math.sqrt((t_1 * (2.0 * C))) / t_0
	elif B_m <= 0.96:
		tmp = -math.sqrt((t_1 * (A + math.hypot(B_m, A)))) / t_0
	else:
		tmp = (math.sqrt(2.0) / B_m) * (math.sqrt((C + math.hypot(C, B_m))) * -math.sqrt(F))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	t_0 = Float64((B_m ^ 2.0) - Float64(C * Float64(A * 4.0)))
	t_1 = Float64(2.0 * Float64(F * t_0))
	tmp = 0.0
	if (B_m <= 2.6e-41)
		tmp = Float64(Float64(-sqrt(Float64(t_1 * Float64(2.0 * C)))) / t_0);
	elseif (B_m <= 0.96)
		tmp = Float64(Float64(-sqrt(Float64(t_1 * Float64(A + hypot(B_m, A))))) / t_0);
	else
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(sqrt(Float64(C + hypot(C, B_m))) * Float64(-sqrt(F))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	t_0 = (B_m ^ 2.0) - (C * (A * 4.0));
	t_1 = 2.0 * (F * t_0);
	tmp = 0.0;
	if (B_m <= 2.6e-41)
		tmp = -sqrt((t_1 * (2.0 * C))) / t_0;
	elseif (B_m <= 0.96)
		tmp = -sqrt((t_1 * (A + hypot(B_m, A)))) / t_0;
	else
		tmp = (sqrt(2.0) / B_m) * (sqrt((C + hypot(C, B_m))) * -sqrt(F));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
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]}, Block[{t$95$1 = N[(2.0 * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 2.6e-41], N[((-N[Sqrt[N[(t$95$1 * N[(2.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], If[LessEqual[B$95$m, 0.96], N[((-N[Sqrt[N[(t$95$1 * N[(A + N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[(N[Sqrt[N[(C + N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if B < 2.5999999999999999e-41

   1. Initial program 19.1%

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

   3. Taylor expanded in A around -inf 19.2%

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

   if 2.5999999999999999e-41 < B < 0.95999999999999996

   1. Initial program 45.0%

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

   3. Taylor expanded in C around 0 33.2%

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

      1. +-commutative 33.2%

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

      2. unpow2 33.2%

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

      3. unpow2 33.2%

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

      4. hypot-def 60.2%

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

   5. Simplified 60.2%

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

   if 0.95999999999999996 < B

   1. Initial program 12.6%

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

   3. Taylor expanded in A around 0 17.3%

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

      1. mul-1-neg 17.3%

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

      2. distribute-rgt-neg-in 17.3%

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

      3. unpow2 17.3%

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

      4. unpow2 17.3%

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

      5. hypot-def 43.5%

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

   5. Simplified 43.5%

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

      1. pow1/2 43.5%

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

      2. *-commutative 43.5%

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

      3. unpow-prod-down 61.9%

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

      4. pow1/2 61.9%

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

      5. hypot-udef 20.1%

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

      6. unpow2 20.1%

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

      7. unpow2 20.1%

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

      8. +-commutative 20.1%

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

      9. unpow2 20.1%

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

      10. unpow2 20.1%

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

      11. hypot-def 61.9%

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

      12. pow1/2 61.9%

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

   7. Applied egg-rr 61.9%

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

4. Final simplification 31.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 2.6 \cdot 10^{-41}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right)\right) \cdot \left(2 \cdot C\right)}}{{B}^{2} - C \cdot \left(A \cdot 4\right)}\\ \mathbf{elif}\;B \leq 0.96:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right)\right) \cdot \left(A + \mathsf{hypot}\left(B, A\right)\right)}}{{B}^{2} - C \cdot \left(A \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 44.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := {B\_m}^{2} - C \cdot \left(A \cdot 4\right)\\ \mathbf{if}\;B\_m \leq 3.8 \cdot 10^{-39}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot t\_0\right)\right) \cdot \left(2 \cdot C\right)}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F} \cdot \left(\sqrt{C + \mathsf{hypot}\left(C, B\_m\right)} \cdot \left(-\frac{\sqrt{2}}{B\_m}\right)\right)\\ \end{array} \end{array} \]

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 3.8e-39)
     (/ (- (sqrt (* (* 2.0 (* F t_0)) (* 2.0 C)))) t_0)
     (* (sqrt F) (* (sqrt (+ C (hypot C B_m))) (- (/ (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) - (C * (A * 4.0));
	double tmp;
	if (B_m <= 3.8e-39) {
		tmp = -sqrt(((2.0 * (F * t_0)) * (2.0 * C))) / t_0;
	} else {
		tmp = sqrt(F) * (sqrt((C + hypot(C, B_m))) * -(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) - (C * (A * 4.0));
	double tmp;
	if (B_m <= 3.8e-39) {
		tmp = -Math.sqrt(((2.0 * (F * t_0)) * (2.0 * C))) / t_0;
	} else {
		tmp = Math.sqrt(F) * (Math.sqrt((C + Math.hypot(C, B_m))) * -(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) - (C * (A * 4.0))
	tmp = 0
	if B_m <= 3.8e-39:
		tmp = -math.sqrt(((2.0 * (F * t_0)) * (2.0 * C))) / t_0
	else:
		tmp = math.sqrt(F) * (math.sqrt((C + math.hypot(C, B_m))) * -(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(C * Float64(A * 4.0)))
	tmp = 0.0
	if (B_m <= 3.8e-39)
		tmp = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(F * t_0)) * Float64(2.0 * C)))) / t_0);
	else
		tmp = Float64(sqrt(F) * Float64(sqrt(Float64(C + hypot(C, B_m))) * 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) - (C * (A * 4.0));
	tmp = 0.0;
	if (B_m <= 3.8e-39)
		tmp = -sqrt(((2.0 * (F * t_0)) * (2.0 * C))) / t_0;
	else
		tmp = sqrt(F) * (sqrt((C + hypot(C, B_m))) * -(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[(C * N[(A * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 3.8e-39], 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[Sqrt[F], $MachinePrecision] * N[(N[Sqrt[N[(C + N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if B < 3.8000000000000002e-39

   1. Initial program 19.1%

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

   3. Taylor expanded in A around -inf 19.2%

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

   if 3.8000000000000002e-39 < B

   1. Initial program 15.7%

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

   3. Taylor expanded in A around 0 16.2%

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

      1. mul-1-neg 16.2%

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

   5. Simplified 16.2%

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

      1. add-cbrt-cube 12.2%

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

      2. pow3 12.2%

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

   7. Applied egg-rr 21.0%

      \[\leadsto -\color{blue}{\sqrt[3]{{\left(\frac{{\left(2 \cdot \left(F \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)\right)\right)}^{0.5}}{B}\right)}^{3}}} \]
   8. Step-by-step derivation

      1. rem-cbrt-cube 40.1%

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

      2. div-inv 40.1%

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

      3. unpow1/2 40.0%

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

      4. *-commutative 40.0%

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

      5. sqrt-prod 40.0%

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

      6. sqrt-prod 56.7%

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

      7. *-commutative 56.7%

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

      8. add-sqr-sqrt 56.6%

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

      9. sqrt-prod 40.0%

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

      10. sqr-neg 40.0%

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

      11. sqrt-unprod 0.5%

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

      12. add-sqr-sqrt 2.4%

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

      13. associate-*r* 2.4%

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

   9. Applied egg-rr 56.7%

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

4. Final simplification 30.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 3.8 \cdot 10^{-39}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right)\right) \cdot \left(2 \cdot C\right)}}{{B}^{2} - C \cdot \left(A \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F} \cdot \left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \left(-\frac{\sqrt{2}}{B}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 44.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := {B\_m}^{2} - C \cdot \left(A \cdot 4\right)\\ \mathbf{if}\;B\_m \leq 2.22 \cdot 10^{-41}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot t\_0\right)\right) \cdot \left(2 \cdot C\right)}}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B\_m} \cdot \left(\sqrt{C + \mathsf{hypot}\left(C, B\_m\right)} \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \end{array} \]

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 2.22e-41)
     (/ (- (sqrt (* (* 2.0 (* F t_0)) (* 2.0 C)))) t_0)
     (* (/ (sqrt 2.0) B_m) (* (sqrt (+ C (hypot C B_m))) (- (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 <= 2.22e-41) {
		tmp = -sqrt(((2.0 * (F * t_0)) * (2.0 * C))) / t_0;
	} else {
		tmp = (sqrt(2.0) / B_m) * (sqrt((C + hypot(C, B_m))) * -sqrt(F));
	}
	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) - (C * (A * 4.0));
	double tmp;
	if (B_m <= 2.22e-41) {
		tmp = -Math.sqrt(((2.0 * (F * t_0)) * (2.0 * C))) / t_0;
	} else {
		tmp = (Math.sqrt(2.0) / B_m) * (Math.sqrt((C + Math.hypot(C, B_m))) * -Math.sqrt(F));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	t_0 = math.pow(B_m, 2.0) - (C * (A * 4.0))
	tmp = 0
	if B_m <= 2.22e-41:
		tmp = -math.sqrt(((2.0 * (F * t_0)) * (2.0 * C))) / t_0
	else:
		tmp = (math.sqrt(2.0) / B_m) * (math.sqrt((C + math.hypot(C, B_m))) * -math.sqrt(F))
	return tmp

Julia

B_m = abs(B)
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 <= 2.22e-41)
		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(C + hypot(C, B_m))) * 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 <= 2.22e-41)
		tmp = -sqrt(((2.0 * (F * t_0)) * (2.0 * C))) / t_0;
	else
		tmp = (sqrt(2.0) / B_m) * (sqrt((C + hypot(C, B_m))) * -sqrt(F));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
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, 2.22e-41], 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[(C + N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if B < 2.2200000000000001e-41

   1. Initial program 19.1%

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

   3. Taylor expanded in A around -inf 19.2%

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

   if 2.2200000000000001e-41 < B

   1. Initial program 15.7%

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

   3. Taylor expanded in A around 0 16.2%

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

      1. mul-1-neg 16.2%

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

      2. distribute-rgt-neg-in 16.2%

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

      3. unpow2 16.2%

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

      4. unpow2 16.2%

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

      5. hypot-def 39.9%

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

   5. Simplified 39.9%

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

      1. pow1/2 40.0%

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

      2. *-commutative 40.0%

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

      3. unpow-prod-down 56.7%

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

      4. pow1/2 56.7%

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

      5. hypot-udef 18.8%

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

      6. unpow2 18.8%

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

      7. unpow2 18.8%

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

      8. +-commutative 18.8%

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

      9. unpow2 18.8%

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

      10. unpow2 18.8%

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

      11. hypot-def 56.7%

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

      12. pow1/2 56.7%

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

   7. Applied egg-rr 56.7%

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

4. Final simplification 30.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 2.22 \cdot 10^{-41}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right)\right) \cdot \left(2 \cdot C\right)}}{{B}^{2} - C \cdot \left(A \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 38.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;F \leq -3.8 \cdot 10^{-184}:\\ \;\;\;\;\frac{\sqrt{2}}{B\_m} \cdot \sqrt{-0.5 \cdot \left(F \cdot \frac{{B\_m}^{2}}{A}\right)}\\ \mathbf{elif}\;F \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(\left(A \cdot -4\right) \cdot \left(C \cdot F\right)\right)\right) \cdot \left(C + \mathsf{hypot}\left(B\_m, C\right)\right)}}{{B\_m}^{2} - C \cdot \left(A \cdot 4\right)}\\ \mathbf{elif}\;F \leq 2.7 \cdot 10^{+68}:\\ \;\;\;\;\sqrt{\left(C + \mathsf{hypot}\left(C, B\_m\right)\right) \cdot \left(2 \cdot F\right)} \cdot \frac{-1}{B\_m}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B\_m}}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= F -3.8e-184)
   (* (/ (sqrt 2.0) B_m) (sqrt (* -0.5 (* F (/ (pow B_m 2.0) A)))))
   (if (<= F -1e-310)
     (/
      (- (sqrt (* (* 2.0 (* (* A -4.0) (* C F))) (+ C (hypot B_m C)))))
      (- (pow B_m 2.0) (* C (* A 4.0))))
     (if (<= F 2.7e+68)
       (* (sqrt (* (+ C (hypot C B_m)) (* 2.0 F))) (/ -1.0 B_m))
       (* (sqrt 2.0) (- (sqrt (/ F B_m))))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= -3.8e-184) {
		tmp = (sqrt(2.0) / B_m) * sqrt((-0.5 * (F * (pow(B_m, 2.0) / A))));
	} else if (F <= -1e-310) {
		tmp = -sqrt(((2.0 * ((A * -4.0) * (C * F))) * (C + hypot(B_m, C)))) / (pow(B_m, 2.0) - (C * (A * 4.0)));
	} else if (F <= 2.7e+68) {
		tmp = sqrt(((C + hypot(C, B_m)) * (2.0 * F))) * (-1.0 / B_m);
	} else {
		tmp = sqrt(2.0) * -sqrt((F / B_m));
	}
	return tmp;
}

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= -3.8e-184) {
		tmp = (Math.sqrt(2.0) / B_m) * Math.sqrt((-0.5 * (F * (Math.pow(B_m, 2.0) / A))));
	} else if (F <= -1e-310) {
		tmp = -Math.sqrt(((2.0 * ((A * -4.0) * (C * F))) * (C + Math.hypot(B_m, C)))) / (Math.pow(B_m, 2.0) - (C * (A * 4.0)));
	} else if (F <= 2.7e+68) {
		tmp = Math.sqrt(((C + Math.hypot(C, B_m)) * (2.0 * F))) * (-1.0 / B_m);
	} else {
		tmp = Math.sqrt(2.0) * -Math.sqrt((F / B_m));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if F <= -3.8e-184:
		tmp = (math.sqrt(2.0) / B_m) * math.sqrt((-0.5 * (F * (math.pow(B_m, 2.0) / A))))
	elif F <= -1e-310:
		tmp = -math.sqrt(((2.0 * ((A * -4.0) * (C * F))) * (C + math.hypot(B_m, C)))) / (math.pow(B_m, 2.0) - (C * (A * 4.0)))
	elif F <= 2.7e+68:
		tmp = math.sqrt(((C + math.hypot(C, B_m)) * (2.0 * F))) * (-1.0 / B_m)
	else:
		tmp = math.sqrt(2.0) * -math.sqrt((F / B_m))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (F <= -3.8e-184)
		tmp = Float64(Float64(sqrt(2.0) / B_m) * sqrt(Float64(-0.5 * Float64(F * Float64((B_m ^ 2.0) / A)))));
	elseif (F <= -1e-310)
		tmp = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(Float64(A * -4.0) * Float64(C * F))) * Float64(C + hypot(B_m, C))))) / Float64((B_m ^ 2.0) - Float64(C * Float64(A * 4.0))));
	elseif (F <= 2.7e+68)
		tmp = Float64(sqrt(Float64(Float64(C + hypot(C, B_m)) * Float64(2.0 * F))) * Float64(-1.0 / B_m));
	else
		tmp = Float64(sqrt(2.0) * Float64(-sqrt(Float64(F / B_m))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (F <= -3.8e-184)
		tmp = (sqrt(2.0) / B_m) * sqrt((-0.5 * (F * ((B_m ^ 2.0) / A))));
	elseif (F <= -1e-310)
		tmp = -sqrt(((2.0 * ((A * -4.0) * (C * F))) * (C + hypot(B_m, C)))) / ((B_m ^ 2.0) - (C * (A * 4.0)));
	elseif (F <= 2.7e+68)
		tmp = sqrt(((C + hypot(C, B_m)) * (2.0 * F))) * (-1.0 / B_m);
	else
		tmp = sqrt(2.0) * -sqrt((F / B_m));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[F, -3.8e-184], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[Sqrt[N[(-0.5 * N[(F * N[(N[Power[B$95$m, 2.0], $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -1e-310], N[((-N[Sqrt[N[(N[(2.0 * N[(N[(A * -4.0), $MachinePrecision] * N[(C * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(C * N[(A * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.7e+68], N[(N[Sqrt[N[(N[(C + N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision] * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / B$95$m), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * (-N[Sqrt[N[(F / B$95$m), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if F < -3.80000000000000017e-184

   1. Initial program 26.3%

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

   3. Taylor expanded in C around 0 0.0%

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

      1. mul-1-neg 0.0%

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

      2. distribute-rgt-neg-in 0.0%

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

      3. +-commutative 0.0%

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

      4. unpow2 0.0%

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

      5. unpow2 0.0%

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

      6. hypot-def 0.0%

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

   5. Simplified 0.0%

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

   6. Taylor expanded in A around -inf 12.6%

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

      1. add-sqr-sqrt 12.6%

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

      2. sqr-neg 12.6%

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

      3. sqrt-unprod 1.3%

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

      4. add-sqr-sqrt 31.8%

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

      5. add-sqr-sqrt 31.8%

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

      6. distribute-rgt-neg-in 31.8%

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

      7. pow1/2 31.8%

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

      8. sqrt-pow1 31.7%

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

      9. associate-/l* 17.5%

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

      10. metadata-eval 17.5%

          \[\leadsto \frac{\sqrt{2}}{B} \cdot \left(-{\left(-0.5 \cdot \frac{{B}^{2}}{\frac{A}{F}}\right)}^{\color{blue}{0.25}} \cdot \left(-\sqrt{\sqrt{-0.5 \cdot \frac{{B}^{2} \cdot F}{A}}}\right)\right) \]

      11. pow1/2 17.5%

          \[\leadsto \frac{\sqrt{2}}{B} \cdot \left(-{\left(-0.5 \cdot \frac{{B}^{2}}{\frac{A}{F}}\right)}^{0.25} \cdot \left(-\sqrt{\color{blue}{{\left(-0.5 \cdot \frac{{B}^{2} \cdot F}{A}\right)}^{0.5}}}\right)\right) \]

      12. sqrt-pow1 17.5%

          \[\leadsto \frac{\sqrt{2}}{B} \cdot \left(-{\left(-0.5 \cdot \frac{{B}^{2}}{\frac{A}{F}}\right)}^{0.25} \cdot \left(-\color{blue}{{\left(-0.5 \cdot \frac{{B}^{2} \cdot F}{A}\right)}^{\left(\frac{0.5}{2}\right)}}\right)\right) \]

   8. Applied egg-rr 17.7%

      \[\leadsto \frac{\sqrt{2}}{B} \cdot \left(-\color{blue}{{\left(-0.5 \cdot \frac{{B}^{2}}{\frac{A}{F}}\right)}^{0.25} \cdot \left(-{\left(-0.5 \cdot \frac{{B}^{2}}{\frac{A}{F}}\right)}^{0.25}\right)}\right) \]
   9. Step-by-step derivation

      1. distribute-rgt-neg-out 17.7%

         \[\leadsto \frac{\sqrt{2}}{B} \cdot \left(-\color{blue}{\left(-{\left(-0.5 \cdot \frac{{B}^{2}}{\frac{A}{F}}\right)}^{0.25} \cdot {\left(-0.5 \cdot \frac{{B}^{2}}{\frac{A}{F}}\right)}^{0.25}\right)}\right) \]

      2. pow-sqr 17.8%

         \[\leadsto \frac{\sqrt{2}}{B} \cdot \left(-\left(-\color{blue}{{\left(-0.5 \cdot \frac{{B}^{2}}{\frac{A}{F}}\right)}^{\left(2 \cdot 0.25\right)}}\right)\right) \]

      3. metadata-eval 17.8%

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

      4. unpow1/2 17.8%

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

      5. associate-/r/ 31.9%

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

      6. *-commutative 31.9%

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

   10. Simplified 31.9%

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

   if -3.80000000000000017e-184 < F < -9.999999999999969e-311

   1. Initial program 30.2%

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

   3. Taylor expanded in A around 0 17.3%

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

      1. unpow2 17.3%

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

      2. unpow2 17.3%

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

      3. hypot-def 38.0%

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

   5. Simplified 38.0%

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

   6. Taylor expanded in B around 0 38.4%

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

      1. associate-*r* 38.4%

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

      2. *-commutative 38.4%

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

      3. *-commutative 38.4%

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

   8. Simplified 38.4%

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

   if -9.999999999999969e-311 < F < 2.69999999999999991e68

   1. Initial program 23.1%

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

   3. Taylor expanded in A around 0 10.7%

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

      1. mul-1-neg 10.7%

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

   5. Simplified 10.7%

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

      1. add-cbrt-cube 8.5%

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

      2. pow3 8.5%

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

   7. Applied egg-rr 14.2%

      \[\leadsto -\color{blue}{\sqrt[3]{{\left(\frac{{\left(2 \cdot \left(F \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)\right)\right)}^{0.5}}{B}\right)}^{3}}} \]
   8. Step-by-step derivation

      1. rem-cbrt-cube 24.5%

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

      2. div-inv 24.5%

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

      3. unpow1/2 24.5%

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

      4. associate-*r* 24.5%

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

   9. Applied egg-rr 24.5%

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

   if 2.69999999999999991e68 < F

   1. Initial program 6.2%

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

   3. Taylor expanded in A around 0 4.7%

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

      1. mul-1-neg 4.7%

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

      2. distribute-rgt-neg-in 4.7%

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

      3. unpow2 4.7%

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

      4. unpow2 4.7%

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

      5. hypot-def 6.1%

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

   5. Simplified 6.1%

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

   6. Taylor expanded in C around 0 16.1%

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

      1. mul-1-neg 16.1%

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

   8. Simplified 16.1%

      \[\leadsto \color{blue}{-\sqrt{\frac{F}{B}} \cdot \sqrt{2}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 23.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3.8 \cdot 10^{-184}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \sqrt{-0.5 \cdot \left(F \cdot \frac{{B}^{2}}{A}\right)}\\ \mathbf{elif}\;F \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(\left(A \cdot -4\right) \cdot \left(C \cdot F\right)\right)\right) \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)}}{{B}^{2} - C \cdot \left(A \cdot 4\right)}\\ \mathbf{elif}\;F \leq 2.7 \cdot 10^{+68}:\\ \;\;\;\;\sqrt{\left(C + \mathsf{hypot}\left(C, B\right)\right) \cdot \left(2 \cdot F\right)} \cdot \frac{-1}{B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 38.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := {B\_m}^{2} - C \cdot \left(A \cdot 4\right)\\ \mathbf{if}\;F \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot t\_0\right)\right) \cdot \left(2 \cdot C\right)}}{t\_0}\\ \mathbf{elif}\;F \leq 8 \cdot 10^{+67}:\\ \;\;\;\;\sqrt{\left(C + \mathsf{hypot}\left(C, B\_m\right)\right) \cdot \left(2 \cdot F\right)} \cdot \frac{-1}{B\_m}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B\_m}}\right)\\ \end{array} \end{array} \]

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 (<= F -1e-310)
     (/ (- (sqrt (* (* 2.0 (* F t_0)) (* 2.0 C)))) t_0)
     (if (<= F 8e+67)
       (* (sqrt (* (+ C (hypot C B_m)) (* 2.0 F))) (/ -1.0 B_m))
       (* (sqrt 2.0) (- (sqrt (/ F 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) - (C * (A * 4.0));
	double tmp;
	if (F <= -1e-310) {
		tmp = -sqrt(((2.0 * (F * t_0)) * (2.0 * C))) / t_0;
	} else if (F <= 8e+67) {
		tmp = sqrt(((C + hypot(C, B_m)) * (2.0 * F))) * (-1.0 / B_m);
	} else {
		tmp = sqrt(2.0) * -sqrt((F / B_m));
	}
	return tmp;
}

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double t_0 = Math.pow(B_m, 2.0) - (C * (A * 4.0));
	double tmp;
	if (F <= -1e-310) {
		tmp = -Math.sqrt(((2.0 * (F * t_0)) * (2.0 * C))) / t_0;
	} else if (F <= 8e+67) {
		tmp = Math.sqrt(((C + Math.hypot(C, B_m)) * (2.0 * F))) * (-1.0 / B_m);
	} else {
		tmp = Math.sqrt(2.0) * -Math.sqrt((F / 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) - (C * (A * 4.0))
	tmp = 0
	if F <= -1e-310:
		tmp = -math.sqrt(((2.0 * (F * t_0)) * (2.0 * C))) / t_0
	elif F <= 8e+67:
		tmp = math.sqrt(((C + math.hypot(C, B_m)) * (2.0 * F))) * (-1.0 / B_m)
	else:
		tmp = math.sqrt(2.0) * -math.sqrt((F / B_m))
	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 (F <= -1e-310)
		tmp = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(F * t_0)) * Float64(2.0 * C)))) / t_0);
	elseif (F <= 8e+67)
		tmp = Float64(sqrt(Float64(Float64(C + hypot(C, B_m)) * Float64(2.0 * F))) * Float64(-1.0 / B_m));
	else
		tmp = Float64(sqrt(2.0) * Float64(-sqrt(Float64(F / 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) - (C * (A * 4.0));
	tmp = 0.0;
	if (F <= -1e-310)
		tmp = -sqrt(((2.0 * (F * t_0)) * (2.0 * C))) / t_0;
	elseif (F <= 8e+67)
		tmp = sqrt(((C + hypot(C, B_m)) * (2.0 * F))) * (-1.0 / B_m);
	else
		tmp = sqrt(2.0) * -sqrt((F / 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[(C * N[(A * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -1e-310], 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], If[LessEqual[F, 8e+67], N[(N[Sqrt[N[(N[(C + N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision] * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / B$95$m), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * (-N[Sqrt[N[(F / B$95$m), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < -9.999999999999969e-311

   1. Initial program 27.9%

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

   3. Taylor expanded in A around -inf 31.3%

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

   if -9.999999999999969e-311 < F < 7.99999999999999986e67

   1. Initial program 23.1%

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

   3. Taylor expanded in A around 0 10.7%

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

      1. mul-1-neg 10.7%

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

   5. Simplified 10.7%

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

      1. add-cbrt-cube 8.5%

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

      2. pow3 8.5%

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

   7. Applied egg-rr 14.2%

      \[\leadsto -\color{blue}{\sqrt[3]{{\left(\frac{{\left(2 \cdot \left(F \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)\right)\right)}^{0.5}}{B}\right)}^{3}}} \]
   8. Step-by-step derivation

      1. rem-cbrt-cube 24.5%

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

      2. div-inv 24.5%

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

      3. unpow1/2 24.5%

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

      4. associate-*r* 24.5%

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

   9. Applied egg-rr 24.5%

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

   if 7.99999999999999986e67 < F

   1. Initial program 6.2%

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

   3. Taylor expanded in A around 0 4.7%

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

      1. mul-1-neg 4.7%

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

      2. distribute-rgt-neg-in 4.7%

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

      3. unpow2 4.7%

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

      4. unpow2 4.7%

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

      5. hypot-def 6.1%

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

   5. Simplified 6.1%

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

   6. Taylor expanded in C around 0 16.1%

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

      1. mul-1-neg 16.1%

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

   8. Simplified 16.1%

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

4. Final simplification 22.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right)\right) \cdot \left(2 \cdot C\right)}}{{B}^{2} - C \cdot \left(A \cdot 4\right)}\\ \mathbf{elif}\;F \leq 8 \cdot 10^{+67}:\\ \;\;\;\;\sqrt{\left(C + \mathsf{hypot}\left(C, B\right)\right) \cdot \left(2 \cdot F\right)} \cdot \frac{-1}{B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 37.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;F \leq -1 \cdot 10^{-278}:\\ \;\;\;\;\sqrt{\left(-0.5 \cdot \frac{{B\_m}^{2}}{\frac{A}{F}}\right) \cdot \frac{2}{{B\_m}^{2}}}\\ \mathbf{elif}\;F \leq 5.2 \cdot 10^{+68}:\\ \;\;\;\;\sqrt{\left(C + \mathsf{hypot}\left(C, B\_m\right)\right) \cdot \left(2 \cdot F\right)} \cdot \frac{-1}{B\_m}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B\_m}}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= F -1e-278)
   (sqrt (* (* -0.5 (/ (pow B_m 2.0) (/ A F))) (/ 2.0 (pow B_m 2.0))))
   (if (<= F 5.2e+68)
     (* (sqrt (* (+ C (hypot C B_m)) (* 2.0 F))) (/ -1.0 B_m))
     (* (sqrt 2.0) (- (sqrt (/ F B_m)))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= -1e-278) {
		tmp = sqrt(((-0.5 * (pow(B_m, 2.0) / (A / F))) * (2.0 / pow(B_m, 2.0))));
	} else if (F <= 5.2e+68) {
		tmp = sqrt(((C + hypot(C, B_m)) * (2.0 * F))) * (-1.0 / B_m);
	} else {
		tmp = sqrt(2.0) * -sqrt((F / B_m));
	}
	return tmp;
}

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= -1e-278) {
		tmp = Math.sqrt(((-0.5 * (Math.pow(B_m, 2.0) / (A / F))) * (2.0 / Math.pow(B_m, 2.0))));
	} else if (F <= 5.2e+68) {
		tmp = Math.sqrt(((C + Math.hypot(C, B_m)) * (2.0 * F))) * (-1.0 / B_m);
	} else {
		tmp = Math.sqrt(2.0) * -Math.sqrt((F / B_m));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if F <= -1e-278:
		tmp = math.sqrt(((-0.5 * (math.pow(B_m, 2.0) / (A / F))) * (2.0 / math.pow(B_m, 2.0))))
	elif F <= 5.2e+68:
		tmp = math.sqrt(((C + math.hypot(C, B_m)) * (2.0 * F))) * (-1.0 / B_m)
	else:
		tmp = math.sqrt(2.0) * -math.sqrt((F / B_m))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (F <= -1e-278)
		tmp = sqrt(Float64(Float64(-0.5 * Float64((B_m ^ 2.0) / Float64(A / F))) * Float64(2.0 / (B_m ^ 2.0))));
	elseif (F <= 5.2e+68)
		tmp = Float64(sqrt(Float64(Float64(C + hypot(C, B_m)) * Float64(2.0 * F))) * Float64(-1.0 / B_m));
	else
		tmp = Float64(sqrt(2.0) * Float64(-sqrt(Float64(F / B_m))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (F <= -1e-278)
		tmp = sqrt(((-0.5 * ((B_m ^ 2.0) / (A / F))) * (2.0 / (B_m ^ 2.0))));
	elseif (F <= 5.2e+68)
		tmp = sqrt(((C + hypot(C, B_m)) * (2.0 * F))) * (-1.0 / B_m);
	else
		tmp = sqrt(2.0) * -sqrt((F / B_m));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[F, -1e-278], N[Sqrt[N[(N[(-0.5 * N[(N[Power[B$95$m, 2.0], $MachinePrecision] / N[(A / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 / N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], If[LessEqual[F, 5.2e+68], N[(N[Sqrt[N[(N[(C + N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision] * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / B$95$m), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * (-N[Sqrt[N[(F / B$95$m), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -9.99999999999999938e-279

   1. Initial program 27.1%

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

   3. Taylor expanded in C around 0 0.4%

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

      1. mul-1-neg 0.4%

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

      2. distribute-rgt-neg-in 0.4%

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

      3. +-commutative 0.4%

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

      4. unpow2 0.4%

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

      5. unpow2 0.4%

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

      6. hypot-def 0.4%

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

   5. Simplified 0.4%

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

   6. Taylor expanded in A around -inf 9.8%

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

      1. add-sqr-sqrt 9.4%

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

      2. sqrt-unprod 19.7%

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

      3. *-commutative 19.7%

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

      4. *-commutative 19.7%

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

      5. swap-sqr 18.2%

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

      6. sqr-neg 18.2%

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

      7. add-sqr-sqrt 18.2%

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

      8. associate-/l* 18.4%

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

      9. frac-times 18.3%

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

      10. rem-square-sqrt 18.5%

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

      11. unpow2 18.5%

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

   8. Applied egg-rr 18.5%

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

   if -9.99999999999999938e-279 < F < 5.1999999999999996e68

   1. Initial program 23.4%

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

   3. Taylor expanded in A around 0 10.5%

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

      1. mul-1-neg 10.5%

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

   5. Simplified 10.5%

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

      1. add-cbrt-cube 8.3%

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

      2. pow3 8.3%

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

   7. Applied egg-rr 13.9%

      \[\leadsto -\color{blue}{\sqrt[3]{{\left(\frac{{\left(2 \cdot \left(F \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)\right)\right)}^{0.5}}{B}\right)}^{3}}} \]
   8. Step-by-step derivation

      1. rem-cbrt-cube 24.0%

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

      2. div-inv 24.0%

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

      3. unpow1/2 24.0%

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

      4. associate-*r* 24.0%

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

   9. Applied egg-rr 24.0%

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

   if 5.1999999999999996e68 < F

   1. Initial program 6.2%

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

   3. Taylor expanded in A around 0 4.7%

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

      1. mul-1-neg 4.7%

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

      2. distribute-rgt-neg-in 4.7%

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

      3. unpow2 4.7%

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

      4. unpow2 4.7%

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

      5. hypot-def 6.1%

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

   5. Simplified 6.1%

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

   6. Taylor expanded in C around 0 16.1%

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

      1. mul-1-neg 16.1%

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

   8. Simplified 16.1%

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

4. Final simplification 20.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1 \cdot 10^{-278}:\\ \;\;\;\;\sqrt{\left(-0.5 \cdot \frac{{B}^{2}}{\frac{A}{F}}\right) \cdot \frac{2}{{B}^{2}}}\\ \mathbf{elif}\;F \leq 5.2 \cdot 10^{+68}:\\ \;\;\;\;\sqrt{\left(C + \mathsf{hypot}\left(C, B\right)\right) \cdot \left(2 \cdot F\right)} \cdot \frac{-1}{B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 38.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;F \leq -8.5 \cdot 10^{-304}:\\ \;\;\;\;\frac{\sqrt{2}}{B\_m} \cdot \sqrt{-0.5 \cdot \left(F \cdot \frac{{B\_m}^{2}}{A}\right)}\\ \mathbf{elif}\;F \leq 1.32 \cdot 10^{+70}:\\ \;\;\;\;\sqrt{\left(C + \mathsf{hypot}\left(C, B\_m\right)\right) \cdot \left(2 \cdot F\right)} \cdot \frac{-1}{B\_m}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B\_m}}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= F -8.5e-304)
   (* (/ (sqrt 2.0) B_m) (sqrt (* -0.5 (* F (/ (pow B_m 2.0) A)))))
   (if (<= F 1.32e+70)
     (* (sqrt (* (+ C (hypot C B_m)) (* 2.0 F))) (/ -1.0 B_m))
     (* (sqrt 2.0) (- (sqrt (/ F B_m)))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= -8.5e-304) {
		tmp = (sqrt(2.0) / B_m) * sqrt((-0.5 * (F * (pow(B_m, 2.0) / A))));
	} else if (F <= 1.32e+70) {
		tmp = sqrt(((C + hypot(C, B_m)) * (2.0 * F))) * (-1.0 / B_m);
	} else {
		tmp = sqrt(2.0) * -sqrt((F / B_m));
	}
	return tmp;
}

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= -8.5e-304) {
		tmp = (Math.sqrt(2.0) / B_m) * Math.sqrt((-0.5 * (F * (Math.pow(B_m, 2.0) / A))));
	} else if (F <= 1.32e+70) {
		tmp = Math.sqrt(((C + Math.hypot(C, B_m)) * (2.0 * F))) * (-1.0 / B_m);
	} else {
		tmp = Math.sqrt(2.0) * -Math.sqrt((F / B_m));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if F <= -8.5e-304:
		tmp = (math.sqrt(2.0) / B_m) * math.sqrt((-0.5 * (F * (math.pow(B_m, 2.0) / A))))
	elif F <= 1.32e+70:
		tmp = math.sqrt(((C + math.hypot(C, B_m)) * (2.0 * F))) * (-1.0 / B_m)
	else:
		tmp = math.sqrt(2.0) * -math.sqrt((F / B_m))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (F <= -8.5e-304)
		tmp = Float64(Float64(sqrt(2.0) / B_m) * sqrt(Float64(-0.5 * Float64(F * Float64((B_m ^ 2.0) / A)))));
	elseif (F <= 1.32e+70)
		tmp = Float64(sqrt(Float64(Float64(C + hypot(C, B_m)) * Float64(2.0 * F))) * Float64(-1.0 / B_m));
	else
		tmp = Float64(sqrt(2.0) * Float64(-sqrt(Float64(F / B_m))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (F <= -8.5e-304)
		tmp = (sqrt(2.0) / B_m) * sqrt((-0.5 * (F * ((B_m ^ 2.0) / A))));
	elseif (F <= 1.32e+70)
		tmp = sqrt(((C + hypot(C, B_m)) * (2.0 * F))) * (-1.0 / B_m);
	else
		tmp = sqrt(2.0) * -sqrt((F / B_m));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[F, -8.5e-304], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[Sqrt[N[(-0.5 * N[(F * N[(N[Power[B$95$m, 2.0], $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.32e+70], N[(N[Sqrt[N[(N[(C + N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision] * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / B$95$m), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * (-N[Sqrt[N[(F / B$95$m), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -8.5e-304

   1. Initial program 28.6%

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

   3. Taylor expanded in C around 0 0.5%

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

      1. mul-1-neg 0.5%

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

      2. distribute-rgt-neg-in 0.5%

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

      3. +-commutative 0.5%

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

      4. unpow2 0.5%

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

      5. unpow2 0.5%

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

      6. hypot-def 0.5%

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

   5. Simplified 0.5%

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

   6. Taylor expanded in A around -inf 9.4%

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

      1. add-sqr-sqrt 9.4%

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

      2. sqr-neg 9.4%

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

      3. sqrt-unprod 2.6%

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

      4. add-sqr-sqrt 21.0%

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

      5. add-sqr-sqrt 21.0%

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

      6. distribute-rgt-neg-in 21.0%

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

      7. pow1/2 21.0%

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

      8. sqrt-pow1 21.0%

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

      9. associate-/l* 12.4%

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

      10. metadata-eval 12.4%

          \[\leadsto \frac{\sqrt{2}}{B} \cdot \left(-{\left(-0.5 \cdot \frac{{B}^{2}}{\frac{A}{F}}\right)}^{\color{blue}{0.25}} \cdot \left(-\sqrt{\sqrt{-0.5 \cdot \frac{{B}^{2} \cdot F}{A}}}\right)\right) \]

      11. pow1/2 12.4%

          \[\leadsto \frac{\sqrt{2}}{B} \cdot \left(-{\left(-0.5 \cdot \frac{{B}^{2}}{\frac{A}{F}}\right)}^{0.25} \cdot \left(-\sqrt{\color{blue}{{\left(-0.5 \cdot \frac{{B}^{2} \cdot F}{A}\right)}^{0.5}}}\right)\right) \]

      12. sqrt-pow1 12.3%

          \[\leadsto \frac{\sqrt{2}}{B} \cdot \left(-{\left(-0.5 \cdot \frac{{B}^{2}}{\frac{A}{F}}\right)}^{0.25} \cdot \left(-\color{blue}{{\left(-0.5 \cdot \frac{{B}^{2} \cdot F}{A}\right)}^{\left(\frac{0.5}{2}\right)}}\right)\right) \]

   8. Applied egg-rr 12.5%

      \[\leadsto \frac{\sqrt{2}}{B} \cdot \left(-\color{blue}{{\left(-0.5 \cdot \frac{{B}^{2}}{\frac{A}{F}}\right)}^{0.25} \cdot \left(-{\left(-0.5 \cdot \frac{{B}^{2}}{\frac{A}{F}}\right)}^{0.25}\right)}\right) \]
   9. Step-by-step derivation

      1. distribute-rgt-neg-out 12.5%

         \[\leadsto \frac{\sqrt{2}}{B} \cdot \left(-\color{blue}{\left(-{\left(-0.5 \cdot \frac{{B}^{2}}{\frac{A}{F}}\right)}^{0.25} \cdot {\left(-0.5 \cdot \frac{{B}^{2}}{\frac{A}{F}}\right)}^{0.25}\right)}\right) \]

      2. pow-sqr 12.5%

         \[\leadsto \frac{\sqrt{2}}{B} \cdot \left(-\left(-\color{blue}{{\left(-0.5 \cdot \frac{{B}^{2}}{\frac{A}{F}}\right)}^{\left(2 \cdot 0.25\right)}}\right)\right) \]

      3. metadata-eval 12.5%

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

      4. unpow1/2 12.5%

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

      5. associate-/r/ 21.1%

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

      6. *-commutative 21.1%

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

   10. Simplified 21.1%

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

   if -8.5e-304 < F < 1.3199999999999999e70

   1. Initial program 23.0%

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

   3. Taylor expanded in A around 0 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%

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

   5. Simplified 10.6%

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

      1. add-cbrt-cube 8.4%

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

      2. pow3 8.4%

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

   7. Applied egg-rr 14.1%

      \[\leadsto -\color{blue}{\sqrt[3]{{\left(\frac{{\left(2 \cdot \left(F \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)\right)\right)}^{0.5}}{B}\right)}^{3}}} \]
   8. Step-by-step derivation

      1. rem-cbrt-cube 24.3%

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

      2. div-inv 24.3%

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

      3. unpow1/2 24.3%

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

      4. associate-*r* 24.3%

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

   9. Applied egg-rr 24.3%

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

   if 1.3199999999999999e70 < F

   1. Initial program 6.2%

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

   3. Taylor expanded in A around 0 4.7%

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

      1. mul-1-neg 4.7%

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

      2. distribute-rgt-neg-in 4.7%

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

      3. unpow2 4.7%

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

      4. unpow2 4.7%

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

      5. hypot-def 6.1%

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

   5. Simplified 6.1%

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

   6. Taylor expanded in C around 0 16.1%

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

      1. mul-1-neg 16.1%

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

   8. Simplified 16.1%

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

4. Final simplification 21.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -8.5 \cdot 10^{-304}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \sqrt{-0.5 \cdot \left(F \cdot \frac{{B}^{2}}{A}\right)}\\ \mathbf{elif}\;F \leq 1.32 \cdot 10^{+70}:\\ \;\;\;\;\sqrt{\left(C + \mathsf{hypot}\left(C, B\right)\right) \cdot \left(2 \cdot F\right)} \cdot \frac{-1}{B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 35.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;F \leq 1.32 \cdot 10^{+70}:\\ \;\;\;\;\sqrt{\left(C + \mathsf{hypot}\left(C, B\_m\right)\right) \cdot \left(2 \cdot F\right)} \cdot \frac{-1}{B\_m}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B\_m}}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= F 1.32e+70)
   (* (sqrt (* (+ C (hypot C B_m)) (* 2.0 F))) (/ -1.0 B_m))
   (* (sqrt 2.0) (- (sqrt (/ F B_m))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= 1.32e+70) {
		tmp = sqrt(((C + hypot(C, B_m)) * (2.0 * F))) * (-1.0 / B_m);
	} else {
		tmp = sqrt(2.0) * -sqrt((F / B_m));
	}
	return tmp;
}

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= 1.32e+70) {
		tmp = Math.sqrt(((C + Math.hypot(C, B_m)) * (2.0 * F))) * (-1.0 / B_m);
	} else {
		tmp = Math.sqrt(2.0) * -Math.sqrt((F / B_m));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if F <= 1.32e+70:
		tmp = math.sqrt(((C + math.hypot(C, B_m)) * (2.0 * F))) * (-1.0 / B_m)
	else:
		tmp = math.sqrt(2.0) * -math.sqrt((F / B_m))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (F <= 1.32e+70)
		tmp = Float64(sqrt(Float64(Float64(C + hypot(C, B_m)) * Float64(2.0 * F))) * Float64(-1.0 / B_m));
	else
		tmp = Float64(sqrt(2.0) * Float64(-sqrt(Float64(F / B_m))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (F <= 1.32e+70)
		tmp = sqrt(((C + hypot(C, B_m)) * (2.0 * F))) * (-1.0 / B_m);
	else
		tmp = sqrt(2.0) * -sqrt((F / B_m));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[F, 1.32e+70], N[(N[Sqrt[N[(N[(C + N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision] * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / B$95$m), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * (-N[Sqrt[N[(F / B$95$m), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < 1.3199999999999999e70

   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 8.6%

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

      1. mul-1-neg 8.6%

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

   5. Simplified 8.6%

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

      1. add-cbrt-cube 6.8%

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

      2. pow3 6.8%

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

   7. Applied egg-rr 11.5%

      \[\leadsto -\color{blue}{\sqrt[3]{{\left(\frac{{\left(2 \cdot \left(F \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)\right)\right)}^{0.5}}{B}\right)}^{3}}} \]
   8. Step-by-step derivation

      1. rem-cbrt-cube 19.7%

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

      2. div-inv 19.7%

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

      3. unpow1/2 19.6%

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

      4. associate-*r* 19.6%

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

   9. Applied egg-rr 19.6%

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

   if 1.3199999999999999e70 < F

   1. Initial program 6.2%

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

   3. Taylor expanded in A around 0 4.7%

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

      1. mul-1-neg 4.7%

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

      2. distribute-rgt-neg-in 4.7%

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

      3. unpow2 4.7%

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

      4. unpow2 4.7%

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

      5. hypot-def 6.1%

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

   5. Simplified 6.1%

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

   6. Taylor expanded in C around 0 16.1%

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

      1. mul-1-neg 16.1%

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

   8. Simplified 16.1%

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

4. Final simplification 18.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 1.32 \cdot 10^{+70}:\\ \;\;\;\;\sqrt{\left(C + \mathsf{hypot}\left(C, B\right)\right) \cdot \left(2 \cdot F\right)} \cdot \frac{-1}{B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 35.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;F \leq 1.55 \cdot 10^{+72}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(F \cdot \left(C + \mathsf{hypot}\left(C, B\_m\right)\right)\right)}}{B\_m}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B\_m}}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= F 1.55e+72)
   (/ (- (sqrt (* 2.0 (* F (+ C (hypot C B_m)))))) B_m)
   (* (sqrt 2.0) (- (sqrt (/ F B_m))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= 1.55e+72) {
		tmp = -sqrt((2.0 * (F * (C + hypot(C, B_m))))) / B_m;
	} else {
		tmp = sqrt(2.0) * -sqrt((F / B_m));
	}
	return tmp;
}

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= 1.55e+72) {
		tmp = -Math.sqrt((2.0 * (F * (C + Math.hypot(C, B_m))))) / B_m;
	} else {
		tmp = Math.sqrt(2.0) * -Math.sqrt((F / B_m));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if F <= 1.55e+72:
		tmp = -math.sqrt((2.0 * (F * (C + math.hypot(C, B_m))))) / B_m
	else:
		tmp = math.sqrt(2.0) * -math.sqrt((F / B_m))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (F <= 1.55e+72)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(F * Float64(C + hypot(C, B_m)))))) / B_m);
	else
		tmp = Float64(sqrt(2.0) * Float64(-sqrt(Float64(F / B_m))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (F <= 1.55e+72)
		tmp = -sqrt((2.0 * (F * (C + hypot(C, B_m))))) / B_m;
	else
		tmp = sqrt(2.0) * -sqrt((F / B_m));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[F, 1.55e+72], N[((-N[Sqrt[N[(2.0 * N[(F * N[(C + N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / B$95$m), $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * (-N[Sqrt[N[(F / B$95$m), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < 1.54999999999999994e72

   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 8.6%

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

      1. mul-1-neg 8.6%

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

      2. distribute-rgt-neg-in 8.6%

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

      3. unpow2 8.6%

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

      4. unpow2 8.6%

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

      5. hypot-def 19.6%

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

   5. Simplified 19.6%

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

   7. Applied egg-rr 3.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-\frac{{\left(2 \cdot \left(F \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)\right)\right)}^{0.5}}{B}\right)} - 1} \]
   8. Step-by-step derivation

      1. expm1-def 16.6%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-\frac{{\left(2 \cdot \left(F \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)\right)\right)}^{0.5}}{B}\right)\right)} \]

      2. expm1-log1p 19.7%

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

      3. distribute-neg-frac 19.7%

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

      4. unpow1/2 19.7%

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

   9. Simplified 19.7%

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

   if 1.54999999999999994e72 < F

   1. Initial program 6.2%

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

   3. Taylor expanded in A around 0 4.7%

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

      1. mul-1-neg 4.7%

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

      2. distribute-rgt-neg-in 4.7%

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

      3. unpow2 4.7%

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

      4. unpow2 4.7%

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

      5. hypot-def 6.1%

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

   5. Simplified 6.1%

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

   6. Taylor expanded in C around 0 16.1%

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

      1. mul-1-neg 16.1%

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

   8. Simplified 16.1%

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

4. Final simplification 18.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 1.55 \cdot 10^{+72}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(F \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)\right)}}{B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 27.8% accurate, 3.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;C \leq 3.8 \cdot 10^{+133}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B\_m}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{F} \cdot \sqrt{C}\right) \cdot \frac{-2}{B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= C 3.8e+133)
   (* (sqrt 2.0) (- (sqrt (/ F B_m))))
   (* (* (sqrt F) (sqrt C)) (/ (- 2.0) B_m))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (C <= 3.8e+133) {
		tmp = sqrt(2.0) * -sqrt((F / B_m));
	} else {
		tmp = (sqrt(F) * sqrt(C)) * (-2.0 / B_m);
	}
	return tmp;
}

Fortran

B_m = abs(B)
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (c <= 3.8d+133) then
        tmp = sqrt(2.0d0) * -sqrt((f / b_m))
    else
        tmp = (sqrt(f) * sqrt(c)) * (-2.0d0 / b_m)
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (C <= 3.8e+133) {
		tmp = Math.sqrt(2.0) * -Math.sqrt((F / B_m));
	} else {
		tmp = (Math.sqrt(F) * Math.sqrt(C)) * (-2.0 / B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if C <= 3.8e+133:
		tmp = math.sqrt(2.0) * -math.sqrt((F / B_m))
	else:
		tmp = (math.sqrt(F) * math.sqrt(C)) * (-2.0 / B_m)
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (C <= 3.8e+133)
		tmp = Float64(sqrt(2.0) * Float64(-sqrt(Float64(F / B_m))));
	else
		tmp = Float64(Float64(sqrt(F) * sqrt(C)) * Float64(Float64(-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 (C <= 3.8e+133)
		tmp = sqrt(2.0) * -sqrt((F / B_m));
	else
		tmp = (sqrt(F) * sqrt(C)) * (-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[C, 3.8e+133], N[(N[Sqrt[2.0], $MachinePrecision] * (-N[Sqrt[N[(F / B$95$m), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[C], $MachinePrecision]), $MachinePrecision] * N[((-2.0) / B$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if C < 3.8000000000000002e133

   1. Initial program 21.4%

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

   3. Taylor expanded in A around 0 8.7%

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

      1. mul-1-neg 8.7%

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

      2. distribute-rgt-neg-in 8.7%

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

      3. unpow2 8.7%

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

      4. unpow2 8.7%

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

      5. hypot-def 14.5%

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

   5. Simplified 14.5%

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

   6. Taylor expanded in C around 0 14.2%

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

      1. mul-1-neg 14.2%

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

   8. Simplified 14.2%

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

   if 3.8000000000000002e133 < C

   1. Initial program 3.8%

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

   3. Taylor expanded in A around 0 1.4%

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

      1. mul-1-neg 1.4%

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

      2. distribute-rgt-neg-in 1.4%

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

      3. unpow2 1.4%

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

      4. unpow2 1.4%

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

      5. hypot-def 17.9%

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

   5. Simplified 17.9%

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

   6. Taylor expanded in B around 0 12.0%

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

      1. mul-1-neg 12.0%

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

      2. *-commutative 12.0%

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

      3. *-commutative 12.0%

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

      4. unpow2 12.0%

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

      5. rem-square-sqrt 12.2%

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

   8. Simplified 12.2%

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

      1. sqrt-prod 16.3%

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

   10. Applied egg-rr 16.3%

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

4. Final simplification 14.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq 3.8 \cdot 10^{+133}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{F} \cdot \sqrt{C}\right) \cdot \frac{-2}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 33.3% accurate, 3.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;F \leq 1.8 \cdot 10^{+57}:\\ \;\;\;\;\frac{\sqrt{2}}{B\_m} \cdot \left(-\sqrt{B\_m \cdot F}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B\_m}}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= F 1.8e+57)
   (* (/ (sqrt 2.0) B_m) (- (sqrt (* B_m F))))
   (* (sqrt 2.0) (- (sqrt (/ F B_m))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= 1.8e+57) {
		tmp = (sqrt(2.0) / B_m) * -sqrt((B_m * F));
	} else {
		tmp = sqrt(2.0) * -sqrt((F / B_m));
	}
	return tmp;
}

Fortran

B_m = abs(B)
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (f <= 1.8d+57) then
        tmp = (sqrt(2.0d0) / b_m) * -sqrt((b_m * f))
    else
        tmp = sqrt(2.0d0) * -sqrt((f / b_m))
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= 1.8e+57) {
		tmp = (Math.sqrt(2.0) / B_m) * -Math.sqrt((B_m * F));
	} else {
		tmp = Math.sqrt(2.0) * -Math.sqrt((F / B_m));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if F <= 1.8e+57:
		tmp = (math.sqrt(2.0) / B_m) * -math.sqrt((B_m * F))
	else:
		tmp = math.sqrt(2.0) * -math.sqrt((F / B_m))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (F <= 1.8e+57)
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(-sqrt(Float64(B_m * F))));
	else
		tmp = Float64(sqrt(2.0) * Float64(-sqrt(Float64(F / B_m))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (F <= 1.8e+57)
		tmp = (sqrt(2.0) / B_m) * -sqrt((B_m * F));
	else
		tmp = sqrt(2.0) * -sqrt((F / B_m));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[F, 1.8e+57], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * (-N[Sqrt[N[(B$95$m * F), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * (-N[Sqrt[N[(F / B$95$m), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < 1.8000000000000001e57

   1. Initial program 24.3%

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

   3. Taylor expanded in A around 0 8.3%

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

      1. mul-1-neg 8.3%

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

      2. distribute-rgt-neg-in 8.3%

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

      3. unpow2 8.3%

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

      4. unpow2 8.3%

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

      5. hypot-def 18.5%

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

   5. Simplified 18.5%

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

   6. Taylor expanded in C around 0 15.1%

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

   if 1.8000000000000001e57 < F

   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 5.4%

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

      1. mul-1-neg 5.4%

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

      2. distribute-rgt-neg-in 5.4%

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

      3. unpow2 5.4%

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

      4. unpow2 5.4%

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

      5. hypot-def 8.9%

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

   5. Simplified 8.9%

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

   6. Taylor expanded in C around 0 16.5%

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

      1. mul-1-neg 16.5%

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

   8. Simplified 16.5%

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

4. Final simplification 15.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 1.8 \cdot 10^{+57}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{B \cdot F}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 26.9% accurate, 3.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;C \leq 9.5 \cdot 10^{+124}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B\_m}}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(C \cdot F\right)}^{0.5} \cdot \frac{-2}{B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= C 9.5e+124)
   (* (sqrt 2.0) (- (sqrt (/ F B_m))))
   (* (pow (* C F) 0.5) (/ (- 2.0) B_m))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (C <= 9.5e+124) {
		tmp = sqrt(2.0) * -sqrt((F / B_m));
	} else {
		tmp = pow((C * F), 0.5) * (-2.0 / B_m);
	}
	return tmp;
}

Fortran

B_m = abs(B)
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (c <= 9.5d+124) then
        tmp = sqrt(2.0d0) * -sqrt((f / b_m))
    else
        tmp = ((c * f) ** 0.5d0) * (-2.0d0 / b_m)
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (C <= 9.5e+124) {
		tmp = Math.sqrt(2.0) * -Math.sqrt((F / B_m));
	} else {
		tmp = Math.pow((C * F), 0.5) * (-2.0 / B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if C <= 9.5e+124:
		tmp = math.sqrt(2.0) * -math.sqrt((F / B_m))
	else:
		tmp = math.pow((C * F), 0.5) * (-2.0 / B_m)
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (C <= 9.5e+124)
		tmp = Float64(sqrt(2.0) * Float64(-sqrt(Float64(F / B_m))));
	else
		tmp = Float64((Float64(C * F) ^ 0.5) * Float64(Float64(-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 (C <= 9.5e+124)
		tmp = sqrt(2.0) * -sqrt((F / B_m));
	else
		tmp = ((C * F) ^ 0.5) * (-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[C, 9.5e+124], N[(N[Sqrt[2.0], $MachinePrecision] * (-N[Sqrt[N[(F / B$95$m), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(N[Power[N[(C * F), $MachinePrecision], 0.5], $MachinePrecision] * N[((-2.0) / B$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if C < 9.50000000000000004e124

   1. Initial program 21.4%

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

   3. Taylor expanded in A around 0 8.7%

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

      1. mul-1-neg 8.7%

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

      2. distribute-rgt-neg-in 8.7%

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

      3. unpow2 8.7%

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

      4. unpow2 8.7%

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

      5. hypot-def 14.5%

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

   5. Simplified 14.5%

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

   6. Taylor expanded in C around 0 14.2%

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

      1. mul-1-neg 14.2%

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

   8. Simplified 14.2%

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

   if 9.50000000000000004e124 < C

   1. Initial program 3.8%

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

   3. Taylor expanded in A around 0 1.4%

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

      1. mul-1-neg 1.4%

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

      2. distribute-rgt-neg-in 1.4%

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

      3. unpow2 1.4%

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

      4. unpow2 1.4%

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

      5. hypot-def 17.9%

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

   5. Simplified 17.9%

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

   6. Taylor expanded in B around 0 12.0%

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

      1. mul-1-neg 12.0%

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

      2. *-commutative 12.0%

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

      3. *-commutative 12.0%

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

      4. unpow2 12.0%

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

      5. rem-square-sqrt 12.2%

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

   8. Simplified 12.2%

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

      1. pow1/2 12.3%

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

      2. *-commutative 12.3%

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

   10. Applied egg-rr 12.3%

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

4. Final simplification 13.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq 9.5 \cdot 10^{+124}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(C \cdot F\right)}^{0.5} \cdot \frac{-2}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 8.4% accurate, 5.6× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := \frac{-2}{B\_m}\\ \mathbf{if}\;A \leq 4.5 \cdot 10^{-174}:\\ \;\;\;\;{\left(C \cdot F\right)}^{0.5} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{A \cdot F} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (/ (- 2.0) B_m)))
   (if (<= A 4.5e-174) (* (pow (* C F) 0.5) t_0) (* (sqrt (* A F)) t_0))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double t_0 = -2.0 / B_m;
	double tmp;
	if (A <= 4.5e-174) {
		tmp = pow((C * F), 0.5) * t_0;
	} else {
		tmp = sqrt((A * F)) * t_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) :: t_0
    real(8) :: tmp
    t_0 = -2.0d0 / b_m
    if (a <= 4.5d-174) then
        tmp = ((c * f) ** 0.5d0) * t_0
    else
        tmp = sqrt((a * f)) * t_0
    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 = -2.0 / B_m;
	double tmp;
	if (A <= 4.5e-174) {
		tmp = Math.pow((C * F), 0.5) * t_0;
	} else {
		tmp = Math.sqrt((A * F)) * t_0;
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	t_0 = -2.0 / B_m
	tmp = 0
	if A <= 4.5e-174:
		tmp = math.pow((C * F), 0.5) * t_0
	else:
		tmp = math.sqrt((A * F)) * t_0
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	t_0 = Float64(Float64(-2.0) / B_m)
	tmp = 0.0
	if (A <= 4.5e-174)
		tmp = Float64((Float64(C * F) ^ 0.5) * t_0);
	else
		tmp = Float64(sqrt(Float64(A * F)) * t_0);
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	t_0 = -2.0 / B_m;
	tmp = 0.0;
	if (A <= 4.5e-174)
		tmp = ((C * F) ^ 0.5) * t_0;
	else
		tmp = sqrt((A * F)) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[((-2.0) / B$95$m), $MachinePrecision]}, If[LessEqual[A, 4.5e-174], N[(N[Power[N[(C * F), $MachinePrecision], 0.5], $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[Sqrt[N[(A * F), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if A < 4.49999999999999964e-174

   1. Initial program 13.4%

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

   3. Taylor expanded in A around 0 7.1%

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

      1. mul-1-neg 7.1%

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

      2. distribute-rgt-neg-in 7.1%

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

      3. unpow2 7.1%

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

      4. unpow2 7.1%

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

      5. hypot-def 17.8%

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

   5. Simplified 17.8%

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

   6. Taylor expanded in B around 0 5.4%

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

      1. mul-1-neg 5.4%

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

      2. *-commutative 5.4%

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

      3. *-commutative 5.4%

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

      4. unpow2 5.4%

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

      5. rem-square-sqrt 5.4%

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

   8. Simplified 5.4%

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

      1. pow1/2 5.6%

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

      2. *-commutative 5.6%

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

   10. Applied egg-rr 5.6%

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

   if 4.49999999999999964e-174 < A

   1. Initial program 24.2%

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

   3. Taylor expanded in C around 0 9.7%

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

      1. mul-1-neg 9.7%

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

      2. distribute-rgt-neg-in 9.7%

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

      3. +-commutative 9.7%

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

      4. unpow2 9.7%

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

      5. unpow2 9.7%

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

      6. hypot-def 17.8%

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

   5. Simplified 17.8%

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

   6. Taylor expanded in B around 0 10.4%

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

      1. mul-1-neg 10.4%

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

      2. *-commutative 10.4%

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

      3. unpow2 10.4%

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

      4. rem-square-sqrt 10.5%

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

      5. *-commutative 10.5%

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

   8. Simplified 10.5%

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

4. Final simplification 7.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq 4.5 \cdot 10^{-174}:\\ \;\;\;\;{\left(C \cdot F\right)}^{0.5} \cdot \frac{-2}{B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{A \cdot F} \cdot \frac{-2}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 8.3% accurate, 5.6× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := \frac{-2}{B\_m}\\ \mathbf{if}\;A \leq 3.2 \cdot 10^{-171}:\\ \;\;\;\;\sqrt{C \cdot F} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{A \cdot F} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (/ (- 2.0) B_m)))
   (if (<= A 3.2e-171) (* (sqrt (* C F)) t_0) (* (sqrt (* A F)) t_0))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double t_0 = -2.0 / B_m;
	double tmp;
	if (A <= 3.2e-171) {
		tmp = sqrt((C * F)) * t_0;
	} else {
		tmp = sqrt((A * F)) * t_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) :: t_0
    real(8) :: tmp
    t_0 = -2.0d0 / b_m
    if (a <= 3.2d-171) then
        tmp = sqrt((c * f)) * t_0
    else
        tmp = sqrt((a * f)) * t_0
    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 = -2.0 / B_m;
	double tmp;
	if (A <= 3.2e-171) {
		tmp = Math.sqrt((C * F)) * t_0;
	} else {
		tmp = Math.sqrt((A * F)) * t_0;
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	t_0 = -2.0 / B_m
	tmp = 0
	if A <= 3.2e-171:
		tmp = math.sqrt((C * F)) * t_0
	else:
		tmp = math.sqrt((A * F)) * t_0
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	t_0 = Float64(Float64(-2.0) / B_m)
	tmp = 0.0
	if (A <= 3.2e-171)
		tmp = Float64(sqrt(Float64(C * F)) * t_0);
	else
		tmp = Float64(sqrt(Float64(A * F)) * t_0);
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	t_0 = -2.0 / B_m;
	tmp = 0.0;
	if (A <= 3.2e-171)
		tmp = sqrt((C * F)) * t_0;
	else
		tmp = sqrt((A * F)) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[((-2.0) / B$95$m), $MachinePrecision]}, If[LessEqual[A, 3.2e-171], N[(N[Sqrt[N[(C * F), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[Sqrt[N[(A * F), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if A < 3.2000000000000001e-171

   1. Initial program 13.4%

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

   3. Taylor expanded in A around 0 7.1%

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

      1. mul-1-neg 7.1%

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

      2. distribute-rgt-neg-in 7.1%

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

      3. unpow2 7.1%

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

      4. unpow2 7.1%

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

      5. hypot-def 17.8%

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

   5. Simplified 17.8%

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

   6. Taylor expanded in B around 0 5.4%

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

      1. mul-1-neg 5.4%

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

      2. *-commutative 5.4%

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

      3. *-commutative 5.4%

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

      4. unpow2 5.4%

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

      5. rem-square-sqrt 5.4%

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

   8. Simplified 5.4%

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

   if 3.2000000000000001e-171 < A

   1. Initial program 24.2%

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

   3. Taylor expanded in C around 0 9.7%

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

      1. mul-1-neg 9.7%

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

      2. distribute-rgt-neg-in 9.7%

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

      3. +-commutative 9.7%

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

      4. unpow2 9.7%

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

      5. unpow2 9.7%

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

      6. hypot-def 17.8%

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

   5. Simplified 17.8%

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

   6. Taylor expanded in B around 0 10.4%

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

      1. mul-1-neg 10.4%

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

      2. *-commutative 10.4%

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

      3. unpow2 10.4%

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

      4. rem-square-sqrt 10.5%

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

      5. *-commutative 10.5%

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

   8. Simplified 10.5%

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

4. Final simplification 7.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq 3.2 \cdot 10^{-171}:\\ \;\;\;\;\sqrt{C \cdot F} \cdot \frac{-2}{B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{A \cdot F} \cdot \frac{-2}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 5.0% accurate, 5.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 18.1%

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

3. Taylor expanded in A around 0 7.3%

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

   1. mul-1-neg 7.3%

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

   2. distribute-rgt-neg-in 7.3%

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

   3. unpow2 7.3%

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

   4. unpow2 7.3%

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

   5. hypot-def 15.1%

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

5. Simplified 15.1%

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

6. Taylor expanded in B around 0 3.4%

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

   1. mul-1-neg 3.4%

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

   2. *-commutative 3.4%

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

   3. *-commutative 3.4%

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

   4. unpow2 3.4%

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

   5. rem-square-sqrt 3.4%

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

8. Simplified 3.4%

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

9. Final simplification 3.4%

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

Alternative 20: 5.0% accurate, 5.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 18.1%

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

3. Taylor expanded in A around 0 7.3%

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

   1. mul-1-neg 7.3%

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

   2. distribute-rgt-neg-in 7.3%

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

   3. unpow2 7.3%

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

   4. unpow2 7.3%

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

   5. hypot-def 15.1%

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

5. Simplified 15.1%

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

   1. pow1/2 15.1%

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

   2. *-commutative 15.1%

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

   3. unpow-prod-down 20.0%

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

   4. pow1/2 20.0%

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

   5. hypot-udef 8.0%

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

   6. unpow2 8.0%

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

   7. unpow2 8.0%

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

   8. +-commutative 8.0%

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

   9. unpow2 8.0%

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

   10. unpow2 8.0%

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

   11. hypot-def 20.0%

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

   12. pow1/2 20.0%

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

7. Applied egg-rr 20.0%

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

8. Taylor expanded in B around 0 3.4%

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

   1. mul-1-neg 3.4%

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

   2. associate-*l/ 3.4%

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

   3. distribute-neg-frac 3.4%

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

   4. unpow2 3.4%

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

   5. rem-square-sqrt 3.4%

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

   6. *-commutative 3.4%

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

   7. distribute-rgt-neg-in 3.4%

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

   8. *-commutative 3.4%

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

   9. metadata-eval 3.4%

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

10. Simplified 3.4%

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

11. Final simplification 3.4%

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

Reproduce

herbie shell --seed 2024027 
(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))))