ABCF->ab-angle a

Percentage Accurate: 19.2% → 52.9%

Time: 27.3s

Alternatives: 13

Speedup: 6.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 19.2% 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: 52.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{+181}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right)\right)} \cdot \left(-\sqrt{A + \left(C + \mathsf{hypot}\left(A - C, B\right)\right)}\right)}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(B, C\right)} \cdot \sqrt{F}\right) \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
(FPCore (A B C F)
 :precision binary64
 (if (<= (pow B 2.0) 1e+181)
   (/
    (*
     (sqrt (* 2.0 (* F (fma B B (* (* A C) -4.0)))))
     (- (sqrt (+ A (+ C (hypot (- A C) B))))))
    (fma B B (* A (* C -4.0))))
   (* (* (sqrt (+ C (hypot B C))) (sqrt F)) (/ (- (sqrt 2.0)) B))))

C

B = abs(B);
double code(double A, double B, double C, double F) {
	double tmp;
	if (pow(B, 2.0) <= 1e+181) {
		tmp = (sqrt((2.0 * (F * fma(B, B, ((A * C) * -4.0))))) * -sqrt((A + (C + hypot((A - C), B))))) / fma(B, B, (A * (C * -4.0)));
	} else {
		tmp = (sqrt((C + hypot(B, C))) * sqrt(F)) * (-sqrt(2.0) / B);
	}
	return tmp;
}

Julia

B = abs(B)
function code(A, B, C, F)
	tmp = 0.0
	if ((B ^ 2.0) <= 1e+181)
		tmp = Float64(Float64(sqrt(Float64(2.0 * Float64(F * fma(B, B, Float64(Float64(A * C) * -4.0))))) * Float64(-sqrt(Float64(A + Float64(C + hypot(Float64(A - C), B)))))) / fma(B, B, Float64(A * Float64(C * -4.0))));
	else
		tmp = Float64(Float64(sqrt(Float64(C + hypot(B, C))) * sqrt(F)) * Float64(Float64(-sqrt(2.0)) / B));
	end
	return tmp
end

Wolfram

NOTE: B should be positive before calling this function
code[A_, B_, C_, F_] := If[LessEqual[N[Power[B, 2.0], $MachinePrecision], 1e+181], N[(N[(N[Sqrt[N[(2.0 * N[(F * N[(B * B + N[(N[(A * C), $MachinePrecision] * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(A + N[(C + N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / N[(B * B + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(C + N[Sqrt[B ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] * N[((-N[Sqrt[2.0], $MachinePrecision]) / B), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B 2) < 9.9999999999999992e180

   1. Initial program 23.8%

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

   3. Simplified 34.2%

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

      1. associate-*r* 34.2%

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

      2. associate-+r+ 32.7%

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

      3. hypot-udef 23.8%

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

      4. unpow2 23.8%

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

      5. unpow2 23.8%

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

      6. +-commutative 23.8%

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

      7. sqrt-prod 27.1%

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

      8. *-commutative 27.1%

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

      9. associate-*r* 27.1%

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

      10. associate-+l+ 27.4%

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

   5. Applied egg-rr 41.2%

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

   if 9.9999999999999992e180 < (pow.f64 B 2)

   1. Initial program 5.8%

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

   2. Taylor expanded in A around 0 5.8%

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

      1. mul-1-neg 5.8%

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

      2. *-commutative 5.8%

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

      3. distribute-rgt-neg-in 5.8%

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

      4. unpow2 5.8%

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

      5. unpow2 5.8%

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

      6. hypot-def 26.0%

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

   4. Simplified 26.0%

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

      1. pow1/2 26.0%

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

      2. *-commutative 26.0%

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

      3. unpow-prod-down 36.1%

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

      4. pow1/2 36.1%

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

      5. pow1/2 36.1%

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

   6. Applied egg-rr 36.1%

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

4. Final simplification 39.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{+181}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, \left(A \cdot C\right) \cdot -4\right)\right)} \cdot \left(-\sqrt{A + \left(C + \mathsf{hypot}\left(A - C, B\right)\right)}\right)}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(B, C\right)} \cdot \sqrt{F}\right) \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 2: 41.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := {B}^{2} - C \cdot \left(A \cdot 4\right)\\ \mathbf{if}\;{B}^{2} \leq 1.5 \cdot 10^{-139}:\\ \;\;\;\;\frac{-\sqrt{\left(F \cdot t_0\right) \cdot \left(2 \cdot \left(A + A\right)\right)}}{t_0}\\ \mathbf{elif}\;{B}^{2} \leq 10^{-18}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot t_1\right)\right) \cdot \left(2 \cdot C\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{F} \cdot \sqrt{\frac{1}{B}}\right) \cdot \left(-\sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (fma B B (* A (* C -4.0)))) (t_1 (- (pow B 2.0) (* C (* A 4.0)))))
   (if (<= (pow B 2.0) 1.5e-139)
     (/ (- (sqrt (* (* F t_0) (* 2.0 (+ A A))))) t_0)
     (if (<= (pow B 2.0) 1e-18)
       (/ (- (sqrt (* (* 2.0 (* F t_1)) (* 2.0 C)))) t_1)
       (* (* (sqrt F) (sqrt (/ 1.0 B))) (- (sqrt 2.0)))))))

C

B = abs(B);
double code(double A, double B, double C, double F) {
	double t_0 = fma(B, B, (A * (C * -4.0)));
	double t_1 = pow(B, 2.0) - (C * (A * 4.0));
	double tmp;
	if (pow(B, 2.0) <= 1.5e-139) {
		tmp = -sqrt(((F * t_0) * (2.0 * (A + A)))) / t_0;
	} else if (pow(B, 2.0) <= 1e-18) {
		tmp = -sqrt(((2.0 * (F * t_1)) * (2.0 * C))) / t_1;
	} else {
		tmp = (sqrt(F) * sqrt((1.0 / B))) * -sqrt(2.0);
	}
	return tmp;
}

Julia

B = abs(B)
function code(A, B, C, F)
	t_0 = fma(B, B, Float64(A * Float64(C * -4.0)))
	t_1 = Float64((B ^ 2.0) - Float64(C * Float64(A * 4.0)))
	tmp = 0.0
	if ((B ^ 2.0) <= 1.5e-139)
		tmp = Float64(Float64(-sqrt(Float64(Float64(F * t_0) * Float64(2.0 * Float64(A + A))))) / t_0);
	elseif ((B ^ 2.0) <= 1e-18)
		tmp = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(F * t_1)) * Float64(2.0 * C)))) / t_1);
	else
		tmp = Float64(Float64(sqrt(F) * sqrt(Float64(1.0 / B))) * Float64(-sqrt(2.0)));
	end
	return tmp
end

Wolfram

NOTE: B should be positive before calling this function
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(B * B + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[B, 2.0], $MachinePrecision] - N[(C * N[(A * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B, 2.0], $MachinePrecision], 1.5e-139], N[((-N[Sqrt[N[(N[(F * t$95$0), $MachinePrecision] * N[(2.0 * N[(A + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], If[LessEqual[N[Power[B, 2.0], $MachinePrecision], 1e-18], N[((-N[Sqrt[N[(N[(2.0 * N[(F * t$95$1), $MachinePrecision]), $MachinePrecision] * N[(2.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision], N[(N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(1.0 / B), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B 2) < 1.5e-139

   1. Initial program 15.9%

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

   3. Simplified 27.4%

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

   4. Taylor expanded in A around inf 26.4%

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

      1. distribute-rgt1-in 26.4%

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

      2. metadata-eval 26.4%

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

      3. mul0-lft 26.4%

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

   6. Simplified 26.4%

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

   if 1.5e-139 < (pow.f64 B 2) < 1.0000000000000001e-18

   1. Initial program 36.2%

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

   2. Taylor expanded in A around -inf 43.7%

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

   if 1.0000000000000001e-18 < (pow.f64 B 2)

   1. Initial program 13.2%

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

   2. Taylor expanded in A around 0 7.3%

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

      1. mul-1-neg 7.3%

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

      2. *-commutative 7.3%

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

      3. distribute-rgt-neg-in 7.3%

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

      4. unpow2 7.3%

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

      5. unpow2 7.3%

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

      6. hypot-def 22.4%

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

   4. Simplified 22.4%

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

   5. Taylor expanded in C around 0 19.1%

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

      1. mul-1-neg 19.1%

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

   7. Simplified 19.1%

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

      1. pow1/2 19.1%

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

      2. div-inv 19.1%

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

      3. unpow-prod-down 26.5%

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

      4. pow1/2 26.5%

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

   9. Applied egg-rr 26.5%

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

       1. unpow1/2 26.5%

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

   11. Simplified 26.5%

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

4. Final simplification 28.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 1.5 \cdot 10^{-139}:\\ \;\;\;\;\frac{-\sqrt{\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot \left(2 \cdot \left(A + A\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;{B}^{2} \leq 10^{-18}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right)\right) \cdot \left(2 \cdot C\right)}}{{B}^{2} - C \cdot \left(A \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{F} \cdot \sqrt{\frac{1}{B}}\right) \cdot \left(-\sqrt{2}\right)\\ \end{array} \]

Alternative 3: 48.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;{B}^{2} \leq 10^{+181}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot t_0\right) \cdot \left(F \cdot \left(A + \left(C + \mathsf{hypot}\left(A - C, B\right)\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(B, C\right)} \cdot \sqrt{F}\right) \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (fma B B (* A (* C -4.0)))))
   (if (<= (pow B 2.0) 1e+181)
     (/ (- (sqrt (* (* 2.0 t_0) (* F (+ A (+ C (hypot (- A C) B))))))) t_0)
     (* (* (sqrt (+ C (hypot B C))) (sqrt F)) (/ (- (sqrt 2.0)) B)))))

C

B = abs(B);
double code(double A, double B, double C, double F) {
	double t_0 = fma(B, B, (A * (C * -4.0)));
	double tmp;
	if (pow(B, 2.0) <= 1e+181) {
		tmp = -sqrt(((2.0 * t_0) * (F * (A + (C + hypot((A - C), B)))))) / t_0;
	} else {
		tmp = (sqrt((C + hypot(B, C))) * sqrt(F)) * (-sqrt(2.0) / B);
	}
	return tmp;
}

Julia

B = abs(B)
function code(A, B, C, F)
	t_0 = fma(B, B, Float64(A * Float64(C * -4.0)))
	tmp = 0.0
	if ((B ^ 2.0) <= 1e+181)
		tmp = Float64(Float64(-sqrt(Float64(Float64(2.0 * t_0) * Float64(F * Float64(A + Float64(C + hypot(Float64(A - C), B))))))) / t_0);
	else
		tmp = Float64(Float64(sqrt(Float64(C + hypot(B, C))) * sqrt(F)) * Float64(Float64(-sqrt(2.0)) / B));
	end
	return tmp
end

Wolfram

NOTE: B should be positive before calling this function
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(B * B + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B, 2.0], $MachinePrecision], 1e+181], N[((-N[Sqrt[N[(N[(2.0 * t$95$0), $MachinePrecision] * N[(F * N[(A + N[(C + N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], N[(N[(N[Sqrt[N[(C + N[Sqrt[B ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] * N[((-N[Sqrt[2.0], $MachinePrecision]) / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B 2) < 9.9999999999999992e180

   1. Initial program 23.8%

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

      1. neg-sub0 23.8%

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

      2. div-sub 23.8%

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

      3. associate-*l* 23.8%

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

   3. Applied egg-rr 34.1%

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

      1. div0 34.1%

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

      2. neg-sub0 34.1%

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

      3. distribute-neg-frac 34.1%

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

   5. Simplified 34.8%

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

   if 9.9999999999999992e180 < (pow.f64 B 2)

   1. Initial program 5.8%

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

   2. Taylor expanded in A around 0 5.8%

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

      1. mul-1-neg 5.8%

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

      2. *-commutative 5.8%

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

      3. distribute-rgt-neg-in 5.8%

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

      4. unpow2 5.8%

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

      5. unpow2 5.8%

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

      6. hypot-def 26.0%

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

   4. Simplified 26.0%

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

      1. pow1/2 26.0%

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

      2. *-commutative 26.0%

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

      3. unpow-prod-down 36.1%

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

      4. pow1/2 36.1%

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

      5. pow1/2 36.1%

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

   6. Applied egg-rr 36.1%

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

4. Final simplification 35.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{+181}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot \left(F \cdot \left(A + \left(C + \mathsf{hypot}\left(A - C, B\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(B, C\right)} \cdot \sqrt{F}\right) \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 4: 45.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ \begin{array}{l} t_0 := {B}^{2} - C \cdot \left(A \cdot 4\right)\\ t_1 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;B \leq 3.9 \cdot 10^{-70}:\\ \;\;\;\;\frac{-\sqrt{\left(F \cdot t_1\right) \cdot \left(2 \cdot \left(A + A\right)\right)}}{t_1}\\ \mathbf{elif}\;B \leq 3.1 \cdot 10^{-22}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot t_0\right)\right) \cdot \left(2 \cdot C\right)}}{t_0}\\ \mathbf{elif}\;B \leq 2 \cdot 10^{+29}:\\ \;\;\;\;\frac{-\sqrt{F \cdot \left(t_1 \cdot \left(2 \cdot \left(\mathsf{hypot}\left(A - C, B\right) + \left(A + C\right)\right)\right)\right)}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(B, C\right)} \cdot \sqrt{F}\right) \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (pow B 2.0) (* C (* A 4.0)))) (t_1 (fma B B (* A (* C -4.0)))))
   (if (<= B 3.9e-70)
     (/ (- (sqrt (* (* F t_1) (* 2.0 (+ A A))))) t_1)
     (if (<= B 3.1e-22)
       (/ (- (sqrt (* (* 2.0 (* F t_0)) (* 2.0 C)))) t_0)
       (if (<= B 2e+29)
         (/ (- (sqrt (* F (* t_1 (* 2.0 (+ (hypot (- A C) B) (+ A C))))))) t_1)
         (* (* (sqrt (+ C (hypot B C))) (sqrt F)) (/ (- (sqrt 2.0)) B)))))))

C

B = abs(B);
double code(double A, double B, double C, double F) {
	double t_0 = pow(B, 2.0) - (C * (A * 4.0));
	double t_1 = fma(B, B, (A * (C * -4.0)));
	double tmp;
	if (B <= 3.9e-70) {
		tmp = -sqrt(((F * t_1) * (2.0 * (A + A)))) / t_1;
	} else if (B <= 3.1e-22) {
		tmp = -sqrt(((2.0 * (F * t_0)) * (2.0 * C))) / t_0;
	} else if (B <= 2e+29) {
		tmp = -sqrt((F * (t_1 * (2.0 * (hypot((A - C), B) + (A + C)))))) / t_1;
	} else {
		tmp = (sqrt((C + hypot(B, C))) * sqrt(F)) * (-sqrt(2.0) / B);
	}
	return tmp;
}

Julia

B = abs(B)
function code(A, B, C, F)
	t_0 = Float64((B ^ 2.0) - Float64(C * Float64(A * 4.0)))
	t_1 = fma(B, B, Float64(A * Float64(C * -4.0)))
	tmp = 0.0
	if (B <= 3.9e-70)
		tmp = Float64(Float64(-sqrt(Float64(Float64(F * t_1) * Float64(2.0 * Float64(A + A))))) / t_1);
	elseif (B <= 3.1e-22)
		tmp = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(F * t_0)) * Float64(2.0 * C)))) / t_0);
	elseif (B <= 2e+29)
		tmp = Float64(Float64(-sqrt(Float64(F * Float64(t_1 * Float64(2.0 * Float64(hypot(Float64(A - C), B) + Float64(A + C))))))) / t_1);
	else
		tmp = Float64(Float64(sqrt(Float64(C + hypot(B, C))) * sqrt(F)) * Float64(Float64(-sqrt(2.0)) / B));
	end
	return tmp
end

Wolfram

NOTE: B should be positive before calling this function
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[Power[B, 2.0], $MachinePrecision] - N[(C * N[(A * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(B * B + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, 3.9e-70], N[((-N[Sqrt[N[(N[(F * t$95$1), $MachinePrecision] * N[(2.0 * N[(A + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision], If[LessEqual[B, 3.1e-22], 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[B, 2e+29], N[((-N[Sqrt[N[(F * N[(t$95$1 * N[(2.0 * N[(N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B ^ 2], $MachinePrecision] + N[(A + C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision], N[(N[(N[Sqrt[N[(C + N[Sqrt[B ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] * N[((-N[Sqrt[2.0], $MachinePrecision]) / B), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if B < 3.90000000000000019e-70

   1. Initial program 17.9%

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

   3. Simplified 25.2%

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

   4. Taylor expanded in A around inf 18.2%

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

      1. distribute-rgt1-in 18.2%

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

      2. metadata-eval 18.2%

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

      3. mul0-lft 18.2%

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

   6. Simplified 18.2%

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

   if 3.90000000000000019e-70 < B < 3.10000000000000013e-22

   1. Initial program 28.3%

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

   2. Taylor expanded in A around -inf 41.4%

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

   if 3.10000000000000013e-22 < B < 1.99999999999999983e29

   1. Initial program 21.9%

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

      1. neg-sub0 21.9%

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

      2. div-sub 21.9%

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

      3. associate-*l* 21.9%

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

   3. Applied egg-rr 40.8%

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

   5. Simplified 41.3%

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

   if 1.99999999999999983e29 < B

   1. Initial program 8.5%

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

   2. Taylor expanded in A around 0 12.6%

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

      1. mul-1-neg 12.6%

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

      2. *-commutative 12.6%

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

      3. distribute-rgt-neg-in 12.6%

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

      4. unpow2 12.6%

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

      5. unpow2 12.6%

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

      6. hypot-def 50.1%

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

   4. Simplified 50.1%

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

      1. pow1/2 50.1%

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

      2. *-commutative 50.1%

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

      3. unpow-prod-down 69.4%

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

      4. pow1/2 69.4%

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

      5. pow1/2 69.4%

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

   6. Applied egg-rr 69.4%

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

4. Final simplification 30.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 3.9 \cdot 10^{-70}:\\ \;\;\;\;\frac{-\sqrt{\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot \left(2 \cdot \left(A + A\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 3.1 \cdot 10^{-22}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right)\right) \cdot \left(2 \cdot C\right)}}{{B}^{2} - C \cdot \left(A \cdot 4\right)}\\ \mathbf{elif}\;B \leq 2 \cdot 10^{+29}:\\ \;\;\;\;\frac{-\sqrt{F \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(2 \cdot \left(\mathsf{hypot}\left(A - C, B\right) + \left(A + C\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(B, C\right)} \cdot \sqrt{F}\right) \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 5: 44.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ \begin{array}{l} t_0 := {B}^{2} - C \cdot \left(A \cdot 4\right)\\ t_1 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;B \leq 3.9 \cdot 10^{-70}:\\ \;\;\;\;\frac{-\sqrt{\left(F \cdot t_1\right) \cdot \left(2 \cdot \left(A + A\right)\right)}}{t_1}\\ \mathbf{elif}\;B \leq 8 \cdot 10^{-23}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot t_0\right)\right) \cdot \left(2 \cdot C\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(B, C\right)} \cdot \sqrt{F}\right) \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (pow B 2.0) (* C (* A 4.0)))) (t_1 (fma B B (* A (* C -4.0)))))
   (if (<= B 3.9e-70)
     (/ (- (sqrt (* (* F t_1) (* 2.0 (+ A A))))) t_1)
     (if (<= B 8e-23)
       (/ (- (sqrt (* (* 2.0 (* F t_0)) (* 2.0 C)))) t_0)
       (* (* (sqrt (+ C (hypot B C))) (sqrt F)) (/ (- (sqrt 2.0)) B))))))

C

B = abs(B);
double code(double A, double B, double C, double F) {
	double t_0 = pow(B, 2.0) - (C * (A * 4.0));
	double t_1 = fma(B, B, (A * (C * -4.0)));
	double tmp;
	if (B <= 3.9e-70) {
		tmp = -sqrt(((F * t_1) * (2.0 * (A + A)))) / t_1;
	} else if (B <= 8e-23) {
		tmp = -sqrt(((2.0 * (F * t_0)) * (2.0 * C))) / t_0;
	} else {
		tmp = (sqrt((C + hypot(B, C))) * sqrt(F)) * (-sqrt(2.0) / B);
	}
	return tmp;
}

Julia

B = abs(B)
function code(A, B, C, F)
	t_0 = Float64((B ^ 2.0) - Float64(C * Float64(A * 4.0)))
	t_1 = fma(B, B, Float64(A * Float64(C * -4.0)))
	tmp = 0.0
	if (B <= 3.9e-70)
		tmp = Float64(Float64(-sqrt(Float64(Float64(F * t_1) * Float64(2.0 * Float64(A + A))))) / t_1);
	elseif (B <= 8e-23)
		tmp = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(F * t_0)) * Float64(2.0 * C)))) / t_0);
	else
		tmp = Float64(Float64(sqrt(Float64(C + hypot(B, C))) * sqrt(F)) * Float64(Float64(-sqrt(2.0)) / B));
	end
	return tmp
end

Wolfram

NOTE: B should be positive before calling this function
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[Power[B, 2.0], $MachinePrecision] - N[(C * N[(A * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(B * B + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, 3.9e-70], N[((-N[Sqrt[N[(N[(F * t$95$1), $MachinePrecision] * N[(2.0 * N[(A + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision], If[LessEqual[B, 8e-23], 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[N[(C + N[Sqrt[B ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] * N[((-N[Sqrt[2.0], $MachinePrecision]) / B), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if B < 3.90000000000000019e-70

   1. Initial program 17.9%

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

   3. Simplified 25.2%

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

   4. Taylor expanded in A around inf 18.2%

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

      1. distribute-rgt1-in 18.2%

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

      2. metadata-eval 18.2%

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

      3. mul0-lft 18.2%

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

   6. Simplified 18.2%

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

   if 3.90000000000000019e-70 < B < 7.99999999999999968e-23

   1. Initial program 28.3%

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

   2. Taylor expanded in A around -inf 41.4%

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

   if 7.99999999999999968e-23 < B

   1. Initial program 10.8%

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

   2. Taylor expanded in A around 0 14.3%

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

      1. mul-1-neg 14.3%

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

      2. *-commutative 14.3%

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

      3. distribute-rgt-neg-in 14.3%

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

      4. unpow2 14.3%

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

      5. unpow2 14.3%

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

      6. hypot-def 47.2%

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

   4. Simplified 47.2%

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

      1. pow1/2 47.2%

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

      2. *-commutative 47.2%

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

      3. unpow-prod-down 63.3%

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

      4. pow1/2 63.3%

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

      5. pow1/2 63.3%

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

   6. Applied egg-rr 63.3%

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

4. Final simplification 30.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 3.9 \cdot 10^{-70}:\\ \;\;\;\;\frac{-\sqrt{\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot \left(2 \cdot \left(A + A\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 8 \cdot 10^{-23}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right)\right) \cdot \left(2 \cdot C\right)}}{{B}^{2} - C \cdot \left(A \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(B, C\right)} \cdot \sqrt{F}\right) \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 6: 39.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ \begin{array}{l} t_0 := {B}^{2} - C \cdot \left(A \cdot 4\right)\\ t_1 := \frac{-\sqrt{\left(2 \cdot \left(F \cdot t_0\right)\right) \cdot \left(2 \cdot C\right)}}{t_0}\\ \mathbf{if}\;B \leq 1.36 \cdot 10^{-155}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;B \leq 3.5 \cdot 10^{-70}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left(C \cdot \left(A \cdot -4\right)\right)\right)\right) \cdot \left(A + \mathsf{hypot}\left(B, A\right)\right)}}{t_0}\\ \mathbf{elif}\;B \leq 4.8 \cdot 10^{-7}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{F} \cdot \sqrt{\frac{1}{B}}\right) \cdot \left(-\sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (pow B 2.0) (* C (* A 4.0))))
        (t_1 (/ (- (sqrt (* (* 2.0 (* F t_0)) (* 2.0 C)))) t_0)))
   (if (<= B 1.36e-155)
     t_1
     (if (<= B 3.5e-70)
       (/ (- (sqrt (* (* 2.0 (* F (* C (* A -4.0)))) (+ A (hypot B A))))) t_0)
       (if (<= B 4.8e-7)
         t_1
         (* (* (sqrt F) (sqrt (/ 1.0 B))) (- (sqrt 2.0))))))))

C

B = abs(B);
double code(double A, double B, double C, double F) {
	double t_0 = pow(B, 2.0) - (C * (A * 4.0));
	double t_1 = -sqrt(((2.0 * (F * t_0)) * (2.0 * C))) / t_0;
	double tmp;
	if (B <= 1.36e-155) {
		tmp = t_1;
	} else if (B <= 3.5e-70) {
		tmp = -sqrt(((2.0 * (F * (C * (A * -4.0)))) * (A + hypot(B, A)))) / t_0;
	} else if (B <= 4.8e-7) {
		tmp = t_1;
	} else {
		tmp = (sqrt(F) * sqrt((1.0 / B))) * -sqrt(2.0);
	}
	return tmp;
}

Java

B = Math.abs(B);
public static double code(double A, double B, double C, double F) {
	double t_0 = Math.pow(B, 2.0) - (C * (A * 4.0));
	double t_1 = -Math.sqrt(((2.0 * (F * t_0)) * (2.0 * C))) / t_0;
	double tmp;
	if (B <= 1.36e-155) {
		tmp = t_1;
	} else if (B <= 3.5e-70) {
		tmp = -Math.sqrt(((2.0 * (F * (C * (A * -4.0)))) * (A + Math.hypot(B, A)))) / t_0;
	} else if (B <= 4.8e-7) {
		tmp = t_1;
	} else {
		tmp = (Math.sqrt(F) * Math.sqrt((1.0 / B))) * -Math.sqrt(2.0);
	}
	return tmp;
}

Python

B = abs(B)
def code(A, B, C, F):
	t_0 = math.pow(B, 2.0) - (C * (A * 4.0))
	t_1 = -math.sqrt(((2.0 * (F * t_0)) * (2.0 * C))) / t_0
	tmp = 0
	if B <= 1.36e-155:
		tmp = t_1
	elif B <= 3.5e-70:
		tmp = -math.sqrt(((2.0 * (F * (C * (A * -4.0)))) * (A + math.hypot(B, A)))) / t_0
	elif B <= 4.8e-7:
		tmp = t_1
	else:
		tmp = (math.sqrt(F) * math.sqrt((1.0 / B))) * -math.sqrt(2.0)
	return tmp

Julia

B = abs(B)
function code(A, B, C, F)
	t_0 = Float64((B ^ 2.0) - Float64(C * Float64(A * 4.0)))
	t_1 = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(F * t_0)) * Float64(2.0 * C)))) / t_0)
	tmp = 0.0
	if (B <= 1.36e-155)
		tmp = t_1;
	elseif (B <= 3.5e-70)
		tmp = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(F * Float64(C * Float64(A * -4.0)))) * Float64(A + hypot(B, A))))) / t_0);
	elseif (B <= 4.8e-7)
		tmp = t_1;
	else
		tmp = Float64(Float64(sqrt(F) * sqrt(Float64(1.0 / B))) * Float64(-sqrt(2.0)));
	end
	return tmp
end

MATLAB

B = abs(B)
function tmp_2 = code(A, B, C, F)
	t_0 = (B ^ 2.0) - (C * (A * 4.0));
	t_1 = -sqrt(((2.0 * (F * t_0)) * (2.0 * C))) / t_0;
	tmp = 0.0;
	if (B <= 1.36e-155)
		tmp = t_1;
	elseif (B <= 3.5e-70)
		tmp = -sqrt(((2.0 * (F * (C * (A * -4.0)))) * (A + hypot(B, A)))) / t_0;
	elseif (B <= 4.8e-7)
		tmp = t_1;
	else
		tmp = (sqrt(F) * sqrt((1.0 / B))) * -sqrt(2.0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: B should be positive before calling this function
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(N[Power[B, 2.0], $MachinePrecision] - N[(C * N[(A * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = 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[B, 1.36e-155], t$95$1, If[LessEqual[B, 3.5e-70], N[((-N[Sqrt[N[(N[(2.0 * N[(F * N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(A + N[Sqrt[B ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], If[LessEqual[B, 4.8e-7], t$95$1, N[(N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(1.0 / B), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if B < 1.35999999999999996e-155 or 3.49999999999999974e-70 < B < 4.79999999999999957e-7

   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. Taylor expanded in A around -inf 17.2%

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

   if 1.35999999999999996e-155 < B < 3.49999999999999974e-70

   1. Initial program 14.0%

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

   2. Taylor expanded in C around 0 25.6%

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

      1. +-commutative 25.6%

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

      2. unpow2 25.6%

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

      3. unpow2 25.6%

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

      4. hypot-def 31.2%

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

   4. Simplified 31.2%

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

   5. Taylor expanded in B around 0 31.2%

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

      1. *-commutative 31.2%

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

      2. metadata-eval 31.2%

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

      3. distribute-rgt-neg-in 31.2%

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

      4. *-commutative 31.2%

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

      5. associate-*r* 31.2%

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

      6. *-commutative 31.2%

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

      7. distribute-lft-neg-out 31.2%

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

      8. *-commutative 31.2%

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

      9. distribute-rgt-neg-in 31.2%

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

      10. metadata-eval 31.2%

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

   7. Simplified 31.2%

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

   if 4.79999999999999957e-7 < B

   1. Initial program 11.0%

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

   2. Taylor expanded in A around 0 14.6%

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

      1. mul-1-neg 14.6%

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

      2. *-commutative 14.6%

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

      3. distribute-rgt-neg-in 14.6%

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

      4. unpow2 14.6%

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

      5. unpow2 14.6%

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

      6. hypot-def 47.0%

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

   4. Simplified 47.0%

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

   5. Taylor expanded in C around 0 41.0%

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

      1. mul-1-neg 41.0%

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

   7. Simplified 41.0%

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

      1. pow1/2 41.0%

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

      2. div-inv 41.0%

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

      3. unpow-prod-down 59.4%

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

      4. pow1/2 59.4%

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

   9. Applied egg-rr 59.4%

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

       1. unpow1/2 59.4%

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

   11. Simplified 59.4%

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

4. Final simplification 27.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.36 \cdot 10^{-155}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right)\right) \cdot \left(2 \cdot C\right)}}{{B}^{2} - C \cdot \left(A \cdot 4\right)}\\ \mathbf{elif}\;B \leq 3.5 \cdot 10^{-70}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left(C \cdot \left(A \cdot -4\right)\right)\right)\right) \cdot \left(A + \mathsf{hypot}\left(B, A\right)\right)}}{{B}^{2} - C \cdot \left(A \cdot 4\right)}\\ \mathbf{elif}\;B \leq 4.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right)\right) \cdot \left(2 \cdot C\right)}}{{B}^{2} - C \cdot \left(A \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{F} \cdot \sqrt{\frac{1}{B}}\right) \cdot \left(-\sqrt{2}\right)\\ \end{array} \]

Alternative 7: 39.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ \begin{array}{l} t_0 := -\sqrt{2}\\ \mathbf{if}\;F \leq 1.9 \cdot 10^{-307}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left(C \cdot \left(A \cdot -4\right)\right)\right)\right) \cdot \left(A + \mathsf{hypot}\left(B, A\right)\right)}}{{B}^{2} - C \cdot \left(A \cdot 4\right)}\\ \mathbf{elif}\;F \leq 2.65 \cdot 10^{+23}:\\ \;\;\;\;\frac{t_0}{B} \cdot \sqrt{F \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{F} \cdot \sqrt{\frac{1}{B}}\right) \cdot t_0\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (sqrt 2.0))))
   (if (<= F 1.9e-307)
     (/
      (- (sqrt (* (* 2.0 (* F (* C (* A -4.0)))) (+ A (hypot B A)))))
      (- (pow B 2.0) (* C (* A 4.0))))
     (if (<= F 2.65e+23)
       (* (/ t_0 B) (sqrt (* F (+ C (hypot B C)))))
       (* (* (sqrt F) (sqrt (/ 1.0 B))) t_0)))))

C

B = abs(B);
double code(double A, double B, double C, double F) {
	double t_0 = -sqrt(2.0);
	double tmp;
	if (F <= 1.9e-307) {
		tmp = -sqrt(((2.0 * (F * (C * (A * -4.0)))) * (A + hypot(B, A)))) / (pow(B, 2.0) - (C * (A * 4.0)));
	} else if (F <= 2.65e+23) {
		tmp = (t_0 / B) * sqrt((F * (C + hypot(B, C))));
	} else {
		tmp = (sqrt(F) * sqrt((1.0 / B))) * t_0;
	}
	return tmp;
}

Java

B = Math.abs(B);
public static double code(double A, double B, double C, double F) {
	double t_0 = -Math.sqrt(2.0);
	double tmp;
	if (F <= 1.9e-307) {
		tmp = -Math.sqrt(((2.0 * (F * (C * (A * -4.0)))) * (A + Math.hypot(B, A)))) / (Math.pow(B, 2.0) - (C * (A * 4.0)));
	} else if (F <= 2.65e+23) {
		tmp = (t_0 / B) * Math.sqrt((F * (C + Math.hypot(B, C))));
	} else {
		tmp = (Math.sqrt(F) * Math.sqrt((1.0 / B))) * t_0;
	}
	return tmp;
}

Python

B = abs(B)
def code(A, B, C, F):
	t_0 = -math.sqrt(2.0)
	tmp = 0
	if F <= 1.9e-307:
		tmp = -math.sqrt(((2.0 * (F * (C * (A * -4.0)))) * (A + math.hypot(B, A)))) / (math.pow(B, 2.0) - (C * (A * 4.0)))
	elif F <= 2.65e+23:
		tmp = (t_0 / B) * math.sqrt((F * (C + math.hypot(B, C))))
	else:
		tmp = (math.sqrt(F) * math.sqrt((1.0 / B))) * t_0
	return tmp

Julia

B = abs(B)
function code(A, B, C, F)
	t_0 = Float64(-sqrt(2.0))
	tmp = 0.0
	if (F <= 1.9e-307)
		tmp = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(F * Float64(C * Float64(A * -4.0)))) * Float64(A + hypot(B, A))))) / Float64((B ^ 2.0) - Float64(C * Float64(A * 4.0))));
	elseif (F <= 2.65e+23)
		tmp = Float64(Float64(t_0 / B) * sqrt(Float64(F * Float64(C + hypot(B, C)))));
	else
		tmp = Float64(Float64(sqrt(F) * sqrt(Float64(1.0 / B))) * t_0);
	end
	return tmp
end

MATLAB

B = abs(B)
function tmp_2 = code(A, B, C, F)
	t_0 = -sqrt(2.0);
	tmp = 0.0;
	if (F <= 1.9e-307)
		tmp = -sqrt(((2.0 * (F * (C * (A * -4.0)))) * (A + hypot(B, A)))) / ((B ^ 2.0) - (C * (A * 4.0)));
	elseif (F <= 2.65e+23)
		tmp = (t_0 / B) * sqrt((F * (C + hypot(B, C))));
	else
		tmp = (sqrt(F) * sqrt((1.0 / B))) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: B should be positive before calling this function
code[A_, B_, C_, F_] := Block[{t$95$0 = (-N[Sqrt[2.0], $MachinePrecision])}, If[LessEqual[F, 1.9e-307], N[((-N[Sqrt[N[(N[(2.0 * N[(F * N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(A + N[Sqrt[B ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(N[Power[B, 2.0], $MachinePrecision] - N[(C * N[(A * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 2.65e+23], N[(N[(t$95$0 / B), $MachinePrecision] * N[Sqrt[N[(F * N[(C + N[Sqrt[B ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(1.0 / B), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < 1.89999999999999993e-307

   1. Initial program 26.8%

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

   2. Taylor expanded in C around 0 20.1%

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

      1. +-commutative 20.1%

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

      2. unpow2 20.1%

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

      3. unpow2 20.1%

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

      4. hypot-def 29.5%

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

   4. Simplified 29.5%

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

   5. Taylor expanded in B around 0 28.4%

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

      1. *-commutative 28.4%

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

      2. metadata-eval 28.4%

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

      3. distribute-rgt-neg-in 28.4%

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

      4. *-commutative 28.4%

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

      5. associate-*r* 28.4%

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

      6. *-commutative 28.4%

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

      7. distribute-lft-neg-out 28.4%

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

      8. *-commutative 28.4%

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

      9. distribute-rgt-neg-in 28.4%

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

      10. metadata-eval 28.4%

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

   7. Simplified 28.4%

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

   if 1.89999999999999993e-307 < F < 2.6500000000000001e23

   1. Initial program 17.0%

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

   2. Taylor expanded in A around 0 7.6%

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

      1. mul-1-neg 7.6%

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

      2. *-commutative 7.6%

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

      3. distribute-rgt-neg-in 7.6%

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

      4. unpow2 7.6%

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

      5. unpow2 7.6%

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

      6. hypot-def 25.3%

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

   4. Simplified 25.3%

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

   if 2.6500000000000001e23 < F

   1. Initial program 12.7%

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

   2. Taylor expanded in A around 0 5.9%

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

      1. mul-1-neg 5.9%

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

      2. *-commutative 5.9%

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

      3. distribute-rgt-neg-in 5.9%

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

      4. unpow2 5.9%

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

      5. unpow2 5.9%

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

      6. hypot-def 6.3%

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

   4. Simplified 6.3%

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

   5. Taylor expanded in C around 0 14.7%

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

      1. mul-1-neg 14.7%

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

   7. Simplified 14.7%

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

      1. pow1/2 15.3%

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

      2. div-inv 15.3%

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

      3. unpow-prod-down 14.8%

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

      4. pow1/2 14.8%

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

   9. Applied egg-rr 14.8%

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

       1. unpow1/2 14.8%

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

   11. Simplified 14.8%

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

4. Final simplification 21.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 1.9 \cdot 10^{-307}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left(C \cdot \left(A \cdot -4\right)\right)\right)\right) \cdot \left(A + \mathsf{hypot}\left(B, A\right)\right)}}{{B}^{2} - C \cdot \left(A \cdot 4\right)}\\ \mathbf{elif}\;F \leq 2.65 \cdot 10^{+23}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{F} \cdot \sqrt{\frac{1}{B}}\right) \cdot \left(-\sqrt{2}\right)\\ \end{array} \]

Alternative 8: 36.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ \begin{array}{l} t_0 := -\sqrt{2}\\ \mathbf{if}\;F \leq 1.55 \cdot 10^{+25}:\\ \;\;\;\;\frac{t_0}{B} \cdot \sqrt{F \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{F} \cdot \sqrt{\frac{1}{B}}\right) \cdot t_0\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (sqrt 2.0))))
   (if (<= F 1.55e+25)
     (* (/ t_0 B) (sqrt (* F (+ C (hypot B C)))))
     (* (* (sqrt F) (sqrt (/ 1.0 B))) t_0))))

C

B = abs(B);
double code(double A, double B, double C, double F) {
	double t_0 = -sqrt(2.0);
	double tmp;
	if (F <= 1.55e+25) {
		tmp = (t_0 / B) * sqrt((F * (C + hypot(B, C))));
	} else {
		tmp = (sqrt(F) * sqrt((1.0 / B))) * t_0;
	}
	return tmp;
}

Java

B = Math.abs(B);
public static double code(double A, double B, double C, double F) {
	double t_0 = -Math.sqrt(2.0);
	double tmp;
	if (F <= 1.55e+25) {
		tmp = (t_0 / B) * Math.sqrt((F * (C + Math.hypot(B, C))));
	} else {
		tmp = (Math.sqrt(F) * Math.sqrt((1.0 / B))) * t_0;
	}
	return tmp;
}

Python

B = abs(B)
def code(A, B, C, F):
	t_0 = -math.sqrt(2.0)
	tmp = 0
	if F <= 1.55e+25:
		tmp = (t_0 / B) * math.sqrt((F * (C + math.hypot(B, C))))
	else:
		tmp = (math.sqrt(F) * math.sqrt((1.0 / B))) * t_0
	return tmp

Julia

B = abs(B)
function code(A, B, C, F)
	t_0 = Float64(-sqrt(2.0))
	tmp = 0.0
	if (F <= 1.55e+25)
		tmp = Float64(Float64(t_0 / B) * sqrt(Float64(F * Float64(C + hypot(B, C)))));
	else
		tmp = Float64(Float64(sqrt(F) * sqrt(Float64(1.0 / B))) * t_0);
	end
	return tmp
end

MATLAB

B = abs(B)
function tmp_2 = code(A, B, C, F)
	t_0 = -sqrt(2.0);
	tmp = 0.0;
	if (F <= 1.55e+25)
		tmp = (t_0 / B) * sqrt((F * (C + hypot(B, C))));
	else
		tmp = (sqrt(F) * sqrt((1.0 / B))) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: B should be positive before calling this function
code[A_, B_, C_, F_] := Block[{t$95$0 = (-N[Sqrt[2.0], $MachinePrecision])}, If[LessEqual[F, 1.55e+25], N[(N[(t$95$0 / B), $MachinePrecision] * N[Sqrt[N[(F * N[(C + N[Sqrt[B ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(1.0 / B), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if F < 1.5499999999999999e25

   1. Initial program 19.5%

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

   2. Taylor expanded in A around 0 5.8%

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

      1. mul-1-neg 5.8%

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

      2. *-commutative 5.8%

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

      3. distribute-rgt-neg-in 5.8%

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

      4. unpow2 5.8%

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

      5. unpow2 5.8%

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

      6. hypot-def 18.9%

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

   4. Simplified 18.9%

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

   if 1.5499999999999999e25 < F

   1. Initial program 12.7%

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

   2. Taylor expanded in A around 0 5.9%

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

      1. mul-1-neg 5.9%

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

      2. *-commutative 5.9%

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

      3. distribute-rgt-neg-in 5.9%

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

      4. unpow2 5.9%

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

      5. unpow2 5.9%

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

      6. hypot-def 6.3%

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

   4. Simplified 6.3%

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

   5. Taylor expanded in C around 0 14.7%

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

      1. mul-1-neg 14.7%

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

   7. Simplified 14.7%

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

      1. pow1/2 15.3%

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

      2. div-inv 15.3%

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

      3. unpow-prod-down 14.8%

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

      4. pow1/2 14.8%

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

   9. Applied egg-rr 14.8%

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

       1. unpow1/2 14.8%

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

   11. Simplified 14.8%

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

4. Final simplification 17.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 1.55 \cdot 10^{+25}:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{F} \cdot \sqrt{\frac{1}{B}}\right) \cdot \left(-\sqrt{2}\right)\\ \end{array} \]

Alternative 9: 35.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ \begin{array}{l} \mathbf{if}\;C \leq 4.7 \cdot 10^{+190}:\\ \;\;\;\;\left(\sqrt{F} \cdot \sqrt{\frac{1}{B}}\right) \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \sqrt{F \cdot C}}{B}\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
(FPCore (A B C F)
 :precision binary64
 (if (<= C 4.7e+190)
   (* (* (sqrt F) (sqrt (/ 1.0 B))) (- (sqrt 2.0)))
   (/ (* -2.0 (sqrt (* F C))) B)))

C

B = abs(B);
double code(double A, double B, double C, double F) {
	double tmp;
	if (C <= 4.7e+190) {
		tmp = (sqrt(F) * sqrt((1.0 / B))) * -sqrt(2.0);
	} else {
		tmp = (-2.0 * sqrt((F * C))) / B;
	}
	return tmp;
}

Fortran

NOTE: B should be positive before calling this function
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) :: tmp
    if (c <= 4.7d+190) then
        tmp = (sqrt(f) * sqrt((1.0d0 / b))) * -sqrt(2.0d0)
    else
        tmp = ((-2.0d0) * sqrt((f * c))) / b
    end if
    code = tmp
end function

Java

B = Math.abs(B);
public static double code(double A, double B, double C, double F) {
	double tmp;
	if (C <= 4.7e+190) {
		tmp = (Math.sqrt(F) * Math.sqrt((1.0 / B))) * -Math.sqrt(2.0);
	} else {
		tmp = (-2.0 * Math.sqrt((F * C))) / B;
	}
	return tmp;
}

Python

B = abs(B)
def code(A, B, C, F):
	tmp = 0
	if C <= 4.7e+190:
		tmp = (math.sqrt(F) * math.sqrt((1.0 / B))) * -math.sqrt(2.0)
	else:
		tmp = (-2.0 * math.sqrt((F * C))) / B
	return tmp

Julia

B = abs(B)
function code(A, B, C, F)
	tmp = 0.0
	if (C <= 4.7e+190)
		tmp = Float64(Float64(sqrt(F) * sqrt(Float64(1.0 / B))) * Float64(-sqrt(2.0)));
	else
		tmp = Float64(Float64(-2.0 * sqrt(Float64(F * C))) / B);
	end
	return tmp
end

MATLAB

B = abs(B)
function tmp_2 = code(A, B, C, F)
	tmp = 0.0;
	if (C <= 4.7e+190)
		tmp = (sqrt(F) * sqrt((1.0 / B))) * -sqrt(2.0);
	else
		tmp = (-2.0 * sqrt((F * C))) / B;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: B should be positive before calling this function
code[A_, B_, C_, F_] := If[LessEqual[C, 4.7e+190], N[(N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(1.0 / B), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], N[(N[(-2.0 * N[Sqrt[N[(F * C), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if C < 4.6999999999999998e190

   1. Initial program 18.9%

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

   2. Taylor expanded in A around 0 6.5%

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

      1. mul-1-neg 6.5%

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

      2. *-commutative 6.5%

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

      3. distribute-rgt-neg-in 6.5%

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

      4. unpow2 6.5%

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

      5. unpow2 6.5%

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

      6. hypot-def 14.1%

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

   4. Simplified 14.1%

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

   5. Taylor expanded in C around 0 12.1%

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

      1. mul-1-neg 12.1%

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

   7. Simplified 12.1%

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

      1. pow1/2 12.4%

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

      2. div-inv 12.4%

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

      3. unpow-prod-down 16.3%

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

      4. pow1/2 16.3%

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

   9. Applied egg-rr 16.3%

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

       1. unpow1/2 16.3%

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

   11. Simplified 16.3%

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

   if 4.6999999999999998e190 < C

   1. Initial program 1.8%

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

   2. Taylor expanded in A around 0 0.9%

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

      1. mul-1-neg 0.9%

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

      2. *-commutative 0.9%

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

      3. distribute-rgt-neg-in 0.9%

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

      4. unpow2 0.9%

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

      5. unpow2 0.9%

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

      6. hypot-def 14.7%

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

   4. Simplified 14.7%

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

      1. pow1/2 15.6%

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

      2. *-commutative 15.6%

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

      3. unpow-prod-down 14.7%

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

      4. pow1/2 14.7%

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

      5. pow1/2 14.7%

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

   6. Applied egg-rr 14.7%

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

   7. Taylor expanded in B around 0 11.5%

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

      1. mul-1-neg 11.5%

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

      2. associate-*l/ 11.5%

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

      3. distribute-neg-frac 11.5%

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

      4. unpow2 11.5%

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

      5. rem-square-sqrt 11.7%

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

      6. distribute-rgt-neg-out 11.7%

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

      7. neg-mul-1 11.7%

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

      8. associate-*r* 11.7%

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

      9. metadata-eval 11.7%

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

      10. *-commutative 11.7%

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

   9. Simplified 11.7%

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

4. Final simplification 15.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq 4.7 \cdot 10^{+190}:\\ \;\;\;\;\left(\sqrt{F} \cdot \sqrt{\frac{1}{B}}\right) \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \sqrt{F \cdot C}}{B}\\ \end{array} \]

Alternative 10: 35.4% accurate, 2.1× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ \begin{array}{l} \mathbf{if}\;C \leq 1.2 \cdot 10^{+192}:\\ \;\;\;\;\frac{\sqrt{F}}{\sqrt{B}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \sqrt{F \cdot C}}{B}\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
(FPCore (A B C F)
 :precision binary64
 (if (<= C 1.2e+192)
   (* (/ (sqrt F) (sqrt B)) (- (sqrt 2.0)))
   (/ (* -2.0 (sqrt (* F C))) B)))

C

B = abs(B);
double code(double A, double B, double C, double F) {
	double tmp;
	if (C <= 1.2e+192) {
		tmp = (sqrt(F) / sqrt(B)) * -sqrt(2.0);
	} else {
		tmp = (-2.0 * sqrt((F * C))) / B;
	}
	return tmp;
}

Fortran

NOTE: B should be positive before calling this function
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) :: tmp
    if (c <= 1.2d+192) then
        tmp = (sqrt(f) / sqrt(b)) * -sqrt(2.0d0)
    else
        tmp = ((-2.0d0) * sqrt((f * c))) / b
    end if
    code = tmp
end function

Java

B = Math.abs(B);
public static double code(double A, double B, double C, double F) {
	double tmp;
	if (C <= 1.2e+192) {
		tmp = (Math.sqrt(F) / Math.sqrt(B)) * -Math.sqrt(2.0);
	} else {
		tmp = (-2.0 * Math.sqrt((F * C))) / B;
	}
	return tmp;
}

Python

B = abs(B)
def code(A, B, C, F):
	tmp = 0
	if C <= 1.2e+192:
		tmp = (math.sqrt(F) / math.sqrt(B)) * -math.sqrt(2.0)
	else:
		tmp = (-2.0 * math.sqrt((F * C))) / B
	return tmp

Julia

B = abs(B)
function code(A, B, C, F)
	tmp = 0.0
	if (C <= 1.2e+192)
		tmp = Float64(Float64(sqrt(F) / sqrt(B)) * Float64(-sqrt(2.0)));
	else
		tmp = Float64(Float64(-2.0 * sqrt(Float64(F * C))) / B);
	end
	return tmp
end

MATLAB

B = abs(B)
function tmp_2 = code(A, B, C, F)
	tmp = 0.0;
	if (C <= 1.2e+192)
		tmp = (sqrt(F) / sqrt(B)) * -sqrt(2.0);
	else
		tmp = (-2.0 * sqrt((F * C))) / B;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: B should be positive before calling this function
code[A_, B_, C_, F_] := If[LessEqual[C, 1.2e+192], N[(N[(N[Sqrt[F], $MachinePrecision] / N[Sqrt[B], $MachinePrecision]), $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], N[(N[(-2.0 * N[Sqrt[N[(F * C), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if C < 1.1999999999999999e192

   1. Initial program 18.9%

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

   2. Taylor expanded in A around 0 6.5%

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

      1. mul-1-neg 6.5%

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

      2. *-commutative 6.5%

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

      3. distribute-rgt-neg-in 6.5%

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

      4. unpow2 6.5%

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

      5. unpow2 6.5%

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

      6. hypot-def 14.1%

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

   4. Simplified 14.1%

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

   5. Taylor expanded in C around 0 12.1%

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

      1. mul-1-neg 12.1%

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

   7. Simplified 12.1%

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

      1. sqrt-div 16.3%

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

   9. Applied egg-rr 16.3%

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

   if 1.1999999999999999e192 < C

   1. Initial program 1.8%

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

   2. Taylor expanded in A around 0 0.9%

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

      1. mul-1-neg 0.9%

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

      2. *-commutative 0.9%

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

      3. distribute-rgt-neg-in 0.9%

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

      4. unpow2 0.9%

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

      5. unpow2 0.9%

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

      6. hypot-def 14.7%

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

   4. Simplified 14.7%

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

      1. pow1/2 15.6%

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

      2. *-commutative 15.6%

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

      3. unpow-prod-down 14.7%

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

      4. pow1/2 14.7%

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

      5. pow1/2 14.7%

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

   6. Applied egg-rr 14.7%

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

   7. Taylor expanded in B around 0 11.5%

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

      1. mul-1-neg 11.5%

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

      2. associate-*l/ 11.5%

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

      3. distribute-neg-frac 11.5%

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

      4. unpow2 11.5%

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

      5. rem-square-sqrt 11.7%

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

      6. distribute-rgt-neg-out 11.7%

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

      7. neg-mul-1 11.7%

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

      8. associate-*r* 11.7%

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

      9. metadata-eval 11.7%

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

      10. *-commutative 11.7%

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

   9. Simplified 11.7%

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

4. Final simplification 15.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq 1.2 \cdot 10^{+192}:\\ \;\;\;\;\frac{\sqrt{F}}{\sqrt{B}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \sqrt{F \cdot C}}{B}\\ \end{array} \]

Alternative 11: 34.3% accurate, 3.0× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ \begin{array}{l} \mathbf{if}\;F \leq 400000:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{B \cdot F}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{2 \cdot F}{B}}\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
(FPCore (A B C F)
 :precision binary64
 (if (<= F 400000.0)
   (* (/ (- (sqrt 2.0)) B) (sqrt (* B F)))
   (- (sqrt (/ (* 2.0 F) B)))))

C

B = abs(B);
double code(double A, double B, double C, double F) {
	double tmp;
	if (F <= 400000.0) {
		tmp = (-sqrt(2.0) / B) * sqrt((B * F));
	} else {
		tmp = -sqrt(((2.0 * F) / B));
	}
	return tmp;
}

Fortran

NOTE: B should be positive before calling this function
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) :: tmp
    if (f <= 400000.0d0) then
        tmp = (-sqrt(2.0d0) / b) * sqrt((b * f))
    else
        tmp = -sqrt(((2.0d0 * f) / b))
    end if
    code = tmp
end function

Java

B = Math.abs(B);
public static double code(double A, double B, double C, double F) {
	double tmp;
	if (F <= 400000.0) {
		tmp = (-Math.sqrt(2.0) / B) * Math.sqrt((B * F));
	} else {
		tmp = -Math.sqrt(((2.0 * F) / B));
	}
	return tmp;
}

Python

B = abs(B)
def code(A, B, C, F):
	tmp = 0
	if F <= 400000.0:
		tmp = (-math.sqrt(2.0) / B) * math.sqrt((B * F))
	else:
		tmp = -math.sqrt(((2.0 * F) / B))
	return tmp

Julia

B = abs(B)
function code(A, B, C, F)
	tmp = 0.0
	if (F <= 400000.0)
		tmp = Float64(Float64(Float64(-sqrt(2.0)) / B) * sqrt(Float64(B * F)));
	else
		tmp = Float64(-sqrt(Float64(Float64(2.0 * F) / B)));
	end
	return tmp
end

MATLAB

B = abs(B)
function tmp_2 = code(A, B, C, F)
	tmp = 0.0;
	if (F <= 400000.0)
		tmp = (-sqrt(2.0) / B) * sqrt((B * F));
	else
		tmp = -sqrt(((2.0 * F) / B));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: B should be positive before calling this function
code[A_, B_, C_, F_] := If[LessEqual[F, 400000.0], N[(N[((-N[Sqrt[2.0], $MachinePrecision]) / B), $MachinePrecision] * N[Sqrt[N[(B * F), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], (-N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] / B), $MachinePrecision]], $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if F < 4e5

   1. Initial program 19.6%

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

   2. Taylor expanded in A around 0 5.9%

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

      1. mul-1-neg 5.9%

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

      2. *-commutative 5.9%

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

      3. distribute-rgt-neg-in 5.9%

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

      4. unpow2 5.9%

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

      5. unpow2 5.9%

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

      6. hypot-def 18.2%

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

   4. Simplified 18.2%

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

   5. Taylor expanded in C around 0 15.6%

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

   if 4e5 < F

   1. Initial program 12.9%

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

   2. Taylor expanded in A around 0 5.7%

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

      1. mul-1-neg 5.7%

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

      2. *-commutative 5.7%

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

      3. distribute-rgt-neg-in 5.7%

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

      4. unpow2 5.7%

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

      5. unpow2 5.7%

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

      6. hypot-def 8.2%

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

   4. Simplified 8.2%

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

   5. Taylor expanded in C around 0 16.0%

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

      1. mul-1-neg 16.0%

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

   7. Simplified 16.0%

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

      1. expm1-log1p-u 15.7%

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

      2. expm1-udef 7.4%

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

      3. pow1/2 8.0%

         \[\leadsto -\left(e^{\mathsf{log1p}\left(\color{blue}{{\left(\frac{F}{B}\right)}^{0.5}} \cdot \sqrt{2}\right)} - 1\right) \]

      4. pow1/2 8.0%

         \[\leadsto -\left(e^{\mathsf{log1p}\left({\left(\frac{F}{B}\right)}^{0.5} \cdot \color{blue}{{2}^{0.5}}\right)} - 1\right) \]

      5. metadata-eval 8.0%

         \[\leadsto -\left(e^{\mathsf{log1p}\left({\left(\frac{F}{B}\right)}^{0.5} \cdot {\color{blue}{\left({2}^{1}\right)}}^{0.5}\right)} - 1\right) \]

      6. metadata-eval 8.0%

         \[\leadsto -\left(e^{\mathsf{log1p}\left({\left(\frac{F}{B}\right)}^{0.5} \cdot {\left({2}^{\color{blue}{\left(\frac{2}{2}\right)}}\right)}^{0.5}\right)} - 1\right) \]

      7. sqrt-pow2 8.0%

         \[\leadsto -\left(e^{\mathsf{log1p}\left({\left(\frac{F}{B}\right)}^{0.5} \cdot {\color{blue}{\left({\left(\sqrt{2}\right)}^{2}\right)}}^{0.5}\right)} - 1\right) \]

      8. pow-prod-down 8.0%

         \[\leadsto -\left(e^{\mathsf{log1p}\left(\color{blue}{{\left(\frac{F}{B} \cdot {\left(\sqrt{2}\right)}^{2}\right)}^{0.5}}\right)} - 1\right) \]

      9. pow1/2 7.4%

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

      10. sqrt-pow2 7.4%

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

      11. metadata-eval 7.4%

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

      12. metadata-eval 7.4%

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

   9. Applied egg-rr 7.4%

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

       1. expm1-def 15.7%

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

       2. expm1-log1p 16.1%

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

       3. associate-*l/ 16.1%

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

   11. Simplified 16.1%

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

4. Final simplification 15.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 400000:\\ \;\;\;\;\frac{-\sqrt{2}}{B} \cdot \sqrt{B \cdot F}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{2 \cdot F}{B}}\\ \end{array} \]

Alternative 12: 27.2% accurate, 5.8× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ \begin{array}{l} \mathbf{if}\;C \leq 9.5 \cdot 10^{+188}:\\ \;\;\;\;-\sqrt{\frac{2 \cdot F}{B}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \sqrt{F \cdot C}}{B}\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
(FPCore (A B C F)
 :precision binary64
 (if (<= C 9.5e+188) (- (sqrt (/ (* 2.0 F) B))) (/ (* -2.0 (sqrt (* F C))) B)))

C

B = abs(B);
double code(double A, double B, double C, double F) {
	double tmp;
	if (C <= 9.5e+188) {
		tmp = -sqrt(((2.0 * F) / B));
	} else {
		tmp = (-2.0 * sqrt((F * C))) / B;
	}
	return tmp;
}

Fortran

NOTE: B should be positive before calling this function
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) :: tmp
    if (c <= 9.5d+188) then
        tmp = -sqrt(((2.0d0 * f) / b))
    else
        tmp = ((-2.0d0) * sqrt((f * c))) / b
    end if
    code = tmp
end function

Java

B = Math.abs(B);
public static double code(double A, double B, double C, double F) {
	double tmp;
	if (C <= 9.5e+188) {
		tmp = -Math.sqrt(((2.0 * F) / B));
	} else {
		tmp = (-2.0 * Math.sqrt((F * C))) / B;
	}
	return tmp;
}

Python

B = abs(B)
def code(A, B, C, F):
	tmp = 0
	if C <= 9.5e+188:
		tmp = -math.sqrt(((2.0 * F) / B))
	else:
		tmp = (-2.0 * math.sqrt((F * C))) / B
	return tmp

Julia

B = abs(B)
function code(A, B, C, F)
	tmp = 0.0
	if (C <= 9.5e+188)
		tmp = Float64(-sqrt(Float64(Float64(2.0 * F) / B)));
	else
		tmp = Float64(Float64(-2.0 * sqrt(Float64(F * C))) / B);
	end
	return tmp
end

MATLAB

B = abs(B)
function tmp_2 = code(A, B, C, F)
	tmp = 0.0;
	if (C <= 9.5e+188)
		tmp = -sqrt(((2.0 * F) / B));
	else
		tmp = (-2.0 * sqrt((F * C))) / B;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: B should be positive before calling this function
code[A_, B_, C_, F_] := If[LessEqual[C, 9.5e+188], (-N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] / B), $MachinePrecision]], $MachinePrecision]), N[(N[(-2.0 * N[Sqrt[N[(F * C), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / B), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if C < 9.4999999999999996e188

   1. Initial program 18.9%

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

   2. Taylor expanded in A around 0 6.5%

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

      1. mul-1-neg 6.5%

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

      2. *-commutative 6.5%

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

      3. distribute-rgt-neg-in 6.5%

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

      4. unpow2 6.5%

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

      5. unpow2 6.5%

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

      6. hypot-def 14.1%

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

   4. Simplified 14.1%

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

   5. Taylor expanded in C around 0 12.1%

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

      1. mul-1-neg 12.1%

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

   7. Simplified 12.1%

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

      1. expm1-log1p-u 11.9%

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

      2. expm1-udef 5.4%

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

      3. pow1/2 5.6%

         \[\leadsto -\left(e^{\mathsf{log1p}\left(\color{blue}{{\left(\frac{F}{B}\right)}^{0.5}} \cdot \sqrt{2}\right)} - 1\right) \]

      4. pow1/2 5.6%

         \[\leadsto -\left(e^{\mathsf{log1p}\left({\left(\frac{F}{B}\right)}^{0.5} \cdot \color{blue}{{2}^{0.5}}\right)} - 1\right) \]

      5. metadata-eval 5.6%

         \[\leadsto -\left(e^{\mathsf{log1p}\left({\left(\frac{F}{B}\right)}^{0.5} \cdot {\color{blue}{\left({2}^{1}\right)}}^{0.5}\right)} - 1\right) \]

      6. metadata-eval 5.6%

         \[\leadsto -\left(e^{\mathsf{log1p}\left({\left(\frac{F}{B}\right)}^{0.5} \cdot {\left({2}^{\color{blue}{\left(\frac{2}{2}\right)}}\right)}^{0.5}\right)} - 1\right) \]

      7. sqrt-pow2 5.6%

         \[\leadsto -\left(e^{\mathsf{log1p}\left({\left(\frac{F}{B}\right)}^{0.5} \cdot {\color{blue}{\left({\left(\sqrt{2}\right)}^{2}\right)}}^{0.5}\right)} - 1\right) \]

      8. pow-prod-down 5.6%

         \[\leadsto -\left(e^{\mathsf{log1p}\left(\color{blue}{{\left(\frac{F}{B} \cdot {\left(\sqrt{2}\right)}^{2}\right)}^{0.5}}\right)} - 1\right) \]

      9. pow1/2 5.4%

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

      10. sqrt-pow2 5.4%

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

      11. metadata-eval 5.4%

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

      12. metadata-eval 5.4%

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

   9. Applied egg-rr 5.4%

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

       1. expm1-def 12.0%

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

       2. expm1-log1p 12.2%

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

       3. associate-*l/ 12.2%

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

   11. Simplified 12.2%

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

   if 9.4999999999999996e188 < C

   1. Initial program 1.8%

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

   2. Taylor expanded in A around 0 0.9%

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

      1. mul-1-neg 0.9%

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

      2. *-commutative 0.9%

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

      3. distribute-rgt-neg-in 0.9%

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

      4. unpow2 0.9%

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

      5. unpow2 0.9%

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

      6. hypot-def 14.7%

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

   4. Simplified 14.7%

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

      1. pow1/2 15.6%

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

      2. *-commutative 15.6%

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

      3. unpow-prod-down 14.7%

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

      4. pow1/2 14.7%

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

      5. pow1/2 14.7%

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

   6. Applied egg-rr 14.7%

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

   7. Taylor expanded in B around 0 11.5%

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

      1. mul-1-neg 11.5%

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

      2. associate-*l/ 11.5%

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

      3. distribute-neg-frac 11.5%

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

      4. unpow2 11.5%

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

      5. rem-square-sqrt 11.7%

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

      6. distribute-rgt-neg-out 11.7%

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

      7. neg-mul-1 11.7%

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

      8. associate-*r* 11.7%

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

      9. metadata-eval 11.7%

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

      10. *-commutative 11.7%

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

   9. Simplified 11.7%

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

4. Final simplification 12.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq 9.5 \cdot 10^{+188}:\\ \;\;\;\;-\sqrt{\frac{2 \cdot F}{B}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2 \cdot \sqrt{F \cdot C}}{B}\\ \end{array} \]

Alternative 13: 26.8% accurate, 6.0× speedup

Math

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

FPCore

NOTE: B should be positive before calling this function
(FPCore (A B C F) :precision binary64 (- (sqrt (/ (* 2.0 F) B))))

C

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

Fortran

NOTE: B should be positive before calling this function
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
    code = -sqrt(((2.0d0 * f) / b))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: B should be positive before calling this function
code[A_, B_, C_, F_] := (-N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] / B), $MachinePrecision]], $MachinePrecision])

Derivation

1. Initial program 17.0%

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

2. Taylor expanded in A around 0 5.8%

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

   1. mul-1-neg 5.8%

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

   2. *-commutative 5.8%

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

   3. distribute-rgt-neg-in 5.8%

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

   4. unpow2 5.8%

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

   5. unpow2 5.8%

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

   6. hypot-def 14.2%

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

4. Simplified 14.2%

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

5. Taylor expanded in C around 0 11.3%

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

   1. mul-1-neg 11.3%

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

7. Simplified 11.3%

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

   1. expm1-log1p-u 11.2%

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

   2. expm1-udef 4.9%

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

   3. pow1/2 5.2%

      \[\leadsto -\left(e^{\mathsf{log1p}\left(\color{blue}{{\left(\frac{F}{B}\right)}^{0.5}} \cdot \sqrt{2}\right)} - 1\right) \]

   4. pow1/2 5.2%

      \[\leadsto -\left(e^{\mathsf{log1p}\left({\left(\frac{F}{B}\right)}^{0.5} \cdot \color{blue}{{2}^{0.5}}\right)} - 1\right) \]

   5. metadata-eval 5.2%

      \[\leadsto -\left(e^{\mathsf{log1p}\left({\left(\frac{F}{B}\right)}^{0.5} \cdot {\color{blue}{\left({2}^{1}\right)}}^{0.5}\right)} - 1\right) \]

   6. metadata-eval 5.2%

      \[\leadsto -\left(e^{\mathsf{log1p}\left({\left(\frac{F}{B}\right)}^{0.5} \cdot {\left({2}^{\color{blue}{\left(\frac{2}{2}\right)}}\right)}^{0.5}\right)} - 1\right) \]

   7. sqrt-pow2 5.2%

      \[\leadsto -\left(e^{\mathsf{log1p}\left({\left(\frac{F}{B}\right)}^{0.5} \cdot {\color{blue}{\left({\left(\sqrt{2}\right)}^{2}\right)}}^{0.5}\right)} - 1\right) \]

   8. pow-prod-down 5.2%

      \[\leadsto -\left(e^{\mathsf{log1p}\left(\color{blue}{{\left(\frac{F}{B} \cdot {\left(\sqrt{2}\right)}^{2}\right)}^{0.5}}\right)} - 1\right) \]

   9. pow1/2 4.9%

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

   10. sqrt-pow2 4.9%

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

   11. metadata-eval 4.9%

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

   12. metadata-eval 4.9%

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

9. Applied egg-rr 4.9%

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

    1. expm1-def 11.2%

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

    2. expm1-log1p 11.4%

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

    3. associate-*l/ 11.4%

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

11. Simplified 11.4%

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

12. Final simplification 11.4%

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

Reproduce

herbie shell --seed 2023310 
(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))))