ABCF->ab-angle b

Percentage Accurate: 18.8% → 46.7%

Time: 27.2s

Alternatives: 11

Speedup: 3.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 18.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 46.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B_m, B_m, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;{B_m}^{2} \leq 5 \cdot 10^{+52}:\\ \;\;\;\;\frac{-\sqrt{t_0 \cdot \left(F \cdot \left(2 \cdot \left(2 \cdot A + \frac{{B_m}^{2} \cdot -0.5}{C}\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B_m, A\right)\right)} \cdot \frac{-1}{\frac{B_m}{\sqrt{2}}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* A (* C -4.0)))))
   (if (<= (pow B_m 2.0) 5e+52)
     (/
      (-
       (sqrt (* t_0 (* F (* 2.0 (+ (* 2.0 A) (/ (* (pow B_m 2.0) -0.5) C)))))))
      t_0)
     (* (sqrt (* F (- A (hypot B_m A)))) (/ -1.0 (/ B_m (sqrt 2.0)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (A * (C * -4.0)));
	double tmp;
	if (pow(B_m, 2.0) <= 5e+52) {
		tmp = -sqrt((t_0 * (F * (2.0 * ((2.0 * A) + ((pow(B_m, 2.0) * -0.5) / C)))))) / t_0;
	} else {
		tmp = sqrt((F * (A - hypot(B_m, A)))) * (-1.0 / (B_m / sqrt(2.0)));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 5e+52)
		tmp = Float64(Float64(-sqrt(Float64(t_0 * Float64(F * Float64(2.0 * Float64(Float64(2.0 * A) + Float64(Float64((B_m ^ 2.0) * -0.5) / C))))))) / t_0);
	else
		tmp = Float64(sqrt(Float64(F * Float64(A - hypot(B_m, A)))) * Float64(-1.0 / Float64(B_m / sqrt(2.0))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e+52], N[((-N[Sqrt[N[(t$95$0 * N[(F * N[(2.0 * N[(N[(2.0 * A), $MachinePrecision] + N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] * -0.5), $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], N[(N[Sqrt[N[(F * N[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / N[(B$95$m / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B 2) < 5e52

   1. Initial program 18.7%

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

   3. Simplified 29.6%

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

   4. Taylor expanded in C around inf 21.2%

      \[\leadsto \frac{-\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(2 \cdot \left(A + \color{blue}{\left(A + -0.5 \cdot \frac{\left({A}^{2} + {B}^{2}\right) - {\left(-1 \cdot A\right)}^{2}}{C}\right)}\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
   5. Step-by-step derivation

      1. associate-*r/ 21.2%

         \[\leadsto \frac{-\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(2 \cdot \left(A + \left(A + \color{blue}{\frac{-0.5 \cdot \left(\left({A}^{2} + {B}^{2}\right) - {\left(-1 \cdot A\right)}^{2}\right)}{C}}\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      2. associate--l+ 21.2%

         \[\leadsto \frac{-\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \left(2 \cdot \left(A + \left(A + \frac{-0.5 \cdot \color{blue}{\left({A}^{2} + \left({B}^{2} - {\left(-1 \cdot A\right)}^{2}\right)\right)}}{C}\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      3. mul-1-neg 21.2%

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

   6. Simplified 21.2%

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

   7. Taylor expanded in A around 0 24.4%

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

      1. distribute-frac-neg 24.4%

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

      2. associate-*l* 26.1%

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

      3. associate-+r+ 26.1%

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

      4. add-log-exp 9.1%

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

      5. add-log-exp 7.0%

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

      6. sum-log 7.0%

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

      7. exp-lft-sqr 7.0%

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

      8. add-log-exp 26.1%

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

   9. Applied egg-rr 26.1%

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

   if 5e52 < (pow.f64 B 2)

   1. Initial program 17.5%

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

   3. Simplified 13.3%

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

   4. Taylor expanded in C around 0 14.5%

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

      1. mul-1-neg 14.5%

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

      2. distribute-rgt-neg-in 14.5%

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

      3. +-commutative 14.5%

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

      4. unpow2 14.5%

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

      5. unpow2 14.5%

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

      6. hypot-def 24.3%

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

   6. Simplified 24.3%

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

      1. clear-num 24.3%

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

      2. inv-pow 24.3%

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

   8. Applied egg-rr 24.3%

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

      1. unpow-1 24.3%

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

   10. Simplified 24.3%

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

4. Final simplification 25.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 5 \cdot 10^{+52}:\\ \;\;\;\;\frac{-\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(F \cdot \left(2 \cdot \left(2 \cdot A + \frac{{B}^{2} \cdot -0.5}{C}\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)} \cdot \frac{-1}{\frac{B}{\sqrt{2}}}\\ \end{array} \]

Alternative 2: 43.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \frac{-\sqrt{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A + A\right)\right)\right)\right)}}{\mathsf{fma}\left(A, C \cdot -4, {B_m}^{2}\right)}\\ \mathbf{if}\;{B_m}^{2} \leq 10^{-285}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;{B_m}^{2} \leq 2 \cdot 10^{-218}:\\ \;\;\;\;\frac{\sqrt{2}}{B_m} \cdot \left(-\sqrt{-0.5 \cdot \frac{{B_m}^{2}}{\frac{C}{F}}}\right)\\ \mathbf{elif}\;{B_m}^{2} \leq 10^{-76}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;{B_m}^{2} \leq 5 \cdot 10^{+52}:\\ \;\;\;\;\sqrt{F \cdot \left(-0.5 \cdot \frac{{B_m}^{2}}{C}\right)} \cdot \frac{-\sqrt{2}}{B_m}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B_m, A\right)\right)} \cdot \frac{-1}{\frac{B_m}{\sqrt{2}}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0
         (/
          (- (sqrt (* -8.0 (* A (* C (* F (+ A A)))))))
          (fma A (* C -4.0) (pow B_m 2.0)))))
   (if (<= (pow B_m 2.0) 1e-285)
     t_0
     (if (<= (pow B_m 2.0) 2e-218)
       (* (/ (sqrt 2.0) B_m) (- (sqrt (* -0.5 (/ (pow B_m 2.0) (/ C F))))))
       (if (<= (pow B_m 2.0) 1e-76)
         t_0
         (if (<= (pow B_m 2.0) 5e+52)
           (* (sqrt (* F (* -0.5 (/ (pow B_m 2.0) C)))) (/ (- (sqrt 2.0)) B_m))
           (*
            (sqrt (* F (- A (hypot B_m A))))
            (/ -1.0 (/ B_m (sqrt 2.0))))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = -sqrt((-8.0 * (A * (C * (F * (A + A)))))) / fma(A, (C * -4.0), pow(B_m, 2.0));
	double tmp;
	if (pow(B_m, 2.0) <= 1e-285) {
		tmp = t_0;
	} else if (pow(B_m, 2.0) <= 2e-218) {
		tmp = (sqrt(2.0) / B_m) * -sqrt((-0.5 * (pow(B_m, 2.0) / (C / F))));
	} else if (pow(B_m, 2.0) <= 1e-76) {
		tmp = t_0;
	} else if (pow(B_m, 2.0) <= 5e+52) {
		tmp = sqrt((F * (-0.5 * (pow(B_m, 2.0) / C)))) * (-sqrt(2.0) / B_m);
	} else {
		tmp = sqrt((F * (A - hypot(B_m, A)))) * (-1.0 / (B_m / sqrt(2.0)));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(Float64(-sqrt(Float64(-8.0 * Float64(A * Float64(C * Float64(F * Float64(A + A))))))) / fma(A, Float64(C * -4.0), (B_m ^ 2.0)))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e-285)
		tmp = t_0;
	elseif ((B_m ^ 2.0) <= 2e-218)
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(-sqrt(Float64(-0.5 * Float64((B_m ^ 2.0) / Float64(C / F))))));
	elseif ((B_m ^ 2.0) <= 1e-76)
		tmp = t_0;
	elseif ((B_m ^ 2.0) <= 5e+52)
		tmp = Float64(sqrt(Float64(F * Float64(-0.5 * Float64((B_m ^ 2.0) / C)))) * Float64(Float64(-sqrt(2.0)) / B_m));
	else
		tmp = Float64(sqrt(Float64(F * Float64(A - hypot(B_m, A)))) * Float64(-1.0 / Float64(B_m / sqrt(2.0))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[((-N[Sqrt[N[(-8.0 * N[(A * N[(C * N[(F * N[(A + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(A * N[(C * -4.0), $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-285], t$95$0, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e-218], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * (-N[Sqrt[N[(-0.5 * N[(N[Power[B$95$m, 2.0], $MachinePrecision] / N[(C / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-76], t$95$0, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e+52], N[(N[Sqrt[N[(F * N[(-0.5 * N[(N[Power[B$95$m, 2.0], $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[((-N[Sqrt[2.0], $MachinePrecision]) / B$95$m), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(F * N[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / N[(B$95$m / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (pow.f64 B 2) < 1.00000000000000007e-285 or 2.0000000000000001e-218 < (pow.f64 B 2) < 9.99999999999999927e-77

   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 22.1%

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

   4. Taylor expanded in C around inf 23.4%

      \[\leadsto \frac{-\sqrt{\color{blue}{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right)\right)}}}{\mathsf{fma}\left(A, C \cdot -4, {B}^{2}\right)} \]

   if 1.00000000000000007e-285 < (pow.f64 B 2) < 2.0000000000000001e-218

   1. Initial program 12.0%

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

   3. Simplified 14.6%

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

   4. Taylor expanded in A around 0 8.4%

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

      1. mul-1-neg 8.4%

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

      2. *-commutative 8.4%

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

      3. distribute-rgt-neg-in 8.4%

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

      4. unpow2 8.4%

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

      5. unpow2 8.4%

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

      6. hypot-def 9.5%

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

   6. Simplified 9.5%

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

   7. Taylor expanded in C around inf 18.2%

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

      1. associate-/l* 18.1%

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

   9. Simplified 18.1%

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

   if 9.99999999999999927e-77 < (pow.f64 B 2) < 5e52

   1. Initial program 25.6%

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

   3. Simplified 23.1%

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

   4. Taylor expanded in A around 0 10.7%

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

      1. mul-1-neg 10.7%

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

      2. *-commutative 10.7%

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

      3. distribute-rgt-neg-in 10.7%

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

      4. unpow2 10.7%

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

      5. unpow2 10.7%

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

      6. hypot-def 10.9%

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

   6. Simplified 10.9%

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

   7. Taylor expanded in C around inf 19.9%

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

   if 5e52 < (pow.f64 B 2)

   1. Initial program 17.5%

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

   3. Simplified 13.3%

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

   4. Taylor expanded in C around 0 14.5%

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

      1. mul-1-neg 14.5%

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

      2. distribute-rgt-neg-in 14.5%

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

      3. +-commutative 14.5%

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

      4. unpow2 14.5%

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

      5. unpow2 14.5%

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

      6. hypot-def 24.3%

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

   6. Simplified 24.3%

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

      1. clear-num 24.3%

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

      2. inv-pow 24.3%

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

   8. Applied egg-rr 24.3%

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

      1. unpow-1 24.3%

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

   10. Simplified 24.3%

       \[\leadsto \color{blue}{\frac{1}{\frac{B}{\sqrt{2}}}} \cdot \left(-\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 23.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{-285}:\\ \;\;\;\;\frac{-\sqrt{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A + A\right)\right)\right)\right)}}{\mathsf{fma}\left(A, C \cdot -4, {B}^{2}\right)}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{-218}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{-0.5 \cdot \frac{{B}^{2}}{\frac{C}{F}}}\right)\\ \mathbf{elif}\;{B}^{2} \leq 10^{-76}:\\ \;\;\;\;\frac{-\sqrt{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A + A\right)\right)\right)\right)}}{\mathsf{fma}\left(A, C \cdot -4, {B}^{2}\right)}\\ \mathbf{elif}\;{B}^{2} \leq 5 \cdot 10^{+52}:\\ \;\;\;\;\sqrt{F \cdot \left(-0.5 \cdot \frac{{B}^{2}}{C}\right)} \cdot \frac{-\sqrt{2}}{B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)} \cdot \frac{-1}{\frac{B}{\sqrt{2}}}\\ \end{array} \]

Alternative 3: 41.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B_m}^{2} \leq 10^{-285}:\\ \;\;\;\;\frac{-\sqrt{\left(A \cdot -8\right) \cdot \left(\left(A + A\right) \cdot \left(C \cdot F\right)\right)}}{\mathsf{fma}\left(A, C \cdot -4, {B_m}^{2}\right)}\\ \mathbf{elif}\;{B_m}^{2} \leq 5 \cdot 10^{+52}:\\ \;\;\;\;\frac{\sqrt{2}}{B_m} \cdot \left(-\sqrt{-0.5 \cdot \frac{{B_m}^{2}}{\frac{C}{F}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B_m, A\right)\right)} \cdot \frac{-1}{\frac{B_m}{\sqrt{2}}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 1e-285)
   (/
    (- (sqrt (* (* A -8.0) (* (+ A A) (* C F)))))
    (fma A (* C -4.0) (pow B_m 2.0)))
   (if (<= (pow B_m 2.0) 5e+52)
     (* (/ (sqrt 2.0) B_m) (- (sqrt (* -0.5 (/ (pow B_m 2.0) (/ C F))))))
     (* (sqrt (* F (- A (hypot B_m A)))) (/ -1.0 (/ B_m (sqrt 2.0)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (pow(B_m, 2.0) <= 1e-285) {
		tmp = -sqrt(((A * -8.0) * ((A + A) * (C * F)))) / fma(A, (C * -4.0), pow(B_m, 2.0));
	} else if (pow(B_m, 2.0) <= 5e+52) {
		tmp = (sqrt(2.0) / B_m) * -sqrt((-0.5 * (pow(B_m, 2.0) / (C / F))));
	} else {
		tmp = sqrt((F * (A - hypot(B_m, A)))) * (-1.0 / (B_m / sqrt(2.0)));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e-285)
		tmp = Float64(Float64(-sqrt(Float64(Float64(A * -8.0) * Float64(Float64(A + A) * Float64(C * F))))) / fma(A, Float64(C * -4.0), (B_m ^ 2.0)));
	elseif ((B_m ^ 2.0) <= 5e+52)
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(-sqrt(Float64(-0.5 * Float64((B_m ^ 2.0) / Float64(C / F))))));
	else
		tmp = Float64(sqrt(Float64(F * Float64(A - hypot(B_m, A)))) * Float64(-1.0 / Float64(B_m / sqrt(2.0))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-285], N[((-N[Sqrt[N[(N[(A * -8.0), $MachinePrecision] * N[(N[(A + A), $MachinePrecision] * N[(C * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(A * N[(C * -4.0), $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e+52], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * (-N[Sqrt[N[(-0.5 * N[(N[Power[B$95$m, 2.0], $MachinePrecision] / N[(C / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[N[(F * N[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / N[(B$95$m / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B 2) < 1.00000000000000007e-285

   1. Initial program 13.7%

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

   3. Simplified 17.4%

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

   4. Taylor expanded in C around inf 21.4%

      \[\leadsto \frac{-\sqrt{\color{blue}{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right)\right)}}}{\mathsf{fma}\left(A, C \cdot -4, {B}^{2}\right)} \]
   5. Step-by-step derivation

      1. associate-*r* 21.4%

         \[\leadsto \frac{-\sqrt{\color{blue}{\left(-8 \cdot A\right) \cdot \left(C \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right)}}}{\mathsf{fma}\left(A, C \cdot -4, {B}^{2}\right)} \]

      2. rem-square-sqrt 0.0%

         \[\leadsto \frac{-\sqrt{\left(\color{blue}{\left(\sqrt{-8} \cdot \sqrt{-8}\right)} \cdot A\right) \cdot \left(C \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right)}}{\mathsf{fma}\left(A, C \cdot -4, {B}^{2}\right)} \]

      3. unpow2 0.0%

         \[\leadsto \frac{-\sqrt{\left(\color{blue}{{\left(\sqrt{-8}\right)}^{2}} \cdot A\right) \cdot \left(C \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right)}}{\mathsf{fma}\left(A, C \cdot -4, {B}^{2}\right)} \]

      4. *-commutative 0.0%

         \[\leadsto \frac{-\sqrt{\color{blue}{\left(A \cdot {\left(\sqrt{-8}\right)}^{2}\right)} \cdot \left(C \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right)}}{\mathsf{fma}\left(A, C \cdot -4, {B}^{2}\right)} \]

      5. unpow2 0.0%

         \[\leadsto \frac{-\sqrt{\left(A \cdot \color{blue}{\left(\sqrt{-8} \cdot \sqrt{-8}\right)}\right) \cdot \left(C \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right)}}{\mathsf{fma}\left(A, C \cdot -4, {B}^{2}\right)} \]

      6. rem-square-sqrt 21.4%

         \[\leadsto \frac{-\sqrt{\left(A \cdot \color{blue}{-8}\right) \cdot \left(C \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right)}}{\mathsf{fma}\left(A, C \cdot -4, {B}^{2}\right)} \]

      7. associate-*r* 17.0%

         \[\leadsto \frac{-\sqrt{\left(A \cdot -8\right) \cdot \color{blue}{\left(\left(C \cdot F\right) \cdot \left(A - -1 \cdot A\right)\right)}}}{\mathsf{fma}\left(A, C \cdot -4, {B}^{2}\right)} \]

      8. *-commutative 17.0%

         \[\leadsto \frac{-\sqrt{\left(A \cdot -8\right) \cdot \left(\color{blue}{\left(F \cdot C\right)} \cdot \left(A - -1 \cdot A\right)\right)}}{\mathsf{fma}\left(A, C \cdot -4, {B}^{2}\right)} \]

      9. sub-neg 17.0%

         \[\leadsto \frac{-\sqrt{\left(A \cdot -8\right) \cdot \left(\left(F \cdot C\right) \cdot \color{blue}{\left(A + \left(--1 \cdot A\right)\right)}\right)}}{\mathsf{fma}\left(A, C \cdot -4, {B}^{2}\right)} \]

      10. mul-1-neg 17.0%

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

      11. remove-double-neg 17.0%

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

   6. Simplified 17.0%

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

   if 1.00000000000000007e-285 < (pow.f64 B 2) < 5e52

   1. Initial program 22.7%

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

   3. Simplified 24.6%

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

   4. Taylor expanded in A around 0 8.1%

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

      1. mul-1-neg 8.1%

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

      2. *-commutative 8.1%

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

      3. distribute-rgt-neg-in 8.1%

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

      4. unpow2 8.1%

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

      5. unpow2 8.1%

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

      6. hypot-def 8.7%

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

   6. Simplified 8.7%

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

   7. Taylor expanded in C around inf 15.2%

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

      1. associate-/l* 15.1%

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

   9. Simplified 15.1%

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

   if 5e52 < (pow.f64 B 2)

   1. Initial program 17.5%

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

   3. Simplified 13.3%

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

   4. Taylor expanded in C around 0 14.5%

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

      1. mul-1-neg 14.5%

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

      2. distribute-rgt-neg-in 14.5%

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

      3. +-commutative 14.5%

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

      4. unpow2 14.5%

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

      5. unpow2 14.5%

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

      6. hypot-def 24.3%

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

   6. Simplified 24.3%

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

      1. clear-num 24.3%

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

      2. inv-pow 24.3%

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

   8. Applied egg-rr 24.3%

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

      1. unpow-1 24.3%

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

   10. Simplified 24.3%

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

4. Final simplification 19.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{-285}:\\ \;\;\;\;\frac{-\sqrt{\left(A \cdot -8\right) \cdot \left(\left(A + A\right) \cdot \left(C \cdot F\right)\right)}}{\mathsf{fma}\left(A, C \cdot -4, {B}^{2}\right)}\\ \mathbf{elif}\;{B}^{2} \leq 5 \cdot 10^{+52}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{-0.5 \cdot \frac{{B}^{2}}{\frac{C}{F}}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)} \cdot \frac{-1}{\frac{B}{\sqrt{2}}}\\ \end{array} \]

Alternative 4: 46.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := {B_m}^{2} - C \cdot \left(A \cdot 4\right)\\ \mathbf{if}\;{B_m}^{2} \leq 10^{-84}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot t_0\right)\right) \cdot \left(A + A\right)}}{t_0}\\ \mathbf{elif}\;{B_m}^{2} \leq 5 \cdot 10^{+52}:\\ \;\;\;\;\sqrt{F \cdot \left(-0.5 \cdot \frac{{B_m}^{2}}{C}\right)} \cdot \frac{-\sqrt{2}}{B_m}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B_m, A\right)\right)} \cdot \frac{-1}{\frac{B_m}{\sqrt{2}}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (- (pow B_m 2.0) (* C (* A 4.0)))))
   (if (<= (pow B_m 2.0) 1e-84)
     (/ (- (sqrt (* (* 2.0 (* F t_0)) (+ A A)))) t_0)
     (if (<= (pow B_m 2.0) 5e+52)
       (* (sqrt (* F (* -0.5 (/ (pow B_m 2.0) C)))) (/ (- (sqrt 2.0)) B_m))
       (* (sqrt (* F (- A (hypot B_m A)))) (/ -1.0 (/ B_m (sqrt 2.0))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = pow(B_m, 2.0) - (C * (A * 4.0));
	double tmp;
	if (pow(B_m, 2.0) <= 1e-84) {
		tmp = -sqrt(((2.0 * (F * t_0)) * (A + A))) / t_0;
	} else if (pow(B_m, 2.0) <= 5e+52) {
		tmp = sqrt((F * (-0.5 * (pow(B_m, 2.0) / C)))) * (-sqrt(2.0) / B_m);
	} else {
		tmp = sqrt((F * (A - hypot(B_m, A)))) * (-1.0 / (B_m / sqrt(2.0)));
	}
	return tmp;
}

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double t_0 = Math.pow(B_m, 2.0) - (C * (A * 4.0));
	double tmp;
	if (Math.pow(B_m, 2.0) <= 1e-84) {
		tmp = -Math.sqrt(((2.0 * (F * t_0)) * (A + A))) / t_0;
	} else if (Math.pow(B_m, 2.0) <= 5e+52) {
		tmp = Math.sqrt((F * (-0.5 * (Math.pow(B_m, 2.0) / C)))) * (-Math.sqrt(2.0) / B_m);
	} else {
		tmp = Math.sqrt((F * (A - Math.hypot(B_m, A)))) * (-1.0 / (B_m / Math.sqrt(2.0)));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	t_0 = math.pow(B_m, 2.0) - (C * (A * 4.0))
	tmp = 0
	if math.pow(B_m, 2.0) <= 1e-84:
		tmp = -math.sqrt(((2.0 * (F * t_0)) * (A + A))) / t_0
	elif math.pow(B_m, 2.0) <= 5e+52:
		tmp = math.sqrt((F * (-0.5 * (math.pow(B_m, 2.0) / C)))) * (-math.sqrt(2.0) / B_m)
	else:
		tmp = math.sqrt((F * (A - math.hypot(B_m, A)))) * (-1.0 / (B_m / math.sqrt(2.0)))
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64((B_m ^ 2.0) - Float64(C * Float64(A * 4.0)))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e-84)
		tmp = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(F * t_0)) * Float64(A + A)))) / t_0);
	elseif ((B_m ^ 2.0) <= 5e+52)
		tmp = Float64(sqrt(Float64(F * Float64(-0.5 * Float64((B_m ^ 2.0) / C)))) * Float64(Float64(-sqrt(2.0)) / B_m));
	else
		tmp = Float64(sqrt(Float64(F * Float64(A - hypot(B_m, A)))) * Float64(-1.0 / Float64(B_m / sqrt(2.0))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	t_0 = (B_m ^ 2.0) - (C * (A * 4.0));
	tmp = 0.0;
	if ((B_m ^ 2.0) <= 1e-84)
		tmp = -sqrt(((2.0 * (F * t_0)) * (A + A))) / t_0;
	elseif ((B_m ^ 2.0) <= 5e+52)
		tmp = sqrt((F * (-0.5 * ((B_m ^ 2.0) / C)))) * (-sqrt(2.0) / B_m);
	else
		tmp = sqrt((F * (A - hypot(B_m, A)))) * (-1.0 / (B_m / sqrt(2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(C * N[(A * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-84], N[((-N[Sqrt[N[(N[(2.0 * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision] * N[(A + A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e+52], N[(N[Sqrt[N[(F * N[(-0.5 * N[(N[Power[B$95$m, 2.0], $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[((-N[Sqrt[2.0], $MachinePrecision]) / B$95$m), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(F * N[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / N[(B$95$m / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B 2) < 1e-84

   1. Initial program 17.2%

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

      1. +-commutative 17.2%

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

      2. unpow2 17.2%

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

      3. unpow2 17.2%

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

      4. hypot-udef 27.7%

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

      5. add-exp-log 26.1%

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

   3. Applied egg-rr 26.1%

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

   4. Taylor expanded in C around inf 25.2%

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

      1. sub-neg 25.2%

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

      2. mul-1-neg 25.2%

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

      3. remove-double-neg 25.2%

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

   6. Simplified 25.2%

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

   if 1e-84 < (pow.f64 B 2) < 5e52

   1. Initial program 24.1%

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

   3. Simplified 22.3%

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

   4. Taylor expanded in A around 0 10.2%

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

      1. mul-1-neg 10.2%

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

      2. *-commutative 10.2%

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

      3. distribute-rgt-neg-in 10.2%

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

      4. unpow2 10.2%

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

      5. unpow2 10.2%

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

      6. hypot-def 10.4%

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

   6. Simplified 10.4%

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

   7. Taylor expanded in C around inf 18.8%

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

   if 5e52 < (pow.f64 B 2)

   1. Initial program 17.5%

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

   3. Simplified 13.3%

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

   4. Taylor expanded in C around 0 14.5%

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

      1. mul-1-neg 14.5%

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

      2. distribute-rgt-neg-in 14.5%

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

      3. +-commutative 14.5%

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

      4. unpow2 14.5%

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

      5. unpow2 14.5%

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

      6. hypot-def 24.3%

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

   6. Simplified 24.3%

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

      1. clear-num 24.3%

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

      2. inv-pow 24.3%

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

   8. Applied egg-rr 24.3%

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

      1. unpow-1 24.3%

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

   10. Simplified 24.3%

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

4. Final simplification 24.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{-84}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \left({B}^{2} - C \cdot \left(A \cdot 4\right)\right)\right)\right) \cdot \left(A + A\right)}}{{B}^{2} - C \cdot \left(A \cdot 4\right)}\\ \mathbf{elif}\;{B}^{2} \leq 5 \cdot 10^{+52}:\\ \;\;\;\;\sqrt{F \cdot \left(-0.5 \cdot \frac{{B}^{2}}{C}\right)} \cdot \frac{-\sqrt{2}}{B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)} \cdot \frac{-1}{\frac{B}{\sqrt{2}}}\\ \end{array} \]

Alternative 5: 29.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;F \leq -2.55 \cdot 10^{+29}:\\ \;\;\;\;\frac{\sqrt{2}}{B_m} \cdot \left(-\sqrt{-0.5 \cdot \frac{{B_m}^{2} \cdot F}{C}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B_m, A\right)\right)} \cdot \frac{-\sqrt{2}}{B_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= F -2.55e+29)
   (* (/ (sqrt 2.0) B_m) (- (sqrt (* -0.5 (/ (* (pow B_m 2.0) F) C)))))
   (* (sqrt (* F (- A (hypot B_m A)))) (/ (- (sqrt 2.0)) B_m))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= -2.55e+29) {
		tmp = (sqrt(2.0) / B_m) * -sqrt((-0.5 * ((pow(B_m, 2.0) * F) / C)));
	} else {
		tmp = sqrt((F * (A - hypot(B_m, A)))) * (-sqrt(2.0) / B_m);
	}
	return tmp;
}

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= -2.55e+29) {
		tmp = (Math.sqrt(2.0) / B_m) * -Math.sqrt((-0.5 * ((Math.pow(B_m, 2.0) * F) / C)));
	} else {
		tmp = Math.sqrt((F * (A - Math.hypot(B_m, A)))) * (-Math.sqrt(2.0) / B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if F <= -2.55e+29:
		tmp = (math.sqrt(2.0) / B_m) * -math.sqrt((-0.5 * ((math.pow(B_m, 2.0) * F) / C)))
	else:
		tmp = math.sqrt((F * (A - math.hypot(B_m, A)))) * (-math.sqrt(2.0) / B_m)
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (F <= -2.55e+29)
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(-sqrt(Float64(-0.5 * Float64(Float64((B_m ^ 2.0) * F) / C)))));
	else
		tmp = Float64(sqrt(Float64(F * Float64(A - hypot(B_m, A)))) * Float64(Float64(-sqrt(2.0)) / B_m));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (F <= -2.55e+29)
		tmp = (sqrt(2.0) / B_m) * -sqrt((-0.5 * (((B_m ^ 2.0) * F) / C)));
	else
		tmp = sqrt((F * (A - hypot(B_m, A)))) * (-sqrt(2.0) / B_m);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[F, -2.55e+29], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * (-N[Sqrt[N[(-0.5 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] * F), $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[N[(F * N[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[((-N[Sqrt[2.0], $MachinePrecision]) / B$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -2.55e29

   1. Initial program 7.1%

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

   3. Simplified 7.3%

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

   4. Taylor expanded in A around 0 5.8%

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]
   5. 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 9.3%

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

   6. Simplified 9.3%

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

   7. Taylor expanded in C around inf 13.7%

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

   if -2.55e29 < F

   1. Initial program 24.9%

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

   3. Simplified 24.3%

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

   4. Taylor expanded in C around 0 12.2%

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

      1. mul-1-neg 12.2%

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

      2. distribute-rgt-neg-in 12.2%

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

      3. +-commutative 12.2%

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

      4. unpow2 12.2%

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

      5. unpow2 12.2%

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

      6. hypot-def 19.5%

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

   6. Simplified 19.5%

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

4. Final simplification 17.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.55 \cdot 10^{+29}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{-0.5 \cdot \frac{{B}^{2} \cdot F}{C}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 6: 29.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;F \leq -2.55 \cdot 10^{+29}:\\ \;\;\;\;\frac{\sqrt{2}}{B_m} \cdot \left(-\sqrt{-0.5 \cdot \frac{{B_m}^{2} \cdot F}{C}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B_m, A\right)\right)} \cdot \frac{-1}{\frac{B_m}{\sqrt{2}}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= F -2.55e+29)
   (* (/ (sqrt 2.0) B_m) (- (sqrt (* -0.5 (/ (* (pow B_m 2.0) F) C)))))
   (* (sqrt (* F (- A (hypot B_m A)))) (/ -1.0 (/ B_m (sqrt 2.0))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= -2.55e+29) {
		tmp = (sqrt(2.0) / B_m) * -sqrt((-0.5 * ((pow(B_m, 2.0) * F) / C)));
	} else {
		tmp = sqrt((F * (A - hypot(B_m, A)))) * (-1.0 / (B_m / sqrt(2.0)));
	}
	return tmp;
}

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= -2.55e+29) {
		tmp = (Math.sqrt(2.0) / B_m) * -Math.sqrt((-0.5 * ((Math.pow(B_m, 2.0) * F) / C)));
	} else {
		tmp = Math.sqrt((F * (A - Math.hypot(B_m, A)))) * (-1.0 / (B_m / Math.sqrt(2.0)));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if F <= -2.55e+29:
		tmp = (math.sqrt(2.0) / B_m) * -math.sqrt((-0.5 * ((math.pow(B_m, 2.0) * F) / C)))
	else:
		tmp = math.sqrt((F * (A - math.hypot(B_m, A)))) * (-1.0 / (B_m / math.sqrt(2.0)))
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (F <= -2.55e+29)
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(-sqrt(Float64(-0.5 * Float64(Float64((B_m ^ 2.0) * F) / C)))));
	else
		tmp = Float64(sqrt(Float64(F * Float64(A - hypot(B_m, A)))) * Float64(-1.0 / Float64(B_m / sqrt(2.0))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (F <= -2.55e+29)
		tmp = (sqrt(2.0) / B_m) * -sqrt((-0.5 * (((B_m ^ 2.0) * F) / C)));
	else
		tmp = sqrt((F * (A - hypot(B_m, A)))) * (-1.0 / (B_m / sqrt(2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[F, -2.55e+29], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * (-N[Sqrt[N[(-0.5 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] * F), $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[N[(F * N[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / N[(B$95$m / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -2.55e29

   1. Initial program 7.1%

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

   3. Simplified 7.3%

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

   4. Taylor expanded in A around 0 5.8%

      \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(C - \sqrt{{B}^{2} + {C}^{2}}\right)}\right)} \]
   5. 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 9.3%

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

   6. Simplified 9.3%

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

   7. Taylor expanded in C around inf 13.7%

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

   if -2.55e29 < F

   1. Initial program 24.9%

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

   3. Simplified 24.3%

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

   4. Taylor expanded in C around 0 12.2%

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

      1. mul-1-neg 12.2%

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

      2. distribute-rgt-neg-in 12.2%

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

      3. +-commutative 12.2%

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

      4. unpow2 12.2%

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

      5. unpow2 12.2%

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

      6. hypot-def 19.5%

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

   6. Simplified 19.5%

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

      1. clear-num 19.5%

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

      2. inv-pow 19.5%

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

   8. Applied egg-rr 19.5%

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

      1. unpow-1 19.5%

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

   10. Simplified 19.5%

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

4. Final simplification 17.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.55 \cdot 10^{+29}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{-0.5 \cdot \frac{{B}^{2} \cdot F}{C}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)} \cdot \frac{-1}{\frac{B}{\sqrt{2}}}\\ \end{array} \]

Alternative 7: 31.4% accurate, 2.0× speedup

Math

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

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (* (sqrt (* F (- A (hypot B_m A)))) (/ (- (sqrt 2.0)) B_m)))

C

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

Java

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

Python

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

Julia

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

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp = code(A, B_m, C, F)
	tmp = sqrt((F * (A - hypot(B_m, A)))) * (-sqrt(2.0) / B_m);
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := N[(N[Sqrt[N[(F * N[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[((-N[Sqrt[2.0], $MachinePrecision]) / B$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 18.2%

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

3. Simplified 17.9%

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

4. Taylor expanded in C around 0 9.6%

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

   1. mul-1-neg 9.6%

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

   2. distribute-rgt-neg-in 9.6%

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

   3. +-commutative 9.6%

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

   4. unpow2 9.6%

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

   5. unpow2 9.6%

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

   6. hypot-def 15.0%

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

6. Simplified 15.0%

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

7. Final simplification 15.0%

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

Alternative 8: 26.6% accurate, 3.0× speedup

Math

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

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (* (/ (- (sqrt 2.0)) B_m) (sqrt (* F (- A B_m)))))

C

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

Fortran

B_m = abs(B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    code = (-sqrt(2.0d0) / b_m) * sqrt((f * (a - b_m)))
end function

Java

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

Python

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

Julia

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

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp = code(A, B_m, C, F)
	tmp = (-sqrt(2.0) / B_m) * sqrt((F * (A - B_m)));
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := N[(N[((-N[Sqrt[2.0], $MachinePrecision]) / B$95$m), $MachinePrecision] * N[Sqrt[N[(F * N[(A - B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 18.2%

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

3. Simplified 17.9%

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

4. Taylor expanded in C around 0 9.6%

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

   1. mul-1-neg 9.6%

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

   2. distribute-rgt-neg-in 9.6%

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

   3. +-commutative 9.6%

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

   4. unpow2 9.6%

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

   5. unpow2 9.6%

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

   6. hypot-def 15.0%

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

6. Simplified 15.0%

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

7. Taylor expanded in A around 0 12.8%

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

   1. mul-1-neg 12.8%

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

   2. unsub-neg 12.8%

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

9. Simplified 12.8%

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

10. Final simplification 12.8%

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

Alternative 9: 26.4% accurate, 3.0× speedup

Math

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

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (* (/ (sqrt 2.0) B_m) (- (sqrt (* B_m (- F))))))

C

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

Fortran

B_m = abs(B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    code = (sqrt(2.0d0) / b_m) * -sqrt((b_m * -f))
end function

Java

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

Python

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

Julia

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

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp = code(A, B_m, C, F)
	tmp = (sqrt(2.0) / B_m) * -sqrt((B_m * -F));
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * (-N[Sqrt[N[(B$95$m * (-F)), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]

Derivation

1. Initial program 18.2%

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

3. Simplified 17.9%

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

4. Taylor expanded in C around 0 9.6%

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

   1. mul-1-neg 9.6%

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

   2. distribute-rgt-neg-in 9.6%

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

   3. +-commutative 9.6%

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

   4. unpow2 9.6%

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

   5. unpow2 9.6%

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

   6. hypot-def 15.0%

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

6. Simplified 15.0%

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

7. Taylor expanded in A around 0 13.3%

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

   1. associate-*r* 13.3%

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

   2. mul-1-neg 13.3%

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

9. Simplified 13.3%

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

10. Final simplification 13.3%

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

Alternative 10: 8.9% accurate, 5.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ -2 \cdot \frac{{\left(A \cdot F\right)}^{0.5}}{B_m} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F) :precision binary64 (* -2.0 (/ (pow (* A F) 0.5) B_m)))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	return -2.0 * (pow((A * F), 0.5) / B_m);
}

Fortran

B_m = abs(B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    code = (-2.0d0) * (((a * f) ** 0.5d0) / b_m)
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	return -2.0 * (Math.pow((A * F), 0.5) / B_m);
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	return -2.0 * (math.pow((A * F), 0.5) / B_m)

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	return Float64(-2.0 * Float64((Float64(A * F) ^ 0.5) / B_m))
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp = code(A, B_m, C, F)
	tmp = -2.0 * (((A * F) ^ 0.5) / B_m);
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := N[(-2.0 * N[(N[Power[N[(A * F), $MachinePrecision], 0.5], $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 18.2%

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

3. Simplified 17.9%

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

4. Taylor expanded in A around -inf 7.9%

   \[\leadsto \frac{-\sqrt{\color{blue}{-16 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right) + 4 \cdot \left(A \cdot \left({B}^{2} \cdot F\right)\right)}}}{\mathsf{fma}\left(A, C \cdot -4, {B}^{2}\right)} \]

5. Taylor expanded in C around 0 3.0%

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

   1. associate-*r/ 3.0%

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

   2. *-rgt-identity 3.0%

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

   3. *-commutative 3.0%

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

7. Simplified 3.0%

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

   1. pow1/2 3.1%

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

   2. *-commutative 3.1%

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

9. Applied egg-rr 3.1%

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

10. Final simplification 3.1%

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

Alternative 11: 8.9% accurate, 5.9× speedup

Math

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

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F) :precision binary64 (* -2.0 (/ (sqrt (* A F)) B_m)))

C

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

Fortran

B_m = abs(B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    code = (-2.0d0) * (sqrt((a * f)) / b_m)
end function

Java

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

Python

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

Julia

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

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp = code(A, B_m, C, F)
	tmp = -2.0 * (sqrt((A * F)) / B_m);
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := N[(-2.0 * N[(N[Sqrt[N[(A * F), $MachinePrecision]], $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 18.2%

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

3. Simplified 17.9%

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

4. Taylor expanded in A around -inf 7.9%

   \[\leadsto \frac{-\sqrt{\color{blue}{-16 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right) + 4 \cdot \left(A \cdot \left({B}^{2} \cdot F\right)\right)}}}{\mathsf{fma}\left(A, C \cdot -4, {B}^{2}\right)} \]

5. Taylor expanded in C around 0 3.0%

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

   1. associate-*r/ 3.0%

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

   2. *-rgt-identity 3.0%

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

   3. *-commutative 3.0%

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

7. Simplified 3.0%

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

8. Final simplification 3.0%

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

Reproduce

herbie shell --seed 2023339 
(FPCore (A B C F)
  :name "ABCF->ab-angle b"
  :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))))