ABCF->ab-angle b

Percentage Accurate: 18.5% → 47.6%

Time: 24.0s

Alternatives: 7

Speedup: 5.4×

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: 18.5% 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: 47.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(A, C \cdot -4, {B_m}^{2}\right)\\ \mathbf{if}\;{B_m}^{2} \leq 10^{-82}:\\ \;\;\;\;\frac{1}{\frac{t_0}{-\sqrt{A \cdot \left(-8 \cdot \left(F \cdot \left(C \cdot \left(2 \cdot A\right)\right)\right)\right)}}}\\ \mathbf{elif}\;{B_m}^{2} \leq 10^{+305}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B_m, A - C\right)\right)\right)} \cdot \left(-\sqrt{2 \cdot t_0}\right)}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B_m} \cdot \left(-\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B_m, A\right)\right)}\right)\\ \end{array} \end{array} \]

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 A (* C -4.0) (pow B_m 2.0))))
   (if (<= (pow B_m 2.0) 1e-82)
     (/ 1.0 (/ t_0 (- (sqrt (* A (* -8.0 (* F (* C (* 2.0 A)))))))))
     (if (<= (pow B_m 2.0) 1e+305)
       (/
        (* (sqrt (* F (+ A (- C (hypot B_m (- A C)))))) (- (sqrt (* 2.0 t_0))))
        t_0)
       (* (/ (sqrt 2.0) B_m) (- (sqrt (* F (- A (hypot B_m A))))))))))

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(A, (C * -4.0), pow(B_m, 2.0));
	double tmp;
	if (pow(B_m, 2.0) <= 1e-82) {
		tmp = 1.0 / (t_0 / -sqrt((A * (-8.0 * (F * (C * (2.0 * A)))))));
	} else if (pow(B_m, 2.0) <= 1e+305) {
		tmp = (sqrt((F * (A + (C - hypot(B_m, (A - C)))))) * -sqrt((2.0 * t_0))) / t_0;
	} else {
		tmp = (sqrt(2.0) / B_m) * -sqrt((F * (A - hypot(B_m, A))));
	}
	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(A, Float64(C * -4.0), (B_m ^ 2.0))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e-82)
		tmp = Float64(1.0 / Float64(t_0 / Float64(-sqrt(Float64(A * Float64(-8.0 * Float64(F * Float64(C * Float64(2.0 * A)))))))));
	elseif ((B_m ^ 2.0) <= 1e+305)
		tmp = Float64(Float64(sqrt(Float64(F * Float64(A + Float64(C - hypot(B_m, Float64(A - C)))))) * Float64(-sqrt(Float64(2.0 * t_0)))) / t_0);
	else
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(-sqrt(Float64(F * Float64(A - hypot(B_m, A))))));
	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[(A * N[(C * -4.0), $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-82], N[(1.0 / N[(t$95$0 / (-N[Sqrt[N[(A * N[(-8.0 * N[(F * N[(C * N[(2.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e+305], N[(N[(N[Sqrt[N[(F * N[(A + N[(C - N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(2.0 * t$95$0), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * (-N[Sqrt[N[(F * N[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B 2) < 1e-82

   1. Initial program 22.6%

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

   3. Simplified 27.6%

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

   4. Taylor expanded in C around inf 23.6%

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

      1. associate-*r* 23.6%

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

      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 23.6%

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

      7. associate-*r* 23.0%

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

      8. *-commutative 23.0%

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

      9. sub-neg 23.0%

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

      10. mul-1-neg 23.0%

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

      11. remove-double-neg 23.0%

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

   6. Simplified 23.0%

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

      1. clear-num 23.0%

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

      2. inv-pow 23.0%

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

      3. associate-*l* 23.0%

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

      4. count-2 23.0%

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

   8. Applied egg-rr 23.0%

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

      1. unpow-1 23.0%

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

      2. associate-*l* 25.6%

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

   10. Simplified 25.6%

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

   if 1e-82 < (pow.f64 B 2) < 9.9999999999999994e304

   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 23.3%

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

      1. pow1/2 23.4%

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

      2. associate-*r* 29.3%

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

      3. unpow-prod-down 46.8%

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

      4. associate-+r- 46.7%

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

      5. pow1/2 46.7%

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

   5. Applied egg-rr 46.7%

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

      1. unpow1/2 46.7%

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

      2. associate-+r- 46.8%

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

      3. +-commutative 46.8%

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

      4. associate-+l- 46.6%

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

   7. Simplified 46.6%

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

   if 9.9999999999999994e304 < (pow.f64 B 2)

   1. Initial program 0.1%

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

   3. Simplified 0.1%

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

   4. Taylor expanded in C around 0 3.1%

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

      1. mul-1-neg 3.1%

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

      2. distribute-rgt-neg-in 3.1%

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

      3. +-commutative 3.1%

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

      4. unpow2 3.1%

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

      5. unpow2 3.1%

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

      6. hypot-def 23.6%

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

   6. Simplified 23.6%

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

4. Final simplification 31.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{-82}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(A, C \cdot -4, {B}^{2}\right)}{-\sqrt{A \cdot \left(-8 \cdot \left(F \cdot \left(C \cdot \left(2 \cdot A\right)\right)\right)\right)}}}\\ \mathbf{elif}\;{B}^{2} \leq 10^{+305}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \left(-\sqrt{2 \cdot \mathsf{fma}\left(A, C \cdot -4, {B}^{2}\right)}\right)}{\mathsf{fma}\left(A, C \cdot -4, {B}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}\right)\\ \end{array} \]

Alternative 2: 46.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B_m}^{2} \leq 2 \cdot 10^{-89}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(A, C \cdot -4, {B_m}^{2}\right)}{-\sqrt{A \cdot \left(-8 \cdot \left(F \cdot \left(C \cdot \left(2 \cdot A\right)\right)\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B_m, A\right)\right)} \cdot \left(-\frac{\sqrt{2}}{B_m}\right)\\ \end{array} \end{array} \]

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) 2e-89)
   (/
    1.0
    (/
     (fma A (* C -4.0) (pow B_m 2.0))
     (- (sqrt (* A (* -8.0 (* F (* C (* 2.0 A)))))))))
   (* (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 (pow(B_m, 2.0) <= 2e-89) {
		tmp = 1.0 / (fma(A, (C * -4.0), pow(B_m, 2.0)) / -sqrt((A * (-8.0 * (F * (C * (2.0 * A)))))));
	} else {
		tmp = sqrt((F * (A - hypot(B_m, A)))) * -(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 ((B_m ^ 2.0) <= 2e-89)
		tmp = Float64(1.0 / Float64(fma(A, Float64(C * -4.0), (B_m ^ 2.0)) / Float64(-sqrt(Float64(A * Float64(-8.0 * Float64(F * Float64(C * Float64(2.0 * A)))))))));
	else
		tmp = Float64(sqrt(Float64(F * Float64(A - hypot(B_m, A)))) * Float64(-Float64(sqrt(2.0) / B_m)));
	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], 2e-89], N[(1.0 / N[(N[(A * N[(C * -4.0), $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision] / (-N[Sqrt[N[(A * N[(-8.0 * N[(F * N[(C * N[(2.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B 2) < 2.00000000000000008e-89

   1. Initial program 23.2%

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

   3. Simplified 28.3%

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

   4. Taylor expanded in C around inf 24.2%

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

      1. associate-*r* 24.2%

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

      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 24.2%

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

      7. associate-*r* 23.6%

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

      8. *-commutative 23.6%

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

      9. sub-neg 23.6%

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

      10. mul-1-neg 23.6%

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

      11. remove-double-neg 23.6%

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

   6. Simplified 23.6%

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

      1. clear-num 23.6%

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

      2. inv-pow 23.6%

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

      3. associate-*l* 23.6%

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

      4. count-2 23.6%

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

   8. Applied egg-rr 23.6%

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

      1. unpow-1 23.6%

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

      2. associate-*l* 26.2%

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

   10. Simplified 26.2%

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

   if 2.00000000000000008e-89 < (pow.f64 B 2)

   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 12.5%

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

   4. Taylor expanded in C around 0 14.2%

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

      1. mul-1-neg 14.2%

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

      2. distribute-rgt-neg-in 14.2%

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

      3. +-commutative 14.2%

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

      4. unpow2 14.2%

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

      5. unpow2 14.2%

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

      6. hypot-def 23.6%

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

   6. Simplified 23.6%

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

4. Final simplification 24.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 2 \cdot 10^{-89}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(A, C \cdot -4, {B}^{2}\right)}{-\sqrt{A \cdot \left(-8 \cdot \left(F \cdot \left(C \cdot \left(2 \cdot A\right)\right)\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)} \cdot \left(-\frac{\sqrt{2}}{B}\right)\\ \end{array} \]

Alternative 3: 46.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B_m}^{2} \leq 2 \cdot 10^{-89}:\\ \;\;\;\;-\frac{\sqrt{A \cdot \left(-8 \cdot \left(F \cdot \left(C \cdot \left(2 \cdot A\right)\right)\right)\right)}}{\mathsf{fma}\left(A, C \cdot -4, {B_m}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B_m, A\right)\right)} \cdot \left(-\frac{\sqrt{2}}{B_m}\right)\\ \end{array} \end{array} \]

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) 2e-89)
   (-
    (/
     (sqrt (* A (* -8.0 (* F (* C (* 2.0 A))))))
     (fma A (* C -4.0) (pow B_m 2.0))))
   (* (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 (pow(B_m, 2.0) <= 2e-89) {
		tmp = -(sqrt((A * (-8.0 * (F * (C * (2.0 * A)))))) / fma(A, (C * -4.0), pow(B_m, 2.0)));
	} else {
		tmp = sqrt((F * (A - hypot(B_m, A)))) * -(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 ((B_m ^ 2.0) <= 2e-89)
		tmp = Float64(-Float64(sqrt(Float64(A * Float64(-8.0 * Float64(F * Float64(C * Float64(2.0 * A)))))) / fma(A, Float64(C * -4.0), (B_m ^ 2.0))));
	else
		tmp = Float64(sqrt(Float64(F * Float64(A - hypot(B_m, A)))) * Float64(-Float64(sqrt(2.0) / B_m)));
	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], 2e-89], (-N[(N[Sqrt[N[(A * N[(-8.0 * N[(F * N[(C * N[(2.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(A * N[(C * -4.0), $MachinePrecision] + N[Power[B$95$m, 2.0], $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 (pow.f64 B 2) < 2.00000000000000008e-89

   1. Initial program 23.2%

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

   3. Simplified 28.3%

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

   4. Taylor expanded in C around inf 24.2%

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

      1. associate-*r* 24.2%

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

      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 24.2%

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

      7. associate-*r* 23.6%

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

      8. *-commutative 23.6%

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

      9. sub-neg 23.6%

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

      10. mul-1-neg 23.6%

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

      11. remove-double-neg 23.6%

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

   6. Simplified 23.6%

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

      1. expm1-log1p-u 22.6%

         \[\leadsto \frac{-\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\left(A \cdot -8\right) \cdot \left(\left(F \cdot C\right) \cdot \left(A + A\right)\right)}\right)\right)}}{\mathsf{fma}\left(A, C \cdot -4, {B}^{2}\right)} \]

      2. expm1-udef 13.9%

         \[\leadsto \frac{-\color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\left(A \cdot -8\right) \cdot \left(\left(F \cdot C\right) \cdot \left(A + A\right)\right)}\right)} - 1\right)}}{\mathsf{fma}\left(A, C \cdot -4, {B}^{2}\right)} \]

      3. associate-*l* 13.9%

         \[\leadsto \frac{-\left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{A \cdot \left(-8 \cdot \left(\left(F \cdot C\right) \cdot \left(A + A\right)\right)\right)}}\right)} - 1\right)}{\mathsf{fma}\left(A, C \cdot -4, {B}^{2}\right)} \]

      4. count-2 13.9%

         \[\leadsto \frac{-\left(e^{\mathsf{log1p}\left(\sqrt{A \cdot \left(-8 \cdot \left(\left(F \cdot C\right) \cdot \color{blue}{\left(2 \cdot A\right)}\right)\right)}\right)} - 1\right)}{\mathsf{fma}\left(A, C \cdot -4, {B}^{2}\right)} \]

   8. Applied egg-rr 13.9%

      \[\leadsto \frac{-\color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{A \cdot \left(-8 \cdot \left(\left(F \cdot C\right) \cdot \left(2 \cdot A\right)\right)\right)}\right)} - 1\right)}}{\mathsf{fma}\left(A, C \cdot -4, {B}^{2}\right)} \]
   9. Step-by-step derivation

      1. expm1-def 22.6%

         \[\leadsto \frac{-\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{A \cdot \left(-8 \cdot \left(\left(F \cdot C\right) \cdot \left(2 \cdot A\right)\right)\right)}\right)\right)}}{\mathsf{fma}\left(A, C \cdot -4, {B}^{2}\right)} \]

      2. expm1-log1p 23.6%

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

      3. associate-*l* 26.2%

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

   10. Simplified 26.2%

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

   if 2.00000000000000008e-89 < (pow.f64 B 2)

   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 12.5%

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

   4. Taylor expanded in C around 0 14.2%

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

      1. mul-1-neg 14.2%

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

      2. distribute-rgt-neg-in 14.2%

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

      3. +-commutative 14.2%

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

      4. unpow2 14.2%

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

      5. unpow2 14.2%

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

      6. hypot-def 23.6%

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

   6. Simplified 23.6%

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

4. Final simplification 24.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 2 \cdot 10^{-89}:\\ \;\;\;\;-\frac{\sqrt{A \cdot \left(-8 \cdot \left(F \cdot \left(C \cdot \left(2 \cdot A\right)\right)\right)\right)}}{\mathsf{fma}\left(A, C \cdot -4, {B}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)} \cdot \left(-\frac{\sqrt{2}}{B}\right)\\ \end{array} \]

Alternative 4: 39.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B_m \leq 3 \cdot 10^{-37}:\\ \;\;\;\;\frac{-\sqrt{\left(A \cdot -8\right) \cdot \left(\left(A + A\right) \cdot \left(C \cdot F\right)\right)}}{{B_m}^{2} + A \cdot \left(C \cdot -4\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(C - \mathsf{hypot}\left(B_m, C\right)\right)} \cdot \frac{-\sqrt{2}}{B_m}\\ \end{array} \end{array} \]

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 (<= B_m 3e-37)
   (/
    (- (sqrt (* (* A -8.0) (* (+ A A) (* C F)))))
    (+ (pow B_m 2.0) (* A (* C -4.0))))
   (* (sqrt (* F (- C (hypot B_m C)))) (/ (- (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 (B_m <= 3e-37) {
		tmp = -sqrt(((A * -8.0) * ((A + A) * (C * F)))) / (pow(B_m, 2.0) + (A * (C * -4.0)));
	} else {
		tmp = sqrt((F * (C - hypot(B_m, C)))) * (-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 (B_m <= 3e-37) {
		tmp = -Math.sqrt(((A * -8.0) * ((A + A) * (C * F)))) / (Math.pow(B_m, 2.0) + (A * (C * -4.0)));
	} else {
		tmp = Math.sqrt((F * (C - Math.hypot(B_m, C)))) * (-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 B_m <= 3e-37:
		tmp = -math.sqrt(((A * -8.0) * ((A + A) * (C * F)))) / (math.pow(B_m, 2.0) + (A * (C * -4.0)))
	else:
		tmp = math.sqrt((F * (C - math.hypot(B_m, C)))) * (-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 (B_m <= 3e-37)
		tmp = Float64(Float64(-sqrt(Float64(Float64(A * -8.0) * Float64(Float64(A + A) * Float64(C * F))))) / Float64((B_m ^ 2.0) + Float64(A * Float64(C * -4.0))));
	else
		tmp = Float64(sqrt(Float64(F * Float64(C - hypot(B_m, C)))) * 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 (B_m <= 3e-37)
		tmp = -sqrt(((A * -8.0) * ((A + A) * (C * F)))) / ((B_m ^ 2.0) + (A * (C * -4.0)));
	else
		tmp = sqrt((F * (C - hypot(B_m, C)))) * (-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[B$95$m, 3e-37], N[((-N[Sqrt[N[(N[(A * -8.0), $MachinePrecision] * N[(N[(A + A), $MachinePrecision] * N[(C * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(F * N[(C - N[Sqrt[B$95$m ^ 2 + C ^ 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 B < 3e-37

   1. Initial program 17.1%

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

   3. Simplified 20.3%

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

   4. Taylor expanded in C around inf 15.1%

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

      1. associate-*r* 15.1%

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

      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 15.1%

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

      7. associate-*r* 14.8%

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

      8. *-commutative 14.8%

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

      9. sub-neg 14.8%

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

      10. mul-1-neg 14.8%

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

      11. remove-double-neg 14.8%

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

   6. Simplified 14.8%

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

      1. fma-udef 14.8%

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

   8. Applied egg-rr 14.8%

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

   if 3e-37 < B

   1. Initial program 19.4%

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

   3. Simplified 16.1%

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

   4. Taylor expanded in A around 0 28.5%

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

      1. mul-1-neg 28.5%

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

      2. *-commutative 28.5%

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

      3. distribute-rgt-neg-in 28.5%

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

      4. unpow2 28.5%

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

      5. unpow2 28.5%

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

      6. hypot-def 48.8%

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

   6. Simplified 48.8%

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

4. Final simplification 24.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 3 \cdot 10^{-37}:\\ \;\;\;\;\frac{-\sqrt{\left(A \cdot -8\right) \cdot \left(\left(A + A\right) \cdot \left(C \cdot F\right)\right)}}{{B}^{2} + A \cdot \left(C \cdot -4\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(C - \mathsf{hypot}\left(B, C\right)\right)} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 5: 42.9% accurate, 2.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B_m \leq 2.4 \cdot 10^{-37}:\\ \;\;\;\;\frac{-\sqrt{\left(A \cdot -8\right) \cdot \left(\left(A + A\right) \cdot \left(C \cdot F\right)\right)}}{{B_m}^{2} + A \cdot \left(C \cdot -4\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B_m, A\right)\right)} \cdot \left(-\frac{\sqrt{2}}{B_m}\right)\\ \end{array} \end{array} \]

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 (<= B_m 2.4e-37)
   (/
    (- (sqrt (* (* A -8.0) (* (+ A A) (* C F)))))
    (+ (pow B_m 2.0) (* A (* C -4.0))))
   (* (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 (B_m <= 2.4e-37) {
		tmp = -sqrt(((A * -8.0) * ((A + A) * (C * F)))) / (pow(B_m, 2.0) + (A * (C * -4.0)));
	} 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 (B_m <= 2.4e-37) {
		tmp = -Math.sqrt(((A * -8.0) * ((A + A) * (C * F)))) / (Math.pow(B_m, 2.0) + (A * (C * -4.0)));
	} 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 B_m <= 2.4e-37:
		tmp = -math.sqrt(((A * -8.0) * ((A + A) * (C * F)))) / (math.pow(B_m, 2.0) + (A * (C * -4.0)))
	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 (B_m <= 2.4e-37)
		tmp = Float64(Float64(-sqrt(Float64(Float64(A * -8.0) * Float64(Float64(A + A) * Float64(C * F))))) / Float64((B_m ^ 2.0) + Float64(A * Float64(C * -4.0))));
	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 (B_m <= 2.4e-37)
		tmp = -sqrt(((A * -8.0) * ((A + A) * (C * F)))) / ((B_m ^ 2.0) + (A * (C * -4.0)));
	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[B$95$m, 2.4e-37], N[((-N[Sqrt[N[(N[(A * -8.0), $MachinePrecision] * N[(N[(A + A), $MachinePrecision] * N[(C * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[(A * N[(C * -4.0), $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 B < 2.39999999999999991e-37

   1. Initial program 17.1%

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

   3. Simplified 20.3%

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

   4. Taylor expanded in C around inf 15.1%

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

      1. associate-*r* 15.1%

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

      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 15.1%

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

      7. associate-*r* 14.8%

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

      8. *-commutative 14.8%

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

      9. sub-neg 14.8%

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

      10. mul-1-neg 14.8%

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

      11. remove-double-neg 14.8%

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

   6. Simplified 14.8%

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

      1. fma-udef 14.8%

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

   8. Applied egg-rr 14.8%

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

   if 2.39999999999999991e-37 < B

   1. Initial program 19.4%

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

   3. Simplified 16.1%

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

   4. Taylor expanded in C around 0 28.5%

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

      1. mul-1-neg 28.5%

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

      2. distribute-rgt-neg-in 28.5%

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

      3. +-commutative 28.5%

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

      4. unpow2 28.5%

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

      5. unpow2 28.5%

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

      6. hypot-def 46.9%

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

   6. Simplified 46.9%

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

4. Final simplification 23.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 2.4 \cdot 10^{-37}:\\ \;\;\;\;\frac{-\sqrt{\left(A \cdot -8\right) \cdot \left(\left(A + A\right) \cdot \left(C \cdot F\right)\right)}}{{B}^{2} + A \cdot \left(C \cdot -4\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)} \cdot \left(-\frac{\sqrt{2}}{B}\right)\\ \end{array} \]

Alternative 6: 21.5% accurate, 5.4× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \frac{-\sqrt{\left(A \cdot -8\right) \cdot \left(\left(A + A\right) \cdot \left(C \cdot F\right)\right)}}{C \cdot \left(A \cdot -4\right)} \end{array} \]

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 (* (* A -8.0) (* (+ A A) (* C F))))) (* C (* A -4.0))))

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) {
	return -sqrt(((A * -8.0) * ((A + A) * (C * F)))) / (C * (A * -4.0));
}

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(((a * (-8.0d0)) * ((a + a) * (c * f)))) / (c * (a * (-4.0d0)))
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(((A * -8.0) * ((A + A) * (C * F)))) / (C * (A * -4.0));
}

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(((A * -8.0) * ((A + A) * (C * F)))) / (C * (A * -4.0))

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(Float64(Float64(A * -8.0) * Float64(Float64(A + A) * Float64(C * F))))) / Float64(C * Float64(A * -4.0)))
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(((A * -8.0) * ((A + A) * (C * F)))) / (C * (A * -4.0));
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[(N[(A * -8.0), $MachinePrecision] * N[(N[(A + A), $MachinePrecision] * N[(C * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 17.7%

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

3. Simplified 19.2%

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

4. Taylor expanded in C around inf 12.0%

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

   1. associate-*r* 12.0%

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

   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 12.0%

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

   7. associate-*r* 11.8%

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

   8. *-commutative 11.8%

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

   9. sub-neg 11.8%

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

   10. mul-1-neg 11.8%

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

   11. remove-double-neg 11.8%

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

6. Simplified 11.8%

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

7. Taylor expanded in A around inf 11.6%

   \[\leadsto \frac{-\sqrt{\left(A \cdot -8\right) \cdot \left(\left(F \cdot C\right) \cdot \left(A + A\right)\right)}}{\color{blue}{-4 \cdot \left(A \cdot C\right)}} \]
8. Step-by-step derivation

   1. associate-*r* 11.6%

      \[\leadsto \frac{-\sqrt{\left(A \cdot -8\right) \cdot \left(\left(F \cdot C\right) \cdot \left(A + A\right)\right)}}{\color{blue}{\left(-4 \cdot A\right) \cdot C}} \]

9. Simplified 11.6%

   \[\leadsto \frac{-\sqrt{\left(A \cdot -8\right) \cdot \left(\left(F \cdot C\right) \cdot \left(A + A\right)\right)}}{\color{blue}{\left(-4 \cdot A\right) \cdot C}} \]

10. Final simplification 11.6%

    \[\leadsto \frac{-\sqrt{\left(A \cdot -8\right) \cdot \left(\left(A + A\right) \cdot \left(C \cdot F\right)\right)}}{C \cdot \left(A \cdot -4\right)} \]

Alternative 7: 1.0% accurate, 5.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ -2 \cdot \left(\frac{1}{B_m} \cdot \sqrt{C \cdot F}\right) \end{array} \]

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
 (* -2.0 (* (/ 1.0 B_m) (sqrt (* C 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 -2.0 * ((1.0 / B_m) * sqrt((C * 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 = (-2.0d0) * ((1.0d0 / b_m) * sqrt((c * 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 -2.0 * ((1.0 / B_m) * Math.sqrt((C * 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 -2.0 * ((1.0 / B_m) * math.sqrt((C * 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(-2.0 * Float64(Float64(1.0 / B_m) * sqrt(Float64(C * 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 = -2.0 * ((1.0 / B_m) * sqrt((C * 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[(-2.0 * N[(N[(1.0 / B$95$m), $MachinePrecision] * N[Sqrt[N[(C * F), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 17.7%

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

3. Simplified 19.2%

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

4. Taylor expanded in C around -inf 11.1%

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

5. Taylor expanded in A around 0 3.0%

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

6. Final simplification 3.0%

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

Reproduce

herbie shell --seed 2023337 
(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))))