ABCF->ab-angle b

Percentage Accurate: 18.9% → 48.5%

Time: 17.9s

Alternatives: 10

Speedup: 13.3×

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: 18.9% 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: 48.5% accurate, 3.8× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;C \leq -1.4 \cdot 10^{-291}:\\ \;\;\;\;\frac{\left(2 \cdot \left(\sqrt{C \cdot -4} \cdot \sqrt{F}\right)\right) \cdot -0.25}{C}\\ \mathbf{elif}\;C \leq 4.7 \cdot 10^{-45}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)} \cdot \sqrt{\left(F \cdot 2\right) \cdot \left(\left(C + A\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right)}\right) \cdot \frac{-1}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.25 \cdot \left(\left(2 \cdot \sqrt{C}\right) \cdot \sqrt{-4 \cdot F}\right)}{C}\\ \end{array} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (if (<= C -1.4e-291)
   (/ (* (* 2.0 (* (sqrt (* C -4.0)) (sqrt F))) -0.25) C)
   (if (<= C 4.7e-45)
     (*
      (*
       (sqrt (fma C (* A -4.0) (* B B)))
       (sqrt (* (* F 2.0) (- (+ C A) (sqrt (fma (- A C) (- A C) (* B B)))))))
      (/ -1.0 (fma B B (* C (* A -4.0)))))
     (/ (* -0.25 (* (* 2.0 (sqrt C)) (sqrt (* -4.0 F)))) C))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	double tmp;
	if (C <= -1.4e-291) {
		tmp = ((2.0 * (sqrt((C * -4.0)) * sqrt(F))) * -0.25) / C;
	} else if (C <= 4.7e-45) {
		tmp = (sqrt(fma(C, (A * -4.0), (B * B))) * sqrt(((F * 2.0) * ((C + A) - sqrt(fma((A - C), (A - C), (B * B))))))) * (-1.0 / fma(B, B, (C * (A * -4.0))));
	} else {
		tmp = (-0.25 * ((2.0 * sqrt(C)) * sqrt((-4.0 * F)))) / C;
	}
	return tmp;
}

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	tmp = 0.0
	if (C <= -1.4e-291)
		tmp = Float64(Float64(Float64(2.0 * Float64(sqrt(Float64(C * -4.0)) * sqrt(F))) * -0.25) / C);
	elseif (C <= 4.7e-45)
		tmp = Float64(Float64(sqrt(fma(C, Float64(A * -4.0), Float64(B * B))) * sqrt(Float64(Float64(F * 2.0) * Float64(Float64(C + A) - sqrt(fma(Float64(A - C), Float64(A - C), Float64(B * B))))))) * Float64(-1.0 / fma(B, B, Float64(C * Float64(A * -4.0)))));
	else
		tmp = Float64(Float64(-0.25 * Float64(Float64(2.0 * sqrt(C)) * sqrt(Float64(-4.0 * F)))) / C);
	end
	return tmp
end

Wolfram

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := If[LessEqual[C, -1.4e-291], N[(N[(N[(2.0 * N[(N[Sqrt[N[(C * -4.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision] / C), $MachinePrecision], If[LessEqual[C, 4.7e-45], N[(N[(N[Sqrt[N[(C * N[(A * -4.0), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(N[(F * 2.0), $MachinePrecision] * N[(N[(C + A), $MachinePrecision] - N[Sqrt[N[(N[(A - C), $MachinePrecision] * N[(A - C), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-1.0 / N[(B * B + N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-0.25 * N[(N[(2.0 * N[Sqrt[C], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(-4.0 * F), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / C), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if C < -1.4e-291

   1. Initial program 18.4%

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

   4. Applied rewrites 5.5%

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

   5. Taylor expanded in A around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. distribute-rgt-out-- N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 14.8

          \[\leadsto -0.25 \cdot \left(\frac{\sqrt{2} \cdot \sqrt{2}}{C} \cdot \sqrt{F \cdot \color{blue}{\left(C \cdot -4\right)}}\right) \]

   7. Applied rewrites 14.8%

      \[\leadsto \color{blue}{-0.25 \cdot \left(\frac{\sqrt{2} \cdot \sqrt{2}}{C} \cdot \sqrt{F \cdot \left(C \cdot -4\right)}\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 15.0%

      \[\leadsto \frac{\left(2 \cdot \sqrt{C \cdot \left(-4 \cdot F\right)}\right) \cdot -0.25}{\color{blue}{C}} \]
   10. Step-by-step derivation

   11. Applied rewrites 18.5%

       \[\leadsto \frac{\left(2 \cdot \left(\sqrt{C \cdot -4} \cdot \sqrt{F}\right)\right) \cdot -0.25}{C} \]

   if -1.4e-291 < C < 4.6999999999999998e-45

   1. Initial program 27.2%

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

   4. Applied rewrites 27.2%

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

   6. Applied rewrites 32.0%

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

   if 4.6999999999999998e-45 < C

   1. Initial program 7.8%

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

   4. Applied rewrites 6.1%

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

   5. Taylor expanded in A around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. distribute-rgt-out-- N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 28.2

          \[\leadsto -0.25 \cdot \left(\frac{\sqrt{2} \cdot \sqrt{2}}{C} \cdot \sqrt{F \cdot \color{blue}{\left(C \cdot -4\right)}}\right) \]

   7. Applied rewrites 28.2%

      \[\leadsto \color{blue}{-0.25 \cdot \left(\frac{\sqrt{2} \cdot \sqrt{2}}{C} \cdot \sqrt{F \cdot \left(C \cdot -4\right)}\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 28.6%

      \[\leadsto \frac{\left(2 \cdot \sqrt{C \cdot \left(-4 \cdot F\right)}\right) \cdot -0.25}{\color{blue}{C}} \]
   10. Step-by-step derivation

   11. Applied rewrites 36.6%

       \[\leadsto \frac{\left(\left(2 \cdot \sqrt{C}\right) \cdot \sqrt{-4 \cdot F}\right) \cdot -0.25}{C} \]
3. Recombined 3 regimes into one program.

4. Final simplification 26.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq -1.4 \cdot 10^{-291}:\\ \;\;\;\;\frac{\left(2 \cdot \left(\sqrt{C \cdot -4} \cdot \sqrt{F}\right)\right) \cdot -0.25}{C}\\ \mathbf{elif}\;C \leq 4.7 \cdot 10^{-45}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)} \cdot \sqrt{\left(F \cdot 2\right) \cdot \left(\left(C + A\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right)}\right) \cdot \frac{-1}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.25 \cdot \left(\left(2 \cdot \sqrt{C}\right) \cdot \sqrt{-4 \cdot F}\right)}{C}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 48.5% accurate, 3.8× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;C \leq -1.4 \cdot 10^{-291}:\\ \;\;\;\;\frac{\left(2 \cdot \left(\sqrt{C \cdot -4} \cdot \sqrt{F}\right)\right) \cdot -0.25}{C}\\ \mathbf{elif}\;C \leq 4.7 \cdot 10^{-45}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \cdot \left(\sqrt{2 \cdot \mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)} \cdot \sqrt{F \cdot \left(\left(C + A\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.25 \cdot \left(\left(2 \cdot \sqrt{C}\right) \cdot \sqrt{-4 \cdot F}\right)}{C}\\ \end{array} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (if (<= C -1.4e-291)
   (/ (* (* 2.0 (* (sqrt (* C -4.0)) (sqrt F))) -0.25) C)
   (if (<= C 4.7e-45)
     (*
      (/ -1.0 (fma B B (* C (* A -4.0))))
      (*
       (sqrt (* 2.0 (fma C (* A -4.0) (* B B))))
       (sqrt (* F (- (+ C A) (sqrt (fma (- A C) (- A C) (* B B))))))))
     (/ (* -0.25 (* (* 2.0 (sqrt C)) (sqrt (* -4.0 F)))) C))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	double tmp;
	if (C <= -1.4e-291) {
		tmp = ((2.0 * (sqrt((C * -4.0)) * sqrt(F))) * -0.25) / C;
	} else if (C <= 4.7e-45) {
		tmp = (-1.0 / fma(B, B, (C * (A * -4.0)))) * (sqrt((2.0 * fma(C, (A * -4.0), (B * B)))) * sqrt((F * ((C + A) - sqrt(fma((A - C), (A - C), (B * B)))))));
	} else {
		tmp = (-0.25 * ((2.0 * sqrt(C)) * sqrt((-4.0 * F)))) / C;
	}
	return tmp;
}

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	tmp = 0.0
	if (C <= -1.4e-291)
		tmp = Float64(Float64(Float64(2.0 * Float64(sqrt(Float64(C * -4.0)) * sqrt(F))) * -0.25) / C);
	elseif (C <= 4.7e-45)
		tmp = Float64(Float64(-1.0 / fma(B, B, Float64(C * Float64(A * -4.0)))) * Float64(sqrt(Float64(2.0 * fma(C, Float64(A * -4.0), Float64(B * B)))) * sqrt(Float64(F * Float64(Float64(C + A) - sqrt(fma(Float64(A - C), Float64(A - C), Float64(B * B))))))));
	else
		tmp = Float64(Float64(-0.25 * Float64(Float64(2.0 * sqrt(C)) * sqrt(Float64(-4.0 * F)))) / C);
	end
	return tmp
end

Wolfram

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := If[LessEqual[C, -1.4e-291], N[(N[(N[(2.0 * N[(N[Sqrt[N[(C * -4.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision] / C), $MachinePrecision], If[LessEqual[C, 4.7e-45], N[(N[(-1.0 / N[(B * B + N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[N[(2.0 * N[(C * N[(A * -4.0), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(F * N[(N[(C + A), $MachinePrecision] - N[Sqrt[N[(N[(A - C), $MachinePrecision] * N[(A - C), $MachinePrecision] + N[(B * B), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(-0.25 * N[(N[(2.0 * N[Sqrt[C], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(-4.0 * F), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / C), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if C < -1.4e-291

   1. Initial program 18.4%

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

   4. Applied rewrites 5.5%

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

   5. Taylor expanded in A around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. distribute-rgt-out-- N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 14.8

          \[\leadsto -0.25 \cdot \left(\frac{\sqrt{2} \cdot \sqrt{2}}{C} \cdot \sqrt{F \cdot \color{blue}{\left(C \cdot -4\right)}}\right) \]

   7. Applied rewrites 14.8%

      \[\leadsto \color{blue}{-0.25 \cdot \left(\frac{\sqrt{2} \cdot \sqrt{2}}{C} \cdot \sqrt{F \cdot \left(C \cdot -4\right)}\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 15.0%

      \[\leadsto \frac{\left(2 \cdot \sqrt{C \cdot \left(-4 \cdot F\right)}\right) \cdot -0.25}{\color{blue}{C}} \]
   10. Step-by-step derivation

   11. Applied rewrites 18.5%

       \[\leadsto \frac{\left(2 \cdot \left(\sqrt{C \cdot -4} \cdot \sqrt{F}\right)\right) \cdot -0.25}{C} \]

   if -1.4e-291 < C < 4.6999999999999998e-45

   1. Initial program 27.2%

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

   4. Applied rewrites 27.2%

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

   6. Applied rewrites 31.9%

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

   if 4.6999999999999998e-45 < C

   1. Initial program 7.8%

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

   4. Applied rewrites 6.1%

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

   5. Taylor expanded in A around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. distribute-rgt-out-- N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 28.2

          \[\leadsto -0.25 \cdot \left(\frac{\sqrt{2} \cdot \sqrt{2}}{C} \cdot \sqrt{F \cdot \color{blue}{\left(C \cdot -4\right)}}\right) \]

   7. Applied rewrites 28.2%

      \[\leadsto \color{blue}{-0.25 \cdot \left(\frac{\sqrt{2} \cdot \sqrt{2}}{C} \cdot \sqrt{F \cdot \left(C \cdot -4\right)}\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 28.6%

      \[\leadsto \frac{\left(2 \cdot \sqrt{C \cdot \left(-4 \cdot F\right)}\right) \cdot -0.25}{\color{blue}{C}} \]
   10. Step-by-step derivation

   11. Applied rewrites 36.6%

       \[\leadsto \frac{\left(\left(2 \cdot \sqrt{C}\right) \cdot \sqrt{-4 \cdot F}\right) \cdot -0.25}{C} \]
3. Recombined 3 regimes into one program.

4. Final simplification 26.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq -1.4 \cdot 10^{-291}:\\ \;\;\;\;\frac{\left(2 \cdot \left(\sqrt{C \cdot -4} \cdot \sqrt{F}\right)\right) \cdot -0.25}{C}\\ \mathbf{elif}\;C \leq 4.7 \cdot 10^{-45}:\\ \;\;\;\;\frac{-1}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)} \cdot \left(\sqrt{2 \cdot \mathsf{fma}\left(C, A \cdot -4, B \cdot B\right)} \cdot \sqrt{F \cdot \left(\left(C + A\right) - \sqrt{\mathsf{fma}\left(A - C, A - C, B \cdot B\right)}\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.25 \cdot \left(\left(2 \cdot \sqrt{C}\right) \cdot \sqrt{-4 \cdot F}\right)}{C}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 45.1% accurate, 6.1× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;C \leq -1.4 \cdot 10^{-291}:\\ \;\;\;\;\frac{\left(2 \cdot \left(\sqrt{C \cdot -4} \cdot \sqrt{F}\right)\right) \cdot -0.25}{C}\\ \mathbf{elif}\;C \leq 7.6 \cdot 10^{-171}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(B, B, A \cdot A\right)}\right)} \cdot \frac{\sqrt{2}}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.25 \cdot \left(\left(2 \cdot \sqrt{C}\right) \cdot \sqrt{-4 \cdot F}\right)}{C}\\ \end{array} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (if (<= C -1.4e-291)
   (/ (* (* 2.0 (* (sqrt (* C -4.0)) (sqrt F))) -0.25) C)
   (if (<= C 7.6e-171)
     (* (sqrt (* F (- A (sqrt (fma B B (* A A)))))) (/ (sqrt 2.0) (- B)))
     (/ (* -0.25 (* (* 2.0 (sqrt C)) (sqrt (* -4.0 F)))) C))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	double tmp;
	if (C <= -1.4e-291) {
		tmp = ((2.0 * (sqrt((C * -4.0)) * sqrt(F))) * -0.25) / C;
	} else if (C <= 7.6e-171) {
		tmp = sqrt((F * (A - sqrt(fma(B, B, (A * A)))))) * (sqrt(2.0) / -B);
	} else {
		tmp = (-0.25 * ((2.0 * sqrt(C)) * sqrt((-4.0 * F)))) / C;
	}
	return tmp;
}

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	tmp = 0.0
	if (C <= -1.4e-291)
		tmp = Float64(Float64(Float64(2.0 * Float64(sqrt(Float64(C * -4.0)) * sqrt(F))) * -0.25) / C);
	elseif (C <= 7.6e-171)
		tmp = Float64(sqrt(Float64(F * Float64(A - sqrt(fma(B, B, Float64(A * A)))))) * Float64(sqrt(2.0) / Float64(-B)));
	else
		tmp = Float64(Float64(-0.25 * Float64(Float64(2.0 * sqrt(C)) * sqrt(Float64(-4.0 * F)))) / C);
	end
	return tmp
end

Wolfram

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := If[LessEqual[C, -1.4e-291], N[(N[(N[(2.0 * N[(N[Sqrt[N[(C * -4.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision] / C), $MachinePrecision], If[LessEqual[C, 7.6e-171], N[(N[Sqrt[N[(F * N[(A - N[Sqrt[N[(B * B + N[(A * A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / (-B)), $MachinePrecision]), $MachinePrecision], N[(N[(-0.25 * N[(N[(2.0 * N[Sqrt[C], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(-4.0 * F), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / C), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if C < -1.4e-291

   1. Initial program 18.4%

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

   4. Applied rewrites 5.5%

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

   5. Taylor expanded in A around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. distribute-rgt-out-- N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 14.8

          \[\leadsto -0.25 \cdot \left(\frac{\sqrt{2} \cdot \sqrt{2}}{C} \cdot \sqrt{F \cdot \color{blue}{\left(C \cdot -4\right)}}\right) \]

   7. Applied rewrites 14.8%

      \[\leadsto \color{blue}{-0.25 \cdot \left(\frac{\sqrt{2} \cdot \sqrt{2}}{C} \cdot \sqrt{F \cdot \left(C \cdot -4\right)}\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 15.0%

      \[\leadsto \frac{\left(2 \cdot \sqrt{C \cdot \left(-4 \cdot F\right)}\right) \cdot -0.25}{\color{blue}{C}} \]
   10. Step-by-step derivation

   11. Applied rewrites 18.5%

       \[\leadsto \frac{\left(2 \cdot \left(\sqrt{C \cdot -4} \cdot \sqrt{F}\right)\right) \cdot -0.25}{C} \]

   if -1.4e-291 < C < 7.60000000000000043e-171

   1. Initial program 28.6%

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

   3. Taylor expanded in C around 0

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

      1. mul-1-neg N/A

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

      2. *-commutative N/A

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

      3. distribute-rgt-neg-in N/A

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

      4. mul-1-neg N/A

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

      5. lower-*.f64 N/A

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

   5. Applied rewrites 15.7%

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

   if 7.60000000000000043e-171 < C

   1. Initial program 10.2%

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

   4. Applied rewrites 7.5%

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

   5. Taylor expanded in A around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. distribute-rgt-out-- N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 27.7

          \[\leadsto -0.25 \cdot \left(\frac{\sqrt{2} \cdot \sqrt{2}}{C} \cdot \sqrt{F \cdot \color{blue}{\left(C \cdot -4\right)}}\right) \]

   7. Applied rewrites 27.7%

      \[\leadsto \color{blue}{-0.25 \cdot \left(\frac{\sqrt{2} \cdot \sqrt{2}}{C} \cdot \sqrt{F \cdot \left(C \cdot -4\right)}\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 28.1%

      \[\leadsto \frac{\left(2 \cdot \sqrt{C \cdot \left(-4 \cdot F\right)}\right) \cdot -0.25}{\color{blue}{C}} \]
   10. Step-by-step derivation

   11. Applied rewrites 35.9%

       \[\leadsto \frac{\left(\left(2 \cdot \sqrt{C}\right) \cdot \sqrt{-4 \cdot F}\right) \cdot -0.25}{C} \]
3. Recombined 3 regimes into one program.

4. Final simplification 24.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq -1.4 \cdot 10^{-291}:\\ \;\;\;\;\frac{\left(2 \cdot \left(\sqrt{C \cdot -4} \cdot \sqrt{F}\right)\right) \cdot -0.25}{C}\\ \mathbf{elif}\;C \leq 7.6 \cdot 10^{-171}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \sqrt{\mathsf{fma}\left(B, B, A \cdot A\right)}\right)} \cdot \frac{\sqrt{2}}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.25 \cdot \left(\left(2 \cdot \sqrt{C}\right) \cdot \sqrt{-4 \cdot F}\right)}{C}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 46.3% accurate, 8.5× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;C \leq -1 \cdot 10^{-291}:\\ \;\;\;\;\frac{\left(2 \cdot \left(\sqrt{C \cdot -4} \cdot \sqrt{F}\right)\right) \cdot -0.25}{C}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.25 \cdot \left(\left(2 \cdot \sqrt{C}\right) \cdot \sqrt{-4 \cdot F}\right)}{C}\\ \end{array} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (if (<= C -1e-291)
   (/ (* (* 2.0 (* (sqrt (* C -4.0)) (sqrt F))) -0.25) C)
   (/ (* -0.25 (* (* 2.0 (sqrt C)) (sqrt (* -4.0 F)))) C)))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	double tmp;
	if (C <= -1e-291) {
		tmp = ((2.0 * (sqrt((C * -4.0)) * sqrt(F))) * -0.25) / C;
	} else {
		tmp = (-0.25 * ((2.0 * sqrt(C)) * sqrt((-4.0 * F)))) / C;
	}
	return tmp;
}

Fortran

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (c <= (-1d-291)) then
        tmp = ((2.0d0 * (sqrt((c * (-4.0d0))) * sqrt(f))) * (-0.25d0)) / c
    else
        tmp = ((-0.25d0) * ((2.0d0 * sqrt(c)) * sqrt(((-4.0d0) * f)))) / c
    end if
    code = tmp
end function

Java

assert A < B && B < C && C < F;
public static double code(double A, double B, double C, double F) {
	double tmp;
	if (C <= -1e-291) {
		tmp = ((2.0 * (Math.sqrt((C * -4.0)) * Math.sqrt(F))) * -0.25) / C;
	} else {
		tmp = (-0.25 * ((2.0 * Math.sqrt(C)) * Math.sqrt((-4.0 * F)))) / C;
	}
	return tmp;
}

Python

[A, B, C, F] = sort([A, B, C, F])
def code(A, B, C, F):
	tmp = 0
	if C <= -1e-291:
		tmp = ((2.0 * (math.sqrt((C * -4.0)) * math.sqrt(F))) * -0.25) / C
	else:
		tmp = (-0.25 * ((2.0 * math.sqrt(C)) * math.sqrt((-4.0 * F)))) / C
	return tmp

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	tmp = 0.0
	if (C <= -1e-291)
		tmp = Float64(Float64(Float64(2.0 * Float64(sqrt(Float64(C * -4.0)) * sqrt(F))) * -0.25) / C);
	else
		tmp = Float64(Float64(-0.25 * Float64(Float64(2.0 * sqrt(C)) * sqrt(Float64(-4.0 * F)))) / C);
	end
	return tmp
end

MATLAB

A, B, C, F = num2cell(sort([A, B, C, F])){:}
function tmp_2 = code(A, B, C, F)
	tmp = 0.0;
	if (C <= -1e-291)
		tmp = ((2.0 * (sqrt((C * -4.0)) * sqrt(F))) * -0.25) / C;
	else
		tmp = (-0.25 * ((2.0 * sqrt(C)) * sqrt((-4.0 * F)))) / C;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := If[LessEqual[C, -1e-291], N[(N[(N[(2.0 * N[(N[Sqrt[N[(C * -4.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision] / C), $MachinePrecision], N[(N[(-0.25 * N[(N[(2.0 * N[Sqrt[C], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(-4.0 * F), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / C), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if C < -9.99999999999999962e-292

   1. Initial program 18.4%

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

   4. Applied rewrites 5.5%

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

   5. Taylor expanded in A around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. distribute-rgt-out-- N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 14.8

          \[\leadsto -0.25 \cdot \left(\frac{\sqrt{2} \cdot \sqrt{2}}{C} \cdot \sqrt{F \cdot \color{blue}{\left(C \cdot -4\right)}}\right) \]

   7. Applied rewrites 14.8%

      \[\leadsto \color{blue}{-0.25 \cdot \left(\frac{\sqrt{2} \cdot \sqrt{2}}{C} \cdot \sqrt{F \cdot \left(C \cdot -4\right)}\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 15.0%

      \[\leadsto \frac{\left(2 \cdot \sqrt{C \cdot \left(-4 \cdot F\right)}\right) \cdot -0.25}{\color{blue}{C}} \]
   10. Step-by-step derivation

   11. Applied rewrites 18.5%

       \[\leadsto \frac{\left(2 \cdot \left(\sqrt{C \cdot -4} \cdot \sqrt{F}\right)\right) \cdot -0.25}{C} \]

   if -9.99999999999999962e-292 < C

   1. Initial program 16.2%

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

   4. Applied rewrites 9.0%

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

   5. Taylor expanded in A around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. distribute-rgt-out-- N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 21.8

          \[\leadsto -0.25 \cdot \left(\frac{\sqrt{2} \cdot \sqrt{2}}{C} \cdot \sqrt{F \cdot \color{blue}{\left(C \cdot -4\right)}}\right) \]

   7. Applied rewrites 21.8%

      \[\leadsto \color{blue}{-0.25 \cdot \left(\frac{\sqrt{2} \cdot \sqrt{2}}{C} \cdot \sqrt{F \cdot \left(C \cdot -4\right)}\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 22.1%

      \[\leadsto \frac{\left(2 \cdot \sqrt{C \cdot \left(-4 \cdot F\right)}\right) \cdot -0.25}{\color{blue}{C}} \]
   10. Step-by-step derivation

   11. Applied rewrites 28.3%

       \[\leadsto \frac{\left(\left(2 \cdot \sqrt{C}\right) \cdot \sqrt{-4 \cdot F}\right) \cdot -0.25}{C} \]
3. Recombined 2 regimes into one program.

4. Final simplification 23.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq -1 \cdot 10^{-291}:\\ \;\;\;\;\frac{\left(2 \cdot \left(\sqrt{C \cdot -4} \cdot \sqrt{F}\right)\right) \cdot -0.25}{C}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.25 \cdot \left(\left(2 \cdot \sqrt{C}\right) \cdot \sqrt{-4 \cdot F}\right)}{C}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 38.7% accurate, 8.5× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;F \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sqrt{C \cdot \left(-4 \cdot F\right)} \cdot -0.5}{C}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(2 \cdot \left(\sqrt{C \cdot -4} \cdot \sqrt{F}\right)\right) \cdot -0.25}{C}\\ \end{array} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (if (<= F -5e-310)
   (/ (* (sqrt (* C (* -4.0 F))) -0.5) C)
   (/ (* (* 2.0 (* (sqrt (* C -4.0)) (sqrt F))) -0.25) C)))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	double tmp;
	if (F <= -5e-310) {
		tmp = (sqrt((C * (-4.0 * F))) * -0.5) / C;
	} else {
		tmp = ((2.0 * (sqrt((C * -4.0)) * sqrt(F))) * -0.25) / C;
	}
	return tmp;
}

Fortran

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (f <= (-5d-310)) then
        tmp = (sqrt((c * ((-4.0d0) * f))) * (-0.5d0)) / c
    else
        tmp = ((2.0d0 * (sqrt((c * (-4.0d0))) * sqrt(f))) * (-0.25d0)) / c
    end if
    code = tmp
end function

Java

assert A < B && B < C && C < F;
public static double code(double A, double B, double C, double F) {
	double tmp;
	if (F <= -5e-310) {
		tmp = (Math.sqrt((C * (-4.0 * F))) * -0.5) / C;
	} else {
		tmp = ((2.0 * (Math.sqrt((C * -4.0)) * Math.sqrt(F))) * -0.25) / C;
	}
	return tmp;
}

Python

[A, B, C, F] = sort([A, B, C, F])
def code(A, B, C, F):
	tmp = 0
	if F <= -5e-310:
		tmp = (math.sqrt((C * (-4.0 * F))) * -0.5) / C
	else:
		tmp = ((2.0 * (math.sqrt((C * -4.0)) * math.sqrt(F))) * -0.25) / C
	return tmp

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	tmp = 0.0
	if (F <= -5e-310)
		tmp = Float64(Float64(sqrt(Float64(C * Float64(-4.0 * F))) * -0.5) / C);
	else
		tmp = Float64(Float64(Float64(2.0 * Float64(sqrt(Float64(C * -4.0)) * sqrt(F))) * -0.25) / C);
	end
	return tmp
end

MATLAB

A, B, C, F = num2cell(sort([A, B, C, F])){:}
function tmp_2 = code(A, B, C, F)
	tmp = 0.0;
	if (F <= -5e-310)
		tmp = (sqrt((C * (-4.0 * F))) * -0.5) / C;
	else
		tmp = ((2.0 * (sqrt((C * -4.0)) * sqrt(F))) * -0.25) / C;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := If[LessEqual[F, -5e-310], N[(N[(N[Sqrt[N[(C * N[(-4.0 * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * -0.5), $MachinePrecision] / C), $MachinePrecision], N[(N[(N[(2.0 * N[(N[Sqrt[N[(C * -4.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision] / C), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -4.999999999999985e-310

   1. Initial program 15.8%

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

   4. Applied rewrites 8.5%

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

   5. Taylor expanded in A around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. distribute-rgt-out-- N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 13.4

          \[\leadsto -0.25 \cdot \left(\frac{\sqrt{2} \cdot \sqrt{2}}{C} \cdot \sqrt{F \cdot \color{blue}{\left(C \cdot -4\right)}}\right) \]

   7. Applied rewrites 13.4%

      \[\leadsto \color{blue}{-0.25 \cdot \left(\frac{\sqrt{2} \cdot \sqrt{2}}{C} \cdot \sqrt{F \cdot \left(C \cdot -4\right)}\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 13.5%

      \[\leadsto \frac{\left(2 \cdot \sqrt{C \cdot \left(-4 \cdot F\right)}\right) \cdot -0.25}{\color{blue}{C}} \]
   10. Step-by-step derivation

   11. Applied rewrites 13.5%

       \[\leadsto \frac{\sqrt{C \cdot \left(-4 \cdot F\right)} \cdot -0.5}{C} \]

   if -4.999999999999985e-310 < F

   1. Initial program 25.6%

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

   4. Applied rewrites 0.0%

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

   5. Taylor expanded in A around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. distribute-rgt-out-- N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 48.2

          \[\leadsto -0.25 \cdot \left(\frac{\sqrt{2} \cdot \sqrt{2}}{C} \cdot \sqrt{F \cdot \color{blue}{\left(C \cdot -4\right)}}\right) \]

   7. Applied rewrites 48.2%

      \[\leadsto \color{blue}{-0.25 \cdot \left(\frac{\sqrt{2} \cdot \sqrt{2}}{C} \cdot \sqrt{F \cdot \left(C \cdot -4\right)}\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 48.8%

      \[\leadsto \frac{\left(2 \cdot \sqrt{C \cdot \left(-4 \cdot F\right)}\right) \cdot -0.25}{\color{blue}{C}} \]
   10. Step-by-step derivation

   11. Applied rewrites 62.0%

       \[\leadsto \frac{\left(2 \cdot \left(\sqrt{C \cdot -4} \cdot \sqrt{F}\right)\right) \cdot -0.25}{C} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 38.7% accurate, 8.5× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;F \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sqrt{C \cdot \left(-4 \cdot F\right)} \cdot -0.5}{C}\\ \mathbf{else}:\\ \;\;\;\;-0.25 \cdot \left(\sqrt{F} \cdot \left(\sqrt{C \cdot -4} \cdot \frac{2}{C}\right)\right)\\ \end{array} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (if (<= F -5e-310)
   (/ (* (sqrt (* C (* -4.0 F))) -0.5) C)
   (* -0.25 (* (sqrt F) (* (sqrt (* C -4.0)) (/ 2.0 C))))))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	double tmp;
	if (F <= -5e-310) {
		tmp = (sqrt((C * (-4.0 * F))) * -0.5) / C;
	} else {
		tmp = -0.25 * (sqrt(F) * (sqrt((C * -4.0)) * (2.0 / C)));
	}
	return tmp;
}

Fortran

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (f <= (-5d-310)) then
        tmp = (sqrt((c * ((-4.0d0) * f))) * (-0.5d0)) / c
    else
        tmp = (-0.25d0) * (sqrt(f) * (sqrt((c * (-4.0d0))) * (2.0d0 / c)))
    end if
    code = tmp
end function

Java

assert A < B && B < C && C < F;
public static double code(double A, double B, double C, double F) {
	double tmp;
	if (F <= -5e-310) {
		tmp = (Math.sqrt((C * (-4.0 * F))) * -0.5) / C;
	} else {
		tmp = -0.25 * (Math.sqrt(F) * (Math.sqrt((C * -4.0)) * (2.0 / C)));
	}
	return tmp;
}

Python

[A, B, C, F] = sort([A, B, C, F])
def code(A, B, C, F):
	tmp = 0
	if F <= -5e-310:
		tmp = (math.sqrt((C * (-4.0 * F))) * -0.5) / C
	else:
		tmp = -0.25 * (math.sqrt(F) * (math.sqrt((C * -4.0)) * (2.0 / C)))
	return tmp

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	tmp = 0.0
	if (F <= -5e-310)
		tmp = Float64(Float64(sqrt(Float64(C * Float64(-4.0 * F))) * -0.5) / C);
	else
		tmp = Float64(-0.25 * Float64(sqrt(F) * Float64(sqrt(Float64(C * -4.0)) * Float64(2.0 / C))));
	end
	return tmp
end

MATLAB

A, B, C, F = num2cell(sort([A, B, C, F])){:}
function tmp_2 = code(A, B, C, F)
	tmp = 0.0;
	if (F <= -5e-310)
		tmp = (sqrt((C * (-4.0 * F))) * -0.5) / C;
	else
		tmp = -0.25 * (sqrt(F) * (sqrt((C * -4.0)) * (2.0 / C)));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := If[LessEqual[F, -5e-310], N[(N[(N[Sqrt[N[(C * N[(-4.0 * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * -0.5), $MachinePrecision] / C), $MachinePrecision], N[(-0.25 * N[(N[Sqrt[F], $MachinePrecision] * N[(N[Sqrt[N[(C * -4.0), $MachinePrecision]], $MachinePrecision] * N[(2.0 / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -4.999999999999985e-310

   1. Initial program 15.8%

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

   4. Applied rewrites 8.5%

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

   5. Taylor expanded in A around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. distribute-rgt-out-- N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 13.4

          \[\leadsto -0.25 \cdot \left(\frac{\sqrt{2} \cdot \sqrt{2}}{C} \cdot \sqrt{F \cdot \color{blue}{\left(C \cdot -4\right)}}\right) \]

   7. Applied rewrites 13.4%

      \[\leadsto \color{blue}{-0.25 \cdot \left(\frac{\sqrt{2} \cdot \sqrt{2}}{C} \cdot \sqrt{F \cdot \left(C \cdot -4\right)}\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 13.5%

      \[\leadsto \frac{\left(2 \cdot \sqrt{C \cdot \left(-4 \cdot F\right)}\right) \cdot -0.25}{\color{blue}{C}} \]
   10. Step-by-step derivation

   11. Applied rewrites 13.5%

       \[\leadsto \frac{\sqrt{C \cdot \left(-4 \cdot F\right)} \cdot -0.5}{C} \]

   if -4.999999999999985e-310 < F

   1. Initial program 25.6%

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

   4. Applied rewrites 0.0%

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

   5. Taylor expanded in A around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. distribute-rgt-out-- N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 48.2

          \[\leadsto -0.25 \cdot \left(\frac{\sqrt{2} \cdot \sqrt{2}}{C} \cdot \sqrt{F \cdot \color{blue}{\left(C \cdot -4\right)}}\right) \]

   7. Applied rewrites 48.2%

      \[\leadsto \color{blue}{-0.25 \cdot \left(\frac{\sqrt{2} \cdot \sqrt{2}}{C} \cdot \sqrt{F \cdot \left(C \cdot -4\right)}\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 61.8%

      \[\leadsto -0.25 \cdot \left(\sqrt{F} \cdot \color{blue}{\left(\sqrt{C \cdot -4} \cdot \frac{2}{C}\right)}\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 37.5% accurate, 9.3× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;C \leq -9 \cdot 10^{-240}:\\ \;\;\;\;\sqrt{C \cdot -4} \cdot \left(\sqrt{F} \cdot \frac{-0.5}{C}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{C \cdot \left(-4 \cdot F\right)} \cdot -0.5}{C}\\ \end{array} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F)
 :precision binary64
 (if (<= C -9e-240)
   (* (sqrt (* C -4.0)) (* (sqrt F) (/ -0.5 C)))
   (/ (* (sqrt (* C (* -4.0 F))) -0.5) C)))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	double tmp;
	if (C <= -9e-240) {
		tmp = sqrt((C * -4.0)) * (sqrt(F) * (-0.5 / C));
	} else {
		tmp = (sqrt((C * (-4.0 * F))) * -0.5) / C;
	}
	return tmp;
}

Fortran

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (c <= (-9d-240)) then
        tmp = sqrt((c * (-4.0d0))) * (sqrt(f) * ((-0.5d0) / c))
    else
        tmp = (sqrt((c * ((-4.0d0) * f))) * (-0.5d0)) / c
    end if
    code = tmp
end function

Java

assert A < B && B < C && C < F;
public static double code(double A, double B, double C, double F) {
	double tmp;
	if (C <= -9e-240) {
		tmp = Math.sqrt((C * -4.0)) * (Math.sqrt(F) * (-0.5 / C));
	} else {
		tmp = (Math.sqrt((C * (-4.0 * F))) * -0.5) / C;
	}
	return tmp;
}

Python

[A, B, C, F] = sort([A, B, C, F])
def code(A, B, C, F):
	tmp = 0
	if C <= -9e-240:
		tmp = math.sqrt((C * -4.0)) * (math.sqrt(F) * (-0.5 / C))
	else:
		tmp = (math.sqrt((C * (-4.0 * F))) * -0.5) / C
	return tmp

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	tmp = 0.0
	if (C <= -9e-240)
		tmp = Float64(sqrt(Float64(C * -4.0)) * Float64(sqrt(F) * Float64(-0.5 / C)));
	else
		tmp = Float64(Float64(sqrt(Float64(C * Float64(-4.0 * F))) * -0.5) / C);
	end
	return tmp
end

MATLAB

A, B, C, F = num2cell(sort([A, B, C, F])){:}
function tmp_2 = code(A, B, C, F)
	tmp = 0.0;
	if (C <= -9e-240)
		tmp = sqrt((C * -4.0)) * (sqrt(F) * (-0.5 / C));
	else
		tmp = (sqrt((C * (-4.0 * F))) * -0.5) / C;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := If[LessEqual[C, -9e-240], N[(N[Sqrt[N[(C * -4.0), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[F], $MachinePrecision] * N[(-0.5 / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[N[(C * N[(-4.0 * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * -0.5), $MachinePrecision] / C), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if C < -9.0000000000000003e-240

   1. Initial program 17.4%

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

   4. Applied rewrites 5.9%

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

   5. Taylor expanded in A around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. distribute-rgt-out-- N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 14.4

          \[\leadsto -0.25 \cdot \left(\frac{\sqrt{2} \cdot \sqrt{2}}{C} \cdot \sqrt{F \cdot \color{blue}{\left(C \cdot -4\right)}}\right) \]

   7. Applied rewrites 14.4%

      \[\leadsto \color{blue}{-0.25 \cdot \left(\frac{\sqrt{2} \cdot \sqrt{2}}{C} \cdot \sqrt{F \cdot \left(C \cdot -4\right)}\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 18.5%

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

   if -9.0000000000000003e-240 < C

   1. Initial program 17.1%

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

   4. Applied rewrites 8.4%

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

   5. Taylor expanded in A around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-sqrt.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. distribute-rgt-out-- N/A

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

      11. metadata-eval N/A

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

      12. lower-*.f64 21.6

          \[\leadsto -0.25 \cdot \left(\frac{\sqrt{2} \cdot \sqrt{2}}{C} \cdot \sqrt{F \cdot \color{blue}{\left(C \cdot -4\right)}}\right) \]

   7. Applied rewrites 21.6%

      \[\leadsto \color{blue}{-0.25 \cdot \left(\frac{\sqrt{2} \cdot \sqrt{2}}{C} \cdot \sqrt{F \cdot \left(C \cdot -4\right)}\right)} \]
   8. Step-by-step derivation

   9. Applied rewrites 21.8%

      \[\leadsto \frac{\left(2 \cdot \sqrt{C \cdot \left(-4 \cdot F\right)}\right) \cdot -0.25}{\color{blue}{C}} \]
   10. Step-by-step derivation

   11. Applied rewrites 21.8%

       \[\leadsto \frac{\sqrt{C \cdot \left(-4 \cdot F\right)} \cdot -0.5}{C} \]
3. Recombined 2 regimes into one program.

4. Final simplification 20.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq -9 \cdot 10^{-240}:\\ \;\;\;\;\sqrt{C \cdot -4} \cdot \left(\sqrt{F} \cdot \frac{-0.5}{C}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{C \cdot \left(-4 \cdot F\right)} \cdot -0.5}{C}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 36.3% accurate, 13.3× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \frac{\sqrt{C \cdot \left(-4 \cdot F\right)} \cdot -0.5}{C} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F) :precision binary64 (/ (* (sqrt (* C (* -4.0 F))) -0.5) C))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	return (sqrt((C * (-4.0 * F))) * -0.5) / C;
}

Fortran

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    code = (sqrt((c * ((-4.0d0) * f))) * (-0.5d0)) / c
end function

Java

assert A < B && B < C && C < F;
public static double code(double A, double B, double C, double F) {
	return (Math.sqrt((C * (-4.0 * F))) * -0.5) / C;
}

Python

[A, B, C, F] = sort([A, B, C, F])
def code(A, B, C, F):
	return (math.sqrt((C * (-4.0 * F))) * -0.5) / C

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	return Float64(Float64(sqrt(Float64(C * Float64(-4.0 * F))) * -0.5) / C)
end

MATLAB

A, B, C, F = num2cell(sort([A, B, C, F])){:}
function tmp = code(A, B, C, F)
	tmp = (sqrt((C * (-4.0 * F))) * -0.5) / C;
end

Wolfram

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := N[(N[(N[Sqrt[N[(C * N[(-4.0 * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * -0.5), $MachinePrecision] / C), $MachinePrecision]

Derivation

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. Add Preprocessing

4. Applied rewrites 7.3%

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

5. Taylor expanded in A around -inf

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-sqrt.f64 N/A

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

   8. lower-sqrt.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. distribute-rgt-out-- N/A

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

   11. metadata-eval N/A

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

   12. lower-*.f64 18.4

       \[\leadsto -0.25 \cdot \left(\frac{\sqrt{2} \cdot \sqrt{2}}{C} \cdot \sqrt{F \cdot \color{blue}{\left(C \cdot -4\right)}}\right) \]

7. Applied rewrites 18.4%

   \[\leadsto \color{blue}{-0.25 \cdot \left(\frac{\sqrt{2} \cdot \sqrt{2}}{C} \cdot \sqrt{F \cdot \left(C \cdot -4\right)}\right)} \]
8. Step-by-step derivation

9. Applied rewrites 18.6%

   \[\leadsto \frac{\left(2 \cdot \sqrt{C \cdot \left(-4 \cdot F\right)}\right) \cdot -0.25}{\color{blue}{C}} \]
10. Step-by-step derivation

11. Applied rewrites 18.6%

    \[\leadsto \frac{\sqrt{C \cdot \left(-4 \cdot F\right)} \cdot -0.5}{C} \]
12. Add Preprocessing

Alternative 9: 36.3% accurate, 13.3× speedup

Math

\[\begin{array}{l} [A, B, C, F] = \mathsf{sort}([A, B, C, F])\\ \\ \sqrt{C \cdot \left(-4 \cdot F\right)} \cdot \frac{-0.5}{C} \end{array} \]

FPCore

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B C F) :precision binary64 (* (sqrt (* C (* -4.0 F))) (/ -0.5 C)))

C

assert(A < B && B < C && C < F);
double code(double A, double B, double C, double F) {
	return sqrt((C * (-4.0 * F))) * (-0.5 / C);
}

Fortran

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    code = sqrt((c * ((-4.0d0) * f))) * ((-0.5d0) / c)
end function

Java

assert A < B && B < C && C < F;
public static double code(double A, double B, double C, double F) {
	return Math.sqrt((C * (-4.0 * F))) * (-0.5 / C);
}

Python

[A, B, C, F] = sort([A, B, C, F])
def code(A, B, C, F):
	return math.sqrt((C * (-4.0 * F))) * (-0.5 / C)

Julia

A, B, C, F = sort([A, B, C, F])
function code(A, B, C, F)
	return Float64(sqrt(Float64(C * Float64(-4.0 * F))) * Float64(-0.5 / C))
end

MATLAB

A, B, C, F = num2cell(sort([A, B, C, F])){:}
function tmp = code(A, B, C, F)
	tmp = sqrt((C * (-4.0 * F))) * (-0.5 / C);
end

Wolfram

NOTE: A, B, C, and F should be sorted in increasing order before calling this function.
code[A_, B_, C_, F_] := N[(N[Sqrt[N[(C * N[(-4.0 * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-0.5 / C), $MachinePrecision]), $MachinePrecision]

Derivation

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. Add Preprocessing

4. Applied rewrites 7.3%

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

5. Taylor expanded in A around -inf

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

   1. lower-*.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-sqrt.f64 N/A

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

   8. lower-sqrt.f64 N/A

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

   9. lower-*.f64 N/A

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

   10. distribute-rgt-out-- N/A

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

   11. metadata-eval N/A

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

   12. lower-*.f64 18.4

       \[\leadsto -0.25 \cdot \left(\frac{\sqrt{2} \cdot \sqrt{2}}{C} \cdot \sqrt{F \cdot \color{blue}{\left(C \cdot -4\right)}}\right) \]

7. Applied rewrites 18.4%

   \[\leadsto \color{blue}{-0.25 \cdot \left(\frac{\sqrt{2} \cdot \sqrt{2}}{C} \cdot \sqrt{F \cdot \left(C \cdot -4\right)}\right)} \]
8. Step-by-step derivation

9. Applied rewrites 18.6%

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

10. Final simplification 18.6%

    \[\leadsto \sqrt{C \cdot \left(-4 \cdot F\right)} \cdot \frac{-0.5}{C} \]
11. Add Preprocessing

Alternative 10: 2.0% accurate, 18.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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. Add Preprocessing

3. Taylor expanded in B around -inf

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. unpow2 N/A

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

   7. rem-square-sqrt N/A

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

   8. lower-*.f64 N/A

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

   9. lower-sqrt.f64 2.0

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

5. Applied rewrites 2.0%

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

7. Applied rewrites 2.0%

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

9. Applied rewrites 2.0%

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

10. Final simplification 2.0%

    \[\leadsto \sqrt{\frac{F \cdot 2}{B}} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024227 
(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))))