ABCF->ab-angle b

Percentage Accurate: 19.0% → 48.7%

Time: 25.7s

Alternatives: 14

Speedup: 3.1×

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {B}^{2} - \left(4 \cdot A\right) \cdot C\\ \frac{-\sqrt{\left(2 \cdot \left(t\_0 \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{t\_0} \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 19.0% 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: 48.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 5 \cdot 10^{-49}:\\ \;\;\;\;\frac{\sqrt{t\_0 \cdot \left(F \cdot \left(A \cdot 4\right)\right)}}{-t\_0}\\ \mathbf{elif}\;{B\_m}^{2} \leq 10^{+308}:\\ \;\;\;\;\sqrt{F \cdot \frac{\left(A + C\right) - \mathsf{hypot}\left(B\_m, A - C\right)}{\mathsf{fma}\left(-4, A \cdot C, {B\_m}^{2}\right)}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{F \cdot \frac{\left(\frac{A}{B\_m} + \frac{C}{B\_m}\right) + -1}{B\_m}}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* A (* C -4.0)))))
   (if (<= (pow B_m 2.0) 5e-49)
     (/ (sqrt (* t_0 (* F (* A 4.0)))) (- t_0))
     (if (<= (pow B_m 2.0) 1e+308)
       (*
        (sqrt
         (*
          F
          (/
           (- (+ A C) (hypot B_m (- A C)))
           (fma -4.0 (* A C) (pow B_m 2.0)))))
        (- (sqrt 2.0)))
       (*
        (sqrt 2.0)
        (- (sqrt (* F (/ (+ (+ (/ A B_m) (/ C B_m)) -1.0) B_m)))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (A * (C * -4.0)));
	double tmp;
	if (pow(B_m, 2.0) <= 5e-49) {
		tmp = sqrt((t_0 * (F * (A * 4.0)))) / -t_0;
	} else if (pow(B_m, 2.0) <= 1e+308) {
		tmp = sqrt((F * (((A + C) - hypot(B_m, (A - C))) / fma(-4.0, (A * C), pow(B_m, 2.0))))) * -sqrt(2.0);
	} else {
		tmp = sqrt(2.0) * -sqrt((F * ((((A / B_m) + (C / B_m)) + -1.0) / B_m)));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 5e-49)
		tmp = Float64(sqrt(Float64(t_0 * Float64(F * Float64(A * 4.0)))) / Float64(-t_0));
	elseif ((B_m ^ 2.0) <= 1e+308)
		tmp = Float64(sqrt(Float64(F * Float64(Float64(Float64(A + C) - hypot(B_m, Float64(A - C))) / fma(-4.0, Float64(A * C), (B_m ^ 2.0))))) * Float64(-sqrt(2.0)));
	else
		tmp = Float64(sqrt(2.0) * Float64(-sqrt(Float64(F * Float64(Float64(Float64(Float64(A / B_m) + Float64(C / B_m)) + -1.0) / B_m)))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e-49], N[(N[Sqrt[N[(t$95$0 * N[(F * N[(A * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e+308], N[(N[Sqrt[N[(F * N[(N[(N[(A + C), $MachinePrecision] - N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] / N[(-4.0 * N[(A * C), $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * (-N[Sqrt[N[(F * N[(N[(N[(N[(A / B$95$m), $MachinePrecision] + N[(C / B$95$m), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 4.9999999999999999e-49

   1. Initial program 22.5%

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

   3. Simplified 32.2%

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

   5. Taylor expanded in C around inf 17.7%

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

      1. mul-1-neg 17.7%

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

   7. Simplified 17.7%

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

   8. Taylor expanded in A around -inf 17.7%

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

      1. associate-*r* 17.7%

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

      2. neg-mul-1 17.7%

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

      3. sub-neg 17.7%

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

      4. metadata-eval 17.7%

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

   10. Simplified 17.7%

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

       1. *-un-lft-identity 17.7%

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

       2. associate-*l* 20.6%

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

       3. *-commutative 20.6%

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

   12. Applied egg-rr 20.6%

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

       1. *-lft-identity 20.6%

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

       2. +-commutative 20.6%

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

   14. Simplified 20.6%

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

   15. Taylor expanded in B around 0 20.3%

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

   if 4.9999999999999999e-49 < (pow.f64 B #s(literal 2 binary64)) < 1e308

   1. Initial program 25.6%

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

   3. Simplified 17.2%

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

      1. div-inv 17.2%

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

   6. Applied egg-rr 31.6%

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

   7. Taylor expanded in F around 0 30.0%

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

      1. mul-1-neg 30.0%

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

   9. Simplified 53.8%

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

   if 1e308 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 0.0%

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

   3. Simplified 0.0%

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

   5. Taylor expanded in B around inf 0.0%

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

   6. Taylor expanded in F around 0 2.9%

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

      1. mul-1-neg 2.9%

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

      2. associate-/l* 4.3%

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

      3. +-commutative 4.3%

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

      4. sub-neg 4.3%

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

      5. metadata-eval 4.3%

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

      6. fma-undefine 4.3%

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

      7. +-commutative 4.3%

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

      8. fma-define 4.3%

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

   8. Simplified 4.3%

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

   9. Taylor expanded in B around inf 32.1%

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

4. Final simplification 31.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 5 \cdot 10^{-49}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(F \cdot \left(A \cdot 4\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;{B}^{2} \leq 10^{+308}:\\ \;\;\;\;\sqrt{F \cdot \frac{\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)}{\mathsf{fma}\left(-4, A \cdot C, {B}^{2}\right)}} \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{F \cdot \frac{\left(\frac{A}{B} + \frac{C}{B}\right) + -1}{B}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 46.9% accurate, 1.5× speedup

Math

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

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* A (* C -4.0)))))
   (if (<= (pow B_m 2.0) 2e-8)
     (/ (sqrt (* t_0 (* F (* A 4.0)))) (- t_0))
     (* (sqrt (* F (- A (hypot B_m A)))) (/ (sqrt 2.0) (- B_m))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (A * (C * -4.0)));
	double tmp;
	if (pow(B_m, 2.0) <= 2e-8) {
		tmp = sqrt((t_0 * (F * (A * 4.0)))) / -t_0;
	} else {
		tmp = sqrt((F * (A - hypot(B_m, A)))) * (sqrt(2.0) / -B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 2e-8)
		tmp = Float64(sqrt(Float64(t_0 * Float64(F * Float64(A * 4.0)))) / Float64(-t_0));
	else
		tmp = Float64(sqrt(Float64(F * Float64(A - hypot(B_m, A)))) * Float64(sqrt(2.0) / Float64(-B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e-8], N[(N[Sqrt[N[(t$95$0 * N[(F * N[(A * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], N[(N[Sqrt[N[(F * N[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 2e-8

   1. Initial program 22.0%

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

   3. Simplified 31.3%

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

   5. Taylor expanded in C around inf 17.2%

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

      1. mul-1-neg 17.2%

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

   7. Simplified 17.2%

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

   8. Taylor expanded in A around -inf 17.2%

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

      1. associate-*r* 17.2%

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

      2. neg-mul-1 17.2%

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

      3. sub-neg 17.2%

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

      4. metadata-eval 17.2%

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

   10. Simplified 17.2%

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

       1. *-un-lft-identity 17.2%

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

       2. associate-*l* 20.1%

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

       3. *-commutative 20.1%

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

   12. Applied egg-rr 20.1%

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

       1. *-lft-identity 20.1%

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

       2. +-commutative 20.1%

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

   14. Simplified 20.1%

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

   15. Taylor expanded in B around 0 19.7%

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

   if 2e-8 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 14.1%

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

   3. Taylor expanded in C around 0 11.5%

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

      1. mul-1-neg 11.5%

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

      2. +-commutative 11.5%

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

      3. unpow2 11.5%

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

      4. unpow2 11.5%

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

      5. hypot-define 24.5%

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

   5. Simplified 24.5%

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

4. Final simplification 22.0%

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

Alternative 3: 38.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\frac{\sqrt{\left(-16 \cdot {A}^{2}\right) \cdot \left(C \cdot F\right)}}{C \cdot \left(A \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B\_m, A\right)\right)} \cdot \frac{\sqrt{2}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 2e-8)
   (/ (sqrt (* (* -16.0 (pow A 2.0)) (* C F))) (* C (* A 4.0)))
   (* (sqrt (* F (- A (hypot B_m A)))) (/ (sqrt 2.0) (- B_m)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (pow(B_m, 2.0) <= 2e-8) {
		tmp = sqrt(((-16.0 * pow(A, 2.0)) * (C * F))) / (C * (A * 4.0));
	} else {
		tmp = sqrt((F * (A - hypot(B_m, A)))) * (sqrt(2.0) / -B_m);
	}
	return tmp;
}

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (Math.pow(B_m, 2.0) <= 2e-8) {
		tmp = Math.sqrt(((-16.0 * Math.pow(A, 2.0)) * (C * F))) / (C * (A * 4.0));
	} else {
		tmp = Math.sqrt((F * (A - Math.hypot(B_m, A)))) * (Math.sqrt(2.0) / -B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if math.pow(B_m, 2.0) <= 2e-8:
		tmp = math.sqrt(((-16.0 * math.pow(A, 2.0)) * (C * F))) / (C * (A * 4.0))
	else:
		tmp = math.sqrt((F * (A - math.hypot(B_m, A)))) * (math.sqrt(2.0) / -B_m)
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 2e-8)
		tmp = Float64(sqrt(Float64(Float64(-16.0 * (A ^ 2.0)) * Float64(C * F))) / Float64(C * Float64(A * 4.0)));
	else
		tmp = Float64(sqrt(Float64(F * Float64(A - hypot(B_m, A)))) * Float64(sqrt(2.0) / Float64(-B_m)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if ((B_m ^ 2.0) <= 2e-8)
		tmp = sqrt(((-16.0 * (A ^ 2.0)) * (C * F))) / (C * (A * 4.0));
	else
		tmp = sqrt((F * (A - hypot(B_m, A)))) * (sqrt(2.0) / -B_m);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e-8], N[(N[Sqrt[N[(N[(-16.0 * N[Power[A, 2.0], $MachinePrecision]), $MachinePrecision] * N[(C * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(C * N[(A * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(F * N[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 2e-8

   1. Initial program 22.0%

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

   3. Simplified 27.5%

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

   5. Taylor expanded in A around -inf 11.5%

      \[\leadsto \frac{\sqrt{\color{blue}{-16 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right)}}}{-\mathsf{fma}\left(C, A \cdot -4, {B}^{2}\right)} \]
   6. Step-by-step derivation

      1. associate-*r* 11.5%

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

   7. Simplified 11.5%

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

   8. Taylor expanded in C around inf 11.4%

      \[\leadsto \frac{\sqrt{\left(-16 \cdot {A}^{2}\right) \cdot \left(C \cdot F\right)}}{\color{blue}{4 \cdot \left(A \cdot C\right)}} \]
   9. Step-by-step derivation

      1. associate-*r* 11.4%

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

   10. Simplified 11.4%

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

   if 2e-8 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 14.1%

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

   3. Taylor expanded in C around 0 11.5%

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

      1. mul-1-neg 11.5%

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

      2. +-commutative 11.5%

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

      3. unpow2 11.5%

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

      4. unpow2 11.5%

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

      5. hypot-define 24.5%

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

   5. Simplified 24.5%

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

4. Final simplification 17.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 2 \cdot 10^{-8}:\\ \;\;\;\;\frac{\sqrt{\left(-16 \cdot {A}^{2}\right) \cdot \left(C \cdot F\right)}}{C \cdot \left(A \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)} \cdot \frac{\sqrt{2}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 46.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;B\_m \leq 0.0058:\\ \;\;\;\;\frac{\sqrt{\left(A \cdot 4\right) \cdot \left(t\_0 \cdot F\right)}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B\_m, A\right)\right)} \cdot \frac{\sqrt{2}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* A (* C -4.0)))))
   (if (<= B_m 0.0058)
     (/ (sqrt (* (* A 4.0) (* t_0 F))) (- t_0))
     (* (sqrt (* F (- A (hypot B_m A)))) (/ (sqrt 2.0) (- B_m))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (A * (C * -4.0)));
	double tmp;
	if (B_m <= 0.0058) {
		tmp = sqrt(((A * 4.0) * (t_0 * F))) / -t_0;
	} else {
		tmp = sqrt((F * (A - hypot(B_m, A)))) * (sqrt(2.0) / -B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	tmp = 0.0
	if (B_m <= 0.0058)
		tmp = Float64(sqrt(Float64(Float64(A * 4.0) * Float64(t_0 * F))) / Float64(-t_0));
	else
		tmp = Float64(sqrt(Float64(F * Float64(A - hypot(B_m, A)))) * Float64(sqrt(2.0) / Float64(-B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 0.0058], N[(N[Sqrt[N[(N[(A * 4.0), $MachinePrecision] * N[(t$95$0 * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], N[(N[Sqrt[N[(F * N[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if B < 0.0058

   1. Initial program 18.5%

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

   3. Simplified 25.7%

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

   5. Taylor expanded in A around -inf 13.2%

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

   if 0.0058 < B

   1. Initial program 17.4%

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

   3. Taylor expanded in C around 0 20.1%

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

      1. mul-1-neg 20.1%

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

      2. +-commutative 20.1%

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

      3. unpow2 20.1%

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

      4. unpow2 20.1%

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

      5. hypot-define 43.1%

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

   5. Simplified 43.1%

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

4. Final simplification 20.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 0.0058:\\ \;\;\;\;\frac{\sqrt{\left(A \cdot 4\right) \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)} \cdot \frac{\sqrt{2}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 44.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 0.000125:\\ \;\;\;\;\frac{\sqrt{\left(A \cdot -8\right) \cdot \left(C \cdot \left(F \cdot \left(A + A\right)\right)\right)}}{-\mathsf{fma}\left(C, A \cdot -4, {B\_m}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B\_m, A\right)\right)} \cdot \frac{\sqrt{2}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 0.000125)
   (/
    (sqrt (* (* A -8.0) (* C (* F (+ A A)))))
    (- (fma C (* A -4.0) (pow B_m 2.0))))
   (* (sqrt (* F (- A (hypot B_m A)))) (/ (sqrt 2.0) (- B_m)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 0.000125) {
		tmp = sqrt(((A * -8.0) * (C * (F * (A + A))))) / -fma(C, (A * -4.0), pow(B_m, 2.0));
	} else {
		tmp = sqrt((F * (A - hypot(B_m, A)))) * (sqrt(2.0) / -B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 0.000125)
		tmp = Float64(sqrt(Float64(Float64(A * -8.0) * Float64(C * Float64(F * Float64(A + A))))) / Float64(-fma(C, Float64(A * -4.0), (B_m ^ 2.0))));
	else
		tmp = Float64(sqrt(Float64(F * Float64(A - hypot(B_m, A)))) * Float64(sqrt(2.0) / Float64(-B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 0.000125], N[(N[Sqrt[N[(N[(A * -8.0), $MachinePrecision] * N[(C * N[(F * N[(A + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-N[(C * N[(A * -4.0), $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[N[(F * N[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 1.25e-4

   1. Initial program 18.5%

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

   3. Simplified 20.9%

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

   5. Taylor expanded in C around inf 13.2%

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

      1. associate-*r* 13.2%

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

      2. mul-1-neg 13.2%

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

   7. Simplified 13.2%

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

   if 1.25e-4 < B

   1. Initial program 17.4%

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

   3. Taylor expanded in C around 0 20.1%

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

      1. mul-1-neg 20.1%

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

      2. +-commutative 20.1%

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

      3. unpow2 20.1%

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

      4. unpow2 20.1%

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

      5. hypot-define 43.1%

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

   5. Simplified 43.1%

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

4. Final simplification 20.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 0.000125:\\ \;\;\;\;\frac{\sqrt{\left(A \cdot -8\right) \cdot \left(C \cdot \left(F \cdot \left(A + A\right)\right)\right)}}{-\mathsf{fma}\left(C, A \cdot -4, {B}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)} \cdot \frac{\sqrt{2}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 35.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 0.0042:\\ \;\;\;\;\frac{\sqrt{\left(-16 \cdot {A}^{2}\right) \cdot \left(C \cdot F\right)}}{C \cdot \left(A \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{F \cdot \frac{\left(\frac{A}{B\_m} + \frac{C}{B\_m}\right) + -1}{B\_m}}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 0.0042)
   (/ (sqrt (* (* -16.0 (pow A 2.0)) (* C F))) (* C (* A 4.0)))
   (* (sqrt 2.0) (- (sqrt (* F (/ (+ (+ (/ A B_m) (/ C B_m)) -1.0) B_m)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 0.0042) {
		tmp = sqrt(((-16.0 * pow(A, 2.0)) * (C * F))) / (C * (A * 4.0));
	} else {
		tmp = sqrt(2.0) * -sqrt((F * ((((A / B_m) + (C / B_m)) + -1.0) / B_m)));
	}
	return tmp;
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (b_m <= 0.0042d0) then
        tmp = sqrt((((-16.0d0) * (a ** 2.0d0)) * (c * f))) / (c * (a * 4.0d0))
    else
        tmp = sqrt(2.0d0) * -sqrt((f * ((((a / b_m) + (c / b_m)) + (-1.0d0)) / b_m)))
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 0.0042) {
		tmp = Math.sqrt(((-16.0 * Math.pow(A, 2.0)) * (C * F))) / (C * (A * 4.0));
	} else {
		tmp = Math.sqrt(2.0) * -Math.sqrt((F * ((((A / B_m) + (C / B_m)) + -1.0) / B_m)));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 0.0042:
		tmp = math.sqrt(((-16.0 * math.pow(A, 2.0)) * (C * F))) / (C * (A * 4.0))
	else:
		tmp = math.sqrt(2.0) * -math.sqrt((F * ((((A / B_m) + (C / B_m)) + -1.0) / B_m)))
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 0.0042)
		tmp = Float64(sqrt(Float64(Float64(-16.0 * (A ^ 2.0)) * Float64(C * F))) / Float64(C * Float64(A * 4.0)));
	else
		tmp = Float64(sqrt(2.0) * Float64(-sqrt(Float64(F * Float64(Float64(Float64(Float64(A / B_m) + Float64(C / B_m)) + -1.0) / B_m)))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (B_m <= 0.0042)
		tmp = sqrt(((-16.0 * (A ^ 2.0)) * (C * F))) / (C * (A * 4.0));
	else
		tmp = sqrt(2.0) * -sqrt((F * ((((A / B_m) + (C / B_m)) + -1.0) / B_m)));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 0.0042], N[(N[Sqrt[N[(N[(-16.0 * N[Power[A, 2.0], $MachinePrecision]), $MachinePrecision] * N[(C * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(C * N[(A * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * (-N[Sqrt[N[(F * N[(N[(N[(N[(A / B$95$m), $MachinePrecision] + N[(C / B$95$m), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 0.00419999999999999974

   1. Initial program 18.5%

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

   3. Simplified 20.9%

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

   5. Taylor expanded in A around -inf 8.8%

      \[\leadsto \frac{\sqrt{\color{blue}{-16 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right)}}}{-\mathsf{fma}\left(C, A \cdot -4, {B}^{2}\right)} \]
   6. Step-by-step derivation

      1. associate-*r* 8.8%

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

   7. Simplified 8.8%

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

   8. Taylor expanded in C around inf 8.6%

      \[\leadsto \frac{\sqrt{\left(-16 \cdot {A}^{2}\right) \cdot \left(C \cdot F\right)}}{\color{blue}{4 \cdot \left(A \cdot C\right)}} \]
   9. Step-by-step derivation

      1. associate-*r* 8.6%

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

   10. Simplified 8.6%

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

   if 0.00419999999999999974 < B

   1. Initial program 17.4%

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

   3. Simplified 13.3%

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

   5. Taylor expanded in B around inf 11.5%

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

   6. Taylor expanded in F around 0 15.7%

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

      1. mul-1-neg 15.7%

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

      2. associate-/l* 17.3%

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

      3. +-commutative 17.3%

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

      4. sub-neg 17.3%

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

      5. metadata-eval 17.3%

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

      6. fma-undefine 17.3%

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

      7. +-commutative 17.3%

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

      8. fma-define 17.3%

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

   8. Simplified 17.3%

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

   9. Taylor expanded in B around inf 41.2%

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

4. Final simplification 17.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 0.0042:\\ \;\;\;\;\frac{\sqrt{\left(-16 \cdot {A}^{2}\right) \cdot \left(C \cdot F\right)}}{C \cdot \left(A \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{F \cdot \frac{\left(\frac{A}{B} + \frac{C}{B}\right) + -1}{B}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 35.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 0.000145:\\ \;\;\;\;\frac{\sqrt{\left(-16 \cdot {A}^{2}\right) \cdot \left(C \cdot F\right)}}{C \cdot \left(A \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F \cdot \frac{A + C}{B\_m} - F}{B\_m}}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 0.000145)
   (/ (sqrt (* (* -16.0 (pow A 2.0)) (* C F))) (* C (* A 4.0)))
   (* (sqrt 2.0) (- (sqrt (/ (- (* F (/ (+ A C) B_m)) F) B_m))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 0.000145) {
		tmp = sqrt(((-16.0 * pow(A, 2.0)) * (C * F))) / (C * (A * 4.0));
	} else {
		tmp = sqrt(2.0) * -sqrt((((F * ((A + C) / B_m)) - F) / B_m));
	}
	return tmp;
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (b_m <= 0.000145d0) then
        tmp = sqrt((((-16.0d0) * (a ** 2.0d0)) * (c * f))) / (c * (a * 4.0d0))
    else
        tmp = sqrt(2.0d0) * -sqrt((((f * ((a + c) / b_m)) - f) / b_m))
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 0.000145) {
		tmp = Math.sqrt(((-16.0 * Math.pow(A, 2.0)) * (C * F))) / (C * (A * 4.0));
	} else {
		tmp = Math.sqrt(2.0) * -Math.sqrt((((F * ((A + C) / B_m)) - F) / B_m));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 0.000145:
		tmp = math.sqrt(((-16.0 * math.pow(A, 2.0)) * (C * F))) / (C * (A * 4.0))
	else:
		tmp = math.sqrt(2.0) * -math.sqrt((((F * ((A + C) / B_m)) - F) / B_m))
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 0.000145)
		tmp = Float64(sqrt(Float64(Float64(-16.0 * (A ^ 2.0)) * Float64(C * F))) / Float64(C * Float64(A * 4.0)));
	else
		tmp = Float64(sqrt(2.0) * Float64(-sqrt(Float64(Float64(Float64(F * Float64(Float64(A + C) / B_m)) - F) / B_m))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (B_m <= 0.000145)
		tmp = sqrt(((-16.0 * (A ^ 2.0)) * (C * F))) / (C * (A * 4.0));
	else
		tmp = sqrt(2.0) * -sqrt((((F * ((A + C) / B_m)) - F) / B_m));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 0.000145], N[(N[Sqrt[N[(N[(-16.0 * N[Power[A, 2.0], $MachinePrecision]), $MachinePrecision] * N[(C * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(C * N[(A * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * (-N[Sqrt[N[(N[(N[(F * N[(N[(A + C), $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision] - F), $MachinePrecision] / B$95$m), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 1.45e-4

   1. Initial program 18.5%

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

   3. Simplified 20.9%

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

   5. Taylor expanded in A around -inf 8.8%

      \[\leadsto \frac{\sqrt{\color{blue}{-16 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right)}}}{-\mathsf{fma}\left(C, A \cdot -4, {B}^{2}\right)} \]
   6. Step-by-step derivation

      1. associate-*r* 8.8%

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

   7. Simplified 8.8%

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

   8. Taylor expanded in C around inf 8.6%

      \[\leadsto \frac{\sqrt{\left(-16 \cdot {A}^{2}\right) \cdot \left(C \cdot F\right)}}{\color{blue}{4 \cdot \left(A \cdot C\right)}} \]
   9. Step-by-step derivation

      1. associate-*r* 8.6%

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

   10. Simplified 8.6%

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

   if 1.45e-4 < B

   1. Initial program 17.4%

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

   3. Simplified 13.3%

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

   5. Taylor expanded in B around inf 11.5%

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

   6. Taylor expanded in F around 0 15.7%

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

      1. mul-1-neg 15.7%

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

      2. associate-/l* 17.3%

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

      3. +-commutative 17.3%

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

      4. sub-neg 17.3%

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

      5. metadata-eval 17.3%

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

      6. fma-undefine 17.3%

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

      7. +-commutative 17.3%

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

      8. fma-define 17.3%

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

   8. Simplified 17.3%

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

   9. Taylor expanded in B around -inf 39.3%

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

       1. mul-1-neg 39.3%

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

       2. mul-1-neg 39.3%

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

       3. associate-/l* 41.3%

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

   11. Simplified 41.3%

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

4. Final simplification 17.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 0.000145:\\ \;\;\;\;\frac{\sqrt{\left(-16 \cdot {A}^{2}\right) \cdot \left(C \cdot F\right)}}{C \cdot \left(A \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F \cdot \frac{A + C}{B} - F}{B}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 35.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 0.0071:\\ \;\;\;\;\frac{\sqrt{\left(-16 \cdot {A}^{2}\right) \cdot \left(C \cdot F\right)}}{C \cdot \left(A \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{-F}{B\_m}}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 0.0071)
   (/ (sqrt (* (* -16.0 (pow A 2.0)) (* C F))) (* C (* A 4.0)))
   (* (sqrt 2.0) (- (sqrt (/ (- F) B_m))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 0.0071) {
		tmp = sqrt(((-16.0 * pow(A, 2.0)) * (C * F))) / (C * (A * 4.0));
	} else {
		tmp = sqrt(2.0) * -sqrt((-F / B_m));
	}
	return tmp;
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (b_m <= 0.0071d0) then
        tmp = sqrt((((-16.0d0) * (a ** 2.0d0)) * (c * f))) / (c * (a * 4.0d0))
    else
        tmp = sqrt(2.0d0) * -sqrt((-f / b_m))
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 0.0071) {
		tmp = Math.sqrt(((-16.0 * Math.pow(A, 2.0)) * (C * F))) / (C * (A * 4.0));
	} else {
		tmp = Math.sqrt(2.0) * -Math.sqrt((-F / B_m));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 0.0071:
		tmp = math.sqrt(((-16.0 * math.pow(A, 2.0)) * (C * F))) / (C * (A * 4.0))
	else:
		tmp = math.sqrt(2.0) * -math.sqrt((-F / B_m))
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 0.0071)
		tmp = Float64(sqrt(Float64(Float64(-16.0 * (A ^ 2.0)) * Float64(C * F))) / Float64(C * Float64(A * 4.0)));
	else
		tmp = Float64(sqrt(2.0) * Float64(-sqrt(Float64(Float64(-F) / B_m))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (B_m <= 0.0071)
		tmp = sqrt(((-16.0 * (A ^ 2.0)) * (C * F))) / (C * (A * 4.0));
	else
		tmp = sqrt(2.0) * -sqrt((-F / B_m));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 0.0071], N[(N[Sqrt[N[(N[(-16.0 * N[Power[A, 2.0], $MachinePrecision]), $MachinePrecision] * N[(C * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(C * N[(A * 4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * (-N[Sqrt[N[((-F) / B$95$m), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 0.0071000000000000004

   1. Initial program 18.5%

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

   3. Simplified 20.9%

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

   5. Taylor expanded in A around -inf 8.8%

      \[\leadsto \frac{\sqrt{\color{blue}{-16 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right)}}}{-\mathsf{fma}\left(C, A \cdot -4, {B}^{2}\right)} \]
   6. Step-by-step derivation

      1. associate-*r* 8.8%

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

   7. Simplified 8.8%

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

   8. Taylor expanded in C around inf 8.6%

      \[\leadsto \frac{\sqrt{\left(-16 \cdot {A}^{2}\right) \cdot \left(C \cdot F\right)}}{\color{blue}{4 \cdot \left(A \cdot C\right)}} \]
   9. Step-by-step derivation

      1. associate-*r* 8.6%

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

   10. Simplified 8.6%

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

   if 0.0071000000000000004 < B

   1. Initial program 17.4%

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

   3. Simplified 13.3%

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

   5. Taylor expanded in B around inf 11.5%

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

   6. Taylor expanded in F around 0 15.7%

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

      1. mul-1-neg 15.7%

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

      2. associate-/l* 17.3%

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

      3. +-commutative 17.3%

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

      4. sub-neg 17.3%

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

      5. metadata-eval 17.3%

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

      6. fma-undefine 17.3%

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

      7. +-commutative 17.3%

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

      8. fma-define 17.3%

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

   8. Simplified 17.3%

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

   9. Taylor expanded in B around inf 40.9%

      \[\leadsto -\sqrt{\color{blue}{-1 \cdot \frac{F}{B}}} \cdot \sqrt{2} \]
   10. Step-by-step derivation

       1. associate-*r/ 40.9%

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

       2. mul-1-neg 40.9%

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

   11. Simplified 40.9%

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

4. Final simplification 16.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 0.0071:\\ \;\;\;\;\frac{\sqrt{\left(-16 \cdot {A}^{2}\right) \cdot \left(C \cdot F\right)}}{C \cdot \left(A \cdot 4\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{-F}{B}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 33.8% accurate, 2.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;F \leq -2.6 \cdot 10^{-68}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{-F}{B\_m}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A - B\_m\right)} \cdot \frac{\sqrt{2}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= F -2.6e-68)
   (* (sqrt 2.0) (- (sqrt (/ (- F) B_m))))
   (* (sqrt (* F (- A B_m))) (/ (sqrt 2.0) (- B_m)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= -2.6e-68) {
		tmp = sqrt(2.0) * -sqrt((-F / B_m));
	} else {
		tmp = sqrt((F * (A - B_m))) * (sqrt(2.0) / -B_m);
	}
	return tmp;
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (f <= (-2.6d-68)) then
        tmp = sqrt(2.0d0) * -sqrt((-f / b_m))
    else
        tmp = sqrt((f * (a - b_m))) * (sqrt(2.0d0) / -b_m)
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= -2.6e-68) {
		tmp = Math.sqrt(2.0) * -Math.sqrt((-F / B_m));
	} else {
		tmp = Math.sqrt((F * (A - B_m))) * (Math.sqrt(2.0) / -B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if F <= -2.6e-68:
		tmp = math.sqrt(2.0) * -math.sqrt((-F / B_m))
	else:
		tmp = math.sqrt((F * (A - B_m))) * (math.sqrt(2.0) / -B_m)
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (F <= -2.6e-68)
		tmp = Float64(sqrt(2.0) * Float64(-sqrt(Float64(Float64(-F) / B_m))));
	else
		tmp = Float64(sqrt(Float64(F * Float64(A - B_m))) * Float64(sqrt(2.0) / Float64(-B_m)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (F <= -2.6e-68)
		tmp = sqrt(2.0) * -sqrt((-F / B_m));
	else
		tmp = sqrt((F * (A - B_m))) * (sqrt(2.0) / -B_m);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[F, -2.6e-68], N[(N[Sqrt[2.0], $MachinePrecision] * (-N[Sqrt[N[((-F) / B$95$m), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[N[(F * N[(A - B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -2.5999999999999998e-68

   1. Initial program 10.1%

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

   3. Simplified 10.6%

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

   5. Taylor expanded in B around inf 3.4%

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

   6. Taylor expanded in F around 0 5.1%

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

      1. mul-1-neg 5.1%

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

      2. associate-/l* 7.0%

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

      3. +-commutative 7.0%

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

      4. sub-neg 7.0%

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

      5. metadata-eval 7.0%

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

      6. fma-undefine 7.0%

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

      7. +-commutative 7.0%

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

      8. fma-define 7.0%

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

   8. Simplified 7.0%

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

   9. Taylor expanded in B around inf 17.0%

      \[\leadsto -\sqrt{\color{blue}{-1 \cdot \frac{F}{B}}} \cdot \sqrt{2} \]
   10. Step-by-step derivation

       1. associate-*r/ 17.0%

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

       2. mul-1-neg 17.0%

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

   11. Simplified 17.0%

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

   if -2.5999999999999998e-68 < F

   1. Initial program 28.5%

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

   3. Simplified 29.5%

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

   5. Taylor expanded in B around inf 8.8%

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

   6. Taylor expanded in C around 0 15.8%

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

      1. mul-1-neg 15.8%

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

      2. mul-1-neg 15.8%

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

   8. Simplified 15.8%

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

4. Final simplification 16.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.6 \cdot 10^{-68}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{-F}{B}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A - B\right)} \cdot \frac{\sqrt{2}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 26.4% accurate, 3.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 0.000118:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{-0.25 \cdot \frac{F}{C}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{-F}{B\_m}}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 0.000118)
   (* (sqrt 2.0) (- (sqrt (* -0.25 (/ F C)))))
   (* (sqrt 2.0) (- (sqrt (/ (- F) B_m))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 0.000118) {
		tmp = sqrt(2.0) * -sqrt((-0.25 * (F / C)));
	} else {
		tmp = sqrt(2.0) * -sqrt((-F / B_m));
	}
	return tmp;
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (b_m <= 0.000118d0) then
        tmp = sqrt(2.0d0) * -sqrt(((-0.25d0) * (f / c)))
    else
        tmp = sqrt(2.0d0) * -sqrt((-f / b_m))
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 0.000118) {
		tmp = Math.sqrt(2.0) * -Math.sqrt((-0.25 * (F / C)));
	} else {
		tmp = Math.sqrt(2.0) * -Math.sqrt((-F / B_m));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 0.000118:
		tmp = math.sqrt(2.0) * -math.sqrt((-0.25 * (F / C)))
	else:
		tmp = math.sqrt(2.0) * -math.sqrt((-F / B_m))
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 0.000118)
		tmp = Float64(sqrt(2.0) * Float64(-sqrt(Float64(-0.25 * Float64(F / C)))));
	else
		tmp = Float64(sqrt(2.0) * Float64(-sqrt(Float64(Float64(-F) / B_m))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (B_m <= 0.000118)
		tmp = sqrt(2.0) * -sqrt((-0.25 * (F / C)));
	else
		tmp = sqrt(2.0) * -sqrt((-F / B_m));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 0.000118], N[(N[Sqrt[2.0], $MachinePrecision] * (-N[Sqrt[N[(-0.25 * N[(F / C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * (-N[Sqrt[N[((-F) / B$95$m), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 1.18e-4

   1. Initial program 18.5%

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

   3. Simplified 20.9%

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

   5. Taylor expanded in B around inf 3.8%

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

   6. Taylor expanded in F around 0 3.4%

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

      1. mul-1-neg 3.4%

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

      2. associate-/l* 4.3%

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

      3. +-commutative 4.3%

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

      4. sub-neg 4.3%

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

      5. metadata-eval 4.3%

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

      6. fma-undefine 4.3%

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

      7. +-commutative 4.3%

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

      8. fma-define 4.3%

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

   8. Simplified 4.3%

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

   9. Taylor expanded in A around inf 5.0%

      \[\leadsto -\sqrt{\color{blue}{-0.25 \cdot \frac{F}{C}}} \cdot \sqrt{2} \]

   if 1.18e-4 < B

   1. Initial program 17.4%

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

   3. Simplified 13.3%

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

   5. Taylor expanded in B around inf 11.5%

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

   6. Taylor expanded in F around 0 15.7%

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

      1. mul-1-neg 15.7%

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

      2. associate-/l* 17.3%

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

      3. +-commutative 17.3%

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

      4. sub-neg 17.3%

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

      5. metadata-eval 17.3%

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

      6. fma-undefine 17.3%

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

      7. +-commutative 17.3%

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

      8. fma-define 17.3%

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

   8. Simplified 17.3%

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

   9. Taylor expanded in B around inf 40.9%

      \[\leadsto -\sqrt{\color{blue}{-1 \cdot \frac{F}{B}}} \cdot \sqrt{2} \]
   10. Step-by-step derivation

       1. associate-*r/ 40.9%

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

       2. mul-1-neg 40.9%

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

   11. Simplified 40.9%

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

4. Final simplification 14.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 0.000118:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{-0.25 \cdot \frac{F}{C}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{-F}{B}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 26.9% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 18.2%

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

3. Simplified 18.9%

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

5. Taylor expanded in B around inf 5.8%

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

6. Taylor expanded in F around 0 6.6%

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

   1. mul-1-neg 6.6%

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

   2. associate-/l* 7.7%

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

   3. +-commutative 7.7%

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

   4. sub-neg 7.7%

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

   5. metadata-eval 7.7%

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

   6. fma-undefine 7.7%

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

   7. +-commutative 7.7%

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

   8. fma-define 7.7%

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

8. Simplified 7.7%

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

9. Taylor expanded in B around inf 12.3%

   \[\leadsto -\sqrt{\color{blue}{-1 \cdot \frac{F}{B}}} \cdot \sqrt{2} \]
10. Step-by-step derivation

    1. associate-*r/ 12.3%

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

    2. mul-1-neg 12.3%

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

11. Simplified 12.3%

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

12. Final simplification 12.3%

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

Alternative 12: 3.3% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 18.2%

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

3. Taylor expanded in B around -inf 0.0%

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

   1. mul-1-neg 0.0%

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

   2. unpow2 0.0%

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

   3. rem-square-sqrt 2.0%

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

5. Simplified 2.0%

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

6. Taylor expanded in F around 0 2.0%

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

   1. pow1 2.0%

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

   2. sqrt-unprod 2.0%

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

8. Applied egg-rr 2.0%

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

   1. unpow1 2.0%

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

   2. associate-*l/ 2.0%

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

10. Simplified 2.0%

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

    1. associate-*l/ 2.0%

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

    2. add-sqr-sqrt 2.0%

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

    3. pow1/2 2.0%

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

    4. pow1/2 2.2%

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

    5. pow-prod-down 3.3%

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

    6. pow2 3.3%

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

    7. associate-*l/ 3.3%

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

    8. associate-/l* 3.3%

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

12. Applied egg-rr 3.3%

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

    1. unpow1/2 3.3%

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

    2. unpow2 3.3%

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

    3. rem-sqrt-square 3.2%

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

14. Simplified 3.2%

    \[\leadsto \sqrt{\color{blue}{\left|F \cdot \frac{2}{B}\right|}} \]
15. Add Preprocessing

Alternative 13: 1.7% accurate, 6.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 18.2%

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

3. Taylor expanded in B around -inf 0.0%

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

   1. mul-1-neg 0.0%

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

   2. unpow2 0.0%

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

   3. rem-square-sqrt 2.0%

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

5. Simplified 2.0%

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

6. Taylor expanded in F around 0 2.0%

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

   1. sqrt-unprod 2.0%

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

   2. pow1/2 2.2%

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

8. Applied egg-rr 2.2%

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

9. Final simplification 2.2%

   \[\leadsto {\left(2 \cdot \frac{F}{B}\right)}^{0.5} \]
10. Add Preprocessing

Alternative 14: 1.6% accurate, 6.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 18.2%

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

3. Taylor expanded in B around -inf 0.0%

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

   1. mul-1-neg 0.0%

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

   2. unpow2 0.0%

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

   3. rem-square-sqrt 2.0%

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

5. Simplified 2.0%

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

6. Taylor expanded in F around 0 2.0%

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

   1. pow1 2.0%

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

   2. sqrt-unprod 2.0%

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

8. Applied egg-rr 2.0%

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

   1. unpow1 2.0%

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

   2. associate-*l/ 2.0%

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

10. Simplified 2.0%

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

11. Final simplification 2.0%

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

Reproduce

herbie shell --seed 2024144 
(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))))