ABCF->ab-angle b

Percentage Accurate: 18.9% → 59.3%

Time: 30.7s

Alternatives: 8

Speedup: 5.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 18.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 59.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := A + \left(C - \mathsf{hypot}\left(B\_m, A - C\right)\right)\\ t_2 := -t\_0\\ t_3 := \left(4 \cdot A\right) \cdot C\\ t_4 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_3\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_3 - {B\_m}^{2}}\\ t_5 := F \cdot t\_0\\ \mathbf{if}\;t\_4 \leq -\infty:\\ \;\;\;\;-\sqrt{\frac{t\_1}{\mathsf{fma}\left(-4, A \cdot C, {B\_m}^{2}\right)} \cdot \left(2 \cdot F\right)}\\ \mathbf{elif}\;t\_4 \leq -5 \cdot 10^{-202}:\\ \;\;\;\;\frac{\sqrt{t\_5 \cdot \left(2 \cdot t\_1\right)}}{t\_2}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;\frac{\sqrt{t\_5 \cdot \left(2 \cdot \left(A + \left(A + -0.5 \cdot \frac{{B\_m}^{2}}{C}\right)\right)\right)}}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{-F}}{\sqrt{B\_m}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \end{array} \]

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))))
        (t_1 (+ A (- C (hypot B_m (- A C)))))
        (t_2 (- t_0))
        (t_3 (* (* 4.0 A) C))
        (t_4
         (/
          (sqrt
           (*
            (* 2.0 (* (- (pow B_m 2.0) t_3) F))
            (- (+ A C) (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))))))
          (- t_3 (pow B_m 2.0))))
        (t_5 (* F t_0)))
   (if (<= t_4 (- INFINITY))
     (- (sqrt (* (/ t_1 (fma -4.0 (* A C) (pow B_m 2.0))) (* 2.0 F))))
     (if (<= t_4 -5e-202)
       (/ (sqrt (* t_5 (* 2.0 t_1))) t_2)
       (if (<= t_4 INFINITY)
         (/
          (sqrt (* t_5 (* 2.0 (+ A (+ A (* -0.5 (/ (pow B_m 2.0) C)))))))
          t_2)
         (* (/ (sqrt (- F)) (sqrt B_m)) (- (sqrt 2.0))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (A * (C * -4.0)));
	double t_1 = A + (C - hypot(B_m, (A - C)));
	double t_2 = -t_0;
	double t_3 = (4.0 * A) * C;
	double t_4 = sqrt(((2.0 * ((pow(B_m, 2.0) - t_3) * F)) * ((A + C) - sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / (t_3 - pow(B_m, 2.0));
	double t_5 = F * t_0;
	double tmp;
	if (t_4 <= -((double) INFINITY)) {
		tmp = -sqrt(((t_1 / fma(-4.0, (A * C), pow(B_m, 2.0))) * (2.0 * F)));
	} else if (t_4 <= -5e-202) {
		tmp = sqrt((t_5 * (2.0 * t_1))) / t_2;
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = sqrt((t_5 * (2.0 * (A + (A + (-0.5 * (pow(B_m, 2.0) / C))))))) / t_2;
	} else {
		tmp = (sqrt(-F) / sqrt(B_m)) * -sqrt(2.0);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	t_1 = Float64(A + Float64(C - hypot(B_m, Float64(A - C))))
	t_2 = Float64(-t_0)
	t_3 = Float64(Float64(4.0 * A) * C)
	t_4 = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_3) * F)) * Float64(Float64(A + C) - sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / Float64(t_3 - (B_m ^ 2.0)))
	t_5 = Float64(F * t_0)
	tmp = 0.0
	if (t_4 <= Float64(-Inf))
		tmp = Float64(-sqrt(Float64(Float64(t_1 / fma(-4.0, Float64(A * C), (B_m ^ 2.0))) * Float64(2.0 * F))));
	elseif (t_4 <= -5e-202)
		tmp = Float64(sqrt(Float64(t_5 * Float64(2.0 * t_1))) / t_2);
	elseif (t_4 <= Inf)
		tmp = Float64(sqrt(Float64(t_5 * Float64(2.0 * Float64(A + Float64(A + Float64(-0.5 * Float64((B_m ^ 2.0) / C))))))) / t_2);
	else
		tmp = Float64(Float64(sqrt(Float64(-F)) / sqrt(B_m)) * Float64(-sqrt(2.0)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(A + N[(C - N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = (-t$95$0)}, Block[{t$95$3 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$3), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$3 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(F * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$4, (-Infinity)], (-N[Sqrt[N[(N[(t$95$1 / N[(-4.0 * N[(A * C), $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), If[LessEqual[t$95$4, -5e-202], N[(N[Sqrt[N[(t$95$5 * N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[(N[Sqrt[N[(t$95$5 * N[(2.0 * N[(A + N[(A + N[(-0.5 * N[(N[Power[B$95$m, 2.0], $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision], N[(N[(N[Sqrt[(-F)], $MachinePrecision] / N[Sqrt[B$95$m], $MachinePrecision]), $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))) < -inf.0

   1. Initial program 3.2%

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

   3. Taylor expanded in F around 0 20.9%

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

      1. mul-1-neg 20.9%

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

      2. distribute-rgt-neg-in 20.9%

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

   5. Simplified 56.7%

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

      1. pow1 56.7%

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

   7. Applied egg-rr 56.0%

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

      1. unpow1 56.0%

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

      2. unpow1/2 56.0%

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

      3. associate-*l* 56.0%

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

      4. associate--l+ 57.2%

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

   9. Simplified 57.2%

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

   if -inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))) < -4.99999999999999973e-202

   1. Initial program 97.3%

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

   3. Simplified 97.4%

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

   if -4.99999999999999973e-202 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))) < +inf.0

   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 25.2%

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

   5. Taylor expanded in C around inf 23.5%

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

   if +inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)))

   1. Initial program 0.0%

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

   3. Taylor expanded in F around 0 0.1%

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

      1. mul-1-neg 0.1%

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

      2. distribute-rgt-neg-in 0.1%

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

   5. Simplified 4.1%

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

   6. Taylor expanded in B around inf 21.6%

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

      1. associate-*r/ 21.6%

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

      2. sqrt-div 24.5%

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

   8. Applied egg-rr 24.5%

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

      1. *-commutative 24.5%

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

      2. neg-mul-1 24.5%

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

   10. Simplified 24.5%

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

4. Final simplification 38.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -\infty:\\ \;\;\;\;-\sqrt{\frac{A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)}{\mathsf{fma}\left(-4, A \cdot C, {B}^{2}\right)} \cdot \left(2 \cdot F\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -5 \cdot 10^{-202}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot \left(2 \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot \left(2 \cdot \left(A + \left(A + -0.5 \cdot \frac{{B}^{2}}{C}\right)\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{-F}}{\sqrt{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 50.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 4 \cdot 10^{-180}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{F \cdot \frac{-0.5}{C}}\right)\\ \mathbf{elif}\;{B\_m}^{2} \leq 2 \cdot 10^{+16}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot t\_0\right) \cdot \left(2 \cdot \left(A + \left(C - \mathsf{hypot}\left(B\_m, A - C\right)\right)\right)\right)}}{-t\_0}\\ \mathbf{elif}\;{B\_m}^{2} \leq 4 \cdot 10^{+146}:\\ \;\;\;\;\sqrt{F \cdot \left(-0.5 \cdot \frac{{B\_m}^{2}}{C}\right)} \cdot \frac{\sqrt{2}}{-B\_m}\\ \mathbf{elif}\;{B\_m}^{2} \leq 5 \cdot 10^{+295}:\\ \;\;\;\;-\sqrt{\left(2 \cdot F\right) \cdot \frac{A - \mathsf{hypot}\left(B\_m, A\right)}{{B\_m}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{-F}}{\sqrt{B\_m}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \end{array} \]

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) 4e-180)
     (* (sqrt 2.0) (- (sqrt (* F (/ -0.5 C)))))
     (if (<= (pow B_m 2.0) 2e+16)
       (/ (sqrt (* (* F t_0) (* 2.0 (+ A (- C (hypot B_m (- A C))))))) (- t_0))
       (if (<= (pow B_m 2.0) 4e+146)
         (* (sqrt (* F (* -0.5 (/ (pow B_m 2.0) C)))) (/ (sqrt 2.0) (- B_m)))
         (if (<= (pow B_m 2.0) 5e+295)
           (- (sqrt (* (* 2.0 F) (/ (- A (hypot B_m A)) (pow B_m 2.0)))))
           (* (/ (sqrt (- F)) (sqrt B_m)) (- (sqrt 2.0)))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (A * (C * -4.0)));
	double tmp;
	if (pow(B_m, 2.0) <= 4e-180) {
		tmp = sqrt(2.0) * -sqrt((F * (-0.5 / C)));
	} else if (pow(B_m, 2.0) <= 2e+16) {
		tmp = sqrt(((F * t_0) * (2.0 * (A + (C - hypot(B_m, (A - C))))))) / -t_0;
	} else if (pow(B_m, 2.0) <= 4e+146) {
		tmp = sqrt((F * (-0.5 * (pow(B_m, 2.0) / C)))) * (sqrt(2.0) / -B_m);
	} else if (pow(B_m, 2.0) <= 5e+295) {
		tmp = -sqrt(((2.0 * F) * ((A - hypot(B_m, A)) / pow(B_m, 2.0))));
	} else {
		tmp = (sqrt(-F) / sqrt(B_m)) * -sqrt(2.0);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 4e-180)
		tmp = Float64(sqrt(2.0) * Float64(-sqrt(Float64(F * Float64(-0.5 / C)))));
	elseif ((B_m ^ 2.0) <= 2e+16)
		tmp = Float64(sqrt(Float64(Float64(F * t_0) * Float64(2.0 * Float64(A + Float64(C - hypot(B_m, Float64(A - C))))))) / Float64(-t_0));
	elseif ((B_m ^ 2.0) <= 4e+146)
		tmp = Float64(sqrt(Float64(F * Float64(-0.5 * Float64((B_m ^ 2.0) / C)))) * Float64(sqrt(2.0) / Float64(-B_m)));
	elseif ((B_m ^ 2.0) <= 5e+295)
		tmp = Float64(-sqrt(Float64(Float64(2.0 * F) * Float64(Float64(A - hypot(B_m, A)) / (B_m ^ 2.0)))));
	else
		tmp = Float64(Float64(sqrt(Float64(-F)) / sqrt(B_m)) * Float64(-sqrt(2.0)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 4e-180], N[(N[Sqrt[2.0], $MachinePrecision] * (-N[Sqrt[N[(F * N[(-0.5 / C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+16], N[(N[Sqrt[N[(N[(F * t$95$0), $MachinePrecision] * N[(2.0 * N[(A + N[(C - N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 4e+146], N[(N[Sqrt[N[(F * N[(-0.5 * N[(N[Power[B$95$m, 2.0], $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e+295], (-N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] * N[(N[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision] / N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), N[(N[(N[Sqrt[(-F)], $MachinePrecision] / N[Sqrt[B$95$m], $MachinePrecision]), $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if (pow.f64 B 2) < 4.0000000000000001e-180

   1. Initial program 14.9%

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

   3. Taylor expanded in F around 0 11.1%

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

      1. mul-1-neg 11.1%

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

      2. distribute-rgt-neg-in 11.1%

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

   5. Simplified 23.9%

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

   6. Taylor expanded in A around -inf 20.0%

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

   if 4.0000000000000001e-180 < (pow.f64 B 2) < 2e16

   1. Initial program 38.1%

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

   3. Simplified 43.2%

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

   if 2e16 < (pow.f64 B 2) < 3.99999999999999973e146

   1. Initial program 29.4%

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

   3. Taylor expanded in A around 0 24.2%

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

      1. mul-1-neg 24.2%

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

      2. *-commutative 24.2%

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

      3. distribute-rgt-neg-in 24.2%

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

      4. unpow2 24.2%

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

      5. unpow2 24.2%

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

      6. hypot-define 24.8%

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

   5. Simplified 24.8%

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

   6. Taylor expanded in C around inf 8.1%

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

   if 3.99999999999999973e146 < (pow.f64 B 2) < 4.99999999999999991e295

   1. Initial program 12.6%

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

   3. Taylor expanded in C around 0 12.5%

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

      1. mul-1-neg 12.5%

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

      2. +-commutative 12.5%

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

   5. Simplified 12.5%

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

      1. add-cbrt-cube 9.3%

         \[\leadsto -\color{blue}{\sqrt[3]{\left(\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{{B}^{2} + {A}^{2}}\right)}\right) \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{{B}^{2} + {A}^{2}}\right)}\right)\right) \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{{B}^{2} + {A}^{2}}\right)}\right)}} \]

      2. pow3 9.3%

         \[\leadsto -\sqrt[3]{\color{blue}{{\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{{B}^{2} + {A}^{2}}\right)}\right)}^{3}}} \]

   7. Applied egg-rr 9.9%

      \[\leadsto -\color{blue}{\sqrt[3]{{\left(\frac{{\left(2 \cdot \left(F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)\right)\right)}^{0.5}}{B}\right)}^{3}}} \]
   8. Step-by-step derivation

      1. rem-cbrt-cube 16.4%

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

      2. add-sqr-sqrt 15.4%

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

      3. sqrt-unprod 40.8%

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

      4. frac-times 40.7%

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

      5. unpow1/2 40.7%

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

      6. unpow1/2 40.7%

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

      7. add-sqr-sqrt 40.9%

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

      8. associate-*r* 40.9%

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

      9. pow2 40.9%

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

   9. Applied egg-rr 40.9%

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

       1. associate-/l* 57.3%

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

       2. *-commutative 57.3%

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

       3. hypot-undefine 50.3%

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

       4. unpow2 50.3%

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

       5. unpow2 50.3%

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

       6. +-commutative 50.3%

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

       7. unpow2 50.3%

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

       8. unpow2 50.3%

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

       9. hypot-define 57.3%

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

   11. Simplified 57.3%

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

   if 4.99999999999999991e295 < (pow.f64 B 2)

   1. Initial program 0.0%

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

   3. Taylor expanded in F around 0 0.0%

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

      1. mul-1-neg 0.0%

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

      2. distribute-rgt-neg-in 0.0%

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

   5. Simplified 4.1%

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

   6. Taylor expanded in B around inf 34.6%

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

      1. associate-*r/ 34.6%

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

      2. sqrt-div 39.5%

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

   8. Applied egg-rr 39.5%

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

      1. *-commutative 39.5%

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

      2. neg-mul-1 39.5%

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

   10. Simplified 39.5%

       \[\leadsto \color{blue}{\frac{\sqrt{-F}}{\sqrt{B}}} \cdot \left(-\sqrt{2}\right) \]
3. Recombined 5 regimes into one program.

4. Final simplification 33.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 4 \cdot 10^{-180}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{F \cdot \frac{-0.5}{C}}\right)\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{+16}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot \left(2 \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;{B}^{2} \leq 4 \cdot 10^{+146}:\\ \;\;\;\;\sqrt{F \cdot \left(-0.5 \cdot \frac{{B}^{2}}{C}\right)} \cdot \frac{\sqrt{2}}{-B}\\ \mathbf{elif}\;{B}^{2} \leq 5 \cdot 10^{+295}:\\ \;\;\;\;-\sqrt{\left(2 \cdot F\right) \cdot \frac{A - \mathsf{hypot}\left(B, A\right)}{{B}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{-F}}{\sqrt{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 54.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ \mathbf{if}\;{B\_m}^{2} \leq 10^{-15}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_0\right) \cdot F\right)\right) \cdot \left(A + A\right)}}{t\_0 - {B\_m}^{2}}\\ \mathbf{elif}\;{B\_m}^{2} \leq 2 \cdot 10^{+260}:\\ \;\;\;\;-\sqrt{\frac{A + \left(C - \mathsf{hypot}\left(B\_m, A - C\right)\right)}{\mathsf{fma}\left(-4, A \cdot C, {B\_m}^{2}\right)} \cdot \left(2 \cdot F\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{-F}}{\sqrt{B\_m}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \end{array} \]

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 (* (* 4.0 A) C)))
   (if (<= (pow B_m 2.0) 1e-15)
     (/
      (sqrt (* (* 2.0 (* (- (pow B_m 2.0) t_0) F)) (+ A A)))
      (- t_0 (pow B_m 2.0)))
     (if (<= (pow B_m 2.0) 2e+260)
       (-
        (sqrt
         (*
          (/ (+ A (- C (hypot B_m (- A C)))) (fma -4.0 (* A C) (pow B_m 2.0)))
          (* 2.0 F))))
       (* (/ (sqrt (- F)) (sqrt B_m)) (- (sqrt 2.0)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = (4.0 * A) * C;
	double tmp;
	if (pow(B_m, 2.0) <= 1e-15) {
		tmp = sqrt(((2.0 * ((pow(B_m, 2.0) - t_0) * F)) * (A + A))) / (t_0 - pow(B_m, 2.0));
	} else if (pow(B_m, 2.0) <= 2e+260) {
		tmp = -sqrt((((A + (C - hypot(B_m, (A - C)))) / fma(-4.0, (A * C), pow(B_m, 2.0))) * (2.0 * F)));
	} else {
		tmp = (sqrt(-F) / sqrt(B_m)) * -sqrt(2.0);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(Float64(4.0 * A) * C)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e-15)
		tmp = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_0) * F)) * Float64(A + A))) / Float64(t_0 - (B_m ^ 2.0)));
	elseif ((B_m ^ 2.0) <= 2e+260)
		tmp = Float64(-sqrt(Float64(Float64(Float64(A + Float64(C - hypot(B_m, Float64(A - C)))) / fma(-4.0, Float64(A * C), (B_m ^ 2.0))) * Float64(2.0 * F))));
	else
		tmp = Float64(Float64(sqrt(Float64(-F)) / sqrt(B_m)) * Float64(-sqrt(2.0)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-15], N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(A + A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+260], (-N[Sqrt[N[(N[(N[(A + N[(C - N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-4.0 * N[(A * C), $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), N[(N[(N[Sqrt[(-F)], $MachinePrecision] / N[Sqrt[B$95$m], $MachinePrecision]), $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B 2) < 1.0000000000000001e-15

   1. Initial program 22.8%

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

   3. Taylor expanded in C around inf 22.7%

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

      1. sub-neg 22.7%

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

      2. mul-1-neg 22.7%

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

      3. remove-double-neg 22.7%

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

   5. Simplified 22.7%

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

   if 1.0000000000000001e-15 < (pow.f64 B 2) < 2.00000000000000013e260

   1. Initial program 20.7%

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

   3. Taylor expanded in F around 0 28.7%

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

      1. mul-1-neg 28.7%

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

      2. distribute-rgt-neg-in 28.7%

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

   5. Simplified 48.7%

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

      1. pow1 48.7%

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

   7. Applied egg-rr 48.4%

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

      1. unpow1 48.4%

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

      2. unpow1/2 48.4%

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

      3. associate-*l* 48.4%

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

      4. associate--l+ 49.0%

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

   9. Simplified 49.0%

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

   if 2.00000000000000013e260 < (pow.f64 B 2)

   1. Initial program 1.4%

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

   3. Taylor expanded in F around 0 3.9%

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

      1. mul-1-neg 3.9%

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

      2. distribute-rgt-neg-in 3.9%

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

   5. Simplified 7.7%

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

   6. Taylor expanded in B around inf 33.9%

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

      1. associate-*r/ 33.8%

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

      2. sqrt-div 38.4%

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

   8. Applied egg-rr 38.4%

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

      1. *-commutative 38.4%

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

      2. neg-mul-1 38.4%

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

   10. Simplified 38.4%

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

4. Final simplification 32.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{-15}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + A\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{+260}:\\ \;\;\;\;-\sqrt{\frac{A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)}{\mathsf{fma}\left(-4, A \cdot C, {B}^{2}\right)} \cdot \left(2 \cdot F\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{-F}}{\sqrt{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 53.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ \mathbf{if}\;{B\_m}^{2} \leq 200:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_0\right) \cdot F\right)\right) \cdot \left(A + A\right)}}{t\_0 - {B\_m}^{2}}\\ \mathbf{elif}\;{B\_m}^{2} \leq 5 \cdot 10^{+295}:\\ \;\;\;\;-\sqrt{\left(2 \cdot F\right) \cdot \frac{A - \mathsf{hypot}\left(B\_m, A\right)}{{B\_m}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{-F}}{\sqrt{B\_m}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \end{array} \]

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 (* (* 4.0 A) C)))
   (if (<= (pow B_m 2.0) 200.0)
     (/
      (sqrt (* (* 2.0 (* (- (pow B_m 2.0) t_0) F)) (+ A A)))
      (- t_0 (pow B_m 2.0)))
     (if (<= (pow B_m 2.0) 5e+295)
       (- (sqrt (* (* 2.0 F) (/ (- A (hypot B_m A)) (pow B_m 2.0)))))
       (* (/ (sqrt (- F)) (sqrt B_m)) (- (sqrt 2.0)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = (4.0 * A) * C;
	double tmp;
	if (pow(B_m, 2.0) <= 200.0) {
		tmp = sqrt(((2.0 * ((pow(B_m, 2.0) - t_0) * F)) * (A + A))) / (t_0 - pow(B_m, 2.0));
	} else if (pow(B_m, 2.0) <= 5e+295) {
		tmp = -sqrt(((2.0 * F) * ((A - hypot(B_m, A)) / pow(B_m, 2.0))));
	} else {
		tmp = (sqrt(-F) / sqrt(B_m)) * -sqrt(2.0);
	}
	return tmp;
}

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double t_0 = (4.0 * A) * C;
	double tmp;
	if (Math.pow(B_m, 2.0) <= 200.0) {
		tmp = Math.sqrt(((2.0 * ((Math.pow(B_m, 2.0) - t_0) * F)) * (A + A))) / (t_0 - Math.pow(B_m, 2.0));
	} else if (Math.pow(B_m, 2.0) <= 5e+295) {
		tmp = -Math.sqrt(((2.0 * F) * ((A - Math.hypot(B_m, A)) / Math.pow(B_m, 2.0))));
	} else {
		tmp = (Math.sqrt(-F) / Math.sqrt(B_m)) * -Math.sqrt(2.0);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	t_0 = (4.0 * A) * C
	tmp = 0
	if math.pow(B_m, 2.0) <= 200.0:
		tmp = math.sqrt(((2.0 * ((math.pow(B_m, 2.0) - t_0) * F)) * (A + A))) / (t_0 - math.pow(B_m, 2.0))
	elif math.pow(B_m, 2.0) <= 5e+295:
		tmp = -math.sqrt(((2.0 * F) * ((A - math.hypot(B_m, A)) / math.pow(B_m, 2.0))))
	else:
		tmp = (math.sqrt(-F) / math.sqrt(B_m)) * -math.sqrt(2.0)
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(Float64(4.0 * A) * C)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 200.0)
		tmp = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_0) * F)) * Float64(A + A))) / Float64(t_0 - (B_m ^ 2.0)));
	elseif ((B_m ^ 2.0) <= 5e+295)
		tmp = Float64(-sqrt(Float64(Float64(2.0 * F) * Float64(Float64(A - hypot(B_m, A)) / (B_m ^ 2.0)))));
	else
		tmp = Float64(Float64(sqrt(Float64(-F)) / sqrt(B_m)) * Float64(-sqrt(2.0)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	t_0 = (4.0 * A) * C;
	tmp = 0.0;
	if ((B_m ^ 2.0) <= 200.0)
		tmp = sqrt(((2.0 * (((B_m ^ 2.0) - t_0) * F)) * (A + A))) / (t_0 - (B_m ^ 2.0));
	elseif ((B_m ^ 2.0) <= 5e+295)
		tmp = -sqrt(((2.0 * F) * ((A - hypot(B_m, A)) / (B_m ^ 2.0))));
	else
		tmp = (sqrt(-F) / sqrt(B_m)) * -sqrt(2.0);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 200.0], N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(A + A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e+295], (-N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] * N[(N[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision] / N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), N[(N[(N[Sqrt[(-F)], $MachinePrecision] / N[Sqrt[B$95$m], $MachinePrecision]), $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B 2) < 200

   1. Initial program 22.0%

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

   3. Taylor expanded in C around inf 21.8%

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

      1. sub-neg 21.8%

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

      2. mul-1-neg 21.8%

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

      3. remove-double-neg 21.8%

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

   5. Simplified 21.8%

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

   if 200 < (pow.f64 B 2) < 4.99999999999999991e295

   1. Initial program 22.6%

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

   3. Taylor expanded in C around 0 16.9%

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

      1. mul-1-neg 16.9%

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

      2. +-commutative 16.9%

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

   5. Simplified 16.9%

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

      1. add-cbrt-cube 15.2%

         \[\leadsto -\color{blue}{\sqrt[3]{\left(\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{{B}^{2} + {A}^{2}}\right)}\right) \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{{B}^{2} + {A}^{2}}\right)}\right)\right) \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{{B}^{2} + {A}^{2}}\right)}\right)}} \]

      2. pow3 15.2%

         \[\leadsto -\sqrt[3]{\color{blue}{{\left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{{B}^{2} + {A}^{2}}\right)}\right)}^{3}}} \]

   7. Applied egg-rr 15.8%

      \[\leadsto -\color{blue}{\sqrt[3]{{\left(\frac{{\left(2 \cdot \left(F \cdot \left(A - \mathsf{hypot}\left(A, B\right)\right)\right)\right)}^{0.5}}{B}\right)}^{3}}} \]
   8. Step-by-step derivation

      1. rem-cbrt-cube 19.4%

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

      2. add-sqr-sqrt 18.6%

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

      3. sqrt-unprod 37.9%

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

      4. frac-times 37.9%

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

      5. unpow1/2 37.9%

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

      6. unpow1/2 37.9%

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

      7. add-sqr-sqrt 38.0%

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

      8. associate-*r* 38.0%

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

      9. pow2 38.0%

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

   9. Applied egg-rr 38.0%

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

       1. associate-/l* 47.0%

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

       2. *-commutative 47.0%

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

       3. hypot-undefine 43.0%

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

       4. unpow2 43.0%

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

       5. unpow2 43.0%

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

       6. +-commutative 43.0%

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

       7. unpow2 43.0%

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

       8. unpow2 43.0%

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

       9. hypot-define 47.0%

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

   11. Simplified 47.0%

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

   if 4.99999999999999991e295 < (pow.f64 B 2)

   1. Initial program 0.0%

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

   3. Taylor expanded in F around 0 0.0%

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

      1. mul-1-neg 0.0%

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

      2. distribute-rgt-neg-in 0.0%

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

   5. Simplified 4.1%

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

   6. Taylor expanded in B around inf 34.6%

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

      1. associate-*r/ 34.6%

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

      2. sqrt-div 39.5%

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

   8. Applied egg-rr 39.5%

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

      1. *-commutative 39.5%

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

      2. neg-mul-1 39.5%

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

   10. Simplified 39.5%

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

4. Final simplification 32.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 200:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + A\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;{B}^{2} \leq 5 \cdot 10^{+295}:\\ \;\;\;\;-\sqrt{\left(2 \cdot F\right) \cdot \frac{A - \mathsf{hypot}\left(B, A\right)}{{B}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{-F}}{\sqrt{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 47.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 4 \cdot 10^{+146}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{F \cdot \frac{-0.5}{C}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{-F}}{\sqrt{B\_m}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 4e+146)
   (* (sqrt 2.0) (- (sqrt (* F (/ -0.5 C)))))
   (* (/ (sqrt (- F)) (sqrt B_m)) (- (sqrt 2.0)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (pow(B_m, 2.0) <= 4e+146) {
		tmp = sqrt(2.0) * -sqrt((F * (-0.5 / C)));
	} else {
		tmp = (sqrt(-F) / sqrt(B_m)) * -sqrt(2.0);
	}
	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 ** 2.0d0) <= 4d+146) then
        tmp = sqrt(2.0d0) * -sqrt((f * ((-0.5d0) / c)))
    else
        tmp = (sqrt(-f) / sqrt(b_m)) * -sqrt(2.0d0)
    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 (Math.pow(B_m, 2.0) <= 4e+146) {
		tmp = Math.sqrt(2.0) * -Math.sqrt((F * (-0.5 / C)));
	} else {
		tmp = (Math.sqrt(-F) / Math.sqrt(B_m)) * -Math.sqrt(2.0);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if math.pow(B_m, 2.0) <= 4e+146:
		tmp = math.sqrt(2.0) * -math.sqrt((F * (-0.5 / C)))
	else:
		tmp = (math.sqrt(-F) / math.sqrt(B_m)) * -math.sqrt(2.0)
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 4e+146)
		tmp = Float64(sqrt(2.0) * Float64(-sqrt(Float64(F * Float64(-0.5 / C)))));
	else
		tmp = Float64(Float64(sqrt(Float64(-F)) / sqrt(B_m)) * Float64(-sqrt(2.0)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if ((B_m ^ 2.0) <= 4e+146)
		tmp = sqrt(2.0) * -sqrt((F * (-0.5 / C)));
	else
		tmp = (sqrt(-F) / sqrt(B_m)) * -sqrt(2.0);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 4e+146], N[(N[Sqrt[2.0], $MachinePrecision] * (-N[Sqrt[N[(F * N[(-0.5 / C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(N[(N[Sqrt[(-F)], $MachinePrecision] / N[Sqrt[B$95$m], $MachinePrecision]), $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B 2) < 3.99999999999999973e146

   1. Initial program 24.0%

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

   3. Taylor expanded in F around 0 18.2%

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

      1. mul-1-neg 18.2%

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

      2. distribute-rgt-neg-in 18.2%

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

   5. Simplified 27.5%

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

   6. Taylor expanded in A around -inf 21.3%

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

   if 3.99999999999999973e146 < (pow.f64 B 2)

   1. Initial program 3.6%

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

   3. Taylor expanded in F around 0 10.2%

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

      1. mul-1-neg 10.2%

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

      2. distribute-rgt-neg-in 10.2%

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

   5. Simplified 20.9%

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

   6. Taylor expanded in B around inf 30.1%

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

      1. associate-*r/ 30.1%

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

      2. sqrt-div 33.5%

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

   8. Applied egg-rr 33.5%

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

      1. *-commutative 33.5%

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

      2. neg-mul-1 33.5%

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

   10. Simplified 33.5%

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

4. Final simplification 26.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 4 \cdot 10^{+146}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{F \cdot \frac{-0.5}{C}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{-F}}{\sqrt{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 40.3% accurate, 3.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 8.8 \cdot 10^{+75}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{F \cdot \frac{-0.5}{C}}\right)\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{2 \cdot \frac{F}{-B\_m}}\\ \end{array} \end{array} \]

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 8.8e+75)
   (* (sqrt 2.0) (- (sqrt (* F (/ -0.5 C)))))
   (- (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) {
	double tmp;
	if (B_m <= 8.8e+75) {
		tmp = sqrt(2.0) * -sqrt((F * (-0.5 / C)));
	} else {
		tmp = -sqrt((2.0 * (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 <= 8.8d+75) then
        tmp = sqrt(2.0d0) * -sqrt((f * ((-0.5d0) / c)))
    else
        tmp = -sqrt((2.0d0 * (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 <= 8.8e+75) {
		tmp = Math.sqrt(2.0) * -Math.sqrt((F * (-0.5 / C)));
	} else {
		tmp = -Math.sqrt((2.0 * (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 <= 8.8e+75:
		tmp = math.sqrt(2.0) * -math.sqrt((F * (-0.5 / C)))
	else:
		tmp = -math.sqrt((2.0 * (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 <= 8.8e+75)
		tmp = Float64(sqrt(2.0) * Float64(-sqrt(Float64(F * Float64(-0.5 / C)))));
	else
		tmp = Float64(-sqrt(Float64(2.0 * Float64(F / 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 <= 8.8e+75)
		tmp = sqrt(2.0) * -sqrt((F * (-0.5 / C)));
	else
		tmp = -sqrt((2.0 * (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, 8.8e+75], N[(N[Sqrt[2.0], $MachinePrecision] * (-N[Sqrt[N[(F * N[(-0.5 / C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], (-N[Sqrt[N[(2.0 * N[(F / (-B$95$m)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if B < 8.80000000000000048e75

   1. Initial program 18.9%

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

   3. Taylor expanded in F around 0 17.0%

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

      1. mul-1-neg 17.0%

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

      2. distribute-rgt-neg-in 17.0%

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

   5. Simplified 26.0%

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

   6. Taylor expanded in A around -inf 17.1%

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

   if 8.80000000000000048e75 < B

   1. Initial program 2.7%

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

   3. Taylor expanded in F around 0 6.7%

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

      1. mul-1-neg 6.7%

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

      2. distribute-rgt-neg-in 6.7%

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

   5. Simplified 19.8%

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

   6. Taylor expanded in B around inf 62.1%

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

      1. distribute-rgt-neg-out 62.1%

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

      2. sqrt-unprod 62.3%

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

   8. Applied egg-rr 62.3%

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

      1. *-commutative 62.3%

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

      2. associate-*r/ 62.3%

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

      3. *-commutative 62.3%

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

      4. associate-*r/ 62.3%

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

      5. mul-1-neg 62.3%

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

      6. distribute-frac-neg2 62.3%

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

   10. Simplified 62.3%

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

4. Final simplification 25.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 8.8 \cdot 10^{+75}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{F \cdot \frac{-0.5}{C}}\right)\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{2 \cdot \frac{F}{-B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 26.9% accurate, 5.9× speedup

Math

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

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 Float64(-sqrt(Float64(2.0 * Float64(F / Float64(-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[(2.0 * N[(F / (-B$95$m)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])

Derivation

1. Initial program 15.8%

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

3. Taylor expanded in F around 0 15.0%

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

   1. mul-1-neg 15.0%

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

   2. distribute-rgt-neg-in 15.0%

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

5. Simplified 24.9%

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

6. Taylor expanded in B around inf 15.5%

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

   1. distribute-rgt-neg-out 15.5%

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

   2. sqrt-unprod 15.5%

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

8. Applied egg-rr 15.5%

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

   1. *-commutative 15.5%

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

   2. associate-*r/ 15.5%

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

   3. *-commutative 15.5%

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

   4. associate-*r/ 15.5%

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

   5. mul-1-neg 15.5%

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

   6. distribute-frac-neg2 15.5%

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

10. Simplified 15.5%

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

11. Final simplification 15.5%

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

Alternative 8: 2.4% accurate, 6.0× speedup

Math

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

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(2.0 * Float64(F / Float64(-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[(2.0 * N[(F / (-B$95$m)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 15.8%

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

3. Taylor expanded in F around 0 15.0%

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

   1. mul-1-neg 15.0%

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

   2. distribute-rgt-neg-in 15.0%

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

5. Simplified 24.9%

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

6. Taylor expanded in B around inf 15.5%

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

   1. add-sqr-sqrt 0.0%

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

   2. sqrt-unprod 1.9%

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

   3. sqr-neg 1.9%

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

   4. sqrt-unprod 1.9%

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

   5. add-sqr-sqrt 1.9%

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

   6. pow1 1.9%

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

   7. sqrt-unprod 1.9%

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

8. Applied egg-rr 1.9%

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

   1. unpow1 1.9%

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

   2. *-commutative 1.9%

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

   3. associate-*r/ 1.9%

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

   4. *-commutative 1.9%

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

   5. associate-*r/ 1.9%

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

   6. mul-1-neg 1.9%

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

   7. distribute-frac-neg2 1.9%

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

10. Simplified 1.9%

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

11. Final simplification 1.9%

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

Reproduce

herbie shell --seed 2024046 
(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))))