ABCF->ab-angle a

Percentage Accurate: 18.5% → 58.9%

Time: 37.4s

Alternatives: 16

Speedup: 6.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 18.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 58.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := -0.5 \cdot \frac{{B\_m}^{2}}{A}\\ t_1 := \left(4 \cdot A\right) \cdot C\\ t_2 := 2 \cdot \left(\left({B\_m}^{2} - t\_1\right) \cdot F\right)\\ t_3 := t\_1 - {B\_m}^{2}\\ t_4 := \frac{\sqrt{t\_2 \cdot \left(\left(A + C\right) + \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_3}\\ \mathbf{if}\;t\_4 \leq -\infty:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(2, C, t\_0\right)} \cdot \left(\sqrt{2 \cdot \mathsf{fma}\left(C, A \cdot -4, {B\_m}^{2}\right)} \cdot \sqrt{F}\right)}{t\_3}\\ \mathbf{elif}\;t\_4 \leq -5 \cdot 10^{-219}:\\ \;\;\;\;\frac{\sqrt{t\_2 \cdot \mathsf{fma}\left({\left(\sqrt[3]{A}\right)}^{2}, \sqrt[3]{A}, C + \mathsf{hypot}\left(A - C, B\_m\right)\right)}}{t\_3}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;\frac{\sqrt{t\_2 \cdot \left(t\_0 + 2 \cdot C\right)}}{t\_1 - B\_m \cdot B\_m}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot F} \cdot \left(-{B\_m}^{-0.5}\right)\\ \end{array} \end{array} \]

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 (* -0.5 (/ (pow B_m 2.0) A)))
        (t_1 (* (* 4.0 A) C))
        (t_2 (* 2.0 (* (- (pow B_m 2.0) t_1) F)))
        (t_3 (- t_1 (pow B_m 2.0)))
        (t_4
         (/
          (sqrt (* t_2 (+ (+ A C) (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))))))
          t_3)))
   (if (<= t_4 (- INFINITY))
     (/
      (*
       (sqrt (fma 2.0 C t_0))
       (* (sqrt (* 2.0 (fma C (* A -4.0) (pow B_m 2.0)))) (sqrt F)))
      t_3)
     (if (<= t_4 -5e-219)
       (/
        (sqrt
         (* t_2 (fma (pow (cbrt A) 2.0) (cbrt A) (+ C (hypot (- A C) B_m)))))
        t_3)
       (if (<= t_4 INFINITY)
         (/ (sqrt (* t_2 (+ t_0 (* 2.0 C)))) (- t_1 (* B_m B_m)))
         (* (sqrt (* 2.0 F)) (- (pow 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) {
	double t_0 = -0.5 * (pow(B_m, 2.0) / A);
	double t_1 = (4.0 * A) * C;
	double t_2 = 2.0 * ((pow(B_m, 2.0) - t_1) * F);
	double t_3 = t_1 - pow(B_m, 2.0);
	double t_4 = sqrt((t_2 * ((A + C) + sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / t_3;
	double tmp;
	if (t_4 <= -((double) INFINITY)) {
		tmp = (sqrt(fma(2.0, C, t_0)) * (sqrt((2.0 * fma(C, (A * -4.0), pow(B_m, 2.0)))) * sqrt(F))) / t_3;
	} else if (t_4 <= -5e-219) {
		tmp = sqrt((t_2 * fma(pow(cbrt(A), 2.0), cbrt(A), (C + hypot((A - C), B_m))))) / t_3;
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = sqrt((t_2 * (t_0 + (2.0 * C)))) / (t_1 - (B_m * B_m));
	} else {
		tmp = sqrt((2.0 * F)) * -pow(B_m, -0.5);
	}
	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(-0.5 * Float64((B_m ^ 2.0) / A))
	t_1 = Float64(Float64(4.0 * A) * C)
	t_2 = Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_1) * F))
	t_3 = Float64(t_1 - (B_m ^ 2.0))
	t_4 = Float64(sqrt(Float64(t_2 * Float64(Float64(A + C) + sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / t_3)
	tmp = 0.0
	if (t_4 <= Float64(-Inf))
		tmp = Float64(Float64(sqrt(fma(2.0, C, t_0)) * Float64(sqrt(Float64(2.0 * fma(C, Float64(A * -4.0), (B_m ^ 2.0)))) * sqrt(F))) / t_3);
	elseif (t_4 <= -5e-219)
		tmp = Float64(sqrt(Float64(t_2 * fma((cbrt(A) ^ 2.0), cbrt(A), Float64(C + hypot(Float64(A - C), B_m))))) / t_3);
	elseif (t_4 <= Inf)
		tmp = Float64(sqrt(Float64(t_2 * Float64(t_0 + Float64(2.0 * C)))) / Float64(t_1 - Float64(B_m * B_m)));
	else
		tmp = Float64(sqrt(Float64(2.0 * F)) * Float64(-(B_m ^ -0.5)));
	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[(-0.5 * N[(N[Power[B$95$m, 2.0], $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$1), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[N[(t$95$2 * 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] / t$95$3), $MachinePrecision]}, If[LessEqual[t$95$4, (-Infinity)], N[(N[(N[Sqrt[N[(2.0 * C + t$95$0), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(2.0 * N[(C * N[(A * -4.0), $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$3), $MachinePrecision], If[LessEqual[t$95$4, -5e-219], N[(N[Sqrt[N[(t$95$2 * N[(N[Power[N[Power[A, 1/3], $MachinePrecision], 2.0], $MachinePrecision] * N[Power[A, 1/3], $MachinePrecision] + N[(C + N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$3), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[(N[Sqrt[N[(t$95$2 * N[(t$95$0 + N[(2.0 * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$1 - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] * (-N[Power[B$95$m, -0.5], $MachinePrecision])), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -inf.0

   1. Initial program 3.0%

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

   3. Taylor expanded in A around -inf 21.8%

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

      1. pow1/2 22.6%

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

      2. *-commutative 22.6%

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

      3. unpow-prod-down 33.6%

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

      4. pow1/2 33.4%

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

      5. +-commutative 33.4%

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

      6. fma-define 33.4%

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

      7. pow1/2 33.4%

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

      8. associate-*r* 33.4%

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

      9. associate-*l* 33.4%

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

   5. Applied egg-rr 33.4%

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

      1. sqrt-prod 35.6%

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

      2. cancel-sign-sub-inv 35.6%

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

      3. metadata-eval 35.6%

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

      4. associate-*l* 35.6%

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

      5. +-commutative 35.6%

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

      6. *-commutative 35.6%

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

      7. fma-define 35.6%

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

      8. *-commutative 35.6%

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

   7. Applied egg-rr 35.6%

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

   if -inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -5.0000000000000002e-219

   1. Initial program 98.4%

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

      1. associate-+l+ 98.5%

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

      2. add-cube-cbrt 98.4%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\color{blue}{\left(\sqrt[3]{A} \cdot \sqrt[3]{A}\right) \cdot \sqrt[3]{A}} + \left(C + \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      3. +-commutative 98.4%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(\sqrt[3]{A} \cdot \sqrt[3]{A}\right) \cdot \sqrt[3]{A} + \left(C + \sqrt{\color{blue}{{B}^{2} + {\left(A - C\right)}^{2}}}\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      4. unpow2 98.4%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(\sqrt[3]{A} \cdot \sqrt[3]{A}\right) \cdot \sqrt[3]{A} + \left(C + \sqrt{\color{blue}{B \cdot B} + {\left(A - C\right)}^{2}}\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      5. unpow2 98.4%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(\sqrt[3]{A} \cdot \sqrt[3]{A}\right) \cdot \sqrt[3]{A} + \left(C + \sqrt{B \cdot B + \color{blue}{\left(A - C\right) \cdot \left(A - C\right)}}\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      6. hypot-undefine 98.4%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(\sqrt[3]{A} \cdot \sqrt[3]{A}\right) \cdot \sqrt[3]{A} + \left(C + \color{blue}{\mathsf{hypot}\left(B, A - C\right)}\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      7. fma-define 98.4%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\mathsf{fma}\left(\sqrt[3]{A} \cdot \sqrt[3]{A}, \sqrt[3]{A}, C + \mathsf{hypot}\left(B, A - C\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      8. pow2 98.4%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left(\color{blue}{{\left(\sqrt[3]{A}\right)}^{2}}, \sqrt[3]{A}, C + \mathsf{hypot}\left(B, A - C\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      9. hypot-undefine 98.4%

         \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left({\left(\sqrt[3]{A}\right)}^{2}, \sqrt[3]{A}, C + \color{blue}{\sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      10. unpow2 98.4%

          \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left({\left(\sqrt[3]{A}\right)}^{2}, \sqrt[3]{A}, C + \sqrt{\color{blue}{{B}^{2}} + \left(A - C\right) \cdot \left(A - C\right)}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      11. unpow2 98.4%

          \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left({\left(\sqrt[3]{A}\right)}^{2}, \sqrt[3]{A}, C + \sqrt{{B}^{2} + \color{blue}{{\left(A - C\right)}^{2}}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      12. +-commutative 98.4%

          \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left({\left(\sqrt[3]{A}\right)}^{2}, \sqrt[3]{A}, C + \sqrt{\color{blue}{{\left(A - C\right)}^{2} + {B}^{2}}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      13. unpow2 98.4%

          \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left({\left(\sqrt[3]{A}\right)}^{2}, \sqrt[3]{A}, C + \sqrt{\color{blue}{\left(A - C\right) \cdot \left(A - C\right)} + {B}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      14. unpow2 98.4%

          \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left({\left(\sqrt[3]{A}\right)}^{2}, \sqrt[3]{A}, C + \sqrt{\left(A - C\right) \cdot \left(A - C\right) + \color{blue}{B \cdot B}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

      15. hypot-define 98.4%

          \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left({\left(\sqrt[3]{A}\right)}^{2}, \sqrt[3]{A}, C + \color{blue}{\mathsf{hypot}\left(A - C, B\right)}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   4. Applied egg-rr 98.4%

      \[\leadsto \frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \color{blue}{\mathsf{fma}\left({\left(\sqrt[3]{A}\right)}^{2}, \sqrt[3]{A}, C + \mathsf{hypot}\left(A - C, B\right)\right)}}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \]

   if -5.0000000000000002e-219 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < +inf.0

   1. Initial program 11.6%

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

   3. Taylor expanded in A around -inf 30.5%

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

      1. unpow2 30.5%

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

   5. Applied egg-rr 30.5%

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

   if +inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) 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 B around inf 21.1%

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

      1. mul-1-neg 21.1%

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

      2. *-commutative 21.1%

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

   5. Simplified 21.1%

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

      1. pow1/2 21.2%

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

      2. div-inv 21.2%

         \[\leadsto -\sqrt{2} \cdot {\color{blue}{\left(F \cdot \frac{1}{B}\right)}}^{0.5} \]

      3. unpow-prod-down 26.8%

         \[\leadsto -\sqrt{2} \cdot \color{blue}{\left({F}^{0.5} \cdot {\left(\frac{1}{B}\right)}^{0.5}\right)} \]

      4. pow1/2 26.8%

         \[\leadsto -\sqrt{2} \cdot \left(\color{blue}{\sqrt{F}} \cdot {\left(\frac{1}{B}\right)}^{0.5}\right) \]

   7. Applied egg-rr 26.8%

      \[\leadsto -\sqrt{2} \cdot \color{blue}{\left(\sqrt{F} \cdot {\left(\frac{1}{B}\right)}^{0.5}\right)} \]
   8. Step-by-step derivation

      1. unpow1/2 26.8%

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

   9. Simplified 26.8%

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

       1. neg-sub0 26.8%

          \[\leadsto \color{blue}{0 - \sqrt{2} \cdot \left(\sqrt{F} \cdot \sqrt{\frac{1}{B}}\right)} \]

       2. associate-*r* 26.8%

          \[\leadsto 0 - \color{blue}{\left(\sqrt{2} \cdot \sqrt{F}\right) \cdot \sqrt{\frac{1}{B}}} \]

       3. pow1/2 26.8%

          \[\leadsto 0 - \left(\color{blue}{{2}^{0.5}} \cdot \sqrt{F}\right) \cdot \sqrt{\frac{1}{B}} \]

       4. pow1/2 26.8%

          \[\leadsto 0 - \left({2}^{0.5} \cdot \color{blue}{{F}^{0.5}}\right) \cdot \sqrt{\frac{1}{B}} \]

       5. pow-prod-down 26.8%

          \[\leadsto 0 - \color{blue}{{\left(2 \cdot F\right)}^{0.5}} \cdot \sqrt{\frac{1}{B}} \]

       6. pow1/2 26.8%

          \[\leadsto 0 - {\left(2 \cdot F\right)}^{0.5} \cdot \color{blue}{{\left(\frac{1}{B}\right)}^{0.5}} \]

       7. inv-pow 26.8%

          \[\leadsto 0 - {\left(2 \cdot F\right)}^{0.5} \cdot {\color{blue}{\left({B}^{-1}\right)}}^{0.5} \]

       8. pow-pow 26.8%

          \[\leadsto 0 - {\left(2 \cdot F\right)}^{0.5} \cdot \color{blue}{{B}^{\left(-1 \cdot 0.5\right)}} \]

       9. metadata-eval 26.8%

          \[\leadsto 0 - {\left(2 \cdot F\right)}^{0.5} \cdot {B}^{\color{blue}{-0.5}} \]

   11. Applied egg-rr 26.8%

       \[\leadsto \color{blue}{0 - {\left(2 \cdot F\right)}^{0.5} \cdot {B}^{-0.5}} \]
   12. Step-by-step derivation

       1. neg-sub0 26.8%

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

       2. *-commutative 26.8%

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

       3. distribute-rgt-neg-in 26.8%

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

       4. unpow1/2 26.8%

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

   13. Simplified 26.8%

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

4. Final simplification 40.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -\infty:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(2, C, -0.5 \cdot \frac{{B}^{2}}{A}\right)} \cdot \left(\sqrt{2 \cdot \mathsf{fma}\left(C, A \cdot -4, {B}^{2}\right)} \cdot \sqrt{F}\right)}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -5 \cdot 10^{-219}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \mathsf{fma}\left({\left(\sqrt[3]{A}\right)}^{2}, \sqrt[3]{A}, C + \mathsf{hypot}\left(A - C, B\right)\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(-0.5 \cdot \frac{{B}^{2}}{A} + 2 \cdot C\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot F} \cdot \left(-{B}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 58.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := -0.5 \cdot \frac{{B\_m}^{2}}{A}\\ t_1 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ t_2 := \left(4 \cdot A\right) \cdot C\\ t_3 := 2 \cdot \left(\left({B\_m}^{2} - t\_2\right) \cdot F\right)\\ t_4 := t\_2 - {B\_m}^{2}\\ t_5 := \frac{\sqrt{t\_3 \cdot \left(\left(A + C\right) + \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_4}\\ \mathbf{if}\;t\_5 \leq -\infty:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(2, C, t\_0\right)} \cdot \left(\sqrt{2 \cdot \mathsf{fma}\left(C, A \cdot -4, {B\_m}^{2}\right)} \cdot \sqrt{F}\right)}{t\_4}\\ \mathbf{elif}\;t\_5 \leq -5 \cdot 10^{-219}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot t\_1\right) \cdot \left(2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B\_m, A - C\right)\right)\right)\right)}}{-t\_1}\\ \mathbf{elif}\;t\_5 \leq \infty:\\ \;\;\;\;\frac{\sqrt{t\_3 \cdot \left(t\_0 + 2 \cdot C\right)}}{t\_2 - B\_m \cdot B\_m}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot F} \cdot \left(-{B\_m}^{-0.5}\right)\\ \end{array} \end{array} \]

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 (* -0.5 (/ (pow B_m 2.0) A)))
        (t_1 (fma B_m B_m (* A (* C -4.0))))
        (t_2 (* (* 4.0 A) C))
        (t_3 (* 2.0 (* (- (pow B_m 2.0) t_2) F)))
        (t_4 (- t_2 (pow B_m 2.0)))
        (t_5
         (/
          (sqrt (* t_3 (+ (+ A C) (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))))))
          t_4)))
   (if (<= t_5 (- INFINITY))
     (/
      (*
       (sqrt (fma 2.0 C t_0))
       (* (sqrt (* 2.0 (fma C (* A -4.0) (pow B_m 2.0)))) (sqrt F)))
      t_4)
     (if (<= t_5 -5e-219)
       (/ (sqrt (* (* F t_1) (* 2.0 (+ A (+ C (hypot B_m (- A C))))))) (- t_1))
       (if (<= t_5 INFINITY)
         (/ (sqrt (* t_3 (+ t_0 (* 2.0 C)))) (- t_2 (* B_m B_m)))
         (* (sqrt (* 2.0 F)) (- (pow 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) {
	double t_0 = -0.5 * (pow(B_m, 2.0) / A);
	double t_1 = fma(B_m, B_m, (A * (C * -4.0)));
	double t_2 = (4.0 * A) * C;
	double t_3 = 2.0 * ((pow(B_m, 2.0) - t_2) * F);
	double t_4 = t_2 - pow(B_m, 2.0);
	double t_5 = sqrt((t_3 * ((A + C) + sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / t_4;
	double tmp;
	if (t_5 <= -((double) INFINITY)) {
		tmp = (sqrt(fma(2.0, C, t_0)) * (sqrt((2.0 * fma(C, (A * -4.0), pow(B_m, 2.0)))) * sqrt(F))) / t_4;
	} else if (t_5 <= -5e-219) {
		tmp = sqrt(((F * t_1) * (2.0 * (A + (C + hypot(B_m, (A - C))))))) / -t_1;
	} else if (t_5 <= ((double) INFINITY)) {
		tmp = sqrt((t_3 * (t_0 + (2.0 * C)))) / (t_2 - (B_m * B_m));
	} else {
		tmp = sqrt((2.0 * F)) * -pow(B_m, -0.5);
	}
	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(-0.5 * Float64((B_m ^ 2.0) / A))
	t_1 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	t_2 = Float64(Float64(4.0 * A) * C)
	t_3 = Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_2) * F))
	t_4 = Float64(t_2 - (B_m ^ 2.0))
	t_5 = Float64(sqrt(Float64(t_3 * Float64(Float64(A + C) + sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / t_4)
	tmp = 0.0
	if (t_5 <= Float64(-Inf))
		tmp = Float64(Float64(sqrt(fma(2.0, C, t_0)) * Float64(sqrt(Float64(2.0 * fma(C, Float64(A * -4.0), (B_m ^ 2.0)))) * sqrt(F))) / t_4);
	elseif (t_5 <= -5e-219)
		tmp = Float64(sqrt(Float64(Float64(F * t_1) * Float64(2.0 * Float64(A + Float64(C + hypot(B_m, Float64(A - C))))))) / Float64(-t_1));
	elseif (t_5 <= Inf)
		tmp = Float64(sqrt(Float64(t_3 * Float64(t_0 + Float64(2.0 * C)))) / Float64(t_2 - Float64(B_m * B_m)));
	else
		tmp = Float64(sqrt(Float64(2.0 * F)) * Float64(-(B_m ^ -0.5)));
	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[(-0.5 * N[(N[Power[B$95$m, 2.0], $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$3 = N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$2), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$2 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[Sqrt[N[(t$95$3 * 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] / t$95$4), $MachinePrecision]}, If[LessEqual[t$95$5, (-Infinity)], N[(N[(N[Sqrt[N[(2.0 * C + t$95$0), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(2.0 * N[(C * N[(A * -4.0), $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$4), $MachinePrecision], If[LessEqual[t$95$5, -5e-219], N[(N[Sqrt[N[(N[(F * t$95$1), $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$1)), $MachinePrecision], If[LessEqual[t$95$5, Infinity], N[(N[Sqrt[N[(t$95$3 * N[(t$95$0 + N[(2.0 * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$2 - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] * (-N[Power[B$95$m, -0.5], $MachinePrecision])), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -inf.0

   1. Initial program 3.0%

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

   3. Taylor expanded in A around -inf 21.8%

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

      1. pow1/2 22.6%

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

      2. *-commutative 22.6%

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

      3. unpow-prod-down 33.6%

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

      4. pow1/2 33.4%

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

      5. +-commutative 33.4%

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

      6. fma-define 33.4%

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

      7. pow1/2 33.4%

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

      8. associate-*r* 33.4%

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

      9. associate-*l* 33.4%

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

   5. Applied egg-rr 33.4%

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

      1. sqrt-prod 35.6%

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

      2. cancel-sign-sub-inv 35.6%

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

      3. metadata-eval 35.6%

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

      4. associate-*l* 35.6%

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

      5. +-commutative 35.6%

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

      6. *-commutative 35.6%

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

      7. fma-define 35.6%

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

      8. *-commutative 35.6%

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

   7. Applied egg-rr 35.6%

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

   if -inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -5.0000000000000002e-219

   1. Initial program 98.4%

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

   3. Simplified 98.5%

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

   if -5.0000000000000002e-219 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < +inf.0

   1. Initial program 11.6%

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

   3. Taylor expanded in A around -inf 30.5%

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

      1. unpow2 30.5%

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

   5. Applied egg-rr 30.5%

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

   if +inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) 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 B around inf 21.1%

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

      1. mul-1-neg 21.1%

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

      2. *-commutative 21.1%

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

   5. Simplified 21.1%

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

      1. pow1/2 21.2%

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

      2. div-inv 21.2%

         \[\leadsto -\sqrt{2} \cdot {\color{blue}{\left(F \cdot \frac{1}{B}\right)}}^{0.5} \]

      3. unpow-prod-down 26.8%

         \[\leadsto -\sqrt{2} \cdot \color{blue}{\left({F}^{0.5} \cdot {\left(\frac{1}{B}\right)}^{0.5}\right)} \]

      4. pow1/2 26.8%

         \[\leadsto -\sqrt{2} \cdot \left(\color{blue}{\sqrt{F}} \cdot {\left(\frac{1}{B}\right)}^{0.5}\right) \]

   7. Applied egg-rr 26.8%

      \[\leadsto -\sqrt{2} \cdot \color{blue}{\left(\sqrt{F} \cdot {\left(\frac{1}{B}\right)}^{0.5}\right)} \]
   8. Step-by-step derivation

      1. unpow1/2 26.8%

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

   9. Simplified 26.8%

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

       1. neg-sub0 26.8%

          \[\leadsto \color{blue}{0 - \sqrt{2} \cdot \left(\sqrt{F} \cdot \sqrt{\frac{1}{B}}\right)} \]

       2. associate-*r* 26.8%

          \[\leadsto 0 - \color{blue}{\left(\sqrt{2} \cdot \sqrt{F}\right) \cdot \sqrt{\frac{1}{B}}} \]

       3. pow1/2 26.8%

          \[\leadsto 0 - \left(\color{blue}{{2}^{0.5}} \cdot \sqrt{F}\right) \cdot \sqrt{\frac{1}{B}} \]

       4. pow1/2 26.8%

          \[\leadsto 0 - \left({2}^{0.5} \cdot \color{blue}{{F}^{0.5}}\right) \cdot \sqrt{\frac{1}{B}} \]

       5. pow-prod-down 26.8%

          \[\leadsto 0 - \color{blue}{{\left(2 \cdot F\right)}^{0.5}} \cdot \sqrt{\frac{1}{B}} \]

       6. pow1/2 26.8%

          \[\leadsto 0 - {\left(2 \cdot F\right)}^{0.5} \cdot \color{blue}{{\left(\frac{1}{B}\right)}^{0.5}} \]

       7. inv-pow 26.8%

          \[\leadsto 0 - {\left(2 \cdot F\right)}^{0.5} \cdot {\color{blue}{\left({B}^{-1}\right)}}^{0.5} \]

       8. pow-pow 26.8%

          \[\leadsto 0 - {\left(2 \cdot F\right)}^{0.5} \cdot \color{blue}{{B}^{\left(-1 \cdot 0.5\right)}} \]

       9. metadata-eval 26.8%

          \[\leadsto 0 - {\left(2 \cdot F\right)}^{0.5} \cdot {B}^{\color{blue}{-0.5}} \]

   11. Applied egg-rr 26.8%

       \[\leadsto \color{blue}{0 - {\left(2 \cdot F\right)}^{0.5} \cdot {B}^{-0.5}} \]
   12. Step-by-step derivation

       1. neg-sub0 26.8%

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

       2. *-commutative 26.8%

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

       3. distribute-rgt-neg-in 26.8%

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

       4. unpow1/2 26.8%

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

   13. Simplified 26.8%

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

4. Final simplification 40.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -\infty:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(2, C, -0.5 \cdot \frac{{B}^{2}}{A}\right)} \cdot \left(\sqrt{2 \cdot \mathsf{fma}\left(C, A \cdot -4, {B}^{2}\right)} \cdot \sqrt{F}\right)}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -5 \cdot 10^{-219}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot \left(2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(-0.5 \cdot \frac{{B}^{2}}{A} + 2 \cdot C\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot F} \cdot \left(-{B}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 56.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ t_1 := 2 \cdot \left(\left({B\_m}^{2} - t\_0\right) \cdot F\right)\\ t_2 := t\_0 - {B\_m}^{2}\\ t_3 := \frac{\sqrt{t\_1 \cdot \left(\left(A + C\right) + \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_2}\\ t_4 := -0.5 \cdot \frac{{B\_m}^{2}}{A}\\ t_5 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(2, C, t\_4\right)} \cdot \sqrt{F \cdot \left(2 \cdot \left({B\_m}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right)}}{t\_2}\\ \mathbf{elif}\;t\_3 \leq -5 \cdot 10^{-219}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot t\_5\right) \cdot \left(2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B\_m, A - C\right)\right)\right)\right)}}{-t\_5}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\frac{\sqrt{t\_1 \cdot \left(t\_4 + 2 \cdot C\right)}}{t\_0 - B\_m \cdot B\_m}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot F} \cdot \left(-{B\_m}^{-0.5}\right)\\ \end{array} \end{array} \]

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))
        (t_1 (* 2.0 (* (- (pow B_m 2.0) t_0) F)))
        (t_2 (- t_0 (pow B_m 2.0)))
        (t_3
         (/
          (sqrt (* t_1 (+ (+ A C) (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))))))
          t_2))
        (t_4 (* -0.5 (/ (pow B_m 2.0) A)))
        (t_5 (fma B_m B_m (* A (* C -4.0)))))
   (if (<= t_3 (- INFINITY))
     (/
      (*
       (sqrt (fma 2.0 C t_4))
       (sqrt (* F (* 2.0 (- (pow B_m 2.0) (* 4.0 (* A C)))))))
      t_2)
     (if (<= t_3 -5e-219)
       (/ (sqrt (* (* F t_5) (* 2.0 (+ A (+ C (hypot B_m (- A C))))))) (- t_5))
       (if (<= t_3 INFINITY)
         (/ (sqrt (* t_1 (+ t_4 (* 2.0 C)))) (- t_0 (* B_m B_m)))
         (* (sqrt (* 2.0 F)) (- (pow 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) {
	double t_0 = (4.0 * A) * C;
	double t_1 = 2.0 * ((pow(B_m, 2.0) - t_0) * F);
	double t_2 = t_0 - pow(B_m, 2.0);
	double t_3 = sqrt((t_1 * ((A + C) + sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / t_2;
	double t_4 = -0.5 * (pow(B_m, 2.0) / A);
	double t_5 = fma(B_m, B_m, (A * (C * -4.0)));
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = (sqrt(fma(2.0, C, t_4)) * sqrt((F * (2.0 * (pow(B_m, 2.0) - (4.0 * (A * C))))))) / t_2;
	} else if (t_3 <= -5e-219) {
		tmp = sqrt(((F * t_5) * (2.0 * (A + (C + hypot(B_m, (A - C))))))) / -t_5;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt((t_1 * (t_4 + (2.0 * C)))) / (t_0 - (B_m * B_m));
	} else {
		tmp = sqrt((2.0 * F)) * -pow(B_m, -0.5);
	}
	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)
	t_1 = Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_0) * F))
	t_2 = Float64(t_0 - (B_m ^ 2.0))
	t_3 = Float64(sqrt(Float64(t_1 * Float64(Float64(A + C) + sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / t_2)
	t_4 = Float64(-0.5 * Float64((B_m ^ 2.0) / A))
	t_5 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(Float64(sqrt(fma(2.0, C, t_4)) * sqrt(Float64(F * Float64(2.0 * Float64((B_m ^ 2.0) - Float64(4.0 * Float64(A * C))))))) / t_2);
	elseif (t_3 <= -5e-219)
		tmp = Float64(sqrt(Float64(Float64(F * t_5) * Float64(2.0 * Float64(A + Float64(C + hypot(B_m, Float64(A - C))))))) / Float64(-t_5));
	elseif (t_3 <= Inf)
		tmp = Float64(sqrt(Float64(t_1 * Float64(t_4 + Float64(2.0 * C)))) / Float64(t_0 - Float64(B_m * B_m)));
	else
		tmp = Float64(sqrt(Float64(2.0 * F)) * Float64(-(B_m ^ -0.5)));
	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]}, Block[{t$95$1 = N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(t$95$1 * 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] / t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(-0.5 * N[(N[Power[B$95$m, 2.0], $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], N[(N[(N[Sqrt[N[(2.0 * C + t$95$4), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(F * N[(2.0 * N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[t$95$3, -5e-219], N[(N[Sqrt[N[(N[(F * t$95$5), $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$5)), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(N[Sqrt[N[(t$95$1 * N[(t$95$4 + N[(2.0 * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] * (-N[Power[B$95$m, -0.5], $MachinePrecision])), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -inf.0

   1. Initial program 3.0%

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

   3. Taylor expanded in A around -inf 21.8%

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

      1. pow1/2 22.6%

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

      2. *-commutative 22.6%

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

      3. unpow-prod-down 33.6%

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

      4. pow1/2 33.4%

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

      5. +-commutative 33.4%

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

      6. fma-define 33.4%

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

      7. pow1/2 33.4%

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

      8. associate-*r* 33.4%

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

      9. associate-*l* 33.4%

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

   5. Applied egg-rr 33.4%

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

   if -inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -5.0000000000000002e-219

   1. Initial program 98.4%

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

   3. Simplified 98.5%

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

   if -5.0000000000000002e-219 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < +inf.0

   1. Initial program 11.6%

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

   3. Taylor expanded in A around -inf 30.5%

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

      1. unpow2 30.5%

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

   5. Applied egg-rr 30.5%

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

   if +inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) 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 B around inf 21.1%

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

      1. mul-1-neg 21.1%

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

      2. *-commutative 21.1%

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

   5. Simplified 21.1%

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

      1. pow1/2 21.2%

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

      2. div-inv 21.2%

         \[\leadsto -\sqrt{2} \cdot {\color{blue}{\left(F \cdot \frac{1}{B}\right)}}^{0.5} \]

      3. unpow-prod-down 26.8%

         \[\leadsto -\sqrt{2} \cdot \color{blue}{\left({F}^{0.5} \cdot {\left(\frac{1}{B}\right)}^{0.5}\right)} \]

      4. pow1/2 26.8%

         \[\leadsto -\sqrt{2} \cdot \left(\color{blue}{\sqrt{F}} \cdot {\left(\frac{1}{B}\right)}^{0.5}\right) \]

   7. Applied egg-rr 26.8%

      \[\leadsto -\sqrt{2} \cdot \color{blue}{\left(\sqrt{F} \cdot {\left(\frac{1}{B}\right)}^{0.5}\right)} \]
   8. Step-by-step derivation

      1. unpow1/2 26.8%

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

   9. Simplified 26.8%

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

       1. neg-sub0 26.8%

          \[\leadsto \color{blue}{0 - \sqrt{2} \cdot \left(\sqrt{F} \cdot \sqrt{\frac{1}{B}}\right)} \]

       2. associate-*r* 26.8%

          \[\leadsto 0 - \color{blue}{\left(\sqrt{2} \cdot \sqrt{F}\right) \cdot \sqrt{\frac{1}{B}}} \]

       3. pow1/2 26.8%

          \[\leadsto 0 - \left(\color{blue}{{2}^{0.5}} \cdot \sqrt{F}\right) \cdot \sqrt{\frac{1}{B}} \]

       4. pow1/2 26.8%

          \[\leadsto 0 - \left({2}^{0.5} \cdot \color{blue}{{F}^{0.5}}\right) \cdot \sqrt{\frac{1}{B}} \]

       5. pow-prod-down 26.8%

          \[\leadsto 0 - \color{blue}{{\left(2 \cdot F\right)}^{0.5}} \cdot \sqrt{\frac{1}{B}} \]

       6. pow1/2 26.8%

          \[\leadsto 0 - {\left(2 \cdot F\right)}^{0.5} \cdot \color{blue}{{\left(\frac{1}{B}\right)}^{0.5}} \]

       7. inv-pow 26.8%

          \[\leadsto 0 - {\left(2 \cdot F\right)}^{0.5} \cdot {\color{blue}{\left({B}^{-1}\right)}}^{0.5} \]

       8. pow-pow 26.8%

          \[\leadsto 0 - {\left(2 \cdot F\right)}^{0.5} \cdot \color{blue}{{B}^{\left(-1 \cdot 0.5\right)}} \]

       9. metadata-eval 26.8%

          \[\leadsto 0 - {\left(2 \cdot F\right)}^{0.5} \cdot {B}^{\color{blue}{-0.5}} \]

   11. Applied egg-rr 26.8%

       \[\leadsto \color{blue}{0 - {\left(2 \cdot F\right)}^{0.5} \cdot {B}^{-0.5}} \]
   12. Step-by-step derivation

       1. neg-sub0 26.8%

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

       2. *-commutative 26.8%

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

       3. distribute-rgt-neg-in 26.8%

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

       4. unpow1/2 26.8%

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

   13. Simplified 26.8%

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

4. Final simplification 39.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -\infty:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(2, C, -0.5 \cdot \frac{{B}^{2}}{A}\right)} \cdot \sqrt{F \cdot \left(2 \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -5 \cdot 10^{-219}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot \left(2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(-0.5 \cdot \frac{{B}^{2}}{A} + 2 \cdot C\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot F} \cdot \left(-{B}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 53.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ t_1 := 2 \cdot \left(\left({B\_m}^{2} - t\_0\right) \cdot F\right)\\ t_2 := t\_0 - {B\_m}^{2}\\ \mathbf{if}\;{B\_m}^{2} \leq 2 \cdot 10^{-126}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(2 \cdot \left({B\_m}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right)} \cdot \sqrt{2 \cdot C}}{t\_2}\\ \mathbf{elif}\;{B\_m}^{2} \leq 10^{+58}:\\ \;\;\;\;\frac{\sqrt{t\_1 \cdot \left(C + \mathsf{hypot}\left(B\_m, C\right)\right)}}{t\_2}\\ \mathbf{elif}\;{B\_m}^{2} \leq 2 \cdot 10^{+148}:\\ \;\;\;\;\frac{\sqrt{t\_1 \cdot \left(-0.5 \cdot \frac{{B\_m}^{2}}{A} + 2 \cdot C\right)}}{t\_0 - B\_m \cdot B\_m}\\ \mathbf{elif}\;{B\_m}^{2} \leq 5 \cdot 10^{+241}:\\ \;\;\;\;\sqrt{2} \cdot \left(\sqrt{F} \cdot \left(-\sqrt{\frac{\left(A + C\right) + \mathsf{hypot}\left(B\_m, A - C\right)}{\mathsf{fma}\left(C, A \cdot -4, {B\_m}^{2}\right)}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot F} \cdot \left(-{B\_m}^{-0.5}\right)\\ \end{array} \end{array} \]

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))
        (t_1 (* 2.0 (* (- (pow B_m 2.0) t_0) F)))
        (t_2 (- t_0 (pow B_m 2.0))))
   (if (<= (pow B_m 2.0) 2e-126)
     (/
      (*
       (sqrt (* F (* 2.0 (- (pow B_m 2.0) (* 4.0 (* A C))))))
       (sqrt (* 2.0 C)))
      t_2)
     (if (<= (pow B_m 2.0) 1e+58)
       (/ (sqrt (* t_1 (+ C (hypot B_m C)))) t_2)
       (if (<= (pow B_m 2.0) 2e+148)
         (/
          (sqrt (* t_1 (+ (* -0.5 (/ (pow B_m 2.0) A)) (* 2.0 C))))
          (- t_0 (* B_m B_m)))
         (if (<= (pow B_m 2.0) 5e+241)
           (*
            (sqrt 2.0)
            (*
             (sqrt F)
             (-
              (sqrt
               (/
                (+ (+ A C) (hypot B_m (- A C)))
                (fma C (* A -4.0) (pow B_m 2.0)))))))
           (* (sqrt (* 2.0 F)) (- (pow 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) {
	double t_0 = (4.0 * A) * C;
	double t_1 = 2.0 * ((pow(B_m, 2.0) - t_0) * F);
	double t_2 = t_0 - pow(B_m, 2.0);
	double tmp;
	if (pow(B_m, 2.0) <= 2e-126) {
		tmp = (sqrt((F * (2.0 * (pow(B_m, 2.0) - (4.0 * (A * C)))))) * sqrt((2.0 * C))) / t_2;
	} else if (pow(B_m, 2.0) <= 1e+58) {
		tmp = sqrt((t_1 * (C + hypot(B_m, C)))) / t_2;
	} else if (pow(B_m, 2.0) <= 2e+148) {
		tmp = sqrt((t_1 * ((-0.5 * (pow(B_m, 2.0) / A)) + (2.0 * C)))) / (t_0 - (B_m * B_m));
	} else if (pow(B_m, 2.0) <= 5e+241) {
		tmp = sqrt(2.0) * (sqrt(F) * -sqrt((((A + C) + hypot(B_m, (A - C))) / fma(C, (A * -4.0), pow(B_m, 2.0)))));
	} else {
		tmp = sqrt((2.0 * F)) * -pow(B_m, -0.5);
	}
	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)
	t_1 = Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_0) * F))
	t_2 = Float64(t_0 - (B_m ^ 2.0))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 2e-126)
		tmp = Float64(Float64(sqrt(Float64(F * Float64(2.0 * Float64((B_m ^ 2.0) - Float64(4.0 * Float64(A * C)))))) * sqrt(Float64(2.0 * C))) / t_2);
	elseif ((B_m ^ 2.0) <= 1e+58)
		tmp = Float64(sqrt(Float64(t_1 * Float64(C + hypot(B_m, C)))) / t_2);
	elseif ((B_m ^ 2.0) <= 2e+148)
		tmp = Float64(sqrt(Float64(t_1 * Float64(Float64(-0.5 * Float64((B_m ^ 2.0) / A)) + Float64(2.0 * C)))) / Float64(t_0 - Float64(B_m * B_m)));
	elseif ((B_m ^ 2.0) <= 5e+241)
		tmp = Float64(sqrt(2.0) * Float64(sqrt(F) * Float64(-sqrt(Float64(Float64(Float64(A + C) + hypot(B_m, Float64(A - C))) / fma(C, Float64(A * -4.0), (B_m ^ 2.0)))))));
	else
		tmp = Float64(sqrt(Float64(2.0 * F)) * Float64(-(B_m ^ -0.5)));
	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]}, Block[{t$95$1 = N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e-126], N[(N[(N[Sqrt[N[(F * N[(2.0 * N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(2.0 * C), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e+58], N[(N[Sqrt[N[(t$95$1 * N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+148], N[(N[Sqrt[N[(t$95$1 * N[(N[(-0.5 * N[(N[Power[B$95$m, 2.0], $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision] + N[(2.0 * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e+241], N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sqrt[F], $MachinePrecision] * (-N[Sqrt[N[(N[(N[(A + C), $MachinePrecision] + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] / N[(C * N[(A * -4.0), $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] * (-N[Power[B$95$m, -0.5], $MachinePrecision])), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 1.9999999999999999e-126

   1. Initial program 21.9%

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

   3. Taylor expanded in A around -inf 23.7%

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

      1. sqrt-prod 27.2%

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

      2. associate-*r* 27.2%

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

      3. associate-*l* 27.2%

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

   5. Applied egg-rr 27.2%

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

   if 1.9999999999999999e-126 < (pow.f64 B #s(literal 2 binary64)) < 9.99999999999999944e57

   1. Initial program 33.0%

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

   3. Taylor expanded in A around 0 29.1%

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

      1. unpow2 29.1%

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

      2. unpow2 29.1%

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

      3. hypot-define 40.3%

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

   5. Simplified 40.3%

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

   if 9.99999999999999944e57 < (pow.f64 B #s(literal 2 binary64)) < 2.0000000000000001e148

   1. Initial program 2.3%

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

   3. Taylor expanded in A around -inf 26.3%

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

      1. unpow2 26.3%

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

   5. Applied egg-rr 26.3%

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

   if 2.0000000000000001e148 < (pow.f64 B #s(literal 2 binary64)) < 5.00000000000000025e241

   1. Initial program 23.8%

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

   3. Taylor expanded in F around 0 30.5%

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

      1. mul-1-neg 30.5%

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

      2. *-commutative 30.5%

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

      3. associate-/l* 30.5%

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

      4. cancel-sign-sub-inv 30.5%

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

      5. metadata-eval 30.5%

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

      6. +-commutative 30.5%

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

   5. Simplified 58.0%

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

      1. pow1/2 58.0%

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

      2. *-commutative 58.0%

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

      3. unpow-prod-down 67.7%

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

      4. pow1/2 67.7%

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

      5. associate-+r+ 66.9%

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

      6. *-commutative 66.9%

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

      7. fma-define 66.9%

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

      8. associate-*l* 66.9%

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

      9. *-commutative 66.9%

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

      10. fma-define 66.9%

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

      11. *-commutative 66.9%

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

      12. pow1/2 66.9%

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

   7. Applied egg-rr 66.9%

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

   if 5.00000000000000025e241 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 6.8%

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

   3. Taylor expanded in B around inf 32.3%

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

      1. mul-1-neg 32.3%

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

      2. *-commutative 32.3%

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

   5. Simplified 32.3%

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

      1. pow1/2 32.3%

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

      2. div-inv 32.3%

         \[\leadsto -\sqrt{2} \cdot {\color{blue}{\left(F \cdot \frac{1}{B}\right)}}^{0.5} \]

      3. unpow-prod-down 41.7%

         \[\leadsto -\sqrt{2} \cdot \color{blue}{\left({F}^{0.5} \cdot {\left(\frac{1}{B}\right)}^{0.5}\right)} \]

      4. pow1/2 41.7%

         \[\leadsto -\sqrt{2} \cdot \left(\color{blue}{\sqrt{F}} \cdot {\left(\frac{1}{B}\right)}^{0.5}\right) \]

   7. Applied egg-rr 41.7%

      \[\leadsto -\sqrt{2} \cdot \color{blue}{\left(\sqrt{F} \cdot {\left(\frac{1}{B}\right)}^{0.5}\right)} \]
   8. Step-by-step derivation

      1. unpow1/2 41.7%

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

   9. Simplified 41.7%

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

       1. neg-sub0 41.7%

          \[\leadsto \color{blue}{0 - \sqrt{2} \cdot \left(\sqrt{F} \cdot \sqrt{\frac{1}{B}}\right)} \]

       2. associate-*r* 41.7%

          \[\leadsto 0 - \color{blue}{\left(\sqrt{2} \cdot \sqrt{F}\right) \cdot \sqrt{\frac{1}{B}}} \]

       3. pow1/2 41.7%

          \[\leadsto 0 - \left(\color{blue}{{2}^{0.5}} \cdot \sqrt{F}\right) \cdot \sqrt{\frac{1}{B}} \]

       4. pow1/2 41.7%

          \[\leadsto 0 - \left({2}^{0.5} \cdot \color{blue}{{F}^{0.5}}\right) \cdot \sqrt{\frac{1}{B}} \]

       5. pow-prod-down 41.7%

          \[\leadsto 0 - \color{blue}{{\left(2 \cdot F\right)}^{0.5}} \cdot \sqrt{\frac{1}{B}} \]

       6. pow1/2 41.7%

          \[\leadsto 0 - {\left(2 \cdot F\right)}^{0.5} \cdot \color{blue}{{\left(\frac{1}{B}\right)}^{0.5}} \]

       7. inv-pow 41.7%

          \[\leadsto 0 - {\left(2 \cdot F\right)}^{0.5} \cdot {\color{blue}{\left({B}^{-1}\right)}}^{0.5} \]

       8. pow-pow 41.8%

          \[\leadsto 0 - {\left(2 \cdot F\right)}^{0.5} \cdot \color{blue}{{B}^{\left(-1 \cdot 0.5\right)}} \]

       9. metadata-eval 41.8%

          \[\leadsto 0 - {\left(2 \cdot F\right)}^{0.5} \cdot {B}^{\color{blue}{-0.5}} \]

   11. Applied egg-rr 41.8%

       \[\leadsto \color{blue}{0 - {\left(2 \cdot F\right)}^{0.5} \cdot {B}^{-0.5}} \]
   12. Step-by-step derivation

       1. neg-sub0 41.8%

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

       2. *-commutative 41.8%

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

       3. distribute-rgt-neg-in 41.8%

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

       4. unpow1/2 41.8%

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

   13. Simplified 41.8%

       \[\leadsto \color{blue}{{B}^{-0.5} \cdot \left(-\sqrt{2 \cdot F}\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 36.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 2 \cdot 10^{-126}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(2 \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right)} \cdot \sqrt{2 \cdot C}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;{B}^{2} \leq 10^{+58}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{+148}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(-0.5 \cdot \frac{{B}^{2}}{A} + 2 \cdot C\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{elif}\;{B}^{2} \leq 5 \cdot 10^{+241}:\\ \;\;\;\;\sqrt{2} \cdot \left(\sqrt{F} \cdot \left(-\sqrt{\frac{\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)}{\mathsf{fma}\left(C, A \cdot -4, {B}^{2}\right)}}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot F} \cdot \left(-{B}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 52.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ t_1 := 2 \cdot \left(\left({B\_m}^{2} - t\_0\right) \cdot F\right)\\ t_2 := t\_0 - {B\_m}^{2}\\ \mathbf{if}\;{B\_m}^{2} \leq 2 \cdot 10^{-126}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(2 \cdot \left({B\_m}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right)} \cdot \sqrt{2 \cdot C}}{t\_2}\\ \mathbf{elif}\;{B\_m}^{2} \leq 10^{+58}:\\ \;\;\;\;\frac{\sqrt{t\_1 \cdot \left(C + \mathsf{hypot}\left(B\_m, C\right)\right)}}{t\_2}\\ \mathbf{elif}\;{B\_m}^{2} \leq 5 \cdot 10^{+143}:\\ \;\;\;\;\frac{\sqrt{t\_1 \cdot \left(-0.5 \cdot \frac{{B\_m}^{2}}{A} + 2 \cdot C\right)}}{t\_0 - B\_m \cdot B\_m}\\ \mathbf{elif}\;{B\_m}^{2} \leq 5 \cdot 10^{+241}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B\_m, B\_m, -4 \cdot \left(A \cdot C\right)\right)\right)} \cdot \left(-\sqrt{A + \left(C + \mathsf{hypot}\left(A - C, B\_m\right)\right)}\right)}{\mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot F} \cdot \left(-{B\_m}^{-0.5}\right)\\ \end{array} \end{array} \]

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))
        (t_1 (* 2.0 (* (- (pow B_m 2.0) t_0) F)))
        (t_2 (- t_0 (pow B_m 2.0))))
   (if (<= (pow B_m 2.0) 2e-126)
     (/
      (*
       (sqrt (* F (* 2.0 (- (pow B_m 2.0) (* 4.0 (* A C))))))
       (sqrt (* 2.0 C)))
      t_2)
     (if (<= (pow B_m 2.0) 1e+58)
       (/ (sqrt (* t_1 (+ C (hypot B_m C)))) t_2)
       (if (<= (pow B_m 2.0) 5e+143)
         (/
          (sqrt (* t_1 (+ (* -0.5 (/ (pow B_m 2.0) A)) (* 2.0 C))))
          (- t_0 (* B_m B_m)))
         (if (<= (pow B_m 2.0) 5e+241)
           (/
            (*
             (sqrt (* 2.0 (* F (fma B_m B_m (* -4.0 (* A C))))))
             (- (sqrt (+ A (+ C (hypot (- A C) B_m))))))
            (fma B_m B_m (* A (* C -4.0))))
           (* (sqrt (* 2.0 F)) (- (pow 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) {
	double t_0 = (4.0 * A) * C;
	double t_1 = 2.0 * ((pow(B_m, 2.0) - t_0) * F);
	double t_2 = t_0 - pow(B_m, 2.0);
	double tmp;
	if (pow(B_m, 2.0) <= 2e-126) {
		tmp = (sqrt((F * (2.0 * (pow(B_m, 2.0) - (4.0 * (A * C)))))) * sqrt((2.0 * C))) / t_2;
	} else if (pow(B_m, 2.0) <= 1e+58) {
		tmp = sqrt((t_1 * (C + hypot(B_m, C)))) / t_2;
	} else if (pow(B_m, 2.0) <= 5e+143) {
		tmp = sqrt((t_1 * ((-0.5 * (pow(B_m, 2.0) / A)) + (2.0 * C)))) / (t_0 - (B_m * B_m));
	} else if (pow(B_m, 2.0) <= 5e+241) {
		tmp = (sqrt((2.0 * (F * fma(B_m, B_m, (-4.0 * (A * C)))))) * -sqrt((A + (C + hypot((A - C), B_m))))) / fma(B_m, B_m, (A * (C * -4.0)));
	} else {
		tmp = sqrt((2.0 * F)) * -pow(B_m, -0.5);
	}
	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)
	t_1 = Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_0) * F))
	t_2 = Float64(t_0 - (B_m ^ 2.0))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 2e-126)
		tmp = Float64(Float64(sqrt(Float64(F * Float64(2.0 * Float64((B_m ^ 2.0) - Float64(4.0 * Float64(A * C)))))) * sqrt(Float64(2.0 * C))) / t_2);
	elseif ((B_m ^ 2.0) <= 1e+58)
		tmp = Float64(sqrt(Float64(t_1 * Float64(C + hypot(B_m, C)))) / t_2);
	elseif ((B_m ^ 2.0) <= 5e+143)
		tmp = Float64(sqrt(Float64(t_1 * Float64(Float64(-0.5 * Float64((B_m ^ 2.0) / A)) + Float64(2.0 * C)))) / Float64(t_0 - Float64(B_m * B_m)));
	elseif ((B_m ^ 2.0) <= 5e+241)
		tmp = Float64(Float64(sqrt(Float64(2.0 * Float64(F * fma(B_m, B_m, Float64(-4.0 * Float64(A * C)))))) * Float64(-sqrt(Float64(A + Float64(C + hypot(Float64(A - C), B_m)))))) / fma(B_m, B_m, Float64(A * Float64(C * -4.0))));
	else
		tmp = Float64(sqrt(Float64(2.0 * F)) * Float64(-(B_m ^ -0.5)));
	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]}, Block[{t$95$1 = N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e-126], N[(N[(N[Sqrt[N[(F * N[(2.0 * N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(2.0 * C), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e+58], N[(N[Sqrt[N[(t$95$1 * N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e+143], N[(N[Sqrt[N[(t$95$1 * N[(N[(-0.5 * N[(N[Power[B$95$m, 2.0], $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision] + N[(2.0 * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e+241], N[(N[(N[Sqrt[N[(2.0 * N[(F * N[(B$95$m * B$95$m + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(A + N[(C + N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] * (-N[Power[B$95$m, -0.5], $MachinePrecision])), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 1.9999999999999999e-126

   1. Initial program 21.9%

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

   3. Taylor expanded in A around -inf 23.7%

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

      1. sqrt-prod 27.2%

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

      2. associate-*r* 27.2%

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

      3. associate-*l* 27.2%

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

   5. Applied egg-rr 27.2%

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

   if 1.9999999999999999e-126 < (pow.f64 B #s(literal 2 binary64)) < 9.99999999999999944e57

   1. Initial program 33.0%

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

   3. Taylor expanded in A around 0 29.1%

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

      1. unpow2 29.1%

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

      2. unpow2 29.1%

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

      3. hypot-define 40.3%

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

   5. Simplified 40.3%

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

   if 9.99999999999999944e57 < (pow.f64 B #s(literal 2 binary64)) < 5.00000000000000012e143

   1. Initial program 2.3%

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

   3. Taylor expanded in A around -inf 27.7%

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

      1. unpow2 27.7%

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

   5. Applied egg-rr 27.7%

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

   if 5.00000000000000012e143 < (pow.f64 B #s(literal 2 binary64)) < 5.00000000000000025e241

   1. Initial program 22.7%

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

   3. Simplified 33.0%

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

      1. associate-*r* 33.0%

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

      2. associate-+r+ 32.8%

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

      3. hypot-undefine 22.7%

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

      4. unpow2 22.7%

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

      5. unpow2 22.7%

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

      6. +-commutative 22.7%

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

      7. sqrt-prod 37.8%

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

      8. *-commutative 37.8%

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

      9. associate-*r* 37.8%

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

      10. associate-+l+ 38.0%

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

   6. Applied egg-rr 63.2%

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

   if 5.00000000000000025e241 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 6.8%

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

   3. Taylor expanded in B around inf 32.3%

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

      1. mul-1-neg 32.3%

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

      2. *-commutative 32.3%

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

   5. Simplified 32.3%

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

      1. pow1/2 32.3%

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

      2. div-inv 32.3%

         \[\leadsto -\sqrt{2} \cdot {\color{blue}{\left(F \cdot \frac{1}{B}\right)}}^{0.5} \]

      3. unpow-prod-down 41.7%

         \[\leadsto -\sqrt{2} \cdot \color{blue}{\left({F}^{0.5} \cdot {\left(\frac{1}{B}\right)}^{0.5}\right)} \]

      4. pow1/2 41.7%

         \[\leadsto -\sqrt{2} \cdot \left(\color{blue}{\sqrt{F}} \cdot {\left(\frac{1}{B}\right)}^{0.5}\right) \]

   7. Applied egg-rr 41.7%

      \[\leadsto -\sqrt{2} \cdot \color{blue}{\left(\sqrt{F} \cdot {\left(\frac{1}{B}\right)}^{0.5}\right)} \]
   8. Step-by-step derivation

      1. unpow1/2 41.7%

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

   9. Simplified 41.7%

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

       1. neg-sub0 41.7%

          \[\leadsto \color{blue}{0 - \sqrt{2} \cdot \left(\sqrt{F} \cdot \sqrt{\frac{1}{B}}\right)} \]

       2. associate-*r* 41.7%

          \[\leadsto 0 - \color{blue}{\left(\sqrt{2} \cdot \sqrt{F}\right) \cdot \sqrt{\frac{1}{B}}} \]

       3. pow1/2 41.7%

          \[\leadsto 0 - \left(\color{blue}{{2}^{0.5}} \cdot \sqrt{F}\right) \cdot \sqrt{\frac{1}{B}} \]

       4. pow1/2 41.7%

          \[\leadsto 0 - \left({2}^{0.5} \cdot \color{blue}{{F}^{0.5}}\right) \cdot \sqrt{\frac{1}{B}} \]

       5. pow-prod-down 41.7%

          \[\leadsto 0 - \color{blue}{{\left(2 \cdot F\right)}^{0.5}} \cdot \sqrt{\frac{1}{B}} \]

       6. pow1/2 41.7%

          \[\leadsto 0 - {\left(2 \cdot F\right)}^{0.5} \cdot \color{blue}{{\left(\frac{1}{B}\right)}^{0.5}} \]

       7. inv-pow 41.7%

          \[\leadsto 0 - {\left(2 \cdot F\right)}^{0.5} \cdot {\color{blue}{\left({B}^{-1}\right)}}^{0.5} \]

       8. pow-pow 41.8%

          \[\leadsto 0 - {\left(2 \cdot F\right)}^{0.5} \cdot \color{blue}{{B}^{\left(-1 \cdot 0.5\right)}} \]

       9. metadata-eval 41.8%

          \[\leadsto 0 - {\left(2 \cdot F\right)}^{0.5} \cdot {B}^{\color{blue}{-0.5}} \]

   11. Applied egg-rr 41.8%

       \[\leadsto \color{blue}{0 - {\left(2 \cdot F\right)}^{0.5} \cdot {B}^{-0.5}} \]
   12. Step-by-step derivation

       1. neg-sub0 41.8%

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

       2. *-commutative 41.8%

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

       3. distribute-rgt-neg-in 41.8%

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

       4. unpow1/2 41.8%

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

   13. Simplified 41.8%

       \[\leadsto \color{blue}{{B}^{-0.5} \cdot \left(-\sqrt{2 \cdot F}\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 36.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 2 \cdot 10^{-126}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(2 \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right)} \cdot \sqrt{2 \cdot C}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;{B}^{2} \leq 10^{+58}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;{B}^{2} \leq 5 \cdot 10^{+143}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(-0.5 \cdot \frac{{B}^{2}}{A} + 2 \cdot C\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{elif}\;{B}^{2} \leq 5 \cdot 10^{+241}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)\right)} \cdot \left(-\sqrt{A + \left(C + \mathsf{hypot}\left(A - C, B\right)\right)}\right)}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot F} \cdot \left(-{B}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 52.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ t_1 := 2 \cdot \left(\left({B\_m}^{2} - t\_0\right) \cdot F\right)\\ t_2 := t\_0 - {B\_m}^{2}\\ \mathbf{if}\;{B\_m}^{2} \leq 2 \cdot 10^{-126}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(2 \cdot \left({B\_m}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right)} \cdot \sqrt{2 \cdot C}}{t\_2}\\ \mathbf{elif}\;{B\_m}^{2} \leq 10^{+58}:\\ \;\;\;\;\frac{\sqrt{t\_1 \cdot \left(C + \mathsf{hypot}\left(B\_m, C\right)\right)}}{t\_2}\\ \mathbf{elif}\;{B\_m}^{2} \leq 2 \cdot 10^{+148}:\\ \;\;\;\;\frac{\sqrt{t\_1 \cdot \left(-0.5 \cdot \frac{{B\_m}^{2}}{A} + 2 \cdot C\right)}}{t\_0 - B\_m \cdot B\_m}\\ \mathbf{elif}\;{B\_m}^{2} \leq 5 \cdot 10^{+241}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{F \cdot \frac{A + \left(C + \mathsf{hypot}\left(B\_m, A - C\right)\right)}{\mathsf{fma}\left(-4, A \cdot C, {B\_m}^{2}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot F} \cdot \left(-{B\_m}^{-0.5}\right)\\ \end{array} \end{array} \]

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))
        (t_1 (* 2.0 (* (- (pow B_m 2.0) t_0) F)))
        (t_2 (- t_0 (pow B_m 2.0))))
   (if (<= (pow B_m 2.0) 2e-126)
     (/
      (*
       (sqrt (* F (* 2.0 (- (pow B_m 2.0) (* 4.0 (* A C))))))
       (sqrt (* 2.0 C)))
      t_2)
     (if (<= (pow B_m 2.0) 1e+58)
       (/ (sqrt (* t_1 (+ C (hypot B_m C)))) t_2)
       (if (<= (pow B_m 2.0) 2e+148)
         (/
          (sqrt (* t_1 (+ (* -0.5 (/ (pow B_m 2.0) A)) (* 2.0 C))))
          (- t_0 (* B_m B_m)))
         (if (<= (pow B_m 2.0) 5e+241)
           (*
            (sqrt 2.0)
            (-
             (sqrt
              (*
               F
               (/
                (+ A (+ C (hypot B_m (- A C))))
                (fma -4.0 (* A C) (pow B_m 2.0)))))))
           (* (sqrt (* 2.0 F)) (- (pow 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) {
	double t_0 = (4.0 * A) * C;
	double t_1 = 2.0 * ((pow(B_m, 2.0) - t_0) * F);
	double t_2 = t_0 - pow(B_m, 2.0);
	double tmp;
	if (pow(B_m, 2.0) <= 2e-126) {
		tmp = (sqrt((F * (2.0 * (pow(B_m, 2.0) - (4.0 * (A * C)))))) * sqrt((2.0 * C))) / t_2;
	} else if (pow(B_m, 2.0) <= 1e+58) {
		tmp = sqrt((t_1 * (C + hypot(B_m, C)))) / t_2;
	} else if (pow(B_m, 2.0) <= 2e+148) {
		tmp = sqrt((t_1 * ((-0.5 * (pow(B_m, 2.0) / A)) + (2.0 * C)))) / (t_0 - (B_m * B_m));
	} else if (pow(B_m, 2.0) <= 5e+241) {
		tmp = sqrt(2.0) * -sqrt((F * ((A + (C + hypot(B_m, (A - C)))) / fma(-4.0, (A * C), pow(B_m, 2.0)))));
	} else {
		tmp = sqrt((2.0 * F)) * -pow(B_m, -0.5);
	}
	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)
	t_1 = Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_0) * F))
	t_2 = Float64(t_0 - (B_m ^ 2.0))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 2e-126)
		tmp = Float64(Float64(sqrt(Float64(F * Float64(2.0 * Float64((B_m ^ 2.0) - Float64(4.0 * Float64(A * C)))))) * sqrt(Float64(2.0 * C))) / t_2);
	elseif ((B_m ^ 2.0) <= 1e+58)
		tmp = Float64(sqrt(Float64(t_1 * Float64(C + hypot(B_m, C)))) / t_2);
	elseif ((B_m ^ 2.0) <= 2e+148)
		tmp = Float64(sqrt(Float64(t_1 * Float64(Float64(-0.5 * Float64((B_m ^ 2.0) / A)) + Float64(2.0 * C)))) / Float64(t_0 - Float64(B_m * B_m)));
	elseif ((B_m ^ 2.0) <= 5e+241)
		tmp = Float64(sqrt(2.0) * Float64(-sqrt(Float64(F * Float64(Float64(A + Float64(C + hypot(B_m, Float64(A - C)))) / fma(-4.0, Float64(A * C), (B_m ^ 2.0)))))));
	else
		tmp = Float64(sqrt(Float64(2.0 * F)) * Float64(-(B_m ^ -0.5)));
	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]}, Block[{t$95$1 = N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e-126], N[(N[(N[Sqrt[N[(F * N[(2.0 * N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(2.0 * C), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e+58], N[(N[Sqrt[N[(t$95$1 * N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+148], N[(N[Sqrt[N[(t$95$1 * N[(N[(-0.5 * N[(N[Power[B$95$m, 2.0], $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision] + N[(2.0 * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e+241], N[(N[Sqrt[2.0], $MachinePrecision] * (-N[Sqrt[N[(F * 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]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] * (-N[Power[B$95$m, -0.5], $MachinePrecision])), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 1.9999999999999999e-126

   1. Initial program 21.9%

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

   3. Taylor expanded in A around -inf 23.7%

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

      1. sqrt-prod 27.2%

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

      2. associate-*r* 27.2%

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

      3. associate-*l* 27.2%

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

   5. Applied egg-rr 27.2%

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

   if 1.9999999999999999e-126 < (pow.f64 B #s(literal 2 binary64)) < 9.99999999999999944e57

   1. Initial program 33.0%

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

   3. Taylor expanded in A around 0 29.1%

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

      1. unpow2 29.1%

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

      2. unpow2 29.1%

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

      3. hypot-define 40.3%

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

   5. Simplified 40.3%

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

   if 9.99999999999999944e57 < (pow.f64 B #s(literal 2 binary64)) < 2.0000000000000001e148

   1. Initial program 2.3%

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

   3. Taylor expanded in A around -inf 26.3%

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

      1. unpow2 26.3%

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

   5. Applied egg-rr 26.3%

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

   if 2.0000000000000001e148 < (pow.f64 B #s(literal 2 binary64)) < 5.00000000000000025e241

   1. Initial program 23.8%

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

   3. Taylor expanded in F around 0 30.5%

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

      1. mul-1-neg 30.5%

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

      2. *-commutative 30.5%

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

      3. associate-/l* 30.5%

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

      4. cancel-sign-sub-inv 30.5%

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

      5. metadata-eval 30.5%

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

      6. +-commutative 30.5%

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

   5. Simplified 58.0%

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

   if 5.00000000000000025e241 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 6.8%

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

   3. Taylor expanded in B around inf 32.3%

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

      1. mul-1-neg 32.3%

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

      2. *-commutative 32.3%

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

   5. Simplified 32.3%

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

      1. pow1/2 32.3%

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

      2. div-inv 32.3%

         \[\leadsto -\sqrt{2} \cdot {\color{blue}{\left(F \cdot \frac{1}{B}\right)}}^{0.5} \]

      3. unpow-prod-down 41.7%

         \[\leadsto -\sqrt{2} \cdot \color{blue}{\left({F}^{0.5} \cdot {\left(\frac{1}{B}\right)}^{0.5}\right)} \]

      4. pow1/2 41.7%

         \[\leadsto -\sqrt{2} \cdot \left(\color{blue}{\sqrt{F}} \cdot {\left(\frac{1}{B}\right)}^{0.5}\right) \]

   7. Applied egg-rr 41.7%

      \[\leadsto -\sqrt{2} \cdot \color{blue}{\left(\sqrt{F} \cdot {\left(\frac{1}{B}\right)}^{0.5}\right)} \]
   8. Step-by-step derivation

      1. unpow1/2 41.7%

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

   9. Simplified 41.7%

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

       1. neg-sub0 41.7%

          \[\leadsto \color{blue}{0 - \sqrt{2} \cdot \left(\sqrt{F} \cdot \sqrt{\frac{1}{B}}\right)} \]

       2. associate-*r* 41.7%

          \[\leadsto 0 - \color{blue}{\left(\sqrt{2} \cdot \sqrt{F}\right) \cdot \sqrt{\frac{1}{B}}} \]

       3. pow1/2 41.7%

          \[\leadsto 0 - \left(\color{blue}{{2}^{0.5}} \cdot \sqrt{F}\right) \cdot \sqrt{\frac{1}{B}} \]

       4. pow1/2 41.7%

          \[\leadsto 0 - \left({2}^{0.5} \cdot \color{blue}{{F}^{0.5}}\right) \cdot \sqrt{\frac{1}{B}} \]

       5. pow-prod-down 41.7%

          \[\leadsto 0 - \color{blue}{{\left(2 \cdot F\right)}^{0.5}} \cdot \sqrt{\frac{1}{B}} \]

       6. pow1/2 41.7%

          \[\leadsto 0 - {\left(2 \cdot F\right)}^{0.5} \cdot \color{blue}{{\left(\frac{1}{B}\right)}^{0.5}} \]

       7. inv-pow 41.7%

          \[\leadsto 0 - {\left(2 \cdot F\right)}^{0.5} \cdot {\color{blue}{\left({B}^{-1}\right)}}^{0.5} \]

       8. pow-pow 41.8%

          \[\leadsto 0 - {\left(2 \cdot F\right)}^{0.5} \cdot \color{blue}{{B}^{\left(-1 \cdot 0.5\right)}} \]

       9. metadata-eval 41.8%

          \[\leadsto 0 - {\left(2 \cdot F\right)}^{0.5} \cdot {B}^{\color{blue}{-0.5}} \]

   11. Applied egg-rr 41.8%

       \[\leadsto \color{blue}{0 - {\left(2 \cdot F\right)}^{0.5} \cdot {B}^{-0.5}} \]
   12. Step-by-step derivation

       1. neg-sub0 41.8%

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

       2. *-commutative 41.8%

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

       3. distribute-rgt-neg-in 41.8%

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

       4. unpow1/2 41.8%

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

   13. Simplified 41.8%

       \[\leadsto \color{blue}{{B}^{-0.5} \cdot \left(-\sqrt{2 \cdot F}\right)} \]
3. Recombined 5 regimes into one program.

4. Final simplification 35.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 2 \cdot 10^{-126}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(2 \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right)} \cdot \sqrt{2 \cdot C}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;{B}^{2} \leq 10^{+58}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{+148}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(-0.5 \cdot \frac{{B}^{2}}{A} + 2 \cdot C\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{elif}\;{B}^{2} \leq 5 \cdot 10^{+241}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{F \cdot \frac{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)}{\mathsf{fma}\left(-4, A \cdot C, {B}^{2}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot F} \cdot \left(-{B}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 52.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ t_1 := 2 \cdot \left(\left({B\_m}^{2} - t\_0\right) \cdot F\right)\\ t_2 := t\_0 - {B\_m}^{2}\\ \mathbf{if}\;{B\_m}^{2} \leq 2 \cdot 10^{-126}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(2 \cdot \left({B\_m}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right)} \cdot \sqrt{2 \cdot C}}{t\_2}\\ \mathbf{elif}\;{B\_m}^{2} \leq 10^{+58}:\\ \;\;\;\;\frac{\sqrt{t\_1 \cdot \left(C + \mathsf{hypot}\left(B\_m, C\right)\right)}}{t\_2}\\ \mathbf{elif}\;{B\_m}^{2} \leq 2 \cdot 10^{+148}:\\ \;\;\;\;\frac{\sqrt{t\_1 \cdot \left(-0.5 \cdot \frac{{B\_m}^{2}}{A} + 2 \cdot C\right)}}{t\_0 - B\_m \cdot B\_m}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot F} \cdot \left(-{B\_m}^{-0.5}\right)\\ \end{array} \end{array} \]

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))
        (t_1 (* 2.0 (* (- (pow B_m 2.0) t_0) F)))
        (t_2 (- t_0 (pow B_m 2.0))))
   (if (<= (pow B_m 2.0) 2e-126)
     (/
      (*
       (sqrt (* F (* 2.0 (- (pow B_m 2.0) (* 4.0 (* A C))))))
       (sqrt (* 2.0 C)))
      t_2)
     (if (<= (pow B_m 2.0) 1e+58)
       (/ (sqrt (* t_1 (+ C (hypot B_m C)))) t_2)
       (if (<= (pow B_m 2.0) 2e+148)
         (/
          (sqrt (* t_1 (+ (* -0.5 (/ (pow B_m 2.0) A)) (* 2.0 C))))
          (- t_0 (* B_m B_m)))
         (* (sqrt (* 2.0 F)) (- (pow 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) {
	double t_0 = (4.0 * A) * C;
	double t_1 = 2.0 * ((pow(B_m, 2.0) - t_0) * F);
	double t_2 = t_0 - pow(B_m, 2.0);
	double tmp;
	if (pow(B_m, 2.0) <= 2e-126) {
		tmp = (sqrt((F * (2.0 * (pow(B_m, 2.0) - (4.0 * (A * C)))))) * sqrt((2.0 * C))) / t_2;
	} else if (pow(B_m, 2.0) <= 1e+58) {
		tmp = sqrt((t_1 * (C + hypot(B_m, C)))) / t_2;
	} else if (pow(B_m, 2.0) <= 2e+148) {
		tmp = sqrt((t_1 * ((-0.5 * (pow(B_m, 2.0) / A)) + (2.0 * C)))) / (t_0 - (B_m * B_m));
	} else {
		tmp = sqrt((2.0 * F)) * -pow(B_m, -0.5);
	}
	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 t_1 = 2.0 * ((Math.pow(B_m, 2.0) - t_0) * F);
	double t_2 = t_0 - Math.pow(B_m, 2.0);
	double tmp;
	if (Math.pow(B_m, 2.0) <= 2e-126) {
		tmp = (Math.sqrt((F * (2.0 * (Math.pow(B_m, 2.0) - (4.0 * (A * C)))))) * Math.sqrt((2.0 * C))) / t_2;
	} else if (Math.pow(B_m, 2.0) <= 1e+58) {
		tmp = Math.sqrt((t_1 * (C + Math.hypot(B_m, C)))) / t_2;
	} else if (Math.pow(B_m, 2.0) <= 2e+148) {
		tmp = Math.sqrt((t_1 * ((-0.5 * (Math.pow(B_m, 2.0) / A)) + (2.0 * C)))) / (t_0 - (B_m * B_m));
	} else {
		tmp = Math.sqrt((2.0 * F)) * -Math.pow(B_m, -0.5);
	}
	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
	t_1 = 2.0 * ((math.pow(B_m, 2.0) - t_0) * F)
	t_2 = t_0 - math.pow(B_m, 2.0)
	tmp = 0
	if math.pow(B_m, 2.0) <= 2e-126:
		tmp = (math.sqrt((F * (2.0 * (math.pow(B_m, 2.0) - (4.0 * (A * C)))))) * math.sqrt((2.0 * C))) / t_2
	elif math.pow(B_m, 2.0) <= 1e+58:
		tmp = math.sqrt((t_1 * (C + math.hypot(B_m, C)))) / t_2
	elif math.pow(B_m, 2.0) <= 2e+148:
		tmp = math.sqrt((t_1 * ((-0.5 * (math.pow(B_m, 2.0) / A)) + (2.0 * C)))) / (t_0 - (B_m * B_m))
	else:
		tmp = math.sqrt((2.0 * F)) * -math.pow(B_m, -0.5)
	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)
	t_1 = Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_0) * F))
	t_2 = Float64(t_0 - (B_m ^ 2.0))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 2e-126)
		tmp = Float64(Float64(sqrt(Float64(F * Float64(2.0 * Float64((B_m ^ 2.0) - Float64(4.0 * Float64(A * C)))))) * sqrt(Float64(2.0 * C))) / t_2);
	elseif ((B_m ^ 2.0) <= 1e+58)
		tmp = Float64(sqrt(Float64(t_1 * Float64(C + hypot(B_m, C)))) / t_2);
	elseif ((B_m ^ 2.0) <= 2e+148)
		tmp = Float64(sqrt(Float64(t_1 * Float64(Float64(-0.5 * Float64((B_m ^ 2.0) / A)) + Float64(2.0 * C)))) / Float64(t_0 - Float64(B_m * B_m)));
	else
		tmp = Float64(sqrt(Float64(2.0 * F)) * Float64(-(B_m ^ -0.5)));
	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;
	t_1 = 2.0 * (((B_m ^ 2.0) - t_0) * F);
	t_2 = t_0 - (B_m ^ 2.0);
	tmp = 0.0;
	if ((B_m ^ 2.0) <= 2e-126)
		tmp = (sqrt((F * (2.0 * ((B_m ^ 2.0) - (4.0 * (A * C)))))) * sqrt((2.0 * C))) / t_2;
	elseif ((B_m ^ 2.0) <= 1e+58)
		tmp = sqrt((t_1 * (C + hypot(B_m, C)))) / t_2;
	elseif ((B_m ^ 2.0) <= 2e+148)
		tmp = sqrt((t_1 * ((-0.5 * ((B_m ^ 2.0) / A)) + (2.0 * C)))) / (t_0 - (B_m * B_m));
	else
		tmp = sqrt((2.0 * F)) * -(B_m ^ -0.5);
	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]}, Block[{t$95$1 = N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e-126], N[(N[(N[Sqrt[N[(F * N[(2.0 * N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(2.0 * C), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e+58], N[(N[Sqrt[N[(t$95$1 * N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+148], N[(N[Sqrt[N[(t$95$1 * N[(N[(-0.5 * N[(N[Power[B$95$m, 2.0], $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision] + N[(2.0 * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] * (-N[Power[B$95$m, -0.5], $MachinePrecision])), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 1.9999999999999999e-126

   1. Initial program 21.9%

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

   3. Taylor expanded in A around -inf 23.7%

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

      1. sqrt-prod 27.2%

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

      2. associate-*r* 27.2%

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

      3. associate-*l* 27.2%

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

   5. Applied egg-rr 27.2%

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

   if 1.9999999999999999e-126 < (pow.f64 B #s(literal 2 binary64)) < 9.99999999999999944e57

   1. Initial program 33.0%

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

   3. Taylor expanded in A around 0 29.1%

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

      1. unpow2 29.1%

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

      2. unpow2 29.1%

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

      3. hypot-define 40.3%

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

   5. Simplified 40.3%

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

   if 9.99999999999999944e57 < (pow.f64 B #s(literal 2 binary64)) < 2.0000000000000001e148

   1. Initial program 2.3%

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

   3. Taylor expanded in A around -inf 26.3%

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

      1. unpow2 26.3%

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

   5. Applied egg-rr 26.3%

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

   if 2.0000000000000001e148 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 10.1%

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

   3. Taylor expanded in B around inf 27.9%

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

      1. mul-1-neg 27.9%

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

      2. *-commutative 27.9%

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

   5. Simplified 27.9%

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

      1. pow1/2 27.9%

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

      2. div-inv 27.9%

         \[\leadsto -\sqrt{2} \cdot {\color{blue}{\left(F \cdot \frac{1}{B}\right)}}^{0.5} \]

      3. unpow-prod-down 36.3%

         \[\leadsto -\sqrt{2} \cdot \color{blue}{\left({F}^{0.5} \cdot {\left(\frac{1}{B}\right)}^{0.5}\right)} \]

      4. pow1/2 36.3%

         \[\leadsto -\sqrt{2} \cdot \left(\color{blue}{\sqrt{F}} \cdot {\left(\frac{1}{B}\right)}^{0.5}\right) \]

   7. Applied egg-rr 36.3%

      \[\leadsto -\sqrt{2} \cdot \color{blue}{\left(\sqrt{F} \cdot {\left(\frac{1}{B}\right)}^{0.5}\right)} \]
   8. Step-by-step derivation

      1. unpow1/2 36.3%

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

   9. Simplified 36.3%

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

       1. neg-sub0 36.3%

          \[\leadsto \color{blue}{0 - \sqrt{2} \cdot \left(\sqrt{F} \cdot \sqrt{\frac{1}{B}}\right)} \]

       2. associate-*r* 36.3%

          \[\leadsto 0 - \color{blue}{\left(\sqrt{2} \cdot \sqrt{F}\right) \cdot \sqrt{\frac{1}{B}}} \]

       3. pow1/2 36.3%

          \[\leadsto 0 - \left(\color{blue}{{2}^{0.5}} \cdot \sqrt{F}\right) \cdot \sqrt{\frac{1}{B}} \]

       4. pow1/2 36.3%

          \[\leadsto 0 - \left({2}^{0.5} \cdot \color{blue}{{F}^{0.5}}\right) \cdot \sqrt{\frac{1}{B}} \]

       5. pow-prod-down 36.4%

          \[\leadsto 0 - \color{blue}{{\left(2 \cdot F\right)}^{0.5}} \cdot \sqrt{\frac{1}{B}} \]

       6. pow1/2 36.4%

          \[\leadsto 0 - {\left(2 \cdot F\right)}^{0.5} \cdot \color{blue}{{\left(\frac{1}{B}\right)}^{0.5}} \]

       7. inv-pow 36.4%

          \[\leadsto 0 - {\left(2 \cdot F\right)}^{0.5} \cdot {\color{blue}{\left({B}^{-1}\right)}}^{0.5} \]

       8. pow-pow 36.4%

          \[\leadsto 0 - {\left(2 \cdot F\right)}^{0.5} \cdot \color{blue}{{B}^{\left(-1 \cdot 0.5\right)}} \]

       9. metadata-eval 36.4%

          \[\leadsto 0 - {\left(2 \cdot F\right)}^{0.5} \cdot {B}^{\color{blue}{-0.5}} \]

   11. Applied egg-rr 36.4%

       \[\leadsto \color{blue}{0 - {\left(2 \cdot F\right)}^{0.5} \cdot {B}^{-0.5}} \]
   12. Step-by-step derivation

       1. neg-sub0 36.4%

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

       2. *-commutative 36.4%

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

       3. distribute-rgt-neg-in 36.4%

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

       4. unpow1/2 36.4%

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

   13. Simplified 36.4%

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

4. Final simplification 32.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 2 \cdot 10^{-126}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(2 \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right)} \cdot \sqrt{2 \cdot C}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;{B}^{2} \leq 10^{+58}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{+148}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(-0.5 \cdot \frac{{B}^{2}}{A} + 2 \cdot C\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot F} \cdot \left(-{B}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 50.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ \mathbf{if}\;{B\_m}^{2} \leq 2 \cdot 10^{+148}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_0\right) \cdot F\right)\right) \cdot \left(-0.5 \cdot \frac{{B\_m}^{2}}{A} + 2 \cdot C\right)}}{t\_0 - B\_m \cdot B\_m}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot F} \cdot \left(-{B\_m}^{-0.5}\right)\\ \end{array} \end{array} \]

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) 2e+148)
     (/
      (sqrt
       (*
        (* 2.0 (* (- (pow B_m 2.0) t_0) F))
        (+ (* -0.5 (/ (pow B_m 2.0) A)) (* 2.0 C))))
      (- t_0 (* B_m B_m)))
     (* (sqrt (* 2.0 F)) (- (pow 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) {
	double t_0 = (4.0 * A) * C;
	double tmp;
	if (pow(B_m, 2.0) <= 2e+148) {
		tmp = sqrt(((2.0 * ((pow(B_m, 2.0) - t_0) * F)) * ((-0.5 * (pow(B_m, 2.0) / A)) + (2.0 * C)))) / (t_0 - (B_m * B_m));
	} else {
		tmp = sqrt((2.0 * F)) * -pow(B_m, -0.5);
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = (4.0d0 * a) * c
    if ((b_m ** 2.0d0) <= 2d+148) then
        tmp = sqrt(((2.0d0 * (((b_m ** 2.0d0) - t_0) * f)) * (((-0.5d0) * ((b_m ** 2.0d0) / a)) + (2.0d0 * c)))) / (t_0 - (b_m * b_m))
    else
        tmp = sqrt((2.0d0 * f)) * -(b_m ** (-0.5d0))
    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 t_0 = (4.0 * A) * C;
	double tmp;
	if (Math.pow(B_m, 2.0) <= 2e+148) {
		tmp = Math.sqrt(((2.0 * ((Math.pow(B_m, 2.0) - t_0) * F)) * ((-0.5 * (Math.pow(B_m, 2.0) / A)) + (2.0 * C)))) / (t_0 - (B_m * B_m));
	} else {
		tmp = Math.sqrt((2.0 * F)) * -Math.pow(B_m, -0.5);
	}
	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) <= 2e+148:
		tmp = math.sqrt(((2.0 * ((math.pow(B_m, 2.0) - t_0) * F)) * ((-0.5 * (math.pow(B_m, 2.0) / A)) + (2.0 * C)))) / (t_0 - (B_m * B_m))
	else:
		tmp = math.sqrt((2.0 * F)) * -math.pow(B_m, -0.5)
	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) <= 2e+148)
		tmp = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_0) * F)) * Float64(Float64(-0.5 * Float64((B_m ^ 2.0) / A)) + Float64(2.0 * C)))) / Float64(t_0 - Float64(B_m * B_m)));
	else
		tmp = Float64(sqrt(Float64(2.0 * F)) * Float64(-(B_m ^ -0.5)));
	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) <= 2e+148)
		tmp = sqrt(((2.0 * (((B_m ^ 2.0) - t_0) * F)) * ((-0.5 * ((B_m ^ 2.0) / A)) + (2.0 * C)))) / (t_0 - (B_m * B_m));
	else
		tmp = sqrt((2.0 * F)) * -(B_m ^ -0.5);
	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], 2e+148], 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[(N[(-0.5 * N[(N[Power[B$95$m, 2.0], $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision] + N[(2.0 * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] * (-N[Power[B$95$m, -0.5], $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 2.0000000000000001e148

   1. Initial program 23.0%

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

   3. Taylor expanded in A around -inf 23.5%

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

      1. unpow2 23.5%

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

   5. Applied egg-rr 23.5%

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

   if 2.0000000000000001e148 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 10.1%

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

   3. Taylor expanded in B around inf 27.9%

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

      1. mul-1-neg 27.9%

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

      2. *-commutative 27.9%

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

   5. Simplified 27.9%

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

      1. pow1/2 27.9%

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

      2. div-inv 27.9%

         \[\leadsto -\sqrt{2} \cdot {\color{blue}{\left(F \cdot \frac{1}{B}\right)}}^{0.5} \]

      3. unpow-prod-down 36.3%

         \[\leadsto -\sqrt{2} \cdot \color{blue}{\left({F}^{0.5} \cdot {\left(\frac{1}{B}\right)}^{0.5}\right)} \]

      4. pow1/2 36.3%

         \[\leadsto -\sqrt{2} \cdot \left(\color{blue}{\sqrt{F}} \cdot {\left(\frac{1}{B}\right)}^{0.5}\right) \]

   7. Applied egg-rr 36.3%

      \[\leadsto -\sqrt{2} \cdot \color{blue}{\left(\sqrt{F} \cdot {\left(\frac{1}{B}\right)}^{0.5}\right)} \]
   8. Step-by-step derivation

      1. unpow1/2 36.3%

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

   9. Simplified 36.3%

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

       1. neg-sub0 36.3%

          \[\leadsto \color{blue}{0 - \sqrt{2} \cdot \left(\sqrt{F} \cdot \sqrt{\frac{1}{B}}\right)} \]

       2. associate-*r* 36.3%

          \[\leadsto 0 - \color{blue}{\left(\sqrt{2} \cdot \sqrt{F}\right) \cdot \sqrt{\frac{1}{B}}} \]

       3. pow1/2 36.3%

          \[\leadsto 0 - \left(\color{blue}{{2}^{0.5}} \cdot \sqrt{F}\right) \cdot \sqrt{\frac{1}{B}} \]

       4. pow1/2 36.3%

          \[\leadsto 0 - \left({2}^{0.5} \cdot \color{blue}{{F}^{0.5}}\right) \cdot \sqrt{\frac{1}{B}} \]

       5. pow-prod-down 36.4%

          \[\leadsto 0 - \color{blue}{{\left(2 \cdot F\right)}^{0.5}} \cdot \sqrt{\frac{1}{B}} \]

       6. pow1/2 36.4%

          \[\leadsto 0 - {\left(2 \cdot F\right)}^{0.5} \cdot \color{blue}{{\left(\frac{1}{B}\right)}^{0.5}} \]

       7. inv-pow 36.4%

          \[\leadsto 0 - {\left(2 \cdot F\right)}^{0.5} \cdot {\color{blue}{\left({B}^{-1}\right)}}^{0.5} \]

       8. pow-pow 36.4%

          \[\leadsto 0 - {\left(2 \cdot F\right)}^{0.5} \cdot \color{blue}{{B}^{\left(-1 \cdot 0.5\right)}} \]

       9. metadata-eval 36.4%

          \[\leadsto 0 - {\left(2 \cdot F\right)}^{0.5} \cdot {B}^{\color{blue}{-0.5}} \]

   11. Applied egg-rr 36.4%

       \[\leadsto \color{blue}{0 - {\left(2 \cdot F\right)}^{0.5} \cdot {B}^{-0.5}} \]
   12. Step-by-step derivation

       1. neg-sub0 36.4%

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

       2. *-commutative 36.4%

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

       3. distribute-rgt-neg-in 36.4%

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

       4. unpow1/2 36.4%

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

   13. Simplified 36.4%

       \[\leadsto \color{blue}{{B}^{-0.5} \cdot \left(-\sqrt{2 \cdot F}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 28.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 2 \cdot 10^{+148}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(-0.5 \cdot \frac{{B}^{2}}{A} + 2 \cdot C\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot F} \cdot \left(-{B}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 50.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ \mathbf{if}\;{B\_m}^{2} \leq 2 \cdot 10^{+14}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_0\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{t\_0 - B\_m \cdot B\_m}\\ \mathbf{elif}\;{B\_m}^{2} \leq 5 \cdot 10^{+199}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{F \cdot \frac{-0.5}{A}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot F} \cdot \left(-{B\_m}^{-0.5}\right)\\ \end{array} \end{array} \]

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) 2e+14)
     (/
      (sqrt (* (* 2.0 (* (- (pow B_m 2.0) t_0) F)) (* 2.0 C)))
      (- t_0 (* B_m B_m)))
     (if (<= (pow B_m 2.0) 5e+199)
       (* (sqrt 2.0) (- (sqrt (* F (/ -0.5 A)))))
       (* (sqrt (* 2.0 F)) (- (pow 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) {
	double t_0 = (4.0 * A) * C;
	double tmp;
	if (pow(B_m, 2.0) <= 2e+14) {
		tmp = sqrt(((2.0 * ((pow(B_m, 2.0) - t_0) * F)) * (2.0 * C))) / (t_0 - (B_m * B_m));
	} else if (pow(B_m, 2.0) <= 5e+199) {
		tmp = sqrt(2.0) * -sqrt((F * (-0.5 / A)));
	} else {
		tmp = sqrt((2.0 * F)) * -pow(B_m, -0.5);
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = (4.0d0 * a) * c
    if ((b_m ** 2.0d0) <= 2d+14) then
        tmp = sqrt(((2.0d0 * (((b_m ** 2.0d0) - t_0) * f)) * (2.0d0 * c))) / (t_0 - (b_m * b_m))
    else if ((b_m ** 2.0d0) <= 5d+199) then
        tmp = sqrt(2.0d0) * -sqrt((f * ((-0.5d0) / a)))
    else
        tmp = sqrt((2.0d0 * f)) * -(b_m ** (-0.5d0))
    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 t_0 = (4.0 * A) * C;
	double tmp;
	if (Math.pow(B_m, 2.0) <= 2e+14) {
		tmp = Math.sqrt(((2.0 * ((Math.pow(B_m, 2.0) - t_0) * F)) * (2.0 * C))) / (t_0 - (B_m * B_m));
	} else if (Math.pow(B_m, 2.0) <= 5e+199) {
		tmp = Math.sqrt(2.0) * -Math.sqrt((F * (-0.5 / A)));
	} else {
		tmp = Math.sqrt((2.0 * F)) * -Math.pow(B_m, -0.5);
	}
	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) <= 2e+14:
		tmp = math.sqrt(((2.0 * ((math.pow(B_m, 2.0) - t_0) * F)) * (2.0 * C))) / (t_0 - (B_m * B_m))
	elif math.pow(B_m, 2.0) <= 5e+199:
		tmp = math.sqrt(2.0) * -math.sqrt((F * (-0.5 / A)))
	else:
		tmp = math.sqrt((2.0 * F)) * -math.pow(B_m, -0.5)
	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) <= 2e+14)
		tmp = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_0) * F)) * Float64(2.0 * C))) / Float64(t_0 - Float64(B_m * B_m)));
	elseif ((B_m ^ 2.0) <= 5e+199)
		tmp = Float64(sqrt(2.0) * Float64(-sqrt(Float64(F * Float64(-0.5 / A)))));
	else
		tmp = Float64(sqrt(Float64(2.0 * F)) * Float64(-(B_m ^ -0.5)));
	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) <= 2e+14)
		tmp = sqrt(((2.0 * (((B_m ^ 2.0) - t_0) * F)) * (2.0 * C))) / (t_0 - (B_m * B_m));
	elseif ((B_m ^ 2.0) <= 5e+199)
		tmp = sqrt(2.0) * -sqrt((F * (-0.5 / A)));
	else
		tmp = sqrt((2.0 * F)) * -(B_m ^ -0.5);
	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], 2e+14], 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[(2.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 - N[(B$95$m * B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e+199], N[(N[Sqrt[2.0], $MachinePrecision] * (-N[Sqrt[N[(F * N[(-0.5 / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] * (-N[Power[B$95$m, -0.5], $MachinePrecision])), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 2e14

   1. Initial program 25.0%

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

   3. Taylor expanded in A around -inf 25.5%

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

      1. unpow2 23.4%

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

   5. Applied egg-rr 25.5%

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

   if 2e14 < (pow.f64 B #s(literal 2 binary64)) < 4.9999999999999998e199

   1. Initial program 16.8%

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

   3. Taylor expanded in F around 0 22.8%

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

      1. mul-1-neg 22.8%

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

      2. *-commutative 22.8%

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

      3. associate-/l* 22.8%

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

      4. cancel-sign-sub-inv 22.8%

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

      5. metadata-eval 22.8%

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

      6. +-commutative 22.8%

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

   5. Simplified 30.7%

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

   6. Taylor expanded in A around -inf 25.9%

      \[\leadsto -\sqrt{2} \cdot \sqrt{F \cdot \color{blue}{\frac{-0.5}{A}}} \]

   if 4.9999999999999998e199 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 8.5%

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

   3. Taylor expanded in B around inf 29.5%

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

      1. mul-1-neg 29.5%

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

      2. *-commutative 29.5%

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

   5. Simplified 29.5%

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

      1. pow1/2 29.5%

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

      2. div-inv 29.5%

         \[\leadsto -\sqrt{2} \cdot {\color{blue}{\left(F \cdot \frac{1}{B}\right)}}^{0.5} \]

      3. unpow-prod-down 38.6%

         \[\leadsto -\sqrt{2} \cdot \color{blue}{\left({F}^{0.5} \cdot {\left(\frac{1}{B}\right)}^{0.5}\right)} \]

      4. pow1/2 38.6%

         \[\leadsto -\sqrt{2} \cdot \left(\color{blue}{\sqrt{F}} \cdot {\left(\frac{1}{B}\right)}^{0.5}\right) \]

   7. Applied egg-rr 38.6%

      \[\leadsto -\sqrt{2} \cdot \color{blue}{\left(\sqrt{F} \cdot {\left(\frac{1}{B}\right)}^{0.5}\right)} \]
   8. Step-by-step derivation

      1. unpow1/2 38.6%

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

   9. Simplified 38.6%

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

       1. neg-sub0 38.6%

          \[\leadsto \color{blue}{0 - \sqrt{2} \cdot \left(\sqrt{F} \cdot \sqrt{\frac{1}{B}}\right)} \]

       2. associate-*r* 38.6%

          \[\leadsto 0 - \color{blue}{\left(\sqrt{2} \cdot \sqrt{F}\right) \cdot \sqrt{\frac{1}{B}}} \]

       3. pow1/2 38.6%

          \[\leadsto 0 - \left(\color{blue}{{2}^{0.5}} \cdot \sqrt{F}\right) \cdot \sqrt{\frac{1}{B}} \]

       4. pow1/2 38.6%

          \[\leadsto 0 - \left({2}^{0.5} \cdot \color{blue}{{F}^{0.5}}\right) \cdot \sqrt{\frac{1}{B}} \]

       5. pow-prod-down 38.6%

          \[\leadsto 0 - \color{blue}{{\left(2 \cdot F\right)}^{0.5}} \cdot \sqrt{\frac{1}{B}} \]

       6. pow1/2 38.6%

          \[\leadsto 0 - {\left(2 \cdot F\right)}^{0.5} \cdot \color{blue}{{\left(\frac{1}{B}\right)}^{0.5}} \]

       7. inv-pow 38.6%

          \[\leadsto 0 - {\left(2 \cdot F\right)}^{0.5} \cdot {\color{blue}{\left({B}^{-1}\right)}}^{0.5} \]

       8. pow-pow 38.7%

          \[\leadsto 0 - {\left(2 \cdot F\right)}^{0.5} \cdot \color{blue}{{B}^{\left(-1 \cdot 0.5\right)}} \]

       9. metadata-eval 38.7%

          \[\leadsto 0 - {\left(2 \cdot F\right)}^{0.5} \cdot {B}^{\color{blue}{-0.5}} \]

   11. Applied egg-rr 38.7%

       \[\leadsto \color{blue}{0 - {\left(2 \cdot F\right)}^{0.5} \cdot {B}^{-0.5}} \]
   12. Step-by-step derivation

       1. neg-sub0 38.7%

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

       2. *-commutative 38.7%

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

       3. distribute-rgt-neg-in 38.7%

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

       4. unpow1/2 38.7%

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

   13. Simplified 38.7%

       \[\leadsto \color{blue}{{B}^{-0.5} \cdot \left(-\sqrt{2 \cdot F}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 30.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 2 \cdot 10^{+14}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{\left(4 \cdot A\right) \cdot C - B \cdot B}\\ \mathbf{elif}\;{B}^{2} \leq 5 \cdot 10^{+199}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{F \cdot \frac{-0.5}{A}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot F} \cdot \left(-{B}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 48.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 2 \cdot 10^{+14}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot C\right) \cdot \left(2 \cdot \left(-4 \cdot \left(A \cdot \left(C \cdot F\right)\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - {B\_m}^{2}}\\ \mathbf{elif}\;{B\_m}^{2} \leq 5 \cdot 10^{+199}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{F \cdot \frac{-0.5}{A}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot F} \cdot \left(-{B\_m}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 2e+14)
   (/
    (sqrt (* (* 2.0 C) (* 2.0 (* -4.0 (* A (* C F))))))
    (- (* (* 4.0 A) C) (pow B_m 2.0)))
   (if (<= (pow B_m 2.0) 5e+199)
     (* (sqrt 2.0) (- (sqrt (* F (/ -0.5 A)))))
     (* (sqrt (* 2.0 F)) (- (pow 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) {
	double tmp;
	if (pow(B_m, 2.0) <= 2e+14) {
		tmp = sqrt(((2.0 * C) * (2.0 * (-4.0 * (A * (C * F)))))) / (((4.0 * A) * C) - pow(B_m, 2.0));
	} else if (pow(B_m, 2.0) <= 5e+199) {
		tmp = sqrt(2.0) * -sqrt((F * (-0.5 / A)));
	} else {
		tmp = sqrt((2.0 * F)) * -pow(B_m, -0.5);
	}
	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) <= 2d+14) then
        tmp = sqrt(((2.0d0 * c) * (2.0d0 * ((-4.0d0) * (a * (c * f)))))) / (((4.0d0 * a) * c) - (b_m ** 2.0d0))
    else if ((b_m ** 2.0d0) <= 5d+199) then
        tmp = sqrt(2.0d0) * -sqrt((f * ((-0.5d0) / a)))
    else
        tmp = sqrt((2.0d0 * f)) * -(b_m ** (-0.5d0))
    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) <= 2e+14) {
		tmp = Math.sqrt(((2.0 * C) * (2.0 * (-4.0 * (A * (C * F)))))) / (((4.0 * A) * C) - Math.pow(B_m, 2.0));
	} else if (Math.pow(B_m, 2.0) <= 5e+199) {
		tmp = Math.sqrt(2.0) * -Math.sqrt((F * (-0.5 / A)));
	} else {
		tmp = Math.sqrt((2.0 * F)) * -Math.pow(B_m, -0.5);
	}
	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+14:
		tmp = math.sqrt(((2.0 * C) * (2.0 * (-4.0 * (A * (C * F)))))) / (((4.0 * A) * C) - math.pow(B_m, 2.0))
	elif math.pow(B_m, 2.0) <= 5e+199:
		tmp = math.sqrt(2.0) * -math.sqrt((F * (-0.5 / A)))
	else:
		tmp = math.sqrt((2.0 * F)) * -math.pow(B_m, -0.5)
	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+14)
		tmp = Float64(sqrt(Float64(Float64(2.0 * C) * Float64(2.0 * Float64(-4.0 * Float64(A * Float64(C * F)))))) / Float64(Float64(Float64(4.0 * A) * C) - (B_m ^ 2.0)));
	elseif ((B_m ^ 2.0) <= 5e+199)
		tmp = Float64(sqrt(2.0) * Float64(-sqrt(Float64(F * Float64(-0.5 / A)))));
	else
		tmp = Float64(sqrt(Float64(2.0 * F)) * Float64(-(B_m ^ -0.5)));
	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+14)
		tmp = sqrt(((2.0 * C) * (2.0 * (-4.0 * (A * (C * F)))))) / (((4.0 * A) * C) - (B_m ^ 2.0));
	elseif ((B_m ^ 2.0) <= 5e+199)
		tmp = sqrt(2.0) * -sqrt((F * (-0.5 / A)));
	else
		tmp = sqrt((2.0 * F)) * -(B_m ^ -0.5);
	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+14], N[(N[Sqrt[N[(N[(2.0 * C), $MachinePrecision] * N[(2.0 * N[(-4.0 * N[(A * N[(C * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision] - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e+199], N[(N[Sqrt[2.0], $MachinePrecision] * (-N[Sqrt[N[(F * N[(-0.5 / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] * (-N[Power[B$95$m, -0.5], $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 2e14

   1. Initial program 25.0%

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

   3. Taylor expanded in A around -inf 25.5%

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

   4. Taylor expanded in B around 0 22.1%

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

   if 2e14 < (pow.f64 B #s(literal 2 binary64)) < 4.9999999999999998e199

   1. Initial program 16.8%

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

   3. Taylor expanded in F around 0 22.8%

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

      1. mul-1-neg 22.8%

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

      2. *-commutative 22.8%

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

      3. associate-/l* 22.8%

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

      4. cancel-sign-sub-inv 22.8%

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

      5. metadata-eval 22.8%

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

      6. +-commutative 22.8%

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

   5. Simplified 30.7%

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

   6. Taylor expanded in A around -inf 25.9%

      \[\leadsto -\sqrt{2} \cdot \sqrt{F \cdot \color{blue}{\frac{-0.5}{A}}} \]

   if 4.9999999999999998e199 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 8.5%

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

   3. Taylor expanded in B around inf 29.5%

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

      1. mul-1-neg 29.5%

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

      2. *-commutative 29.5%

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

   5. Simplified 29.5%

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

      1. pow1/2 29.5%

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

      2. div-inv 29.5%

         \[\leadsto -\sqrt{2} \cdot {\color{blue}{\left(F \cdot \frac{1}{B}\right)}}^{0.5} \]

      3. unpow-prod-down 38.6%

         \[\leadsto -\sqrt{2} \cdot \color{blue}{\left({F}^{0.5} \cdot {\left(\frac{1}{B}\right)}^{0.5}\right)} \]

      4. pow1/2 38.6%

         \[\leadsto -\sqrt{2} \cdot \left(\color{blue}{\sqrt{F}} \cdot {\left(\frac{1}{B}\right)}^{0.5}\right) \]

   7. Applied egg-rr 38.6%

      \[\leadsto -\sqrt{2} \cdot \color{blue}{\left(\sqrt{F} \cdot {\left(\frac{1}{B}\right)}^{0.5}\right)} \]
   8. Step-by-step derivation

      1. unpow1/2 38.6%

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

   9. Simplified 38.6%

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

       1. neg-sub0 38.6%

          \[\leadsto \color{blue}{0 - \sqrt{2} \cdot \left(\sqrt{F} \cdot \sqrt{\frac{1}{B}}\right)} \]

       2. associate-*r* 38.6%

          \[\leadsto 0 - \color{blue}{\left(\sqrt{2} \cdot \sqrt{F}\right) \cdot \sqrt{\frac{1}{B}}} \]

       3. pow1/2 38.6%

          \[\leadsto 0 - \left(\color{blue}{{2}^{0.5}} \cdot \sqrt{F}\right) \cdot \sqrt{\frac{1}{B}} \]

       4. pow1/2 38.6%

          \[\leadsto 0 - \left({2}^{0.5} \cdot \color{blue}{{F}^{0.5}}\right) \cdot \sqrt{\frac{1}{B}} \]

       5. pow-prod-down 38.6%

          \[\leadsto 0 - \color{blue}{{\left(2 \cdot F\right)}^{0.5}} \cdot \sqrt{\frac{1}{B}} \]

       6. pow1/2 38.6%

          \[\leadsto 0 - {\left(2 \cdot F\right)}^{0.5} \cdot \color{blue}{{\left(\frac{1}{B}\right)}^{0.5}} \]

       7. inv-pow 38.6%

          \[\leadsto 0 - {\left(2 \cdot F\right)}^{0.5} \cdot {\color{blue}{\left({B}^{-1}\right)}}^{0.5} \]

       8. pow-pow 38.7%

          \[\leadsto 0 - {\left(2 \cdot F\right)}^{0.5} \cdot \color{blue}{{B}^{\left(-1 \cdot 0.5\right)}} \]

       9. metadata-eval 38.7%

          \[\leadsto 0 - {\left(2 \cdot F\right)}^{0.5} \cdot {B}^{\color{blue}{-0.5}} \]

   11. Applied egg-rr 38.7%

       \[\leadsto \color{blue}{0 - {\left(2 \cdot F\right)}^{0.5} \cdot {B}^{-0.5}} \]
   12. Step-by-step derivation

       1. neg-sub0 38.7%

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

       2. *-commutative 38.7%

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

       3. distribute-rgt-neg-in 38.7%

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

       4. unpow1/2 38.7%

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

   13. Simplified 38.7%

       \[\leadsto \color{blue}{{B}^{-0.5} \cdot \left(-\sqrt{2 \cdot F}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 28.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 2 \cdot 10^{+14}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot C\right) \cdot \left(2 \cdot \left(-4 \cdot \left(A \cdot \left(C \cdot F\right)\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;{B}^{2} \leq 5 \cdot 10^{+199}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{F \cdot \frac{-0.5}{A}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot F} \cdot \left(-{B}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 47.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 5 \cdot 10^{+199}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{F \cdot \frac{-0.5}{A}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot F} \cdot \left(-{B\_m}^{-0.5}\right)\\ \end{array} \end{array} \]

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) 5e+199)
   (* (sqrt 2.0) (- (sqrt (* F (/ -0.5 A)))))
   (* (sqrt (* 2.0 F)) (- (pow 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) {
	double tmp;
	if (pow(B_m, 2.0) <= 5e+199) {
		tmp = sqrt(2.0) * -sqrt((F * (-0.5 / A)));
	} else {
		tmp = sqrt((2.0 * F)) * -pow(B_m, -0.5);
	}
	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) <= 5d+199) then
        tmp = sqrt(2.0d0) * -sqrt((f * ((-0.5d0) / a)))
    else
        tmp = sqrt((2.0d0 * f)) * -(b_m ** (-0.5d0))
    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) <= 5e+199) {
		tmp = Math.sqrt(2.0) * -Math.sqrt((F * (-0.5 / A)));
	} else {
		tmp = Math.sqrt((2.0 * F)) * -Math.pow(B_m, -0.5);
	}
	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) <= 5e+199:
		tmp = math.sqrt(2.0) * -math.sqrt((F * (-0.5 / A)))
	else:
		tmp = math.sqrt((2.0 * F)) * -math.pow(B_m, -0.5)
	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) <= 5e+199)
		tmp = Float64(sqrt(2.0) * Float64(-sqrt(Float64(F * Float64(-0.5 / A)))));
	else
		tmp = Float64(sqrt(Float64(2.0 * F)) * Float64(-(B_m ^ -0.5)));
	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) <= 5e+199)
		tmp = sqrt(2.0) * -sqrt((F * (-0.5 / A)));
	else
		tmp = sqrt((2.0 * F)) * -(B_m ^ -0.5);
	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], 5e+199], N[(N[Sqrt[2.0], $MachinePrecision] * (-N[Sqrt[N[(F * N[(-0.5 / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] * (-N[Power[B$95$m, -0.5], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 4.9999999999999998e199

   1. Initial program 23.4%

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

   3. Taylor expanded in F around 0 17.4%

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

      1. mul-1-neg 17.4%

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

      2. *-commutative 17.4%

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

      3. associate-/l* 17.5%

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

      4. cancel-sign-sub-inv 17.5%

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

      5. metadata-eval 17.5%

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

      6. +-commutative 17.5%

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

   5. Simplified 22.5%

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

   6. Taylor expanded in A around -inf 19.5%

      \[\leadsto -\sqrt{2} \cdot \sqrt{F \cdot \color{blue}{\frac{-0.5}{A}}} \]

   if 4.9999999999999998e199 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 8.5%

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

   3. Taylor expanded in B around inf 29.5%

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

      1. mul-1-neg 29.5%

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

      2. *-commutative 29.5%

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

   5. Simplified 29.5%

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

      1. pow1/2 29.5%

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

      2. div-inv 29.5%

         \[\leadsto -\sqrt{2} \cdot {\color{blue}{\left(F \cdot \frac{1}{B}\right)}}^{0.5} \]

      3. unpow-prod-down 38.6%

         \[\leadsto -\sqrt{2} \cdot \color{blue}{\left({F}^{0.5} \cdot {\left(\frac{1}{B}\right)}^{0.5}\right)} \]

      4. pow1/2 38.6%

         \[\leadsto -\sqrt{2} \cdot \left(\color{blue}{\sqrt{F}} \cdot {\left(\frac{1}{B}\right)}^{0.5}\right) \]

   7. Applied egg-rr 38.6%

      \[\leadsto -\sqrt{2} \cdot \color{blue}{\left(\sqrt{F} \cdot {\left(\frac{1}{B}\right)}^{0.5}\right)} \]
   8. Step-by-step derivation

      1. unpow1/2 38.6%

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

   9. Simplified 38.6%

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

       1. neg-sub0 38.6%

          \[\leadsto \color{blue}{0 - \sqrt{2} \cdot \left(\sqrt{F} \cdot \sqrt{\frac{1}{B}}\right)} \]

       2. associate-*r* 38.6%

          \[\leadsto 0 - \color{blue}{\left(\sqrt{2} \cdot \sqrt{F}\right) \cdot \sqrt{\frac{1}{B}}} \]

       3. pow1/2 38.6%

          \[\leadsto 0 - \left(\color{blue}{{2}^{0.5}} \cdot \sqrt{F}\right) \cdot \sqrt{\frac{1}{B}} \]

       4. pow1/2 38.6%

          \[\leadsto 0 - \left({2}^{0.5} \cdot \color{blue}{{F}^{0.5}}\right) \cdot \sqrt{\frac{1}{B}} \]

       5. pow-prod-down 38.6%

          \[\leadsto 0 - \color{blue}{{\left(2 \cdot F\right)}^{0.5}} \cdot \sqrt{\frac{1}{B}} \]

       6. pow1/2 38.6%

          \[\leadsto 0 - {\left(2 \cdot F\right)}^{0.5} \cdot \color{blue}{{\left(\frac{1}{B}\right)}^{0.5}} \]

       7. inv-pow 38.6%

          \[\leadsto 0 - {\left(2 \cdot F\right)}^{0.5} \cdot {\color{blue}{\left({B}^{-1}\right)}}^{0.5} \]

       8. pow-pow 38.7%

          \[\leadsto 0 - {\left(2 \cdot F\right)}^{0.5} \cdot \color{blue}{{B}^{\left(-1 \cdot 0.5\right)}} \]

       9. metadata-eval 38.7%

          \[\leadsto 0 - {\left(2 \cdot F\right)}^{0.5} \cdot {B}^{\color{blue}{-0.5}} \]

   11. Applied egg-rr 38.7%

       \[\leadsto \color{blue}{0 - {\left(2 \cdot F\right)}^{0.5} \cdot {B}^{-0.5}} \]
   12. Step-by-step derivation

       1. neg-sub0 38.7%

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

       2. *-commutative 38.7%

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

       3. distribute-rgt-neg-in 38.7%

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

       4. unpow1/2 38.7%

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

   13. Simplified 38.7%

       \[\leadsto \color{blue}{{B}^{-0.5} \cdot \left(-\sqrt{2 \cdot F}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 26.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 5 \cdot 10^{+199}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{F \cdot \frac{-0.5}{A}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot F} \cdot \left(-{B}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 35.8% accurate, 3.1× speedup

Math

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

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 F)) (- (pow 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 sqrt((2.0 * F)) * -pow(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 = sqrt((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.sqrt((2.0 * F)) * -Math.pow(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.sqrt((2.0 * F)) * -math.pow(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(sqrt(Float64(2.0 * F)) * Float64(-(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 = sqrt((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[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] * (-N[Power[B$95$m, -0.5], $MachinePrecision])), $MachinePrecision]

Derivation

1. Initial program 18.3%

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

3. Taylor expanded in B around inf 13.0%

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

   1. mul-1-neg 13.0%

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

   2. *-commutative 13.0%

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

5. Simplified 13.0%

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

   1. pow1/2 13.2%

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

   2. div-inv 13.2%

      \[\leadsto -\sqrt{2} \cdot {\color{blue}{\left(F \cdot \frac{1}{B}\right)}}^{0.5} \]

   3. unpow-prod-down 16.4%

      \[\leadsto -\sqrt{2} \cdot \color{blue}{\left({F}^{0.5} \cdot {\left(\frac{1}{B}\right)}^{0.5}\right)} \]

   4. pow1/2 16.4%

      \[\leadsto -\sqrt{2} \cdot \left(\color{blue}{\sqrt{F}} \cdot {\left(\frac{1}{B}\right)}^{0.5}\right) \]

7. Applied egg-rr 16.4%

   \[\leadsto -\sqrt{2} \cdot \color{blue}{\left(\sqrt{F} \cdot {\left(\frac{1}{B}\right)}^{0.5}\right)} \]
8. Step-by-step derivation

   1. unpow1/2 16.4%

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

9. Simplified 16.4%

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

    1. neg-sub0 16.4%

       \[\leadsto \color{blue}{0 - \sqrt{2} \cdot \left(\sqrt{F} \cdot \sqrt{\frac{1}{B}}\right)} \]

    2. associate-*r* 16.4%

       \[\leadsto 0 - \color{blue}{\left(\sqrt{2} \cdot \sqrt{F}\right) \cdot \sqrt{\frac{1}{B}}} \]

    3. pow1/2 16.4%

       \[\leadsto 0 - \left(\color{blue}{{2}^{0.5}} \cdot \sqrt{F}\right) \cdot \sqrt{\frac{1}{B}} \]

    4. pow1/2 16.4%

       \[\leadsto 0 - \left({2}^{0.5} \cdot \color{blue}{{F}^{0.5}}\right) \cdot \sqrt{\frac{1}{B}} \]

    5. pow-prod-down 16.4%

       \[\leadsto 0 - \color{blue}{{\left(2 \cdot F\right)}^{0.5}} \cdot \sqrt{\frac{1}{B}} \]

    6. pow1/2 16.4%

       \[\leadsto 0 - {\left(2 \cdot F\right)}^{0.5} \cdot \color{blue}{{\left(\frac{1}{B}\right)}^{0.5}} \]

    7. inv-pow 16.4%

       \[\leadsto 0 - {\left(2 \cdot F\right)}^{0.5} \cdot {\color{blue}{\left({B}^{-1}\right)}}^{0.5} \]

    8. pow-pow 16.4%

       \[\leadsto 0 - {\left(2 \cdot F\right)}^{0.5} \cdot \color{blue}{{B}^{\left(-1 \cdot 0.5\right)}} \]

    9. metadata-eval 16.4%

       \[\leadsto 0 - {\left(2 \cdot F\right)}^{0.5} \cdot {B}^{\color{blue}{-0.5}} \]

11. Applied egg-rr 16.4%

    \[\leadsto \color{blue}{0 - {\left(2 \cdot F\right)}^{0.5} \cdot {B}^{-0.5}} \]
12. Step-by-step derivation

    1. neg-sub0 16.4%

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

    2. *-commutative 16.4%

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

    3. distribute-rgt-neg-in 16.4%

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

    4. unpow1/2 16.4%

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

13. Simplified 16.4%

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

14. Final simplification 16.4%

    \[\leadsto \sqrt{2 \cdot F} \cdot \left(-{B}^{-0.5}\right) \]
15. Add Preprocessing

Alternative 13: 35.8% accurate, 3.1× speedup

Math

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

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 F) (- (sqrt (/ 2.0 B_m)))))

C

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 18.3%

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

3. Taylor expanded in B around inf 13.0%

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

   1. mul-1-neg 13.0%

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

   2. *-commutative 13.0%

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

5. Simplified 13.0%

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

   1. *-commutative 13.0%

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

   2. pow1/2 13.2%

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

   3. pow1/2 13.2%

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

   4. pow-prod-down 13.3%

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

7. Applied egg-rr 13.3%

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

   1. unpow1/2 13.0%

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

9. Simplified 13.0%

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

    1. associate-*l/ 13.0%

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

11. Applied egg-rr 13.0%

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

    1. associate-/l* 13.0%

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

13. Simplified 13.0%

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

    1. pow1/2 13.3%

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

    2. *-commutative 13.3%

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

    3. unpow-prod-down 16.4%

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

    4. pow1/2 16.4%

       \[\leadsto -\color{blue}{\sqrt{\frac{2}{B}}} \cdot {F}^{0.5} \]

    5. pow1/2 16.4%

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

15. Applied egg-rr 16.4%

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

16. Final simplification 16.4%

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

Alternative 14: 27.8% accurate, 5.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;C \leq 7.2 \cdot 10^{+128}:\\ \;\;\;\;-\sqrt{F \cdot \frac{2}{B\_m}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-0.5 \cdot \left(B\_m \cdot \sqrt{\frac{1}{C \cdot F}}\right)}\\ \end{array} \end{array} \]

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 (<= C 7.2e+128)
   (- (sqrt (* F (/ 2.0 B_m))))
   (/ 1.0 (* -0.5 (* B_m (sqrt (/ 1.0 (* C F))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (C <= 7.2e+128) {
		tmp = -sqrt((F * (2.0 / B_m)));
	} else {
		tmp = 1.0 / (-0.5 * (B_m * sqrt((1.0 / (C * F)))));
	}
	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 (c <= 7.2d+128) then
        tmp = -sqrt((f * (2.0d0 / b_m)))
    else
        tmp = 1.0d0 / ((-0.5d0) * (b_m * sqrt((1.0d0 / (c * f)))))
    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 (C <= 7.2e+128) {
		tmp = -Math.sqrt((F * (2.0 / B_m)));
	} else {
		tmp = 1.0 / (-0.5 * (B_m * Math.sqrt((1.0 / (C * F)))));
	}
	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 C <= 7.2e+128:
		tmp = -math.sqrt((F * (2.0 / B_m)))
	else:
		tmp = 1.0 / (-0.5 * (B_m * math.sqrt((1.0 / (C * F)))))
	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 (C <= 7.2e+128)
		tmp = Float64(-sqrt(Float64(F * Float64(2.0 / B_m))));
	else
		tmp = Float64(1.0 / Float64(-0.5 * Float64(B_m * sqrt(Float64(1.0 / Float64(C * F))))));
	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 (C <= 7.2e+128)
		tmp = -sqrt((F * (2.0 / B_m)));
	else
		tmp = 1.0 / (-0.5 * (B_m * sqrt((1.0 / (C * F)))));
	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[C, 7.2e+128], (-N[Sqrt[N[(F * N[(2.0 / B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), N[(1.0 / N[(-0.5 * N[(B$95$m * N[Sqrt[N[(1.0 / N[(C * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if C < 7.20000000000000054e128

   1. Initial program 19.8%

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

   3. Taylor expanded in B around inf 14.2%

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

      1. mul-1-neg 14.2%

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

      2. *-commutative 14.2%

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

   5. Simplified 14.2%

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

      1. *-commutative 14.2%

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

      2. pow1/2 14.5%

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

      3. pow1/2 14.5%

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

      4. pow-prod-down 14.6%

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

   7. Applied egg-rr 14.6%

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

      1. unpow1/2 14.3%

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

   9. Simplified 14.3%

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

       1. associate-*l/ 14.2%

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

   11. Applied egg-rr 14.2%

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

       1. associate-/l* 14.3%

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

   13. Simplified 14.3%

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

   if 7.20000000000000054e128 < C

   1. Initial program 9.7%

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

   3. Taylor expanded in A around -inf 34.3%

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

      1. clear-num 34.3%

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

      2. inv-pow 34.3%

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

      3. associate-*l* 34.3%

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

      4. associate-*l* 34.3%

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

      5. associate-*l* 34.3%

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

   5. Applied egg-rr 34.3%

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

      1. unpow-1 34.3%

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

      2. cancel-sign-sub-inv 34.3%

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

      3. metadata-eval 34.3%

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

      4. associate-*r* 34.3%

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

      5. cancel-sign-sub-inv 34.3%

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

      6. metadata-eval 34.3%

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

      7. associate-*r* 34.3%

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

   7. Simplified 34.3%

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

   8. Taylor expanded in B around inf 5.8%

      \[\leadsto \frac{1}{\color{blue}{-0.5 \cdot \left(B \cdot \sqrt{\frac{1}{C \cdot F}}\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 27.7% accurate, 5.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;C \leq 3.8 \cdot 10^{+128}:\\ \;\;\;\;-\sqrt{F \cdot \frac{2}{B\_m}}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\sqrt{C \cdot F}}{B\_m}\\ \end{array} \end{array} \]

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 (<= C 3.8e+128)
   (- (sqrt (* F (/ 2.0 B_m))))
   (* -2.0 (/ (sqrt (* C 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 (C <= 3.8e+128) {
		tmp = -sqrt((F * (2.0 / B_m)));
	} else {
		tmp = -2.0 * (sqrt((C * 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 (c <= 3.8d+128) then
        tmp = -sqrt((f * (2.0d0 / b_m)))
    else
        tmp = (-2.0d0) * (sqrt((c * 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 (C <= 3.8e+128) {
		tmp = -Math.sqrt((F * (2.0 / B_m)));
	} else {
		tmp = -2.0 * (Math.sqrt((C * 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 C <= 3.8e+128:
		tmp = -math.sqrt((F * (2.0 / B_m)))
	else:
		tmp = -2.0 * (math.sqrt((C * 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 (C <= 3.8e+128)
		tmp = Float64(-sqrt(Float64(F * Float64(2.0 / B_m))));
	else
		tmp = Float64(-2.0 * Float64(sqrt(Float64(C * 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 (C <= 3.8e+128)
		tmp = -sqrt((F * (2.0 / B_m)));
	else
		tmp = -2.0 * (sqrt((C * 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[C, 3.8e+128], (-N[Sqrt[N[(F * N[(2.0 / B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), N[(-2.0 * N[(N[Sqrt[N[(C * F), $MachinePrecision]], $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if C < 3.7999999999999999e128

   1. Initial program 19.8%

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

   3. Taylor expanded in B around inf 14.2%

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

      1. mul-1-neg 14.2%

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

      2. *-commutative 14.2%

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

   5. Simplified 14.2%

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

      1. *-commutative 14.2%

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

      2. pow1/2 14.5%

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

      3. pow1/2 14.5%

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

      4. pow-prod-down 14.6%

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

   7. Applied egg-rr 14.6%

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

      1. unpow1/2 14.3%

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

   9. Simplified 14.3%

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

       1. associate-*l/ 14.2%

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

   11. Applied egg-rr 14.2%

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

       1. associate-/l* 14.3%

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

   13. Simplified 14.3%

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

   if 3.7999999999999999e128 < C

   1. Initial program 9.7%

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

   3. Taylor expanded in A around -inf 34.3%

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

      1. clear-num 34.3%

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

      2. inv-pow 34.3%

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

      3. associate-*l* 34.3%

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

      4. associate-*l* 34.3%

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

      5. associate-*l* 34.3%

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

   5. Applied egg-rr 34.3%

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

      1. unpow-1 34.3%

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

      2. cancel-sign-sub-inv 34.3%

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

      3. metadata-eval 34.3%

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

      4. associate-*r* 34.3%

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

      5. cancel-sign-sub-inv 34.3%

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

      6. metadata-eval 34.3%

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

      7. associate-*r* 34.3%

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

   7. Simplified 34.3%

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

   8. Taylor expanded in A around 0 5.2%

      \[\leadsto \frac{1}{\color{blue}{-0.5 \cdot \left(B \cdot \sqrt{\frac{1}{C \cdot F}}\right) + \frac{A}{B} \cdot \sqrt{\frac{C}{F}}}} \]

   9. Taylor expanded in B around inf 5.5%

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

       1. associate-*l/ 5.5%

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

       2. *-lft-identity 5.5%

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

       3. *-commutative 5.5%

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

   11. Simplified 5.5%

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

4. Final simplification 12.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq 3.8 \cdot 10^{+128}:\\ \;\;\;\;-\sqrt{F \cdot \frac{2}{B}}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \frac{\sqrt{C \cdot F}}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 27.1% accurate, 6.0× speedup

Math

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

FPCore

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

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	return -sqrt((F * (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((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((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((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 Float64(-sqrt(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((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[(F * N[(2.0 / B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])

Derivation

1. Initial program 18.3%

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

3. Taylor expanded in B around inf 13.0%

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

   1. mul-1-neg 13.0%

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

   2. *-commutative 13.0%

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

5. Simplified 13.0%

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

   1. *-commutative 13.0%

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

   2. pow1/2 13.2%

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

   3. pow1/2 13.2%

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

   4. pow-prod-down 13.3%

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

7. Applied egg-rr 13.3%

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

   1. unpow1/2 13.0%

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

9. Simplified 13.0%

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

    1. associate-*l/ 13.0%

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

11. Applied egg-rr 13.0%

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

    1. associate-/l* 13.0%

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

13. Simplified 13.0%

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

Reproduce

herbie shell --seed 2024110 
(FPCore (A B C F)
  :name "ABCF->ab-angle a"
  :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))))