ABCF->ab-angle a

Percentage Accurate: 18.8% → 54.7%

Time: 48.9s

Alternatives: 21

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: 18.8% 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: 54.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := A + \left(C + \mathsf{hypot}\left(B\_m, A - C\right)\right)\\ t_1 := A \cdot \left(C \cdot -4\right)\\ t_2 := \mathsf{fma}\left(B\_m, B\_m, t\_1\right)\\ t_3 := \left(4 \cdot A\right) \cdot C\\ t_4 := t\_3 - {B\_m}^{2}\\ t_5 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_3\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_4}\\ t_6 := 4 \cdot \left(A \cdot C\right) - {B\_m}^{2}\\ \mathbf{if}\;t\_5 \leq -\infty:\\ \;\;\;\;\frac{\sqrt{t\_0} \cdot \left(\mathsf{hypot}\left(\sqrt{t\_1}, B\_m\right) \cdot \sqrt{2 \cdot F}\right)}{t\_4}\\ \mathbf{elif}\;t\_5 \leq -2 \cdot 10^{-167}:\\ \;\;\;\;\frac{\sqrt{F \cdot t\_0} \cdot \sqrt{2 \cdot \mathsf{fma}\left(A, C \cdot -4, {B\_m}^{2}\right)}}{t\_6}\\ \mathbf{elif}\;t\_5 \leq 0:\\ \;\;\;\;\frac{\left|\sqrt{A \cdot -16} \cdot \left(C \cdot \sqrt{F}\right)\right|}{t\_6}\\ \mathbf{elif}\;t\_5 \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(A + C\right) + \mathsf{hypot}\left(A - C, B\_m\right)} \cdot \sqrt{\left(2 \cdot F\right) \cdot t\_2}}{-t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B\_m} \cdot \left(\sqrt{F} \cdot \left(-\sqrt{C + \mathsf{hypot}\left(B\_m, C\right)}\right)\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (+ A (+ C (hypot B_m (- A C)))))
        (t_1 (* A (* C -4.0)))
        (t_2 (fma B_m B_m t_1))
        (t_3 (* (* 4.0 A) C))
        (t_4 (- t_3 (pow B_m 2.0)))
        (t_5
         (/
          (sqrt
           (*
            (* 2.0 (* (- (pow B_m 2.0) t_3) F))
            (+ (+ A C) (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))))))
          t_4))
        (t_6 (- (* 4.0 (* A C)) (pow B_m 2.0))))
   (if (<= t_5 (- INFINITY))
     (/ (* (sqrt t_0) (* (hypot (sqrt t_1) B_m) (sqrt (* 2.0 F)))) t_4)
     (if (<= t_5 -2e-167)
       (/
        (* (sqrt (* F t_0)) (sqrt (* 2.0 (fma A (* C -4.0) (pow B_m 2.0)))))
        t_6)
       (if (<= t_5 0.0)
         (/ (fabs (* (sqrt (* A -16.0)) (* C (sqrt F)))) t_6)
         (if (<= t_5 INFINITY)
           (/
            (* (sqrt (+ (+ A C) (hypot (- A C) B_m))) (sqrt (* (* 2.0 F) t_2)))
            (- t_2))
           (*
            (/ (sqrt 2.0) B_m)
            (* (sqrt F) (- (sqrt (+ C (hypot B_m C))))))))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double t_0 = A + (C + hypot(B_m, (A - C)));
	double t_1 = A * (C * -4.0);
	double t_2 = fma(B_m, B_m, t_1);
	double t_3 = (4.0 * A) * C;
	double t_4 = t_3 - pow(B_m, 2.0);
	double t_5 = sqrt(((2.0 * ((pow(B_m, 2.0) - t_3) * F)) * ((A + C) + sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / t_4;
	double t_6 = (4.0 * (A * C)) - pow(B_m, 2.0);
	double tmp;
	if (t_5 <= -((double) INFINITY)) {
		tmp = (sqrt(t_0) * (hypot(sqrt(t_1), B_m) * sqrt((2.0 * F)))) / t_4;
	} else if (t_5 <= -2e-167) {
		tmp = (sqrt((F * t_0)) * sqrt((2.0 * fma(A, (C * -4.0), pow(B_m, 2.0))))) / t_6;
	} else if (t_5 <= 0.0) {
		tmp = fabs((sqrt((A * -16.0)) * (C * sqrt(F)))) / t_6;
	} else if (t_5 <= ((double) INFINITY)) {
		tmp = (sqrt(((A + C) + hypot((A - C), B_m))) * sqrt(((2.0 * F) * t_2))) / -t_2;
	} else {
		tmp = (sqrt(2.0) / B_m) * (sqrt(F) * -sqrt((C + hypot(B_m, C))));
	}
	return tmp;
}

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	t_0 = Float64(A + Float64(C + hypot(B_m, Float64(A - C))))
	t_1 = Float64(A * Float64(C * -4.0))
	t_2 = fma(B_m, B_m, t_1)
	t_3 = Float64(Float64(4.0 * A) * C)
	t_4 = Float64(t_3 - (B_m ^ 2.0))
	t_5 = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_3) * F)) * Float64(Float64(A + C) + sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / t_4)
	t_6 = Float64(Float64(4.0 * Float64(A * C)) - (B_m ^ 2.0))
	tmp = 0.0
	if (t_5 <= Float64(-Inf))
		tmp = Float64(Float64(sqrt(t_0) * Float64(hypot(sqrt(t_1), B_m) * sqrt(Float64(2.0 * F)))) / t_4);
	elseif (t_5 <= -2e-167)
		tmp = Float64(Float64(sqrt(Float64(F * t_0)) * sqrt(Float64(2.0 * fma(A, Float64(C * -4.0), (B_m ^ 2.0))))) / t_6);
	elseif (t_5 <= 0.0)
		tmp = Float64(abs(Float64(sqrt(Float64(A * -16.0)) * Float64(C * sqrt(F)))) / t_6);
	elseif (t_5 <= Inf)
		tmp = Float64(Float64(sqrt(Float64(Float64(A + C) + hypot(Float64(A - C), B_m))) * sqrt(Float64(Float64(2.0 * F) * t_2))) / Float64(-t_2));
	else
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(sqrt(F) * Float64(-sqrt(Float64(C + hypot(B_m, C))))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(A + N[(C + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(B$95$m * B$95$m + t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$3), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$4), $MachinePrecision]}, Block[{t$95$6 = N[(N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision] - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$5, (-Infinity)], N[(N[(N[Sqrt[t$95$0], $MachinePrecision] * N[(N[Sqrt[N[Sqrt[t$95$1], $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision] * N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$4), $MachinePrecision], If[LessEqual[t$95$5, -2e-167], N[(N[(N[Sqrt[N[(F * t$95$0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(2.0 * N[(A * N[(C * -4.0), $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$6), $MachinePrecision], If[LessEqual[t$95$5, 0.0], N[(N[Abs[N[(N[Sqrt[N[(A * -16.0), $MachinePrecision]], $MachinePrecision] * N[(C * N[Sqrt[F], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$6), $MachinePrecision], If[LessEqual[t$95$5, Infinity], N[(N[(N[Sqrt[N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] * t$95$2), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / (-t$95$2)), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[(N[Sqrt[F], $MachinePrecision] * (-N[Sqrt[N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 3.4%

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

      1. pow1/2 3.4%

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

      2. *-commutative 3.4%

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

      3. +-commutative 3.4%

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

      4. unpow2 3.4%

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

      5. unpow2 3.4%

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

      6. hypot-undefine 15.5%

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

      7. unpow-prod-down 41.4%

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

   4. Applied egg-rr 41.4%

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

   6. Simplified 43.1%

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

      1. sqrt-prod 67.3%

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

      2. fma-undefine 67.3%

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

      3. add-sqr-sqrt 57.3%

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

      4. unpow2 57.3%

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

      5. hypot-define 57.3%

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

   8. Applied egg-rr 57.3%

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

   if -inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))) < -2e-167

   1. Initial program 96.0%

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

   3. Simplified 88.4%

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

      1. pow1/2 88.4%

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

      2. associate-*r* 97.0%

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

      3. unpow-prod-down 98.6%

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

      4. +-commutative 98.6%

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

      5. hypot-undefine 98.6%

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

      6. unpow2 98.6%

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

      7. unpow2 98.6%

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

      8. +-commutative 98.6%

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

      9. unpow2 98.6%

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

      10. unpow2 98.6%

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

      11. hypot-define 98.6%

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

      12. pow1/2 98.6%

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

   6. Applied egg-rr 98.6%

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

      1. unpow1/2 98.6%

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

      2. +-commutative 98.6%

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

      3. hypot-undefine 98.6%

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

      4. unpow2 98.6%

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

      5. unpow2 98.6%

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

      6. +-commutative 98.6%

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

      7. associate-+r+ 98.6%

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

      8. unpow2 98.6%

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

      9. unpow2 98.6%

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

      10. hypot-undefine 98.6%

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

   8. Simplified 98.6%

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

   if -2e-167 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))) < -0.0

   1. Initial program 3.7%

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

   3. Simplified 7.2%

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

   5. Taylor expanded in A around -inf 19.7%

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

      1. associate-*r* 19.7%

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

   7. Simplified 19.7%

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

      1. add-sqr-sqrt 19.7%

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

      2. rem-sqrt-square 19.7%

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

      3. sqrt-prod 19.3%

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

      4. sqrt-prod 24.8%

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

      5. unpow2 24.8%

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

      6. sqrt-prod 11.3%

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

      7. add-sqr-sqrt 30.1%

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

   9. Applied egg-rr 30.1%

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

   if -0.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))) < +inf.0

   1. Initial program 31.5%

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

   3. Simplified 63.4%

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

      1. pow1/2 63.4%

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

      2. *-commutative 63.4%

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

      3. unpow-prod-down 83.4%

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

      4. pow1/2 83.4%

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

      5. +-commutative 83.4%

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

      6. hypot-undefine 35.5%

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

      7. unpow2 35.5%

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

      8. unpow2 35.5%

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

      9. +-commutative 35.5%

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

      10. unpow2 35.5%

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

      11. unpow2 35.5%

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

      12. hypot-define 83.4%

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

      13. pow1/2 83.4%

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

      14. *-commutative 83.4%

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

   6. Applied egg-rr 83.4%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in A around 0 1.6%

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

      1. mul-1-neg 1.6%

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

      2. distribute-rgt-neg-in 1.6%

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

      3. unpow2 1.6%

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

      4. unpow2 1.6%

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

      5. hypot-define 19.7%

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

   5. Simplified 19.7%

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

      1. pow1/2 19.8%

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

      2. *-commutative 19.8%

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

      3. unpow-prod-down 27.4%

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

      4. pow1/2 27.4%

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

      5. pow1/2 27.4%

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

   7. Applied egg-rr 27.4%

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

4. Final simplification 48.6%

   \[\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{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(\mathsf{hypot}\left(\sqrt{A \cdot \left(C \cdot -4\right)}, B\right) \cdot \sqrt{2 \cdot 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 -2 \cdot 10^{-167}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(A, C \cdot -4, {B}^{2}\right)}}{4 \cdot \left(A \cdot C\right) - {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 0:\\ \;\;\;\;\frac{\left|\sqrt{A \cdot -16} \cdot \left(C \cdot \sqrt{F}\right)\right|}{4 \cdot \left(A \cdot C\right) - {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(A + C\right) + \mathsf{hypot}\left(A - C, B\right)} \cdot \sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{F} \cdot \left(-\sqrt{C + \mathsf{hypot}\left(B, C\right)}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 49.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := \mathsf{fma}\left(A, C \cdot -4, {B\_m}^{2}\right)\\ t_1 := A + \left(C + \mathsf{hypot}\left(B\_m, A - C\right)\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 1.5 \cdot 10^{-176}:\\ \;\;\;\;\frac{\sqrt{t\_1} \cdot \sqrt{\left(2 \cdot F\right) \cdot t\_0}}{\left(A \cdot C\right) \cdot \left(--4\right)}\\ \mathbf{elif}\;{B\_m}^{2} \leq 2 \cdot 10^{-58}:\\ \;\;\;\;\frac{B\_m \cdot {\left(\sqrt{\sqrt{2 \cdot \left(F \cdot \left(A + \mathsf{hypot}\left(B\_m, A\right)\right)\right)}}\right)}^{2}}{\left(4 \cdot A\right) \cdot C - {B\_m}^{2}}\\ \mathbf{elif}\;{B\_m}^{2} \leq 10^{+96}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\left(F \cdot t\_1\right) \cdot t\_0\right)}}{4 \cdot \left(A \cdot C\right) - {B\_m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B\_m} \cdot \left(\sqrt{F} \cdot \left(-\sqrt{C + \mathsf{hypot}\left(B\_m, C\right)}\right)\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma A (* C -4.0) (pow B_m 2.0)))
        (t_1 (+ A (+ C (hypot B_m (- A C))))))
   (if (<= (pow B_m 2.0) 1.5e-176)
     (/ (* (sqrt t_1) (sqrt (* (* 2.0 F) t_0))) (* (* A C) (- -4.0)))
     (if (<= (pow B_m 2.0) 2e-58)
       (/
        (* B_m (pow (sqrt (sqrt (* 2.0 (* F (+ A (hypot B_m A)))))) 2.0))
        (- (* (* 4.0 A) C) (pow B_m 2.0)))
       (if (<= (pow B_m 2.0) 1e+96)
         (/ (sqrt (* 2.0 (* (* F t_1) t_0))) (- (* 4.0 (* A C)) (pow B_m 2.0)))
         (*
          (/ (sqrt 2.0) B_m)
          (* (sqrt F) (- (sqrt (+ C (hypot B_m C)))))))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(A, (C * -4.0), pow(B_m, 2.0));
	double t_1 = A + (C + hypot(B_m, (A - C)));
	double tmp;
	if (pow(B_m, 2.0) <= 1.5e-176) {
		tmp = (sqrt(t_1) * sqrt(((2.0 * F) * t_0))) / ((A * C) * -(-4.0));
	} else if (pow(B_m, 2.0) <= 2e-58) {
		tmp = (B_m * pow(sqrt(sqrt((2.0 * (F * (A + hypot(B_m, A)))))), 2.0)) / (((4.0 * A) * C) - pow(B_m, 2.0));
	} else if (pow(B_m, 2.0) <= 1e+96) {
		tmp = sqrt((2.0 * ((F * t_1) * t_0))) / ((4.0 * (A * C)) - pow(B_m, 2.0));
	} else {
		tmp = (sqrt(2.0) / B_m) * (sqrt(F) * -sqrt((C + hypot(B_m, C))));
	}
	return tmp;
}

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	t_0 = fma(A, Float64(C * -4.0), (B_m ^ 2.0))
	t_1 = Float64(A + Float64(C + hypot(B_m, Float64(A - C))))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1.5e-176)
		tmp = Float64(Float64(sqrt(t_1) * sqrt(Float64(Float64(2.0 * F) * t_0))) / Float64(Float64(A * C) * Float64(-(-4.0))));
	elseif ((B_m ^ 2.0) <= 2e-58)
		tmp = Float64(Float64(B_m * (sqrt(sqrt(Float64(2.0 * Float64(F * Float64(A + hypot(B_m, A)))))) ^ 2.0)) / Float64(Float64(Float64(4.0 * A) * C) - (B_m ^ 2.0)));
	elseif ((B_m ^ 2.0) <= 1e+96)
		tmp = Float64(sqrt(Float64(2.0 * Float64(Float64(F * t_1) * t_0))) / Float64(Float64(4.0 * Float64(A * C)) - (B_m ^ 2.0)));
	else
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(sqrt(F) * Float64(-sqrt(Float64(C + hypot(B_m, C))))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(A * N[(C * -4.0), $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(A + N[(C + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1.5e-176], N[(N[(N[Sqrt[t$95$1], $MachinePrecision] * N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[(A * C), $MachinePrecision] * (--4.0)), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e-58], N[(N[(B$95$m * N[Power[N[Sqrt[N[Sqrt[N[(2.0 * N[(F * N[(A + N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision], 2.0], $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], 1e+96], N[(N[Sqrt[N[(2.0 * N[(N[(F * t$95$1), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision] - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[(N[Sqrt[F], $MachinePrecision] * (-N[Sqrt[N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (pow.f64 B 2) < 1.5e-176

   1. Initial program 17.0%

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

      1. pow1/2 17.2%

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

      2. *-commutative 17.2%

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

      3. +-commutative 17.2%

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

      4. unpow2 17.2%

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

      5. unpow2 17.2%

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

      6. hypot-undefine 28.3%

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

      7. unpow-prod-down 35.6%

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

   4. Applied egg-rr 35.7%

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

   6. Simplified 36.6%

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

   7. Taylor expanded in B around 0 33.4%

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

   if 1.5e-176 < (pow.f64 B 2) < 2.0000000000000001e-58

   1. Initial program 26.3%

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

   3. Taylor expanded in C around 0 14.4%

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

      1. associate-*l* 14.3%

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

      2. +-commutative 14.3%

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

      3. unpow2 14.3%

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

      4. unpow2 14.3%

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

      5. hypot-define 14.8%

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

   5. Simplified 14.8%

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

      1. add-sqr-sqrt 14.7%

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

      2. pow2 14.7%

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

      3. sqrt-unprod 14.7%

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

   7. Applied egg-rr 14.7%

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

   if 2.0000000000000001e-58 < (pow.f64 B 2) < 1.00000000000000005e96

   1. Initial program 40.8%

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

   3. Simplified 40.9%

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

   5. Taylor expanded in F around 0 40.6%

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

      1. associate-*r* 40.7%

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

      2. unpow2 40.7%

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

      3. unpow2 40.7%

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

      4. hypot-undefine 47.6%

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

      5. *-commutative 47.6%

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

      6. associate-*r* 47.6%

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

      7. fma-undefine 47.6%

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

   7. Simplified 47.6%

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

   if 1.00000000000000005e96 < (pow.f64 B 2)

   1. Initial program 11.3%

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

   3. Taylor expanded in A around 0 7.5%

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

      1. mul-1-neg 7.5%

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

      2. distribute-rgt-neg-in 7.5%

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

      3. unpow2 7.5%

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

      4. unpow2 7.5%

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

      5. hypot-define 26.6%

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

   5. Simplified 26.6%

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

      1. pow1/2 26.7%

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

      2. *-commutative 26.7%

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

      3. unpow-prod-down 38.1%

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

      4. pow1/2 38.1%

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

      5. pow1/2 38.1%

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

   7. Applied egg-rr 38.1%

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

4. Final simplification 35.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 1.5 \cdot 10^{-176}:\\ \;\;\;\;\frac{\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(A, C \cdot -4, {B}^{2}\right)}}{\left(A \cdot C\right) \cdot \left(--4\right)}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{-58}:\\ \;\;\;\;\frac{B \cdot {\left(\sqrt{\sqrt{2 \cdot \left(F \cdot \left(A + \mathsf{hypot}\left(B, A\right)\right)\right)}}\right)}^{2}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;{B}^{2} \leq 10^{+96}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\left(F \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right) \cdot \mathsf{fma}\left(A, C \cdot -4, {B}^{2}\right)\right)}}{4 \cdot \left(A \cdot C\right) - {B}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{F} \cdot \left(-\sqrt{C + \mathsf{hypot}\left(B, C\right)}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 52.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 10^{+96}:\\ \;\;\;\;\frac{\sqrt{A + \left(C + \mathsf{hypot}\left(B\_m, A - C\right)\right)} \cdot \sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(A, C \cdot -4, {B\_m}^{2}\right)}}{\left(4 \cdot A\right) \cdot C - {B\_m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B\_m} \cdot \left(\sqrt{F} \cdot \left(-\sqrt{C + \mathsf{hypot}\left(B\_m, C\right)}\right)\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 1e+96)
   (/
    (*
     (sqrt (+ A (+ C (hypot B_m (- A C)))))
     (sqrt (* (* 2.0 F) (fma A (* C -4.0) (pow B_m 2.0)))))
    (- (* (* 4.0 A) C) (pow B_m 2.0)))
   (* (/ (sqrt 2.0) B_m) (* (sqrt F) (- (sqrt (+ C (hypot B_m C))))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (pow(B_m, 2.0) <= 1e+96) {
		tmp = (sqrt((A + (C + hypot(B_m, (A - C))))) * sqrt(((2.0 * F) * fma(A, (C * -4.0), pow(B_m, 2.0))))) / (((4.0 * A) * C) - pow(B_m, 2.0));
	} else {
		tmp = (sqrt(2.0) / B_m) * (sqrt(F) * -sqrt((C + hypot(B_m, C))));
	}
	return tmp;
}

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e+96)
		tmp = Float64(Float64(sqrt(Float64(A + Float64(C + hypot(B_m, Float64(A - C))))) * sqrt(Float64(Float64(2.0 * F) * fma(A, Float64(C * -4.0), (B_m ^ 2.0))))) / Float64(Float64(Float64(4.0 * A) * C) - (B_m ^ 2.0)));
	else
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(sqrt(F) * Float64(-sqrt(Float64(C + hypot(B_m, C))))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e+96], N[(N[(N[Sqrt[N[(A + N[(C + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] * N[(A * N[(C * -4.0), $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision] - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[(N[Sqrt[F], $MachinePrecision] * (-N[Sqrt[N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B 2) < 1.00000000000000005e96

   1. Initial program 23.5%

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

      1. pow1/2 23.8%

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

      2. *-commutative 23.8%

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

      3. +-commutative 23.8%

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

      4. unpow2 23.8%

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

      5. unpow2 23.8%

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

      6. hypot-undefine 31.9%

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

      7. unpow-prod-down 39.1%

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

   4. Applied egg-rr 39.1%

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

   6. Simplified 39.9%

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

   if 1.00000000000000005e96 < (pow.f64 B 2)

   1. Initial program 11.3%

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

   3. Taylor expanded in A around 0 7.5%

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

      1. mul-1-neg 7.5%

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

      2. distribute-rgt-neg-in 7.5%

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

      3. unpow2 7.5%

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

      4. unpow2 7.5%

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

      5. hypot-define 26.6%

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

   5. Simplified 26.6%

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

      1. pow1/2 26.7%

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

      2. *-commutative 26.7%

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

      3. unpow-prod-down 38.1%

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

      4. pow1/2 38.1%

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

      5. pow1/2 38.1%

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

   7. Applied egg-rr 38.1%

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

4. Final simplification 39.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{+96}:\\ \;\;\;\;\frac{\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(A, C \cdot -4, {B}^{2}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{F} \cdot \left(-\sqrt{C + \mathsf{hypot}\left(B, C\right)}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 50.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := A + \left(C + \mathsf{hypot}\left(B\_m, A - C\right)\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 5 \cdot 10^{-268}:\\ \;\;\;\;\frac{\sqrt{t\_1} \cdot \sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(A, C \cdot -4, {B\_m}^{2}\right)}}{\left(A \cdot C\right) \cdot \left(--4\right)}\\ \mathbf{elif}\;{B\_m}^{2} \leq 10^{+96}:\\ \;\;\;\;\frac{\sqrt{t\_1 \cdot \left(\left(2 \cdot F\right) \cdot t\_0\right)}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B\_m} \cdot \left(\sqrt{F} \cdot \left(-\sqrt{C + \mathsf{hypot}\left(B\_m, C\right)}\right)\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* A (* C -4.0))))
        (t_1 (+ A (+ C (hypot B_m (- A C))))))
   (if (<= (pow B_m 2.0) 5e-268)
     (/
      (* (sqrt t_1) (sqrt (* (* 2.0 F) (fma A (* C -4.0) (pow B_m 2.0)))))
      (* (* A C) (- -4.0)))
     (if (<= (pow B_m 2.0) 1e+96)
       (/ (sqrt (* t_1 (* (* 2.0 F) t_0))) (- t_0))
       (* (/ (sqrt 2.0) B_m) (* (sqrt F) (- (sqrt (+ C (hypot B_m C))))))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (A * (C * -4.0)));
	double t_1 = A + (C + hypot(B_m, (A - C)));
	double tmp;
	if (pow(B_m, 2.0) <= 5e-268) {
		tmp = (sqrt(t_1) * sqrt(((2.0 * F) * fma(A, (C * -4.0), pow(B_m, 2.0))))) / ((A * C) * -(-4.0));
	} else if (pow(B_m, 2.0) <= 1e+96) {
		tmp = sqrt((t_1 * ((2.0 * F) * t_0))) / -t_0;
	} else {
		tmp = (sqrt(2.0) / B_m) * (sqrt(F) * -sqrt((C + hypot(B_m, C))));
	}
	return tmp;
}

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	t_1 = Float64(A + Float64(C + hypot(B_m, Float64(A - C))))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 5e-268)
		tmp = Float64(Float64(sqrt(t_1) * sqrt(Float64(Float64(2.0 * F) * fma(A, Float64(C * -4.0), (B_m ^ 2.0))))) / Float64(Float64(A * C) * Float64(-(-4.0))));
	elseif ((B_m ^ 2.0) <= 1e+96)
		tmp = Float64(sqrt(Float64(t_1 * Float64(Float64(2.0 * F) * t_0))) / Float64(-t_0));
	else
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(sqrt(F) * Float64(-sqrt(Float64(C + hypot(B_m, C))))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(A + N[(C + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e-268], N[(N[(N[Sqrt[t$95$1], $MachinePrecision] * N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] * N[(A * N[(C * -4.0), $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[(A * C), $MachinePrecision] * (--4.0)), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e+96], N[(N[Sqrt[N[(t$95$1 * N[(N[(2.0 * F), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[(N[Sqrt[F], $MachinePrecision] * (-N[Sqrt[N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B 2) < 4.9999999999999999e-268

   1. Initial program 14.0%

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

      1. pow1/2 14.3%

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

      2. *-commutative 14.3%

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

      3. +-commutative 14.3%

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

      4. unpow2 14.3%

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

      5. unpow2 14.3%

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

      6. hypot-undefine 26.4%

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

      7. unpow-prod-down 34.2%

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

   4. Applied egg-rr 34.3%

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

   6. Simplified 35.0%

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

   7. Taylor expanded in B around 0 33.7%

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

   if 4.9999999999999999e-268 < (pow.f64 B 2) < 1.00000000000000005e96

   1. Initial program 33.5%

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

   3. Simplified 37.6%

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

      1. add-cbrt-cube 28.5%

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

      2. add-sqr-sqrt 28.6%

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

      3. pow1 28.6%

         \[\leadsto \frac{\sqrt[3]{\color{blue}{{\left(\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(F \cdot 2\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{1}} \cdot \sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(F \cdot 2\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      4. pow1/2 28.6%

         \[\leadsto \frac{\sqrt[3]{{\left(\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(F \cdot 2\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{1} \cdot \color{blue}{{\left(\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(F \cdot 2\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{0.5}}}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      5. pow-prod-up 28.5%

         \[\leadsto \frac{\sqrt[3]{\color{blue}{{\left(\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(F \cdot 2\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{\left(1 + 0.5\right)}}}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   6. Applied egg-rr 28.5%

      \[\leadsto \frac{\color{blue}{\sqrt[3]{{\left(\left(\mathsf{hypot}\left(A - C, B\right) + \left(A + C\right)\right) \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(2 \cdot F\right)\right)\right)}^{1.5}}}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 28.5%

         \[\leadsto \color{blue}{1 \cdot \frac{\sqrt[3]{{\left(\left(\mathsf{hypot}\left(A - C, B\right) + \left(A + C\right)\right) \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(2 \cdot F\right)\right)\right)}^{1.5}}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}} \]

      2. pow1/3 26.5%

         \[\leadsto 1 \cdot \frac{\color{blue}{{\left({\left(\left(\mathsf{hypot}\left(A - C, B\right) + \left(A + C\right)\right) \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(2 \cdot F\right)\right)\right)}^{1.5}\right)}^{0.3333333333333333}}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      3. pow-pow 37.6%

         \[\leadsto 1 \cdot \frac{\color{blue}{{\left(\left(\mathsf{hypot}\left(A - C, B\right) + \left(A + C\right)\right) \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(2 \cdot F\right)\right)\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      4. metadata-eval 37.6%

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

      5. pow1/2 37.6%

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

      6. *-commutative 37.6%

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

   8. Applied egg-rr 37.6%

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

      1. *-lft-identity 37.6%

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

      2. +-commutative 37.6%

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

      3. hypot-undefine 33.5%

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

      4. unpow2 33.5%

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

      5. unpow2 33.5%

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

      6. +-commutative 33.5%

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

      7. unpow2 33.5%

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

      8. unpow2 33.5%

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

      9. hypot-undefine 37.6%

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

      10. associate-+l+ 38.8%

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

   10. Simplified 38.8%

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

   if 1.00000000000000005e96 < (pow.f64 B 2)

   1. Initial program 11.3%

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

   3. Taylor expanded in A around 0 7.5%

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

      1. mul-1-neg 7.5%

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

      2. distribute-rgt-neg-in 7.5%

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

      3. unpow2 7.5%

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

      4. unpow2 7.5%

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

      5. hypot-define 26.6%

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

   5. Simplified 26.6%

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

      1. pow1/2 26.7%

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

      2. *-commutative 26.7%

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

      3. unpow-prod-down 38.1%

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

      4. pow1/2 38.1%

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

      5. pow1/2 38.1%

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

   7. Applied egg-rr 38.1%

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

4. Final simplification 37.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 5 \cdot 10^{-268}:\\ \;\;\;\;\frac{\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(A, C \cdot -4, {B}^{2}\right)}}{\left(A \cdot C\right) \cdot \left(--4\right)}\\ \mathbf{elif}\;{B}^{2} \leq 10^{+96}:\\ \;\;\;\;\frac{\sqrt{\left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{F} \cdot \left(-\sqrt{C + \mathsf{hypot}\left(B, C\right)}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 52.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 2 \cdot 10^{+94}:\\ \;\;\;\;\frac{\sqrt{\left(A + C\right) + \mathsf{hypot}\left(A - C, B\_m\right)} \cdot \sqrt{\left(2 \cdot F\right) \cdot t\_0}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B\_m} \cdot \left(\sqrt{F} \cdot \left(-\sqrt{C + \mathsf{hypot}\left(B\_m, C\right)}\right)\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* A (* C -4.0)))))
   (if (<= (pow B_m 2.0) 2e+94)
     (/
      (* (sqrt (+ (+ A C) (hypot (- A C) B_m))) (sqrt (* (* 2.0 F) t_0)))
      (- t_0))
     (* (/ (sqrt 2.0) B_m) (* (sqrt F) (- (sqrt (+ C (hypot B_m C)))))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (A * (C * -4.0)));
	double tmp;
	if (pow(B_m, 2.0) <= 2e+94) {
		tmp = (sqrt(((A + C) + hypot((A - C), B_m))) * sqrt(((2.0 * F) * t_0))) / -t_0;
	} else {
		tmp = (sqrt(2.0) / B_m) * (sqrt(F) * -sqrt((C + hypot(B_m, C))));
	}
	return tmp;
}

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 2e+94)
		tmp = Float64(Float64(sqrt(Float64(Float64(A + C) + hypot(Float64(A - C), B_m))) * sqrt(Float64(Float64(2.0 * F) * t_0))) / Float64(-t_0));
	else
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(sqrt(F) * Float64(-sqrt(Float64(C + hypot(B_m, C))))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+94], N[(N[(N[Sqrt[N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / (-t$95$0)), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[(N[Sqrt[F], $MachinePrecision] * (-N[Sqrt[N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B 2) < 2e94

   1. Initial program 23.6%

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

   3. Simplified 32.1%

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

      1. pow1/2 32.1%

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

      2. *-commutative 32.1%

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

      3. unpow-prod-down 39.4%

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

      4. pow1/2 39.4%

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

      5. +-commutative 39.4%

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

      6. hypot-undefine 25.7%

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

      7. unpow2 25.7%

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

      8. unpow2 25.7%

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

      9. +-commutative 25.7%

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

      10. unpow2 25.7%

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

      11. unpow2 25.7%

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

      12. hypot-define 39.4%

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

      13. pow1/2 39.4%

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

      14. *-commutative 39.4%

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

   6. Applied egg-rr 39.4%

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

   if 2e94 < (pow.f64 B 2)

   1. Initial program 11.2%

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

   3. Taylor expanded in A around 0 7.5%

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

      1. mul-1-neg 7.5%

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

      2. distribute-rgt-neg-in 7.5%

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

      3. unpow2 7.5%

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

      4. unpow2 7.5%

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

      5. hypot-define 26.4%

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

   5. Simplified 26.4%

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

      1. pow1/2 26.5%

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

      2. *-commutative 26.5%

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

      3. unpow-prod-down 37.8%

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

      4. pow1/2 37.8%

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

      5. pow1/2 37.8%

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

   7. Applied egg-rr 37.8%

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

4. Final simplification 38.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 2 \cdot 10^{+94}:\\ \;\;\;\;\frac{\sqrt{\left(A + C\right) + \mathsf{hypot}\left(A - C, B\right)} \cdot \sqrt{\left(2 \cdot F\right) \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{F} \cdot \left(-\sqrt{C + \mathsf{hypot}\left(B, C\right)}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 49.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 10^{+96}:\\ \;\;\;\;\frac{-1}{\frac{t\_0}{\sqrt{\left(A + \left(C + \mathsf{hypot}\left(B\_m, A - C\right)\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot t\_0\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B\_m} \cdot \left(\sqrt{F} \cdot \left(-\sqrt{C + \mathsf{hypot}\left(B\_m, C\right)}\right)\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* A (* C -4.0)))))
   (if (<= (pow B_m 2.0) 1e+96)
     (/
      -1.0
      (/ t_0 (sqrt (* (+ A (+ C (hypot B_m (- A C)))) (* (* 2.0 F) t_0)))))
     (* (/ (sqrt 2.0) B_m) (* (sqrt F) (- (sqrt (+ C (hypot B_m C)))))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (A * (C * -4.0)));
	double tmp;
	if (pow(B_m, 2.0) <= 1e+96) {
		tmp = -1.0 / (t_0 / sqrt(((A + (C + hypot(B_m, (A - C)))) * ((2.0 * F) * t_0))));
	} else {
		tmp = (sqrt(2.0) / B_m) * (sqrt(F) * -sqrt((C + hypot(B_m, C))));
	}
	return tmp;
}

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e+96)
		tmp = Float64(-1.0 / Float64(t_0 / sqrt(Float64(Float64(A + Float64(C + hypot(B_m, Float64(A - C)))) * Float64(Float64(2.0 * F) * t_0)))));
	else
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(sqrt(F) * Float64(-sqrt(Float64(C + hypot(B_m, C))))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e+96], N[(-1.0 / N[(t$95$0 / N[Sqrt[N[(N[(A + N[(C + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 * F), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[(N[Sqrt[F], $MachinePrecision] * (-N[Sqrt[N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B 2) < 1.00000000000000005e96

   1. Initial program 23.5%

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

   3. Simplified 31.9%

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

      1. add-cbrt-cube 23.9%

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

      2. add-sqr-sqrt 24.0%

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

      3. pow1 24.0%

         \[\leadsto \frac{\sqrt[3]{\color{blue}{{\left(\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(F \cdot 2\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{1}} \cdot \sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(F \cdot 2\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      4. pow1/2 24.0%

         \[\leadsto \frac{\sqrt[3]{{\left(\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(F \cdot 2\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{1} \cdot \color{blue}{{\left(\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(F \cdot 2\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{0.5}}}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      5. pow-prod-up 24.0%

         \[\leadsto \frac{\sqrt[3]{\color{blue}{{\left(\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(F \cdot 2\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{\left(1 + 0.5\right)}}}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   6. Applied egg-rr 24.0%

      \[\leadsto \frac{\color{blue}{\sqrt[3]{{\left(\left(\mathsf{hypot}\left(A - C, B\right) + \left(A + C\right)\right) \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(2 \cdot F\right)\right)\right)}^{1.5}}}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
   7. Step-by-step derivation

      1. clear-num 24.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}{\sqrt[3]{{\left(\left(\mathsf{hypot}\left(A - C, B\right) + \left(A + C\right)\right) \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(2 \cdot F\right)\right)\right)}^{1.5}}}}} \]

      2. inv-pow 24.0%

         \[\leadsto \color{blue}{{\left(\frac{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}{\sqrt[3]{{\left(\left(\mathsf{hypot}\left(A - C, B\right) + \left(A + C\right)\right) \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(2 \cdot F\right)\right)\right)}^{1.5}}}\right)}^{-1}} \]

      3. pow1/3 22.3%

         \[\leadsto {\left(\frac{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}{\color{blue}{{\left({\left(\left(\mathsf{hypot}\left(A - C, B\right) + \left(A + C\right)\right) \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(2 \cdot F\right)\right)\right)}^{1.5}\right)}^{0.3333333333333333}}}\right)}^{-1} \]

      4. pow-pow 32.0%

         \[\leadsto {\left(\frac{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}{\color{blue}{{\left(\left(\mathsf{hypot}\left(A - C, B\right) + \left(A + C\right)\right) \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(2 \cdot F\right)\right)\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}}}\right)}^{-1} \]

      5. metadata-eval 32.0%

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

      6. pow1/2 32.0%

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

      7. *-commutative 32.0%

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

   8. Applied egg-rr 32.0%

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

      1. unpow-1 32.0%

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

      2. +-commutative 32.0%

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

      3. hypot-undefine 23.6%

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

      4. unpow2 23.6%

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

      5. unpow2 23.6%

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

      6. +-commutative 23.6%

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

      7. unpow2 23.6%

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

      8. unpow2 23.6%

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

      9. hypot-undefine 32.0%

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

      10. associate-+l+ 33.2%

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

   10. Simplified 33.2%

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

   if 1.00000000000000005e96 < (pow.f64 B 2)

   1. Initial program 11.3%

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

   3. Taylor expanded in A around 0 7.5%

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

      1. mul-1-neg 7.5%

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

      2. distribute-rgt-neg-in 7.5%

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

      3. unpow2 7.5%

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

      4. unpow2 7.5%

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

      5. hypot-define 26.6%

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

   5. Simplified 26.6%

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

      1. pow1/2 26.7%

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

      2. *-commutative 26.7%

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

      3. unpow-prod-down 38.1%

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

      4. pow1/2 38.1%

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

      5. pow1/2 38.1%

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

   7. Applied egg-rr 38.1%

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

4. Final simplification 35.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{+96}:\\ \;\;\;\;\frac{-1}{\frac{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}{\sqrt{\left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{F} \cdot \left(-\sqrt{C + \mathsf{hypot}\left(B, C\right)}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 48.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 2 \cdot 10^{+94}:\\ \;\;\;\;\frac{\sqrt{\left(\left(2 \cdot F\right) \cdot t\_0\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B\_m, A - C\right)\right)}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B\_m} \cdot \left(\sqrt{F} \cdot \left(-\sqrt{C + \mathsf{hypot}\left(B\_m, C\right)}\right)\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* A (* C -4.0)))))
   (if (<= (pow B_m 2.0) 2e+94)
     (/ (sqrt (* (* (* 2.0 F) t_0) (+ (+ A C) (hypot B_m (- A C))))) (- t_0))
     (* (/ (sqrt 2.0) B_m) (* (sqrt F) (- (sqrt (+ C (hypot B_m C)))))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (A * (C * -4.0)));
	double tmp;
	if (pow(B_m, 2.0) <= 2e+94) {
		tmp = sqrt((((2.0 * F) * t_0) * ((A + C) + hypot(B_m, (A - C))))) / -t_0;
	} else {
		tmp = (sqrt(2.0) / B_m) * (sqrt(F) * -sqrt((C + hypot(B_m, C))));
	}
	return tmp;
}

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 2e+94)
		tmp = Float64(sqrt(Float64(Float64(Float64(2.0 * F) * t_0) * Float64(Float64(A + C) + hypot(B_m, Float64(A - C))))) / Float64(-t_0));
	else
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(sqrt(F) * Float64(-sqrt(Float64(C + hypot(B_m, C))))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+94], N[(N[Sqrt[N[(N[(N[(2.0 * F), $MachinePrecision] * t$95$0), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[(N[Sqrt[F], $MachinePrecision] * (-N[Sqrt[N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B 2) < 2e94

   1. Initial program 23.6%

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

   3. Simplified 32.1%

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

   if 2e94 < (pow.f64 B 2)

   1. Initial program 11.2%

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

   3. Taylor expanded in A around 0 7.5%

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

      1. mul-1-neg 7.5%

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

      2. distribute-rgt-neg-in 7.5%

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

      3. unpow2 7.5%

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

      4. unpow2 7.5%

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

      5. hypot-define 26.4%

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

   5. Simplified 26.4%

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

      1. pow1/2 26.5%

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

      2. *-commutative 26.5%

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

      3. unpow-prod-down 37.8%

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

      4. pow1/2 37.8%

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

      5. pow1/2 37.8%

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

   7. Applied egg-rr 37.8%

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

4. Final simplification 34.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 2 \cdot 10^{+94}:\\ \;\;\;\;\frac{\sqrt{\left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{F} \cdot \left(-\sqrt{C + \mathsf{hypot}\left(B, C\right)}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 49.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 10^{+96}:\\ \;\;\;\;\frac{\sqrt{\left(A + \left(C + \mathsf{hypot}\left(B\_m, A - C\right)\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot t\_0\right)}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B\_m} \cdot \left(\sqrt{F} \cdot \left(-\sqrt{C + \mathsf{hypot}\left(B\_m, C\right)}\right)\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* A (* C -4.0)))))
   (if (<= (pow B_m 2.0) 1e+96)
     (/ (sqrt (* (+ A (+ C (hypot B_m (- A C)))) (* (* 2.0 F) t_0))) (- t_0))
     (* (/ (sqrt 2.0) B_m) (* (sqrt F) (- (sqrt (+ C (hypot B_m C)))))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (A * (C * -4.0)));
	double tmp;
	if (pow(B_m, 2.0) <= 1e+96) {
		tmp = sqrt(((A + (C + hypot(B_m, (A - C)))) * ((2.0 * F) * t_0))) / -t_0;
	} else {
		tmp = (sqrt(2.0) / B_m) * (sqrt(F) * -sqrt((C + hypot(B_m, C))));
	}
	return tmp;
}

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e+96)
		tmp = Float64(sqrt(Float64(Float64(A + Float64(C + hypot(B_m, Float64(A - C)))) * Float64(Float64(2.0 * F) * t_0))) / Float64(-t_0));
	else
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(sqrt(F) * Float64(-sqrt(Float64(C + hypot(B_m, C))))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e+96], N[(N[Sqrt[N[(N[(A + N[(C + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 * F), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[(N[Sqrt[F], $MachinePrecision] * (-N[Sqrt[N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B 2) < 1.00000000000000005e96

   1. Initial program 23.5%

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

   3. Simplified 31.9%

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

      1. add-cbrt-cube 23.9%

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

      2. add-sqr-sqrt 24.0%

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

      3. pow1 24.0%

         \[\leadsto \frac{\sqrt[3]{\color{blue}{{\left(\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(F \cdot 2\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{1}} \cdot \sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(F \cdot 2\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      4. pow1/2 24.0%

         \[\leadsto \frac{\sqrt[3]{{\left(\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(F \cdot 2\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{1} \cdot \color{blue}{{\left(\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(F \cdot 2\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{0.5}}}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      5. pow-prod-up 24.0%

         \[\leadsto \frac{\sqrt[3]{\color{blue}{{\left(\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(F \cdot 2\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)\right)}^{\left(1 + 0.5\right)}}}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   6. Applied egg-rr 24.0%

      \[\leadsto \frac{\color{blue}{\sqrt[3]{{\left(\left(\mathsf{hypot}\left(A - C, B\right) + \left(A + C\right)\right) \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(2 \cdot F\right)\right)\right)}^{1.5}}}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 24.0%

         \[\leadsto \color{blue}{1 \cdot \frac{\sqrt[3]{{\left(\left(\mathsf{hypot}\left(A - C, B\right) + \left(A + C\right)\right) \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(2 \cdot F\right)\right)\right)}^{1.5}}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}} \]

      2. pow1/3 22.3%

         \[\leadsto 1 \cdot \frac{\color{blue}{{\left({\left(\left(\mathsf{hypot}\left(A - C, B\right) + \left(A + C\right)\right) \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(2 \cdot F\right)\right)\right)}^{1.5}\right)}^{0.3333333333333333}}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      3. pow-pow 31.9%

         \[\leadsto 1 \cdot \frac{\color{blue}{{\left(\left(\mathsf{hypot}\left(A - C, B\right) + \left(A + C\right)\right) \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(2 \cdot F\right)\right)\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      4. metadata-eval 31.9%

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

      5. pow1/2 31.9%

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

      6. *-commutative 31.9%

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

   8. Applied egg-rr 31.9%

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

      1. *-lft-identity 31.9%

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

      2. +-commutative 31.9%

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

      3. hypot-undefine 23.5%

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

      4. unpow2 23.5%

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

      5. unpow2 23.5%

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

      6. +-commutative 23.5%

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

      7. unpow2 23.5%

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

      8. unpow2 23.5%

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

      9. hypot-undefine 31.9%

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

      10. associate-+l+ 33.1%

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

   10. Simplified 33.1%

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

   if 1.00000000000000005e96 < (pow.f64 B 2)

   1. Initial program 11.3%

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

   3. Taylor expanded in A around 0 7.5%

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

      1. mul-1-neg 7.5%

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

      2. distribute-rgt-neg-in 7.5%

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

      3. unpow2 7.5%

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

      4. unpow2 7.5%

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

      5. hypot-define 26.6%

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

   5. Simplified 26.6%

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

      1. pow1/2 26.7%

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

      2. *-commutative 26.7%

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

      3. unpow-prod-down 38.1%

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

      4. pow1/2 38.1%

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

      5. pow1/2 38.1%

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

   7. Applied egg-rr 38.1%

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

4. Final simplification 35.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{+96}:\\ \;\;\;\;\frac{\sqrt{\left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{F} \cdot \left(-\sqrt{C + \mathsf{hypot}\left(B, C\right)}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 39.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 5 \cdot 10^{-226}:\\ \;\;\;\;\frac{\sqrt{\left(A \cdot -16\right) \cdot \left(F \cdot {C}^{2}\right)}}{4 \cdot \left(A \cdot C\right) - {B\_m}^{2}}\\ \mathbf{elif}\;{B\_m}^{2} \leq 5 \cdot 10^{+90}:\\ \;\;\;\;B\_m \cdot \frac{\sqrt{2 \cdot \left(F \cdot \left(A + \mathsf{hypot}\left(B\_m, A\right)\right)\right)}}{-\mathsf{fma}\left(B\_m, B\_m, C \cdot \left(A \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \frac{-1}{\frac{\sqrt{B\_m}}{\sqrt{F}}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 5e-226)
   (/
    (sqrt (* (* A -16.0) (* F (pow C 2.0))))
    (- (* 4.0 (* A C)) (pow B_m 2.0)))
   (if (<= (pow B_m 2.0) 5e+90)
     (*
      B_m
      (/
       (sqrt (* 2.0 (* F (+ A (hypot B_m A)))))
       (- (fma B_m B_m (* C (* A -4.0))))))
     (* (sqrt 2.0) (/ -1.0 (/ (sqrt B_m) (sqrt F)))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (pow(B_m, 2.0) <= 5e-226) {
		tmp = sqrt(((A * -16.0) * (F * pow(C, 2.0)))) / ((4.0 * (A * C)) - pow(B_m, 2.0));
	} else if (pow(B_m, 2.0) <= 5e+90) {
		tmp = B_m * (sqrt((2.0 * (F * (A + hypot(B_m, A))))) / -fma(B_m, B_m, (C * (A * -4.0))));
	} else {
		tmp = sqrt(2.0) * (-1.0 / (sqrt(B_m) / sqrt(F)));
	}
	return tmp;
}

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 5e-226)
		tmp = Float64(sqrt(Float64(Float64(A * -16.0) * Float64(F * (C ^ 2.0)))) / Float64(Float64(4.0 * Float64(A * C)) - (B_m ^ 2.0)));
	elseif ((B_m ^ 2.0) <= 5e+90)
		tmp = Float64(B_m * Float64(sqrt(Float64(2.0 * Float64(F * Float64(A + hypot(B_m, A))))) / Float64(-fma(B_m, B_m, Float64(C * Float64(A * -4.0))))));
	else
		tmp = Float64(sqrt(2.0) * Float64(-1.0 / Float64(sqrt(B_m) / sqrt(F))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e-226], N[(N[Sqrt[N[(N[(A * -16.0), $MachinePrecision] * N[(F * N[Power[C, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision] - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e+90], N[(B$95$m * N[(N[Sqrt[N[(2.0 * N[(F * N[(A + N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-N[(B$95$m * B$95$m + N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * N[(-1.0 / N[(N[Sqrt[B$95$m], $MachinePrecision] / N[Sqrt[F], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B 2) < 4.9999999999999998e-226

   1. Initial program 16.1%

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

   3. Simplified 24.0%

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

   5. Taylor expanded in A around -inf 17.1%

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

      1. associate-*r* 17.0%

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

   7. Simplified 17.0%

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

   if 4.9999999999999998e-226 < (pow.f64 B 2) < 5.0000000000000004e90

   1. Initial program 32.9%

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

   3. Taylor expanded in C around 0 16.8%

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

      1. associate-*l* 16.8%

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

      2. +-commutative 16.8%

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

      3. unpow2 16.8%

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

      4. unpow2 16.8%

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

      5. hypot-define 17.2%

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

   5. Simplified 17.2%

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

      1. *-un-lft-identity 17.2%

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

      2. distribute-lft-neg-in 17.2%

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

      3. sqrt-unprod 17.2%

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

      4. *-commutative 17.2%

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

      5. *-commutative 17.2%

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

   7. Applied egg-rr 17.2%

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

      1. *-lft-identity 17.2%

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

      2. neg-mul-1 17.2%

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

      3. associate-/l* 17.2%

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

      4. neg-mul-1 17.2%

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

      5. sub-neg 17.2%

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

      6. unpow2 17.2%

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

      7. fma-undefine 17.2%

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

      8. distribute-rgt-neg-in 17.2%

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

      9. distribute-rgt-neg-in 17.2%

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

      10. metadata-eval 17.2%

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

   9. Simplified 17.2%

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

   if 5.0000000000000004e90 < (pow.f64 B 2)

   1. Initial program 12.1%

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

   3. Taylor expanded in A around 0 7.4%

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

      1. mul-1-neg 7.4%

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

      2. distribute-rgt-neg-in 7.4%

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

      3. unpow2 7.4%

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

      4. unpow2 7.4%

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

      5. hypot-define 26.1%

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

   5. Simplified 26.1%

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

   6. Taylor expanded in C around 0 22.3%

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

      1. mul-1-neg 22.3%

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

   8. Simplified 22.3%

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

      1. sqrt-div 33.7%

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

      2. clear-num 33.7%

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

   10. Applied egg-rr 33.7%

       \[\leadsto -\color{blue}{\frac{1}{\frac{\sqrt{B}}{\sqrt{F}}}} \cdot \sqrt{2} \]
3. Recombined 3 regimes into one program.

4. Final simplification 23.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 5 \cdot 10^{-226}:\\ \;\;\;\;\frac{\sqrt{\left(A \cdot -16\right) \cdot \left(F \cdot {C}^{2}\right)}}{4 \cdot \left(A \cdot C\right) - {B}^{2}}\\ \mathbf{elif}\;{B}^{2} \leq 5 \cdot 10^{+90}:\\ \;\;\;\;B \cdot \frac{\sqrt{2 \cdot \left(F \cdot \left(A + \mathsf{hypot}\left(B, A\right)\right)\right)}}{-\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \frac{-1}{\frac{\sqrt{B}}{\sqrt{F}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 42.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 1.5 \cdot 10^{-220}:\\ \;\;\;\;\frac{\sqrt{\left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B\_m, A - C\right)\right)}}{A \cdot \left(C \cdot \left(--4\right)\right)}\\ \mathbf{elif}\;{B\_m}^{2} \leq 5 \cdot 10^{+90}:\\ \;\;\;\;B\_m \cdot \frac{\sqrt{2 \cdot \left(F \cdot \left(A + \mathsf{hypot}\left(B\_m, A\right)\right)\right)}}{-\mathsf{fma}\left(B\_m, B\_m, C \cdot \left(A \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \frac{-1}{\frac{\sqrt{B\_m}}{\sqrt{F}}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 1.5e-220)
   (/
    (sqrt
     (*
      (* (* 2.0 F) (fma B_m B_m (* A (* C -4.0))))
      (+ (+ A C) (hypot B_m (- A C)))))
    (* A (* C (- -4.0))))
   (if (<= (pow B_m 2.0) 5e+90)
     (*
      B_m
      (/
       (sqrt (* 2.0 (* F (+ A (hypot B_m A)))))
       (- (fma B_m B_m (* C (* A -4.0))))))
     (* (sqrt 2.0) (/ -1.0 (/ (sqrt B_m) (sqrt F)))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (pow(B_m, 2.0) <= 1.5e-220) {
		tmp = sqrt((((2.0 * F) * fma(B_m, B_m, (A * (C * -4.0)))) * ((A + C) + hypot(B_m, (A - C))))) / (A * (C * -(-4.0)));
	} else if (pow(B_m, 2.0) <= 5e+90) {
		tmp = B_m * (sqrt((2.0 * (F * (A + hypot(B_m, A))))) / -fma(B_m, B_m, (C * (A * -4.0))));
	} else {
		tmp = sqrt(2.0) * (-1.0 / (sqrt(B_m) / sqrt(F)));
	}
	return tmp;
}

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1.5e-220)
		tmp = Float64(sqrt(Float64(Float64(Float64(2.0 * F) * fma(B_m, B_m, Float64(A * Float64(C * -4.0)))) * Float64(Float64(A + C) + hypot(B_m, Float64(A - C))))) / Float64(A * Float64(C * Float64(-(-4.0)))));
	elseif ((B_m ^ 2.0) <= 5e+90)
		tmp = Float64(B_m * Float64(sqrt(Float64(2.0 * Float64(F * Float64(A + hypot(B_m, A))))) / Float64(-fma(B_m, B_m, Float64(C * Float64(A * -4.0))))));
	else
		tmp = Float64(sqrt(2.0) * Float64(-1.0 / Float64(sqrt(B_m) / sqrt(F))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1.5e-220], N[(N[Sqrt[N[(N[(N[(2.0 * F), $MachinePrecision] * N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(A * N[(C * (--4.0)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e+90], N[(B$95$m * N[(N[Sqrt[N[(2.0 * N[(F * N[(A + N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-N[(B$95$m * B$95$m + N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * N[(-1.0 / N[(N[Sqrt[B$95$m], $MachinePrecision] / N[Sqrt[F], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B 2) < 1.50000000000000009e-220

   1. Initial program 17.0%

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

   3. Simplified 28.2%

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

   5. Taylor expanded in B around 0 26.9%

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

      1. *-commutative 26.9%

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

      2. associate-*r* 26.9%

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

   7. Simplified 26.9%

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

   if 1.50000000000000009e-220 < (pow.f64 B 2) < 5.0000000000000004e90

   1. Initial program 31.8%

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

   3. Taylor expanded in C around 0 17.0%

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

      1. associate-*l* 17.0%

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

      2. +-commutative 17.0%

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

      3. unpow2 17.0%

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

      4. unpow2 17.0%

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

      5. hypot-define 17.5%

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

   5. Simplified 17.5%

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

      1. *-un-lft-identity 17.5%

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

      2. distribute-lft-neg-in 17.5%

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

      3. sqrt-unprod 17.5%

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

      4. *-commutative 17.5%

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

      5. *-commutative 17.5%

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

   7. Applied egg-rr 17.5%

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

      1. *-lft-identity 17.5%

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

      2. neg-mul-1 17.5%

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

      3. associate-/l* 17.5%

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

      4. neg-mul-1 17.5%

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

      5. sub-neg 17.5%

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

      6. unpow2 17.5%

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

      7. fma-undefine 17.5%

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

      8. distribute-rgt-neg-in 17.5%

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

      9. distribute-rgt-neg-in 17.5%

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

      10. metadata-eval 17.5%

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

   9. Simplified 17.5%

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

   if 5.0000000000000004e90 < (pow.f64 B 2)

   1. Initial program 12.1%

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

   3. Taylor expanded in A around 0 7.4%

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

      1. mul-1-neg 7.4%

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

      2. distribute-rgt-neg-in 7.4%

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

      3. unpow2 7.4%

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

      4. unpow2 7.4%

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

      5. hypot-define 26.1%

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

   5. Simplified 26.1%

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

   6. Taylor expanded in C around 0 22.3%

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

      1. mul-1-neg 22.3%

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

   8. Simplified 22.3%

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

      1. sqrt-div 33.7%

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

      2. clear-num 33.7%

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

   10. Applied egg-rr 33.7%

       \[\leadsto -\color{blue}{\frac{1}{\frac{\sqrt{B}}{\sqrt{F}}}} \cdot \sqrt{2} \]
3. Recombined 3 regimes into one program.

4. Final simplification 27.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 1.5 \cdot 10^{-220}:\\ \;\;\;\;\frac{\sqrt{\left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}{A \cdot \left(C \cdot \left(--4\right)\right)}\\ \mathbf{elif}\;{B}^{2} \leq 5 \cdot 10^{+90}:\\ \;\;\;\;B \cdot \frac{\sqrt{2 \cdot \left(F \cdot \left(A + \mathsf{hypot}\left(B, A\right)\right)\right)}}{-\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \frac{-1}{\frac{\sqrt{B}}{\sqrt{F}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 45.1% accurate, 1.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 10^{-249}:\\ \;\;\;\;\frac{\sqrt{\left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B\_m, A - C\right)\right)}}{A \cdot \left(C \cdot \left(--4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B\_m} \cdot \left(\sqrt{F} \cdot \left(-\sqrt{C + \mathsf{hypot}\left(B\_m, C\right)}\right)\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 1e-249)
   (/
    (sqrt
     (*
      (* (* 2.0 F) (fma B_m B_m (* A (* C -4.0))))
      (+ (+ A C) (hypot B_m (- A C)))))
    (* A (* C (- -4.0))))
   (* (/ (sqrt 2.0) B_m) (* (sqrt F) (- (sqrt (+ C (hypot B_m C))))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (pow(B_m, 2.0) <= 1e-249) {
		tmp = sqrt((((2.0 * F) * fma(B_m, B_m, (A * (C * -4.0)))) * ((A + C) + hypot(B_m, (A - C))))) / (A * (C * -(-4.0)));
	} else {
		tmp = (sqrt(2.0) / B_m) * (sqrt(F) * -sqrt((C + hypot(B_m, C))));
	}
	return tmp;
}

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e-249)
		tmp = Float64(sqrt(Float64(Float64(Float64(2.0 * F) * fma(B_m, B_m, Float64(A * Float64(C * -4.0)))) * Float64(Float64(A + C) + hypot(B_m, Float64(A - C))))) / Float64(A * Float64(C * Float64(-(-4.0)))));
	else
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(sqrt(F) * Float64(-sqrt(Float64(C + hypot(B_m, C))))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-249], N[(N[Sqrt[N[(N[(N[(2.0 * F), $MachinePrecision] * N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(A * N[(C * (--4.0)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[(N[Sqrt[F], $MachinePrecision] * (-N[Sqrt[N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B 2) < 1.00000000000000005e-249

   1. Initial program 14.8%

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

   3. Simplified 26.9%

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

   5. Taylor expanded in B around 0 26.7%

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

      1. *-commutative 26.7%

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

      2. associate-*r* 26.7%

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

   7. Simplified 26.7%

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

   if 1.00000000000000005e-249 < (pow.f64 B 2)

   1. Initial program 20.5%

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

   3. Taylor expanded in A around 0 10.9%

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

      1. mul-1-neg 10.9%

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

      2. distribute-rgt-neg-in 10.9%

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

      3. unpow2 10.9%

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

      4. unpow2 10.9%

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

      5. hypot-define 22.3%

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

   5. Simplified 22.3%

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

      1. pow1/2 22.4%

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

      2. *-commutative 22.4%

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

      3. unpow-prod-down 29.6%

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

      4. pow1/2 29.6%

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

      5. pow1/2 29.6%

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

   7. Applied egg-rr 29.6%

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

4. Final simplification 28.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{-249}:\\ \;\;\;\;\frac{\sqrt{\left(\left(2 \cdot F\right) \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}{A \cdot \left(C \cdot \left(--4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{F} \cdot \left(-\sqrt{C + \mathsf{hypot}\left(B, C\right)}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 46.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;F \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sqrt{A + \left(C + \mathsf{hypot}\left(B\_m, A - C\right)\right)} \cdot \sqrt{-8 \cdot \left(A \cdot \left(C \cdot F\right)\right)}}{\left(4 \cdot A\right) \cdot C - {B\_m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B\_m} \cdot \left(\sqrt{F} \cdot \left(-\sqrt{C + \mathsf{hypot}\left(B\_m, C\right)}\right)\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= F -2e-310)
   (/
    (* (sqrt (+ A (+ C (hypot B_m (- A C))))) (sqrt (* -8.0 (* A (* C F)))))
    (- (* (* 4.0 A) C) (pow B_m 2.0)))
   (* (/ (sqrt 2.0) B_m) (* (sqrt F) (- (sqrt (+ C (hypot B_m C))))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= -2e-310) {
		tmp = (sqrt((A + (C + hypot(B_m, (A - C))))) * sqrt((-8.0 * (A * (C * F))))) / (((4.0 * A) * C) - pow(B_m, 2.0));
	} else {
		tmp = (sqrt(2.0) / B_m) * (sqrt(F) * -sqrt((C + hypot(B_m, C))));
	}
	return tmp;
}

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= -2e-310) {
		tmp = (Math.sqrt((A + (C + Math.hypot(B_m, (A - C))))) * Math.sqrt((-8.0 * (A * (C * F))))) / (((4.0 * A) * C) - Math.pow(B_m, 2.0));
	} else {
		tmp = (Math.sqrt(2.0) / B_m) * (Math.sqrt(F) * -Math.sqrt((C + Math.hypot(B_m, C))));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if F <= -2e-310:
		tmp = (math.sqrt((A + (C + math.hypot(B_m, (A - C))))) * math.sqrt((-8.0 * (A * (C * F))))) / (((4.0 * A) * C) - math.pow(B_m, 2.0))
	else:
		tmp = (math.sqrt(2.0) / B_m) * (math.sqrt(F) * -math.sqrt((C + math.hypot(B_m, C))))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (F <= -2e-310)
		tmp = Float64(Float64(sqrt(Float64(A + Float64(C + hypot(B_m, Float64(A - C))))) * sqrt(Float64(-8.0 * Float64(A * Float64(C * F))))) / Float64(Float64(Float64(4.0 * A) * C) - (B_m ^ 2.0)));
	else
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(sqrt(F) * Float64(-sqrt(Float64(C + hypot(B_m, C))))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (F <= -2e-310)
		tmp = (sqrt((A + (C + hypot(B_m, (A - C))))) * sqrt((-8.0 * (A * (C * F))))) / (((4.0 * A) * C) - (B_m ^ 2.0));
	else
		tmp = (sqrt(2.0) / B_m) * (sqrt(F) * -sqrt((C + hypot(B_m, C))));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[F, -2e-310], N[(N[(N[Sqrt[N[(A + N[(C + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(-8.0 * N[(A * N[(C * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision] - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[(N[Sqrt[F], $MachinePrecision] * (-N[Sqrt[N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -1.999999999999994e-310

   1. Initial program 21.7%

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

      1. pow1/2 21.7%

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

      2. *-commutative 21.7%

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

      3. +-commutative 21.7%

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

      4. unpow2 21.7%

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

      5. unpow2 21.7%

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

      6. hypot-undefine 43.6%

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

      7. unpow-prod-down 57.3%

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

   4. Applied egg-rr 57.3%

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

   6. Simplified 57.3%

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

   7. Taylor expanded in A around inf 48.4%

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

   if -1.999999999999994e-310 < F

   1. Initial program 18.2%

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

   3. Taylor expanded in A around 0 9.3%

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

      1. mul-1-neg 9.3%

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

      2. distribute-rgt-neg-in 9.3%

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

      3. unpow2 9.3%

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

      4. unpow2 9.3%

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

      5. hypot-define 19.0%

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

   5. Simplified 19.0%

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

      1. pow1/2 19.0%

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

      2. *-commutative 19.0%

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

      3. unpow-prod-down 24.9%

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

      4. pow1/2 24.9%

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

      5. pow1/2 24.9%

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

   7. Applied egg-rr 24.9%

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

4. Final simplification 28.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{-8 \cdot \left(A \cdot \left(C \cdot F\right)\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{F} \cdot \left(-\sqrt{C + \mathsf{hypot}\left(B, C\right)}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 38.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 0:\\ \;\;\;\;\frac{\sqrt{\left(A \cdot -16\right) \cdot \left(F \cdot {C}^{2}\right)}}{4 \cdot \left(A \cdot C\right) - {B\_m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \frac{-1}{\frac{\sqrt{B\_m}}{\sqrt{F}}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 0.0)
   (/
    (sqrt (* (* A -16.0) (* F (pow C 2.0))))
    (- (* 4.0 (* A C)) (pow B_m 2.0)))
   (* (sqrt 2.0) (/ -1.0 (/ (sqrt B_m) (sqrt F))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (pow(B_m, 2.0) <= 0.0) {
		tmp = sqrt(((A * -16.0) * (F * pow(C, 2.0)))) / ((4.0 * (A * C)) - pow(B_m, 2.0));
	} else {
		tmp = sqrt(2.0) * (-1.0 / (sqrt(B_m) / sqrt(F)));
	}
	return tmp;
}

Fortran

B_m = abs(B)
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) <= 0.0d0) then
        tmp = sqrt(((a * (-16.0d0)) * (f * (c ** 2.0d0)))) / ((4.0d0 * (a * c)) - (b_m ** 2.0d0))
    else
        tmp = sqrt(2.0d0) * ((-1.0d0) / (sqrt(b_m) / sqrt(f)))
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (Math.pow(B_m, 2.0) <= 0.0) {
		tmp = Math.sqrt(((A * -16.0) * (F * Math.pow(C, 2.0)))) / ((4.0 * (A * C)) - Math.pow(B_m, 2.0));
	} else {
		tmp = Math.sqrt(2.0) * (-1.0 / (Math.sqrt(B_m) / Math.sqrt(F)));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if math.pow(B_m, 2.0) <= 0.0:
		tmp = math.sqrt(((A * -16.0) * (F * math.pow(C, 2.0)))) / ((4.0 * (A * C)) - math.pow(B_m, 2.0))
	else:
		tmp = math.sqrt(2.0) * (-1.0 / (math.sqrt(B_m) / math.sqrt(F)))
	return tmp

Julia

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

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if ((B_m ^ 2.0) <= 0.0)
		tmp = sqrt(((A * -16.0) * (F * (C ^ 2.0)))) / ((4.0 * (A * C)) - (B_m ^ 2.0));
	else
		tmp = sqrt(2.0) * (-1.0 / (sqrt(B_m) / sqrt(F)));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 0.0], N[(N[Sqrt[N[(N[(A * -16.0), $MachinePrecision] * N[(F * N[Power[C, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision] - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * N[(-1.0 / N[(N[Sqrt[B$95$m], $MachinePrecision] / N[Sqrt[F], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B 2) < 0.0

   1. Initial program 15.5%

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

   3. Simplified 27.5%

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

   5. Taylor expanded in A around -inf 18.2%

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

      1. associate-*r* 18.1%

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

   7. Simplified 18.1%

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

   if 0.0 < (pow.f64 B 2)

   1. Initial program 19.7%

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

   3. Taylor expanded in A around 0 10.1%

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

      1. mul-1-neg 10.1%

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

      2. distribute-rgt-neg-in 10.1%

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

      3. unpow2 10.1%

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

      4. unpow2 10.1%

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

      5. hypot-define 20.4%

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

   5. Simplified 20.4%

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

   6. Taylor expanded in C around 0 17.8%

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

      1. mul-1-neg 17.8%

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

   8. Simplified 17.8%

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

      1. sqrt-div 24.0%

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

      2. clear-num 23.9%

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

   10. Applied egg-rr 23.9%

       \[\leadsto -\color{blue}{\frac{1}{\frac{\sqrt{B}}{\sqrt{F}}}} \cdot \sqrt{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 22.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 0:\\ \;\;\;\;\frac{\sqrt{\left(A \cdot -16\right) \cdot \left(F \cdot {C}^{2}\right)}}{4 \cdot \left(A \cdot C\right) - {B}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \frac{-1}{\frac{\sqrt{B}}{\sqrt{F}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 40.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;F \leq -2.2 \cdot 10^{-301}:\\ \;\;\;\;\frac{\sqrt{\left(\left(A + C\right) + \mathsf{hypot}\left(B\_m, A - C\right)\right) \cdot \left(-8 \cdot \left(A \cdot \left(C \cdot F\right)\right)\right)}}{-\mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \frac{-1}{\frac{\sqrt{B\_m}}{\sqrt{F}}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= F -2.2e-301)
   (/
    (sqrt (* (+ (+ A C) (hypot B_m (- A C))) (* -8.0 (* A (* C F)))))
    (- (fma B_m B_m (* A (* C -4.0)))))
   (* (sqrt 2.0) (/ -1.0 (/ (sqrt B_m) (sqrt F))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= -2.2e-301) {
		tmp = sqrt((((A + C) + hypot(B_m, (A - C))) * (-8.0 * (A * (C * F))))) / -fma(B_m, B_m, (A * (C * -4.0)));
	} else {
		tmp = sqrt(2.0) * (-1.0 / (sqrt(B_m) / sqrt(F)));
	}
	return tmp;
}

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (F <= -2.2e-301)
		tmp = Float64(sqrt(Float64(Float64(Float64(A + C) + hypot(B_m, Float64(A - C))) * Float64(-8.0 * Float64(A * Float64(C * F))))) / Float64(-fma(B_m, B_m, Float64(A * Float64(C * -4.0)))));
	else
		tmp = Float64(sqrt(2.0) * Float64(-1.0 / Float64(sqrt(B_m) / sqrt(F))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[F, -2.2e-301], N[(N[Sqrt[N[(N[(N[(A + C), $MachinePrecision] + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[(-8.0 * N[(A * N[(C * F), $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[2.0], $MachinePrecision] * N[(-1.0 / N[(N[Sqrt[B$95$m], $MachinePrecision] / N[Sqrt[F], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -2.2e-301

   1. Initial program 22.3%

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

   3. Simplified 44.8%

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

   5. Taylor expanded in B around 0 39.6%

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

   if -2.2e-301 < F

   1. Initial program 18.1%

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

   3. Taylor expanded in A around 0 9.3%

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

      1. mul-1-neg 9.3%

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

      2. distribute-rgt-neg-in 9.3%

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

      3. unpow2 9.3%

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

      4. unpow2 9.3%

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

      5. hypot-define 18.9%

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

   5. Simplified 18.9%

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

   6. Taylor expanded in C around 0 15.7%

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

      1. mul-1-neg 15.7%

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

   8. Simplified 15.7%

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

      1. sqrt-div 21.6%

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

      2. clear-num 21.6%

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

   10. Applied egg-rr 21.6%

       \[\leadsto -\color{blue}{\frac{1}{\frac{\sqrt{B}}{\sqrt{F}}}} \cdot \sqrt{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 24.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.2 \cdot 10^{-301}:\\ \;\;\;\;\frac{\sqrt{\left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right) \cdot \left(-8 \cdot \left(A \cdot \left(C \cdot F\right)\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \frac{-1}{\frac{\sqrt{B}}{\sqrt{F}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 35.6% accurate, 2.1× speedup

Math

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

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (* (sqrt 2.0) (* (sqrt (/ 1.0 B_m)) (- (sqrt F)))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	return sqrt(2.0) * (sqrt((1.0 / B_m)) * -sqrt(F));
}

Fortran

B_m = abs(B)
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    code = sqrt(2.0d0) * (sqrt((1.0d0 / b_m)) * -sqrt(f))
end function

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	return Math.sqrt(2.0) * (Math.sqrt((1.0 / B_m)) * -Math.sqrt(F));
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	return math.sqrt(2.0) * (math.sqrt((1.0 / B_m)) * -math.sqrt(F))

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	return Float64(sqrt(2.0) * Float64(sqrt(Float64(1.0 / B_m)) * Float64(-sqrt(F))))
end

MATLAB

B_m = abs(B);
function tmp = code(A, B_m, C, F)
	tmp = sqrt(2.0) * (sqrt((1.0 / B_m)) * -sqrt(F));
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sqrt[N[(1.0 / B$95$m), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 18.6%

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

3. Taylor expanded in A around 0 8.1%

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

   1. mul-1-neg 8.1%

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

   2. distribute-rgt-neg-in 8.1%

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

   3. unpow2 8.1%

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

   4. unpow2 8.1%

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

   5. hypot-define 16.4%

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

5. Simplified 16.4%

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

6. Taylor expanded in C around 0 13.7%

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

   1. mul-1-neg 13.7%

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

8. Simplified 13.7%

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

   1. pow1/2 13.9%

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

   2. div-inv 13.9%

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

   3. unpow-prod-down 18.7%

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

   4. pow1/2 18.7%

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

10. Applied egg-rr 18.7%

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

    1. unpow1/2 18.7%

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

12. Simplified 18.7%

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

13. Final simplification 18.7%

    \[\leadsto \sqrt{2} \cdot \left(\sqrt{\frac{1}{B}} \cdot \left(-\sqrt{F}\right)\right) \]
14. Add Preprocessing

Alternative 16: 35.5% accurate, 2.1× speedup

Math

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

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (* (sqrt 2.0) (* (sqrt F) (/ -1.0 (sqrt B_m)))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	return sqrt(2.0) * (sqrt(F) * (-1.0 / sqrt(B_m)));
}

Fortran

B_m = abs(B)
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    code = sqrt(2.0d0) * (sqrt(f) * ((-1.0d0) / sqrt(b_m)))
end function

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	return Math.sqrt(2.0) * (Math.sqrt(F) * (-1.0 / Math.sqrt(B_m)));
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	return math.sqrt(2.0) * (math.sqrt(F) * (-1.0 / math.sqrt(B_m)))

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	return Float64(sqrt(2.0) * Float64(sqrt(F) * Float64(-1.0 / sqrt(B_m))))
end

MATLAB

B_m = abs(B);
function tmp = code(A, B_m, C, F)
	tmp = sqrt(2.0) * (sqrt(F) * (-1.0 / sqrt(B_m)));
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sqrt[F], $MachinePrecision] * N[(-1.0 / N[Sqrt[B$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 18.6%

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

3. Taylor expanded in A around 0 8.1%

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

   1. mul-1-neg 8.1%

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

   2. distribute-rgt-neg-in 8.1%

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

   3. unpow2 8.1%

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

   4. unpow2 8.1%

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

   5. hypot-define 16.4%

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

5. Simplified 16.4%

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

6. Taylor expanded in C around 0 13.7%

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

   1. mul-1-neg 13.7%

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

8. Simplified 13.7%

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

   1. sqrt-div 18.8%

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

   2. div-inv 18.8%

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

10. Applied egg-rr 18.8%

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

11. Final simplification 18.8%

    \[\leadsto \sqrt{2} \cdot \left(\sqrt{F} \cdot \frac{-1}{\sqrt{B}}\right) \]
12. Add Preprocessing

Alternative 17: 35.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \sqrt{2} \cdot \frac{-1}{\frac{\sqrt{B\_m}}{\sqrt{F}}} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (* (sqrt 2.0) (/ -1.0 (/ (sqrt B_m) (sqrt F)))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	return sqrt(2.0) * (-1.0 / (sqrt(B_m) / sqrt(F)));
}

Fortran

B_m = abs(B)
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) * ((-1.0d0) / (sqrt(b_m) / sqrt(f)))
end function

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	return Math.sqrt(2.0) * (-1.0 / (Math.sqrt(B_m) / Math.sqrt(F)));
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	return math.sqrt(2.0) * (-1.0 / (math.sqrt(B_m) / math.sqrt(F)))

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	return Float64(sqrt(2.0) * Float64(-1.0 / Float64(sqrt(B_m) / sqrt(F))))
end

MATLAB

B_m = abs(B);
function tmp = code(A, B_m, C, F)
	tmp = sqrt(2.0) * (-1.0 / (sqrt(B_m) / sqrt(F)));
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := N[(N[Sqrt[2.0], $MachinePrecision] * N[(-1.0 / N[(N[Sqrt[B$95$m], $MachinePrecision] / N[Sqrt[F], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 18.6%

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

3. Taylor expanded in A around 0 8.1%

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

   1. mul-1-neg 8.1%

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

   2. distribute-rgt-neg-in 8.1%

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

   3. unpow2 8.1%

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

   4. unpow2 8.1%

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

   5. hypot-define 16.4%

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

5. Simplified 16.4%

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

6. Taylor expanded in C around 0 13.7%

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

   1. mul-1-neg 13.7%

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

8. Simplified 13.7%

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

   1. sqrt-div 18.8%

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

   2. clear-num 18.7%

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

10. Applied egg-rr 18.7%

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

11. Final simplification 18.7%

    \[\leadsto \sqrt{2} \cdot \frac{-1}{\frac{\sqrt{B}}{\sqrt{F}}} \]
12. Add Preprocessing

Alternative 18: 35.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \sqrt{2} \cdot \frac{-\sqrt{F}}{\sqrt{B\_m}} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (* (sqrt 2.0) (/ (- (sqrt F)) (sqrt B_m))))

C

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

Fortran

B_m = abs(B)
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    code = sqrt(2.0d0) * (-sqrt(f) / sqrt(b_m))
end function

Java

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

Python

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

Julia

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

MATLAB

B_m = abs(B);
function tmp = code(A, B_m, C, F)
	tmp = sqrt(2.0) * (-sqrt(F) / sqrt(B_m));
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := N[(N[Sqrt[2.0], $MachinePrecision] * N[((-N[Sqrt[F], $MachinePrecision]) / N[Sqrt[B$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 18.6%

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

3. Taylor expanded in A around 0 8.1%

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

   1. mul-1-neg 8.1%

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

   2. distribute-rgt-neg-in 8.1%

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

   3. unpow2 8.1%

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

   4. unpow2 8.1%

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

   5. hypot-define 16.4%

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

5. Simplified 16.4%

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

6. Taylor expanded in C around 0 13.7%

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

   1. mul-1-neg 13.7%

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

8. Simplified 13.7%

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

   1. sqrt-div 18.8%

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

   2. div-inv 18.8%

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

10. Applied egg-rr 18.8%

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

    1. associate-*r/ 18.8%

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

    2. *-rgt-identity 18.8%

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

12. Simplified 18.8%

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

13. Final simplification 18.8%

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

Alternative 19: 33.8% accurate, 3.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;F \leq 5.4 \cdot 10^{+82}:\\ \;\;\;\;\sqrt{B\_m \cdot F} \cdot \frac{\sqrt{2}}{-B\_m}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \frac{-1}{\sqrt{\frac{B\_m}{F}}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= F 5.4e+82)
   (* (sqrt (* B_m F)) (/ (sqrt 2.0) (- B_m)))
   (* (sqrt 2.0) (/ -1.0 (sqrt (/ B_m F))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= 5.4e+82) {
		tmp = sqrt((B_m * F)) * (sqrt(2.0) / -B_m);
	} else {
		tmp = sqrt(2.0) * (-1.0 / sqrt((B_m / F)));
	}
	return tmp;
}

Fortran

B_m = abs(B)
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (f <= 5.4d+82) then
        tmp = sqrt((b_m * f)) * (sqrt(2.0d0) / -b_m)
    else
        tmp = sqrt(2.0d0) * ((-1.0d0) / sqrt((b_m / f)))
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= 5.4e+82) {
		tmp = Math.sqrt((B_m * F)) * (Math.sqrt(2.0) / -B_m);
	} else {
		tmp = Math.sqrt(2.0) * (-1.0 / Math.sqrt((B_m / F)));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if F <= 5.4e+82:
		tmp = math.sqrt((B_m * F)) * (math.sqrt(2.0) / -B_m)
	else:
		tmp = math.sqrt(2.0) * (-1.0 / math.sqrt((B_m / F)))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (F <= 5.4e+82)
		tmp = Float64(sqrt(Float64(B_m * F)) * Float64(sqrt(2.0) / Float64(-B_m)));
	else
		tmp = Float64(sqrt(2.0) * Float64(-1.0 / sqrt(Float64(B_m / F))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (F <= 5.4e+82)
		tmp = sqrt((B_m * F)) * (sqrt(2.0) / -B_m);
	else
		tmp = sqrt(2.0) * (-1.0 / sqrt((B_m / F)));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[F, 5.4e+82], N[(N[Sqrt[N[(B$95$m * F), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * N[(-1.0 / N[Sqrt[N[(B$95$m / F), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < 5.3999999999999999e82

   1. Initial program 20.4%

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

   3. Taylor expanded in A around 0 8.1%

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

      1. mul-1-neg 8.1%

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

      2. distribute-rgt-neg-in 8.1%

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

      3. unpow2 8.1%

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

      4. unpow2 8.1%

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

      5. hypot-define 18.9%

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

   5. Simplified 18.9%

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

   6. Taylor expanded in C around 0 17.7%

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

   if 5.3999999999999999e82 < F

   1. Initial program 14.8%

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

   3. Taylor expanded in A around 0 8.1%

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

      1. mul-1-neg 8.1%

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

      2. distribute-rgt-neg-in 8.1%

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

      3. unpow2 8.1%

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

      4. unpow2 8.1%

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

      5. hypot-define 11.0%

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

   5. Simplified 11.0%

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

   6. Taylor expanded in C around 0 20.4%

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

      1. mul-1-neg 20.4%

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

   8. Simplified 20.4%

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

      1. clear-num 20.4%

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

      2. sqrt-div 20.4%

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

      3. metadata-eval 20.4%

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

   10. Applied egg-rr 20.4%

       \[\leadsto -\color{blue}{\frac{1}{\sqrt{\frac{B}{F}}}} \cdot \sqrt{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 18.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 5.4 \cdot 10^{+82}:\\ \;\;\;\;\sqrt{B \cdot F} \cdot \frac{\sqrt{2}}{-B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \frac{-1}{\sqrt{\frac{B}{F}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 20: 27.2% accurate, 5.9× speedup

Math

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

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F) :precision binary64 (- (pow (* 2.0 (/ F B_m)) 0.5)))

C

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

Fortran

B_m = abs(B)
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    code = -((2.0d0 * (f / b_m)) ** 0.5d0)
end function

Java

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

Python

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

Julia

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

MATLAB

B_m = abs(B);
function tmp = code(A, B_m, C, F)
	tmp = -((2.0 * (F / B_m)) ^ 0.5);
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := (-N[Power[N[(2.0 * N[(F / B$95$m), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision])

Derivation

1. Initial program 18.6%

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

3. Taylor expanded in A around 0 8.1%

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

   1. mul-1-neg 8.1%

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

   2. distribute-rgt-neg-in 8.1%

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

   3. unpow2 8.1%

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

   4. unpow2 8.1%

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

   5. hypot-define 16.4%

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

5. Simplified 16.4%

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

6. Taylor expanded in C around 0 13.7%

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

   1. mul-1-neg 13.7%

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

8. Simplified 13.7%

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

   1. *-commutative 13.7%

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

   2. pow1/2 13.7%

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

   3. pow1/2 13.9%

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

   4. pow-prod-down 13.9%

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

10. Applied egg-rr 13.9%

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

11. Final simplification 13.9%

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

Alternative 21: 27.2% accurate, 6.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ -\sqrt{2 \cdot \frac{F}{B\_m}} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F) :precision binary64 (- (sqrt (* 2.0 (/ F B_m)))))

C

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

Fortran

B_m = abs(B)
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    code = -sqrt((2.0d0 * (f / b_m)))
end function

Java

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

Python

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

Julia

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

MATLAB

B_m = abs(B);
function tmp = code(A, B_m, C, F)
	tmp = -sqrt((2.0 * (F / B_m)));
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := (-N[Sqrt[N[(2.0 * N[(F / B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])

Derivation

1. Initial program 18.6%

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

3. Taylor expanded in A around 0 8.1%

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

   1. mul-1-neg 8.1%

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

   2. distribute-rgt-neg-in 8.1%

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

   3. unpow2 8.1%

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

   4. unpow2 8.1%

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

   5. hypot-define 16.4%

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

5. Simplified 16.4%

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

6. Taylor expanded in C around 0 13.7%

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

   1. mul-1-neg 13.7%

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

8. Simplified 13.7%

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

   1. pow1 13.7%

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

   2. *-commutative 13.7%

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

   3. sqrt-unprod 13.8%

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

10. Applied egg-rr 13.8%

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

    1. unpow1 13.8%

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

12. Simplified 13.8%

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

13. Final simplification 13.8%

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

Reproduce

herbie shell --seed 2024052 
(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))))