ABCF->ab-angle a

Percentage Accurate: 18.7% → 56.7%

Time: 39.5s

Alternatives: 17

Speedup: 3.0×

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: 18.7% 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: 56.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := \left(4 \cdot A\right) \cdot C\\ t_2 := 2 \cdot \left(\left({B\_m}^{2} - t\_1\right) \cdot F\right)\\ t_3 := t\_1 - {B\_m}^{2}\\ t_4 := \frac{\sqrt{t\_2 \cdot \left(\left(A + C\right) + \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_3}\\ \mathbf{if}\;t\_4 \leq -1 \cdot 10^{-211}:\\ \;\;\;\;\frac{\left(\sqrt{C + \left(A + \mathsf{hypot}\left(B\_m, A - C\right)\right)} \cdot \sqrt{F}\right) \cdot \sqrt{2 \cdot \mathsf{fma}\left(A, C \cdot -4, {B\_m}^{2}\right)}}{4 \cdot \left(A \cdot C\right) - {B\_m}^{2}}\\ \mathbf{elif}\;t\_4 \leq 4 \cdot 10^{+185}:\\ \;\;\;\;\frac{\sqrt{t\_2 \cdot \left(-0.5 \cdot \frac{{B\_m}^{2}}{A} + 2 \cdot C\right)}}{t\_3}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(A + C\right) + \mathsf{hypot}\left(A - C, B\_m\right)} \cdot \sqrt{t\_0 \cdot \left(2 \cdot F\right)}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B\_m} \cdot \left(\sqrt{C + \mathsf{hypot}\left(B\_m, C\right)} \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \end{array} \]

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 (* (* 4.0 A) C))
        (t_2 (* 2.0 (* (- (pow B_m 2.0) t_1) F)))
        (t_3 (- t_1 (pow B_m 2.0)))
        (t_4
         (/
          (sqrt (* t_2 (+ (+ A C) (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))))))
          t_3)))
   (if (<= t_4 -1e-211)
     (/
      (*
       (* (sqrt (+ C (+ A (hypot B_m (- A C))))) (sqrt F))
       (sqrt (* 2.0 (fma A (* C -4.0) (pow B_m 2.0)))))
      (- (* 4.0 (* A C)) (pow B_m 2.0)))
     (if (<= t_4 4e+185)
       (/ (sqrt (* t_2 (+ (* -0.5 (/ (pow B_m 2.0) A)) (* 2.0 C)))) t_3)
       (if (<= t_4 INFINITY)
         (/
          (* (sqrt (+ (+ A C) (hypot (- A C) B_m))) (sqrt (* t_0 (* 2.0 F))))
          (- t_0))
         (*
          (/ (sqrt 2.0) B_m)
          (* (sqrt (+ C (hypot B_m C))) (- (sqrt F)))))))))

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 = (4.0 * A) * C;
	double t_2 = 2.0 * ((pow(B_m, 2.0) - t_1) * F);
	double t_3 = t_1 - pow(B_m, 2.0);
	double t_4 = sqrt((t_2 * ((A + C) + sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / t_3;
	double tmp;
	if (t_4 <= -1e-211) {
		tmp = ((sqrt((C + (A + hypot(B_m, (A - C))))) * sqrt(F)) * sqrt((2.0 * fma(A, (C * -4.0), pow(B_m, 2.0))))) / ((4.0 * (A * C)) - pow(B_m, 2.0));
	} else if (t_4 <= 4e+185) {
		tmp = sqrt((t_2 * ((-0.5 * (pow(B_m, 2.0) / A)) + (2.0 * C)))) / t_3;
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = (sqrt(((A + C) + hypot((A - C), B_m))) * sqrt((t_0 * (2.0 * F)))) / -t_0;
	} else {
		tmp = (sqrt(2.0) / B_m) * (sqrt((C + hypot(B_m, C))) * -sqrt(F));
	}
	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(Float64(4.0 * A) * C)
	t_2 = Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_1) * F))
	t_3 = Float64(t_1 - (B_m ^ 2.0))
	t_4 = Float64(sqrt(Float64(t_2 * Float64(Float64(A + C) + sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / t_3)
	tmp = 0.0
	if (t_4 <= -1e-211)
		tmp = Float64(Float64(Float64(sqrt(Float64(C + Float64(A + hypot(B_m, Float64(A - C))))) * sqrt(F)) * sqrt(Float64(2.0 * fma(A, Float64(C * -4.0), (B_m ^ 2.0))))) / Float64(Float64(4.0 * Float64(A * C)) - (B_m ^ 2.0)));
	elseif (t_4 <= 4e+185)
		tmp = Float64(sqrt(Float64(t_2 * Float64(Float64(-0.5 * Float64((B_m ^ 2.0) / A)) + Float64(2.0 * C)))) / t_3);
	elseif (t_4 <= Inf)
		tmp = Float64(Float64(sqrt(Float64(Float64(A + C) + hypot(Float64(A - C), B_m))) * sqrt(Float64(t_0 * Float64(2.0 * F)))) / Float64(-t_0));
	else
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(sqrt(Float64(C + hypot(B_m, C))) * Float64(-sqrt(F))));
	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[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$1), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[N[(t$95$2 * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$3), $MachinePrecision]}, If[LessEqual[t$95$4, -1e-211], N[(N[(N[(N[Sqrt[N[(C + N[(A + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $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] / N[(N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision] - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 4e+185], N[(N[Sqrt[N[(t$95$2 * N[(N[(-0.5 * N[(N[Power[B$95$m, 2.0], $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision] + N[(2.0 * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$3), $MachinePrecision], If[LessEqual[t$95$4, 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[(t$95$0 * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / (-t$95$0)), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[(N[Sqrt[N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))) < -1.00000000000000009e-211

   1. Initial program 48.1%

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

   3. Simplified 43.3%

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

      1. pow1/2 43.3%

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

      2. associate-*r* 56.1%

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

      3. unpow-prod-down 69.5%

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

      4. +-commutative 69.5%

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

      5. hypot-undefine 54.5%

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

      6. unpow2 54.5%

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

      7. unpow2 54.5%

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

      8. +-commutative 54.5%

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

      9. unpow2 54.5%

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

      10. unpow2 54.5%

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

      11. hypot-define 69.5%

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

      12. pow1/2 69.5%

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

   6. Applied egg-rr 69.5%

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

      1. unpow1/2 69.5%

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

      2. +-commutative 69.5%

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

      3. +-commutative 69.5%

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

      4. hypot-undefine 54.5%

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

      5. unpow2 54.5%

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

      6. unpow2 54.5%

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

      7. +-commutative 54.5%

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

      8. unpow2 54.5%

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

      9. unpow2 54.5%

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

      10. hypot-undefine 69.5%

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

   8. Simplified 69.5%

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

      1. pow1/2 69.5%

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

      2. *-commutative 69.5%

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

      3. unpow-prod-down 76.6%

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

      4. pow1/2 76.6%

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

      5. associate-+l+ 77.2%

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

      6. pow1/2 77.2%

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

   10. Applied egg-rr 77.2%

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

   if -1.00000000000000009e-211 < (/.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))) < 3.9999999999999999e185

   1. Initial program 21.3%

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

   3. Taylor expanded in A around -inf 50.9%

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

   if 3.9999999999999999e185 < (/.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 9.3%

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

   3. Simplified 54.4%

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

      1. pow1/2 54.4%

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

      2. *-commutative 54.4%

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

      3. unpow-prod-down 71.1%

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

      4. pow1/2 71.1%

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

      5. +-commutative 71.1%

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

      6. hypot-undefine 9.4%

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

      7. unpow2 9.4%

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

      8. unpow2 9.4%

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

      9. +-commutative 9.4%

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

      10. unpow2 9.4%

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

      11. unpow2 9.4%

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

      12. hypot-define 71.1%

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

      13. pow1/2 71.1%

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

      14. *-commutative 71.1%

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

   6. Applied egg-rr 71.1%

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

   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 2.1%

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

      1. mul-1-neg 2.1%

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

      2. unpow2 2.1%

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

      3. unpow2 2.1%

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

      4. hypot-define 20.2%

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

   5. Simplified 20.2%

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

      1. pow1/2 20.2%

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

      2. *-commutative 20.2%

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

      3. unpow-prod-down 31.6%

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

      4. pow1/2 31.6%

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

      5. pow1/2 31.6%

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

   7. Applied egg-rr 31.6%

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

4. Final simplification 53.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -1 \cdot 10^{-211}:\\ \;\;\;\;\frac{\left(\sqrt{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \sqrt{F}\right) \cdot \sqrt{2 \cdot \mathsf{fma}\left(A, C \cdot -4, {B}^{2}\right)}}{4 \cdot \left(A \cdot C\right) - {B}^{2}}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq 4 \cdot 10^{+185}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(-0.5 \cdot \frac{{B}^{2}}{A} + 2 \cdot C\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(A + C\right) + \mathsf{hypot}\left(A - C, B\right)} \cdot \sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(2 \cdot F\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{C + \mathsf{hypot}\left(B, C\right)} \cdot \left(-\sqrt{F}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 45.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ t_1 := t\_0 - {B\_m}^{2}\\ t_2 := 2 \cdot \left({B\_m}^{2} + -4 \cdot \left(A \cdot C\right)\right)\\ t_3 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_0\right) \cdot F\right)\right) \cdot \left(2 \cdot A\right)}}{t\_1}\\ \mathbf{if}\;{B\_m}^{2} \leq 0:\\ \;\;\;\;\frac{\sqrt{\left|t\_2 \cdot \left(F \cdot \left(2 \cdot C\right)\right)\right|}}{t\_1}\\ \mathbf{elif}\;{B\_m}^{2} \leq 10^{-219}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;{B\_m}^{2} \leq 2 \cdot 10^{-128}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot F\right) \cdot t\_2} \cdot \sqrt{C}}{t\_1}\\ \mathbf{elif}\;{B\_m}^{2} \leq 10^{-127}:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{F} \cdot \sqrt{A + \mathsf{hypot}\left(B\_m, A\right)}\right) \cdot \frac{\sqrt{2}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (* (* 4.0 A) C))
        (t_1 (- t_0 (pow B_m 2.0)))
        (t_2 (* 2.0 (+ (pow B_m 2.0) (* -4.0 (* A C)))))
        (t_3 (/ (sqrt (* (* 2.0 (* (- (pow B_m 2.0) t_0) F)) (* 2.0 A))) t_1)))
   (if (<= (pow B_m 2.0) 0.0)
     (/ (sqrt (fabs (* t_2 (* F (* 2.0 C))))) t_1)
     (if (<= (pow B_m 2.0) 1e-219)
       t_3
       (if (<= (pow B_m 2.0) 2e-128)
         (/ (* (sqrt (* (* 2.0 F) t_2)) (sqrt C)) t_1)
         (if (<= (pow B_m 2.0) 1e-127)
           t_3
           (*
            (* (sqrt F) (sqrt (+ A (hypot B_m A))))
            (/ (sqrt 2.0) (- B_m)))))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double t_0 = (4.0 * A) * C;
	double t_1 = t_0 - pow(B_m, 2.0);
	double t_2 = 2.0 * (pow(B_m, 2.0) + (-4.0 * (A * C)));
	double t_3 = sqrt(((2.0 * ((pow(B_m, 2.0) - t_0) * F)) * (2.0 * A))) / t_1;
	double tmp;
	if (pow(B_m, 2.0) <= 0.0) {
		tmp = sqrt(fabs((t_2 * (F * (2.0 * C))))) / t_1;
	} else if (pow(B_m, 2.0) <= 1e-219) {
		tmp = t_3;
	} else if (pow(B_m, 2.0) <= 2e-128) {
		tmp = (sqrt(((2.0 * F) * t_2)) * sqrt(C)) / t_1;
	} else if (pow(B_m, 2.0) <= 1e-127) {
		tmp = t_3;
	} else {
		tmp = (sqrt(F) * sqrt((A + hypot(B_m, A)))) * (sqrt(2.0) / -B_m);
	}
	return tmp;
}

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double t_0 = (4.0 * A) * C;
	double t_1 = t_0 - Math.pow(B_m, 2.0);
	double t_2 = 2.0 * (Math.pow(B_m, 2.0) + (-4.0 * (A * C)));
	double t_3 = Math.sqrt(((2.0 * ((Math.pow(B_m, 2.0) - t_0) * F)) * (2.0 * A))) / t_1;
	double tmp;
	if (Math.pow(B_m, 2.0) <= 0.0) {
		tmp = Math.sqrt(Math.abs((t_2 * (F * (2.0 * C))))) / t_1;
	} else if (Math.pow(B_m, 2.0) <= 1e-219) {
		tmp = t_3;
	} else if (Math.pow(B_m, 2.0) <= 2e-128) {
		tmp = (Math.sqrt(((2.0 * F) * t_2)) * Math.sqrt(C)) / t_1;
	} else if (Math.pow(B_m, 2.0) <= 1e-127) {
		tmp = t_3;
	} else {
		tmp = (Math.sqrt(F) * Math.sqrt((A + Math.hypot(B_m, A)))) * (Math.sqrt(2.0) / -B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	t_0 = (4.0 * A) * C
	t_1 = t_0 - math.pow(B_m, 2.0)
	t_2 = 2.0 * (math.pow(B_m, 2.0) + (-4.0 * (A * C)))
	t_3 = math.sqrt(((2.0 * ((math.pow(B_m, 2.0) - t_0) * F)) * (2.0 * A))) / t_1
	tmp = 0
	if math.pow(B_m, 2.0) <= 0.0:
		tmp = math.sqrt(math.fabs((t_2 * (F * (2.0 * C))))) / t_1
	elif math.pow(B_m, 2.0) <= 1e-219:
		tmp = t_3
	elif math.pow(B_m, 2.0) <= 2e-128:
		tmp = (math.sqrt(((2.0 * F) * t_2)) * math.sqrt(C)) / t_1
	elif math.pow(B_m, 2.0) <= 1e-127:
		tmp = t_3
	else:
		tmp = (math.sqrt(F) * math.sqrt((A + math.hypot(B_m, A)))) * (math.sqrt(2.0) / -B_m)
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	t_0 = Float64(Float64(4.0 * A) * C)
	t_1 = Float64(t_0 - (B_m ^ 2.0))
	t_2 = Float64(2.0 * Float64((B_m ^ 2.0) + Float64(-4.0 * Float64(A * C))))
	t_3 = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_0) * F)) * Float64(2.0 * A))) / t_1)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 0.0)
		tmp = Float64(sqrt(abs(Float64(t_2 * Float64(F * Float64(2.0 * C))))) / t_1);
	elseif ((B_m ^ 2.0) <= 1e-219)
		tmp = t_3;
	elseif ((B_m ^ 2.0) <= 2e-128)
		tmp = Float64(Float64(sqrt(Float64(Float64(2.0 * F) * t_2)) * sqrt(C)) / t_1);
	elseif ((B_m ^ 2.0) <= 1e-127)
		tmp = t_3;
	else
		tmp = Float64(Float64(sqrt(F) * sqrt(Float64(A + hypot(B_m, A)))) * Float64(sqrt(2.0) / Float64(-B_m)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	t_0 = (4.0 * A) * C;
	t_1 = t_0 - (B_m ^ 2.0);
	t_2 = 2.0 * ((B_m ^ 2.0) + (-4.0 * (A * C)));
	t_3 = sqrt(((2.0 * (((B_m ^ 2.0) - t_0) * F)) * (2.0 * A))) / t_1;
	tmp = 0.0;
	if ((B_m ^ 2.0) <= 0.0)
		tmp = sqrt(abs((t_2 * (F * (2.0 * C))))) / t_1;
	elseif ((B_m ^ 2.0) <= 1e-219)
		tmp = t_3;
	elseif ((B_m ^ 2.0) <= 2e-128)
		tmp = (sqrt(((2.0 * F) * t_2)) * sqrt(C)) / t_1;
	elseif ((B_m ^ 2.0) <= 1e-127)
		tmp = t_3;
	else
		tmp = (sqrt(F) * sqrt((A + hypot(B_m, A)))) * (sqrt(2.0) / -B_m);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 * N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(2.0 * A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 0.0], N[(N[Sqrt[N[Abs[N[(t$95$2 * N[(F * N[(2.0 * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-219], t$95$3, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e-128], N[(N[(N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] * t$95$2), $MachinePrecision]], $MachinePrecision] * N[Sqrt[C], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-127], t$95$3, N[(N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(A + N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 20.2%

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

   3. Taylor expanded in A around -inf 34.3%

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

      1. add-sqr-sqrt 34.3%

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

      2. pow1/2 34.3%

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

      3. pow1/2 34.4%

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

      4. pow-prod-down 18.1%

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

      5. pow2 18.1%

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

      6. associate-*r* 18.1%

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

      7. *-commutative 18.1%

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

      8. *-commutative 18.1%

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

   5. Applied egg-rr 18.1%

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

      1. unpow1/2 18.1%

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

      2. unpow2 18.1%

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

      3. rem-sqrt-square 34.9%

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

      4. associate-*l* 36.7%

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

      5. *-commutative 36.7%

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

      6. *-commutative 36.7%

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

      7. associate-*r* 36.7%

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

      8. cancel-sign-sub-inv 36.7%

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

      9. metadata-eval 36.7%

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

   7. Simplified 36.7%

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

   if 0.0 < (pow.f64 B 2) < 1e-219 or 2.00000000000000011e-128 < (pow.f64 B 2) < 1e-127

   1. Initial program 31.9%

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

   3. Taylor expanded in A around inf 58.3%

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

   if 1e-219 < (pow.f64 B 2) < 2.00000000000000011e-128

   1. Initial program 28.8%

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

   3. Taylor expanded in A around -inf 20.8%

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

      1. pow1/2 20.9%

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

      2. associate-*r* 20.9%

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

      3. unpow-prod-down 34.2%

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

      4. associate-*r* 34.2%

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

      5. *-commutative 34.2%

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

      6. *-commutative 34.2%

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

      7. pow1/2 34.2%

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

   5. Applied egg-rr 34.2%

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

      1. unpow1/2 34.2%

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

      2. associate-*l* 34.2%

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

      3. *-commutative 34.2%

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

      4. *-commutative 34.2%

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

      5. associate-*r* 34.2%

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

      6. cancel-sign-sub-inv 34.2%

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

      7. metadata-eval 34.2%

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

   7. Simplified 34.2%

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

   if 1e-127 < (pow.f64 B 2)

   1. Initial program 18.7%

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

   3. Taylor expanded in C around 0 13.4%

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

      1. mul-1-neg 13.4%

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

      2. +-commutative 13.4%

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

      3. unpow2 13.4%

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

      4. unpow2 13.4%

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

      5. hypot-define 27.5%

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

   5. Simplified 27.5%

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

      1. pow1/2 27.6%

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

      2. *-commutative 27.6%

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

      3. unpow-prod-down 38.5%

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

      4. pow1/2 38.5%

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

      5. pow1/2 38.5%

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

   7. Applied egg-rr 38.5%

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

4. Final simplification 39.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 0:\\ \;\;\;\;\frac{\sqrt{\left|\left(2 \cdot \left({B}^{2} + -4 \cdot \left(A \cdot C\right)\right)\right) \cdot \left(F \cdot \left(2 \cdot C\right)\right)\right|}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;{B}^{2} \leq 10^{-219}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot A\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{-128}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot F\right) \cdot \left(2 \cdot \left({B}^{2} + -4 \cdot \left(A \cdot C\right)\right)\right)} \cdot \sqrt{C}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;{B}^{2} \leq 10^{-127}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot A\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{F} \cdot \sqrt{A + \mathsf{hypot}\left(B, A\right)}\right) \cdot \frac{\sqrt{2}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 45.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ t_1 := {B\_m}^{2} - t\_0\\ t_2 := t\_0 - {B\_m}^{2}\\ t_3 := \frac{\sqrt{\left(2 \cdot \left(t\_1 \cdot F\right)\right) \cdot \left(2 \cdot A\right)}}{t\_2}\\ \mathbf{if}\;{B\_m}^{2} \leq 0:\\ \;\;\;\;\frac{\sqrt{\left|\left(2 \cdot \left({B\_m}^{2} + -4 \cdot \left(A \cdot C\right)\right)\right) \cdot \left(F \cdot \left(2 \cdot C\right)\right)\right|}}{t\_2}\\ \mathbf{elif}\;{B\_m}^{2} \leq 10^{-219}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;{B\_m}^{2} \leq 2 \cdot 10^{-128}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(2 \cdot t\_1\right)} \cdot \sqrt{2 \cdot C}}{t\_2}\\ \mathbf{elif}\;{B\_m}^{2} \leq 10^{-127}:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{F} \cdot \sqrt{A + \mathsf{hypot}\left(B\_m, A\right)}\right) \cdot \frac{\sqrt{2}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (* (* 4.0 A) C))
        (t_1 (- (pow B_m 2.0) t_0))
        (t_2 (- t_0 (pow B_m 2.0)))
        (t_3 (/ (sqrt (* (* 2.0 (* t_1 F)) (* 2.0 A))) t_2)))
   (if (<= (pow B_m 2.0) 0.0)
     (/
      (sqrt
       (fabs (* (* 2.0 (+ (pow B_m 2.0) (* -4.0 (* A C)))) (* F (* 2.0 C)))))
      t_2)
     (if (<= (pow B_m 2.0) 1e-219)
       t_3
       (if (<= (pow B_m 2.0) 2e-128)
         (/ (* (sqrt (* F (* 2.0 t_1))) (sqrt (* 2.0 C))) t_2)
         (if (<= (pow B_m 2.0) 1e-127)
           t_3
           (*
            (* (sqrt F) (sqrt (+ A (hypot B_m A))))
            (/ (sqrt 2.0) (- B_m)))))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double t_0 = (4.0 * A) * C;
	double t_1 = pow(B_m, 2.0) - t_0;
	double t_2 = t_0 - pow(B_m, 2.0);
	double t_3 = sqrt(((2.0 * (t_1 * F)) * (2.0 * A))) / t_2;
	double tmp;
	if (pow(B_m, 2.0) <= 0.0) {
		tmp = sqrt(fabs(((2.0 * (pow(B_m, 2.0) + (-4.0 * (A * C)))) * (F * (2.0 * C))))) / t_2;
	} else if (pow(B_m, 2.0) <= 1e-219) {
		tmp = t_3;
	} else if (pow(B_m, 2.0) <= 2e-128) {
		tmp = (sqrt((F * (2.0 * t_1))) * sqrt((2.0 * C))) / t_2;
	} else if (pow(B_m, 2.0) <= 1e-127) {
		tmp = t_3;
	} else {
		tmp = (sqrt(F) * sqrt((A + hypot(B_m, A)))) * (sqrt(2.0) / -B_m);
	}
	return tmp;
}

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double t_0 = (4.0 * A) * C;
	double t_1 = Math.pow(B_m, 2.0) - t_0;
	double t_2 = t_0 - Math.pow(B_m, 2.0);
	double t_3 = Math.sqrt(((2.0 * (t_1 * F)) * (2.0 * A))) / t_2;
	double tmp;
	if (Math.pow(B_m, 2.0) <= 0.0) {
		tmp = Math.sqrt(Math.abs(((2.0 * (Math.pow(B_m, 2.0) + (-4.0 * (A * C)))) * (F * (2.0 * C))))) / t_2;
	} else if (Math.pow(B_m, 2.0) <= 1e-219) {
		tmp = t_3;
	} else if (Math.pow(B_m, 2.0) <= 2e-128) {
		tmp = (Math.sqrt((F * (2.0 * t_1))) * Math.sqrt((2.0 * C))) / t_2;
	} else if (Math.pow(B_m, 2.0) <= 1e-127) {
		tmp = t_3;
	} else {
		tmp = (Math.sqrt(F) * Math.sqrt((A + Math.hypot(B_m, A)))) * (Math.sqrt(2.0) / -B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	t_0 = (4.0 * A) * C
	t_1 = math.pow(B_m, 2.0) - t_0
	t_2 = t_0 - math.pow(B_m, 2.0)
	t_3 = math.sqrt(((2.0 * (t_1 * F)) * (2.0 * A))) / t_2
	tmp = 0
	if math.pow(B_m, 2.0) <= 0.0:
		tmp = math.sqrt(math.fabs(((2.0 * (math.pow(B_m, 2.0) + (-4.0 * (A * C)))) * (F * (2.0 * C))))) / t_2
	elif math.pow(B_m, 2.0) <= 1e-219:
		tmp = t_3
	elif math.pow(B_m, 2.0) <= 2e-128:
		tmp = (math.sqrt((F * (2.0 * t_1))) * math.sqrt((2.0 * C))) / t_2
	elif math.pow(B_m, 2.0) <= 1e-127:
		tmp = t_3
	else:
		tmp = (math.sqrt(F) * math.sqrt((A + math.hypot(B_m, A)))) * (math.sqrt(2.0) / -B_m)
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	t_0 = Float64(Float64(4.0 * A) * C)
	t_1 = Float64((B_m ^ 2.0) - t_0)
	t_2 = Float64(t_0 - (B_m ^ 2.0))
	t_3 = Float64(sqrt(Float64(Float64(2.0 * Float64(t_1 * F)) * Float64(2.0 * A))) / t_2)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 0.0)
		tmp = Float64(sqrt(abs(Float64(Float64(2.0 * Float64((B_m ^ 2.0) + Float64(-4.0 * Float64(A * C)))) * Float64(F * Float64(2.0 * C))))) / t_2);
	elseif ((B_m ^ 2.0) <= 1e-219)
		tmp = t_3;
	elseif ((B_m ^ 2.0) <= 2e-128)
		tmp = Float64(Float64(sqrt(Float64(F * Float64(2.0 * t_1))) * sqrt(Float64(2.0 * C))) / t_2);
	elseif ((B_m ^ 2.0) <= 1e-127)
		tmp = t_3;
	else
		tmp = Float64(Float64(sqrt(F) * sqrt(Float64(A + hypot(B_m, A)))) * Float64(sqrt(2.0) / Float64(-B_m)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	t_0 = (4.0 * A) * C;
	t_1 = (B_m ^ 2.0) - t_0;
	t_2 = t_0 - (B_m ^ 2.0);
	t_3 = sqrt(((2.0 * (t_1 * F)) * (2.0 * A))) / t_2;
	tmp = 0.0;
	if ((B_m ^ 2.0) <= 0.0)
		tmp = sqrt(abs(((2.0 * ((B_m ^ 2.0) + (-4.0 * (A * C)))) * (F * (2.0 * C))))) / t_2;
	elseif ((B_m ^ 2.0) <= 1e-219)
		tmp = t_3;
	elseif ((B_m ^ 2.0) <= 2e-128)
		tmp = (sqrt((F * (2.0 * t_1))) * sqrt((2.0 * C))) / t_2;
	elseif ((B_m ^ 2.0) <= 1e-127)
		tmp = t_3;
	else
		tmp = (sqrt(F) * sqrt((A + hypot(B_m, A)))) * (sqrt(2.0) / -B_m);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$1 = N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(N[(2.0 * N[(t$95$1 * F), $MachinePrecision]), $MachinePrecision] * N[(2.0 * A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 0.0], N[(N[Sqrt[N[Abs[N[(N[(2.0 * N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(F * N[(2.0 * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-219], t$95$3, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e-128], N[(N[(N[Sqrt[N[(F * N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(2.0 * C), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-127], t$95$3, N[(N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(A + N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 20.2%

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

   3. Taylor expanded in A around -inf 34.3%

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

      1. add-sqr-sqrt 34.3%

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

      2. pow1/2 34.3%

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

      3. pow1/2 34.4%

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

      4. pow-prod-down 18.1%

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

      5. pow2 18.1%

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

      6. associate-*r* 18.1%

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

      7. *-commutative 18.1%

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

      8. *-commutative 18.1%

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

   5. Applied egg-rr 18.1%

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

      1. unpow1/2 18.1%

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

      2. unpow2 18.1%

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

      3. rem-sqrt-square 34.9%

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

      4. associate-*l* 36.7%

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

      5. *-commutative 36.7%

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

      6. *-commutative 36.7%

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

      7. associate-*r* 36.7%

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

      8. cancel-sign-sub-inv 36.7%

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

      9. metadata-eval 36.7%

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

   7. Simplified 36.7%

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

   if 0.0 < (pow.f64 B 2) < 1e-219 or 2.00000000000000011e-128 < (pow.f64 B 2) < 1e-127

   1. Initial program 31.9%

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

   3. Taylor expanded in A around inf 58.3%

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

   if 1e-219 < (pow.f64 B 2) < 2.00000000000000011e-128

   1. Initial program 28.8%

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

   3. Taylor expanded in A around -inf 20.8%

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

      1. sqrt-prod 34.3%

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

      2. associate-*r* 34.3%

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

      3. *-commutative 34.3%

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

      4. *-commutative 34.3%

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

   5. Applied egg-rr 34.3%

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

   if 1e-127 < (pow.f64 B 2)

   1. Initial program 18.7%

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

   3. Taylor expanded in C around 0 13.4%

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

      1. mul-1-neg 13.4%

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

      2. +-commutative 13.4%

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

      3. unpow2 13.4%

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

      4. unpow2 13.4%

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

      5. hypot-define 27.5%

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

   5. Simplified 27.5%

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

      1. pow1/2 27.6%

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

      2. *-commutative 27.6%

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

      3. unpow-prod-down 38.5%

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

      4. pow1/2 38.5%

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

      5. pow1/2 38.5%

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

   7. Applied egg-rr 38.5%

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

4. Final simplification 39.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 0:\\ \;\;\;\;\frac{\sqrt{\left|\left(2 \cdot \left({B}^{2} + -4 \cdot \left(A \cdot C\right)\right)\right) \cdot \left(F \cdot \left(2 \cdot C\right)\right)\right|}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;{B}^{2} \leq 10^{-219}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot A\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{-128}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(2 \cdot \left({B}^{2} - \left(4 \cdot A\right) \cdot C\right)\right)} \cdot \sqrt{2 \cdot C}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;{B}^{2} \leq 10^{-127}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot A\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{F} \cdot \sqrt{A + \mathsf{hypot}\left(B, A\right)}\right) \cdot \frac{\sqrt{2}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 47.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 0:\\ \;\;\;\;\frac{\sqrt{\left|\left(2 \cdot \left({B\_m}^{2} + -4 \cdot \left(A \cdot C\right)\right)\right) \cdot \left(F \cdot \left(2 \cdot C\right)\right)\right|}}{\left(4 \cdot A\right) \cdot C - {B\_m}^{2}}\\ \mathbf{elif}\;{B\_m}^{2} \leq 2 \cdot 10^{+117}:\\ \;\;\;\;\frac{\sqrt{\left(t\_0 \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B\_m, A - C\right)\right)}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{F} \cdot \sqrt{A + \mathsf{hypot}\left(B\_m, A\right)}\right) \cdot \frac{\sqrt{2}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(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) 0.0)
     (/
      (sqrt
       (fabs (* (* 2.0 (+ (pow B_m 2.0) (* -4.0 (* A C)))) (* F (* 2.0 C)))))
      (- (* (* 4.0 A) C) (pow B_m 2.0)))
     (if (<= (pow B_m 2.0) 2e+117)
       (/ (sqrt (* (* t_0 (* 2.0 F)) (+ (+ A C) (hypot B_m (- A C))))) (- t_0))
       (* (* (sqrt F) (sqrt (+ A (hypot B_m A)))) (/ (sqrt 2.0) (- B_m)))))))

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) <= 0.0) {
		tmp = sqrt(fabs(((2.0 * (pow(B_m, 2.0) + (-4.0 * (A * C)))) * (F * (2.0 * C))))) / (((4.0 * A) * C) - pow(B_m, 2.0));
	} else if (pow(B_m, 2.0) <= 2e+117) {
		tmp = sqrt(((t_0 * (2.0 * F)) * ((A + C) + hypot(B_m, (A - C))))) / -t_0;
	} else {
		tmp = (sqrt(F) * sqrt((A + hypot(B_m, A)))) * (sqrt(2.0) / -B_m);
	}
	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) <= 0.0)
		tmp = Float64(sqrt(abs(Float64(Float64(2.0 * Float64((B_m ^ 2.0) + Float64(-4.0 * Float64(A * C)))) * Float64(F * Float64(2.0 * C))))) / Float64(Float64(Float64(4.0 * A) * C) - (B_m ^ 2.0)));
	elseif ((B_m ^ 2.0) <= 2e+117)
		tmp = Float64(sqrt(Float64(Float64(t_0 * Float64(2.0 * F)) * Float64(Float64(A + C) + hypot(B_m, Float64(A - C))))) / Float64(-t_0));
	else
		tmp = Float64(Float64(sqrt(F) * sqrt(Float64(A + hypot(B_m, A)))) * Float64(sqrt(2.0) / Float64(-B_m)));
	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], 0.0], N[(N[Sqrt[N[Abs[N[(N[(2.0 * N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(F * N[(2.0 * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / N[(N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision] - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+117], N[(N[Sqrt[N[(N[(t$95$0 * N[(2.0 * F), $MachinePrecision]), $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[F], $MachinePrecision] * N[Sqrt[N[(A + N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 20.2%

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

   3. Taylor expanded in A around -inf 34.3%

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

      1. add-sqr-sqrt 34.3%

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

      2. pow1/2 34.3%

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

      3. pow1/2 34.4%

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

      4. pow-prod-down 18.1%

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

      5. pow2 18.1%

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

      6. associate-*r* 18.1%

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

      7. *-commutative 18.1%

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

      8. *-commutative 18.1%

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

   5. Applied egg-rr 18.1%

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

      1. unpow1/2 18.1%

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

      2. unpow2 18.1%

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

      3. rem-sqrt-square 34.9%

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

      4. associate-*l* 36.7%

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

      5. *-commutative 36.7%

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

      6. *-commutative 36.7%

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

      7. associate-*r* 36.7%

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

      8. cancel-sign-sub-inv 36.7%

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

      9. metadata-eval 36.7%

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

   7. Simplified 36.7%

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

   if 0.0 < (pow.f64 B 2) < 2.0000000000000001e117

   1. Initial program 37.3%

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

   3. Simplified 45.3%

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

   if 2.0000000000000001e117 < (pow.f64 B 2)

   1. Initial program 7.9%

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

   3. Taylor expanded in C around 0 11.1%

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

      1. mul-1-neg 11.1%

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

      2. +-commutative 11.1%

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

      3. unpow2 11.1%

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

      4. unpow2 11.1%

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

      5. hypot-define 30.3%

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

   5. Simplified 30.3%

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

      1. pow1/2 30.3%

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

      2. *-commutative 30.3%

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

      3. unpow-prod-down 46.1%

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

      4. pow1/2 46.1%

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

      5. pow1/2 46.2%

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

   7. Applied egg-rr 46.2%

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

4. Final simplification 43.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 0:\\ \;\;\;\;\frac{\sqrt{\left|\left(2 \cdot \left({B}^{2} + -4 \cdot \left(A \cdot C\right)\right)\right) \cdot \left(F \cdot \left(2 \cdot C\right)\right)\right|}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{+117}:\\ \;\;\;\;\frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{F} \cdot \sqrt{A + \mathsf{hypot}\left(B, A\right)}\right) \cdot \frac{\sqrt{2}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 51.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 2 \cdot 10^{+117}:\\ \;\;\;\;\frac{\sqrt{\left(A + C\right) + \mathsf{hypot}\left(A - C, B\_m\right)} \cdot \sqrt{t\_0 \cdot \left(2 \cdot F\right)}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{F} \cdot \sqrt{A + \mathsf{hypot}\left(B\_m, A\right)}\right) \cdot \frac{\sqrt{2}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(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+117)
     (/
      (* (sqrt (+ (+ A C) (hypot (- A C) B_m))) (sqrt (* t_0 (* 2.0 F))))
      (- t_0))
     (* (* (sqrt F) (sqrt (+ A (hypot B_m A)))) (/ (sqrt 2.0) (- B_m))))))

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+117) {
		tmp = (sqrt(((A + C) + hypot((A - C), B_m))) * sqrt((t_0 * (2.0 * F)))) / -t_0;
	} else {
		tmp = (sqrt(F) * sqrt((A + hypot(B_m, A)))) * (sqrt(2.0) / -B_m);
	}
	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+117)
		tmp = Float64(Float64(sqrt(Float64(Float64(A + C) + hypot(Float64(A - C), B_m))) * sqrt(Float64(t_0 * Float64(2.0 * F)))) / Float64(-t_0));
	else
		tmp = Float64(Float64(sqrt(F) * sqrt(Float64(A + hypot(B_m, A)))) * Float64(sqrt(2.0) / Float64(-B_m)));
	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+117], 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[(t$95$0 * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / (-t$95$0)), $MachinePrecision], N[(N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(A + N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B 2) < 2.0000000000000001e117

   1. Initial program 30.7%

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

   3. Simplified 39.0%

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

      1. pow1/2 39.0%

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

      2. *-commutative 39.0%

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

      3. unpow-prod-down 45.9%

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

      4. pow1/2 45.9%

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

      5. +-commutative 45.9%

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

      6. hypot-undefine 33.4%

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

      7. unpow2 33.4%

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

      8. unpow2 33.4%

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

      9. +-commutative 33.4%

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

      10. unpow2 33.4%

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

      11. unpow2 33.4%

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

      12. hypot-define 45.9%

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

      13. pow1/2 45.9%

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

      14. *-commutative 45.9%

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

   6. Applied egg-rr 45.9%

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

   if 2.0000000000000001e117 < (pow.f64 B 2)

   1. Initial program 7.9%

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

   3. Taylor expanded in C around 0 11.1%

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

      1. mul-1-neg 11.1%

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

      2. +-commutative 11.1%

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

      3. unpow2 11.1%

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

      4. unpow2 11.1%

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

      5. hypot-define 30.3%

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

   5. Simplified 30.3%

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

      1. pow1/2 30.3%

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

      2. *-commutative 30.3%

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

      3. unpow-prod-down 46.1%

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

      4. pow1/2 46.1%

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

      5. pow1/2 46.2%

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

   7. Applied egg-rr 46.2%

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

4. Final simplification 46.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 2 \cdot 10^{+117}:\\ \;\;\;\;\frac{\sqrt{\left(A + C\right) + \mathsf{hypot}\left(A - C, B\right)} \cdot \sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(2 \cdot F\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{F} \cdot \sqrt{A + \mathsf{hypot}\left(B, A\right)}\right) \cdot \frac{\sqrt{2}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 44.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ t_1 := t\_0 - {B\_m}^{2}\\ \mathbf{if}\;{B\_m}^{2} \leq 0:\\ \;\;\;\;\frac{\sqrt{4 \cdot \left(\left({B\_m}^{2} + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(C \cdot F\right)\right)}}{t\_1}\\ \mathbf{elif}\;{B\_m}^{2} \leq 10^{-127}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_0\right) \cdot F\right)\right) \cdot \left(2 \cdot A\right)}}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{F} \cdot \sqrt{A + \mathsf{hypot}\left(B\_m, A\right)}\right) \cdot \frac{\sqrt{2}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (* (* 4.0 A) C)) (t_1 (- t_0 (pow B_m 2.0))))
   (if (<= (pow B_m 2.0) 0.0)
     (/ (sqrt (* 4.0 (* (+ (pow B_m 2.0) (* -4.0 (* A C))) (* C F)))) t_1)
     (if (<= (pow B_m 2.0) 1e-127)
       (/ (sqrt (* (* 2.0 (* (- (pow B_m 2.0) t_0) F)) (* 2.0 A))) t_1)
       (* (* (sqrt F) (sqrt (+ A (hypot B_m A)))) (/ (sqrt 2.0) (- B_m)))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double t_0 = (4.0 * A) * C;
	double t_1 = t_0 - pow(B_m, 2.0);
	double tmp;
	if (pow(B_m, 2.0) <= 0.0) {
		tmp = sqrt((4.0 * ((pow(B_m, 2.0) + (-4.0 * (A * C))) * (C * F)))) / t_1;
	} else if (pow(B_m, 2.0) <= 1e-127) {
		tmp = sqrt(((2.0 * ((pow(B_m, 2.0) - t_0) * F)) * (2.0 * A))) / t_1;
	} else {
		tmp = (sqrt(F) * sqrt((A + hypot(B_m, A)))) * (sqrt(2.0) / -B_m);
	}
	return tmp;
}

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double t_0 = (4.0 * A) * C;
	double t_1 = t_0 - Math.pow(B_m, 2.0);
	double tmp;
	if (Math.pow(B_m, 2.0) <= 0.0) {
		tmp = Math.sqrt((4.0 * ((Math.pow(B_m, 2.0) + (-4.0 * (A * C))) * (C * F)))) / t_1;
	} else if (Math.pow(B_m, 2.0) <= 1e-127) {
		tmp = Math.sqrt(((2.0 * ((Math.pow(B_m, 2.0) - t_0) * F)) * (2.0 * A))) / t_1;
	} else {
		tmp = (Math.sqrt(F) * Math.sqrt((A + Math.hypot(B_m, A)))) * (Math.sqrt(2.0) / -B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	t_0 = (4.0 * A) * C
	t_1 = t_0 - math.pow(B_m, 2.0)
	tmp = 0
	if math.pow(B_m, 2.0) <= 0.0:
		tmp = math.sqrt((4.0 * ((math.pow(B_m, 2.0) + (-4.0 * (A * C))) * (C * F)))) / t_1
	elif math.pow(B_m, 2.0) <= 1e-127:
		tmp = math.sqrt(((2.0 * ((math.pow(B_m, 2.0) - t_0) * F)) * (2.0 * A))) / t_1
	else:
		tmp = (math.sqrt(F) * math.sqrt((A + math.hypot(B_m, A)))) * (math.sqrt(2.0) / -B_m)
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	t_0 = Float64(Float64(4.0 * A) * C)
	t_1 = Float64(t_0 - (B_m ^ 2.0))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 0.0)
		tmp = Float64(sqrt(Float64(4.0 * Float64(Float64((B_m ^ 2.0) + Float64(-4.0 * Float64(A * C))) * Float64(C * F)))) / t_1);
	elseif ((B_m ^ 2.0) <= 1e-127)
		tmp = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_0) * F)) * Float64(2.0 * A))) / t_1);
	else
		tmp = Float64(Float64(sqrt(F) * sqrt(Float64(A + hypot(B_m, A)))) * Float64(sqrt(2.0) / Float64(-B_m)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	t_0 = (4.0 * A) * C;
	t_1 = t_0 - (B_m ^ 2.0);
	tmp = 0.0;
	if ((B_m ^ 2.0) <= 0.0)
		tmp = sqrt((4.0 * (((B_m ^ 2.0) + (-4.0 * (A * C))) * (C * F)))) / t_1;
	elseif ((B_m ^ 2.0) <= 1e-127)
		tmp = sqrt(((2.0 * (((B_m ^ 2.0) - t_0) * F)) * (2.0 * A))) / t_1;
	else
		tmp = (sqrt(F) * sqrt((A + hypot(B_m, A)))) * (sqrt(2.0) / -B_m);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 0.0], N[(N[Sqrt[N[(4.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(C * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-127], N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(2.0 * A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(A + N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 20.2%

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

   3. Taylor expanded in A around -inf 34.3%

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

   4. Taylor expanded in F around 0 34.1%

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

      1. associate-*r* 35.9%

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

      2. *-commutative 35.9%

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

      3. cancel-sign-sub-inv 35.9%

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

      4. metadata-eval 35.9%

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

   6. Simplified 35.9%

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

   if 0.0 < (pow.f64 B 2) < 1e-127

   1. Initial program 30.3%

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

   3. Taylor expanded in A around inf 34.9%

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

   if 1e-127 < (pow.f64 B 2)

   1. Initial program 18.7%

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

   3. Taylor expanded in C around 0 13.4%

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

      1. mul-1-neg 13.4%

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

      2. +-commutative 13.4%

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

      3. unpow2 13.4%

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

      4. unpow2 13.4%

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

      5. hypot-define 27.5%

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

   5. Simplified 27.5%

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

      1. pow1/2 27.6%

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

      2. *-commutative 27.6%

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

      3. unpow-prod-down 38.5%

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

      4. pow1/2 38.5%

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

      5. pow1/2 38.5%

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

   7. Applied egg-rr 38.5%

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

4. Final simplification 37.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 0:\\ \;\;\;\;\frac{\sqrt{4 \cdot \left(\left({B}^{2} + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(C \cdot F\right)\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;{B}^{2} \leq 10^{-127}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot A\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{F} \cdot \sqrt{A + \mathsf{hypot}\left(B, A\right)}\right) \cdot \frac{\sqrt{2}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 44.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ t_1 := t\_0 - {B\_m}^{2}\\ \mathbf{if}\;{B\_m}^{2} \leq 0:\\ \;\;\;\;\frac{\sqrt{\left|\left(2 \cdot \left({B\_m}^{2} + -4 \cdot \left(A \cdot C\right)\right)\right) \cdot \left(F \cdot \left(2 \cdot C\right)\right)\right|}}{t\_1}\\ \mathbf{elif}\;{B\_m}^{2} \leq 10^{-127}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_0\right) \cdot F\right)\right) \cdot \left(2 \cdot A\right)}}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{F} \cdot \sqrt{A + \mathsf{hypot}\left(B\_m, A\right)}\right) \cdot \frac{\sqrt{2}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (* (* 4.0 A) C)) (t_1 (- t_0 (pow B_m 2.0))))
   (if (<= (pow B_m 2.0) 0.0)
     (/
      (sqrt
       (fabs (* (* 2.0 (+ (pow B_m 2.0) (* -4.0 (* A C)))) (* F (* 2.0 C)))))
      t_1)
     (if (<= (pow B_m 2.0) 1e-127)
       (/ (sqrt (* (* 2.0 (* (- (pow B_m 2.0) t_0) F)) (* 2.0 A))) t_1)
       (* (* (sqrt F) (sqrt (+ A (hypot B_m A)))) (/ (sqrt 2.0) (- B_m)))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double t_0 = (4.0 * A) * C;
	double t_1 = t_0 - pow(B_m, 2.0);
	double tmp;
	if (pow(B_m, 2.0) <= 0.0) {
		tmp = sqrt(fabs(((2.0 * (pow(B_m, 2.0) + (-4.0 * (A * C)))) * (F * (2.0 * C))))) / t_1;
	} else if (pow(B_m, 2.0) <= 1e-127) {
		tmp = sqrt(((2.0 * ((pow(B_m, 2.0) - t_0) * F)) * (2.0 * A))) / t_1;
	} else {
		tmp = (sqrt(F) * sqrt((A + hypot(B_m, A)))) * (sqrt(2.0) / -B_m);
	}
	return tmp;
}

Java

B_m = Math.abs(B);
public static double code(double A, double B_m, double C, double F) {
	double t_0 = (4.0 * A) * C;
	double t_1 = t_0 - Math.pow(B_m, 2.0);
	double tmp;
	if (Math.pow(B_m, 2.0) <= 0.0) {
		tmp = Math.sqrt(Math.abs(((2.0 * (Math.pow(B_m, 2.0) + (-4.0 * (A * C)))) * (F * (2.0 * C))))) / t_1;
	} else if (Math.pow(B_m, 2.0) <= 1e-127) {
		tmp = Math.sqrt(((2.0 * ((Math.pow(B_m, 2.0) - t_0) * F)) * (2.0 * A))) / t_1;
	} else {
		tmp = (Math.sqrt(F) * Math.sqrt((A + Math.hypot(B_m, A)))) * (Math.sqrt(2.0) / -B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	t_0 = (4.0 * A) * C
	t_1 = t_0 - math.pow(B_m, 2.0)
	tmp = 0
	if math.pow(B_m, 2.0) <= 0.0:
		tmp = math.sqrt(math.fabs(((2.0 * (math.pow(B_m, 2.0) + (-4.0 * (A * C)))) * (F * (2.0 * C))))) / t_1
	elif math.pow(B_m, 2.0) <= 1e-127:
		tmp = math.sqrt(((2.0 * ((math.pow(B_m, 2.0) - t_0) * F)) * (2.0 * A))) / t_1
	else:
		tmp = (math.sqrt(F) * math.sqrt((A + math.hypot(B_m, A)))) * (math.sqrt(2.0) / -B_m)
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	t_0 = Float64(Float64(4.0 * A) * C)
	t_1 = Float64(t_0 - (B_m ^ 2.0))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 0.0)
		tmp = Float64(sqrt(abs(Float64(Float64(2.0 * Float64((B_m ^ 2.0) + Float64(-4.0 * Float64(A * C)))) * Float64(F * Float64(2.0 * C))))) / t_1);
	elseif ((B_m ^ 2.0) <= 1e-127)
		tmp = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_0) * F)) * Float64(2.0 * A))) / t_1);
	else
		tmp = Float64(Float64(sqrt(F) * sqrt(Float64(A + hypot(B_m, A)))) * Float64(sqrt(2.0) / Float64(-B_m)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	t_0 = (4.0 * A) * C;
	t_1 = t_0 - (B_m ^ 2.0);
	tmp = 0.0;
	if ((B_m ^ 2.0) <= 0.0)
		tmp = sqrt(abs(((2.0 * ((B_m ^ 2.0) + (-4.0 * (A * C)))) * (F * (2.0 * C))))) / t_1;
	elseif ((B_m ^ 2.0) <= 1e-127)
		tmp = sqrt(((2.0 * (((B_m ^ 2.0) - t_0) * F)) * (2.0 * A))) / t_1;
	else
		tmp = (sqrt(F) * sqrt((A + hypot(B_m, A)))) * (sqrt(2.0) / -B_m);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 0.0], N[(N[Sqrt[N[Abs[N[(N[(2.0 * N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(F * N[(2.0 * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-127], N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(2.0 * A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(A + N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 20.2%

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

   3. Taylor expanded in A around -inf 34.3%

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

      1. add-sqr-sqrt 34.3%

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

      2. pow1/2 34.3%

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

      3. pow1/2 34.4%

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

      4. pow-prod-down 18.1%

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

      5. pow2 18.1%

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

      6. associate-*r* 18.1%

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

      7. *-commutative 18.1%

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

      8. *-commutative 18.1%

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

   5. Applied egg-rr 18.1%

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

      1. unpow1/2 18.1%

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

      2. unpow2 18.1%

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

      3. rem-sqrt-square 34.9%

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

      4. associate-*l* 36.7%

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

      5. *-commutative 36.7%

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

      6. *-commutative 36.7%

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

      7. associate-*r* 36.7%

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

      8. cancel-sign-sub-inv 36.7%

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

      9. metadata-eval 36.7%

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

   7. Simplified 36.7%

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

   if 0.0 < (pow.f64 B 2) < 1e-127

   1. Initial program 30.3%

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

   3. Taylor expanded in A around inf 34.9%

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

   if 1e-127 < (pow.f64 B 2)

   1. Initial program 18.7%

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

   3. Taylor expanded in C around 0 13.4%

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

      1. mul-1-neg 13.4%

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

      2. +-commutative 13.4%

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

      3. unpow2 13.4%

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

      4. unpow2 13.4%

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

      5. hypot-define 27.5%

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

   5. Simplified 27.5%

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

      1. pow1/2 27.6%

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

      2. *-commutative 27.6%

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

      3. unpow-prod-down 38.5%

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

      4. pow1/2 38.5%

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

      5. pow1/2 38.5%

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

   7. Applied egg-rr 38.5%

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

4. Final simplification 37.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 0:\\ \;\;\;\;\frac{\sqrt{\left|\left(2 \cdot \left({B}^{2} + -4 \cdot \left(A \cdot C\right)\right)\right) \cdot \left(F \cdot \left(2 \cdot C\right)\right)\right|}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;{B}^{2} \leq 10^{-127}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot A\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{F} \cdot \sqrt{A + \mathsf{hypot}\left(B, A\right)}\right) \cdot \frac{\sqrt{2}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 40.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;F \leq -4.3 \cdot 10^{-296}:\\ \;\;\;\;\frac{\sqrt{\left(\left(A + C\right) + \mathsf{hypot}\left(B\_m, A - C\right)\right) \cdot \left(\left(C \cdot F\right) \cdot \left(A \cdot -8\right)\right)}}{-\mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;F \leq 175000:\\ \;\;\;\;\frac{{\left(2 \cdot \left(F \cdot \left(A + \mathsf{hypot}\left(B\_m, A\right)\right)\right)\right)}^{0.5}}{-B\_m}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B\_m}}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= F -4.3e-296)
   (/
    (sqrt (* (+ (+ A C) (hypot B_m (- A C))) (* (* C F) (* A -8.0))))
    (- (fma B_m B_m (* A (* C -4.0)))))
   (if (<= F 175000.0)
     (/ (pow (* 2.0 (* F (+ A (hypot B_m A)))) 0.5) (- B_m))
     (* (sqrt 2.0) (- (sqrt (/ F B_m)))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= -4.3e-296) {
		tmp = sqrt((((A + C) + hypot(B_m, (A - C))) * ((C * F) * (A * -8.0)))) / -fma(B_m, B_m, (A * (C * -4.0)));
	} else if (F <= 175000.0) {
		tmp = pow((2.0 * (F * (A + hypot(B_m, A)))), 0.5) / -B_m;
	} else {
		tmp = sqrt(2.0) * -sqrt((F / B_m));
	}
	return tmp;
}

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (F <= -4.3e-296)
		tmp = Float64(sqrt(Float64(Float64(Float64(A + C) + hypot(B_m, Float64(A - C))) * Float64(Float64(C * F) * Float64(A * -8.0)))) / Float64(-fma(B_m, B_m, Float64(A * Float64(C * -4.0)))));
	elseif (F <= 175000.0)
		tmp = Float64((Float64(2.0 * Float64(F * Float64(A + hypot(B_m, A)))) ^ 0.5) / Float64(-B_m));
	else
		tmp = Float64(sqrt(2.0) * Float64(-sqrt(Float64(F / B_m))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[F, -4.3e-296], N[(N[Sqrt[N[(N[(N[(A + C), $MachinePrecision] + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(C * F), $MachinePrecision] * N[(A * -8.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], If[LessEqual[F, 175000.0], N[(N[Power[N[(2.0 * N[(F * N[(A + N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision] / (-B$95$m)), $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * (-N[Sqrt[N[(F / B$95$m), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -4.29999999999999978e-296

   1. Initial program 21.8%

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

   3. Simplified 43.7%

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

   5. Taylor expanded in B around 0 35.7%

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

      1. associate-*r* 35.7%

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

   7. Simplified 35.7%

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

   if -4.29999999999999978e-296 < F < 175000

   1. Initial program 20.7%

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

   3. Taylor expanded in C around 0 12.3%

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

      1. mul-1-neg 12.3%

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

      2. +-commutative 12.3%

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

      3. unpow2 12.3%

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

      4. unpow2 12.3%

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

      5. hypot-define 28.0%

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

   5. Simplified 28.0%

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

      1. associate-*l/ 28.0%

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

      2. pow1/2 28.0%

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

      3. pow1/2 28.0%

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

      4. pow-prod-down 28.2%

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

   7. Applied egg-rr 28.2%

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

   if 175000 < F

   1. Initial program 20.8%

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

   3. Taylor expanded in C around 0 13.0%

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

      1. mul-1-neg 13.0%

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

      2. +-commutative 13.0%

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

      3. unpow2 13.0%

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

      4. unpow2 13.0%

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

      5. hypot-define 18.4%

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

   5. Simplified 18.4%

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

   6. Taylor expanded in A around 0 30.6%

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

4. Final simplification 30.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -4.3 \cdot 10^{-296}:\\ \;\;\;\;\frac{\sqrt{\left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right) \cdot \left(\left(C \cdot F\right) \cdot \left(A \cdot -8\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;F \leq 175000:\\ \;\;\;\;\frac{{\left(2 \cdot \left(F \cdot \left(A + \mathsf{hypot}\left(B, A\right)\right)\right)\right)}^{0.5}}{-B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 38.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;F \leq -4.3 \cdot 10^{-296}:\\ \;\;\;\;\frac{\sqrt{4 \cdot \left(\left({B\_m}^{2} + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(C \cdot F\right)\right)}}{\left(4 \cdot A\right) \cdot C - {B\_m}^{2}}\\ \mathbf{elif}\;F \leq 500000:\\ \;\;\;\;\frac{{\left(2 \cdot \left(F \cdot \left(A + \mathsf{hypot}\left(B\_m, A\right)\right)\right)\right)}^{0.5}}{-B\_m}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B\_m}}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= F -4.3e-296)
   (/
    (sqrt (* 4.0 (* (+ (pow B_m 2.0) (* -4.0 (* A C))) (* C F))))
    (- (* (* 4.0 A) C) (pow B_m 2.0)))
   (if (<= F 500000.0)
     (/ (pow (* 2.0 (* F (+ A (hypot B_m A)))) 0.5) (- B_m))
     (* (sqrt 2.0) (- (sqrt (/ F B_m)))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= -4.3e-296) {
		tmp = sqrt((4.0 * ((pow(B_m, 2.0) + (-4.0 * (A * C))) * (C * F)))) / (((4.0 * A) * C) - pow(B_m, 2.0));
	} else if (F <= 500000.0) {
		tmp = pow((2.0 * (F * (A + hypot(B_m, A)))), 0.5) / -B_m;
	} else {
		tmp = sqrt(2.0) * -sqrt((F / B_m));
	}
	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 <= -4.3e-296) {
		tmp = Math.sqrt((4.0 * ((Math.pow(B_m, 2.0) + (-4.0 * (A * C))) * (C * F)))) / (((4.0 * A) * C) - Math.pow(B_m, 2.0));
	} else if (F <= 500000.0) {
		tmp = Math.pow((2.0 * (F * (A + Math.hypot(B_m, A)))), 0.5) / -B_m;
	} else {
		tmp = Math.sqrt(2.0) * -Math.sqrt((F / B_m));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if F <= -4.3e-296:
		tmp = math.sqrt((4.0 * ((math.pow(B_m, 2.0) + (-4.0 * (A * C))) * (C * F)))) / (((4.0 * A) * C) - math.pow(B_m, 2.0))
	elif F <= 500000.0:
		tmp = math.pow((2.0 * (F * (A + math.hypot(B_m, A)))), 0.5) / -B_m
	else:
		tmp = math.sqrt(2.0) * -math.sqrt((F / B_m))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (F <= -4.3e-296)
		tmp = Float64(sqrt(Float64(4.0 * Float64(Float64((B_m ^ 2.0) + Float64(-4.0 * Float64(A * C))) * Float64(C * F)))) / Float64(Float64(Float64(4.0 * A) * C) - (B_m ^ 2.0)));
	elseif (F <= 500000.0)
		tmp = Float64((Float64(2.0 * Float64(F * Float64(A + hypot(B_m, A)))) ^ 0.5) / Float64(-B_m));
	else
		tmp = Float64(sqrt(2.0) * Float64(-sqrt(Float64(F / B_m))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (F <= -4.3e-296)
		tmp = sqrt((4.0 * (((B_m ^ 2.0) + (-4.0 * (A * C))) * (C * F)))) / (((4.0 * A) * C) - (B_m ^ 2.0));
	elseif (F <= 500000.0)
		tmp = ((2.0 * (F * (A + hypot(B_m, A)))) ^ 0.5) / -B_m;
	else
		tmp = sqrt(2.0) * -sqrt((F / B_m));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[F, -4.3e-296], N[(N[Sqrt[N[(4.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(C * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision] - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 500000.0], N[(N[Power[N[(2.0 * N[(F * N[(A + N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision] / (-B$95$m)), $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * (-N[Sqrt[N[(F / B$95$m), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -4.29999999999999978e-296

   1. Initial program 21.8%

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

   3. Taylor expanded in A around -inf 22.2%

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

   4. Taylor expanded in F around 0 22.2%

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

      1. associate-*r* 25.2%

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

      2. *-commutative 25.2%

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

      3. cancel-sign-sub-inv 25.2%

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

      4. metadata-eval 25.2%

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

   6. Simplified 25.2%

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

   if -4.29999999999999978e-296 < F < 5e5

   1. Initial program 20.7%

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

   3. Taylor expanded in C around 0 12.3%

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

      1. mul-1-neg 12.3%

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

      2. +-commutative 12.3%

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

      3. unpow2 12.3%

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

      4. unpow2 12.3%

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

      5. hypot-define 28.0%

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

   5. Simplified 28.0%

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

      1. associate-*l/ 28.0%

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

      2. pow1/2 28.0%

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

      3. pow1/2 28.0%

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

      4. pow-prod-down 28.2%

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

   7. Applied egg-rr 28.2%

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

   if 5e5 < F

   1. Initial program 20.8%

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

   3. Taylor expanded in C around 0 13.0%

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

      1. mul-1-neg 13.0%

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

      2. +-commutative 13.0%

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

      3. unpow2 13.0%

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

      4. unpow2 13.0%

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

      5. hypot-define 18.4%

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

   5. Simplified 18.4%

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

   6. Taylor expanded in A around 0 30.6%

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

4. Final simplification 28.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -4.3 \cdot 10^{-296}:\\ \;\;\;\;\frac{\sqrt{4 \cdot \left(\left({B}^{2} + -4 \cdot \left(A \cdot C\right)\right) \cdot \left(C \cdot F\right)\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;F \leq 500000:\\ \;\;\;\;\frac{{\left(2 \cdot \left(F \cdot \left(A + \mathsf{hypot}\left(B, A\right)\right)\right)\right)}^{0.5}}{-B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 38.7% accurate, 2.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;F \leq -4.3 \cdot 10^{-296}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot C\right) \cdot \left(2 \cdot \left(\left(C \cdot F\right) \cdot \left(A \cdot -4\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - {B\_m}^{2}}\\ \mathbf{elif}\;F \leq 530000:\\ \;\;\;\;\frac{{\left(2 \cdot \left(F \cdot \left(A + \mathsf{hypot}\left(B\_m, A\right)\right)\right)\right)}^{0.5}}{-B\_m}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B\_m}}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= F -4.3e-296)
   (/
    (sqrt (* (* 2.0 C) (* 2.0 (* (* C F) (* A -4.0)))))
    (- (* (* 4.0 A) C) (pow B_m 2.0)))
   (if (<= F 530000.0)
     (/ (pow (* 2.0 (* F (+ A (hypot B_m A)))) 0.5) (- B_m))
     (* (sqrt 2.0) (- (sqrt (/ F B_m)))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= -4.3e-296) {
		tmp = sqrt(((2.0 * C) * (2.0 * ((C * F) * (A * -4.0))))) / (((4.0 * A) * C) - pow(B_m, 2.0));
	} else if (F <= 530000.0) {
		tmp = pow((2.0 * (F * (A + hypot(B_m, A)))), 0.5) / -B_m;
	} else {
		tmp = sqrt(2.0) * -sqrt((F / B_m));
	}
	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 <= -4.3e-296) {
		tmp = Math.sqrt(((2.0 * C) * (2.0 * ((C * F) * (A * -4.0))))) / (((4.0 * A) * C) - Math.pow(B_m, 2.0));
	} else if (F <= 530000.0) {
		tmp = Math.pow((2.0 * (F * (A + Math.hypot(B_m, A)))), 0.5) / -B_m;
	} else {
		tmp = Math.sqrt(2.0) * -Math.sqrt((F / B_m));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if F <= -4.3e-296:
		tmp = math.sqrt(((2.0 * C) * (2.0 * ((C * F) * (A * -4.0))))) / (((4.0 * A) * C) - math.pow(B_m, 2.0))
	elif F <= 530000.0:
		tmp = math.pow((2.0 * (F * (A + math.hypot(B_m, A)))), 0.5) / -B_m
	else:
		tmp = math.sqrt(2.0) * -math.sqrt((F / B_m))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (F <= -4.3e-296)
		tmp = Float64(sqrt(Float64(Float64(2.0 * C) * Float64(2.0 * Float64(Float64(C * F) * Float64(A * -4.0))))) / Float64(Float64(Float64(4.0 * A) * C) - (B_m ^ 2.0)));
	elseif (F <= 530000.0)
		tmp = Float64((Float64(2.0 * Float64(F * Float64(A + hypot(B_m, A)))) ^ 0.5) / Float64(-B_m));
	else
		tmp = Float64(sqrt(2.0) * Float64(-sqrt(Float64(F / B_m))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (F <= -4.3e-296)
		tmp = sqrt(((2.0 * C) * (2.0 * ((C * F) * (A * -4.0))))) / (((4.0 * A) * C) - (B_m ^ 2.0));
	elseif (F <= 530000.0)
		tmp = ((2.0 * (F * (A + hypot(B_m, A)))) ^ 0.5) / -B_m;
	else
		tmp = sqrt(2.0) * -sqrt((F / B_m));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[F, -4.3e-296], N[(N[Sqrt[N[(N[(2.0 * C), $MachinePrecision] * N[(2.0 * N[(N[(C * F), $MachinePrecision] * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision] - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 530000.0], N[(N[Power[N[(2.0 * N[(F * N[(A + N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision] / (-B$95$m)), $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * (-N[Sqrt[N[(F / B$95$m), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -4.29999999999999978e-296

   1. Initial program 21.8%

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

   3. Taylor expanded in A around -inf 22.2%

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

   4. Taylor expanded in B around 0 22.6%

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

      1. associate-*r* 22.6%

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

      2. *-commutative 22.6%

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

   6. Simplified 22.6%

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

   if -4.29999999999999978e-296 < F < 5.3e5

   1. Initial program 20.7%

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

   3. Taylor expanded in C around 0 12.3%

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

      1. mul-1-neg 12.3%

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

      2. +-commutative 12.3%

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

      3. unpow2 12.3%

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

      4. unpow2 12.3%

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

      5. hypot-define 28.0%

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

   5. Simplified 28.0%

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

      1. associate-*l/ 28.0%

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

      2. pow1/2 28.0%

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

      3. pow1/2 28.0%

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

      4. pow-prod-down 28.2%

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

   7. Applied egg-rr 28.2%

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

   if 5.3e5 < F

   1. Initial program 20.8%

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

   3. Taylor expanded in C around 0 13.0%

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

      1. mul-1-neg 13.0%

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

      2. +-commutative 13.0%

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

      3. unpow2 13.0%

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

      4. unpow2 13.0%

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

      5. hypot-define 18.4%

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

   5. Simplified 18.4%

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

   6. Taylor expanded in A around 0 30.6%

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

4. Final simplification 28.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -4.3 \cdot 10^{-296}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot C\right) \cdot \left(2 \cdot \left(\left(C \cdot F\right) \cdot \left(A \cdot -4\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;F \leq 530000:\\ \;\;\;\;\frac{{\left(2 \cdot \left(F \cdot \left(A + \mathsf{hypot}\left(B, A\right)\right)\right)\right)}^{0.5}}{-B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 35.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;F \leq 680:\\ \;\;\;\;\frac{{\left(2 \cdot \left(F \cdot \left(A + \mathsf{hypot}\left(B\_m, A\right)\right)\right)\right)}^{0.5}}{-B\_m}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B\_m}}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= F 680.0)
   (/ (pow (* 2.0 (* F (+ A (hypot B_m A)))) 0.5) (- B_m))
   (* (sqrt 2.0) (- (sqrt (/ F B_m))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= 680.0) {
		tmp = pow((2.0 * (F * (A + hypot(B_m, A)))), 0.5) / -B_m;
	} else {
		tmp = sqrt(2.0) * -sqrt((F / B_m));
	}
	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 <= 680.0) {
		tmp = Math.pow((2.0 * (F * (A + Math.hypot(B_m, A)))), 0.5) / -B_m;
	} else {
		tmp = Math.sqrt(2.0) * -Math.sqrt((F / B_m));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if F <= 680.0:
		tmp = math.pow((2.0 * (F * (A + math.hypot(B_m, A)))), 0.5) / -B_m
	else:
		tmp = math.sqrt(2.0) * -math.sqrt((F / B_m))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (F <= 680.0)
		tmp = Float64((Float64(2.0 * Float64(F * Float64(A + hypot(B_m, A)))) ^ 0.5) / Float64(-B_m));
	else
		tmp = Float64(sqrt(2.0) * Float64(-sqrt(Float64(F / B_m))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (F <= 680.0)
		tmp = ((2.0 * (F * (A + hypot(B_m, A)))) ^ 0.5) / -B_m;
	else
		tmp = sqrt(2.0) * -sqrt((F / B_m));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[F, 680.0], N[(N[Power[N[(2.0 * N[(F * N[(A + N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision] / (-B$95$m)), $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * (-N[Sqrt[N[(F / B$95$m), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < 680

   1. Initial program 21.0%

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

   3. Taylor expanded in C around 0 9.4%

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

      1. mul-1-neg 9.4%

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

      2. +-commutative 9.4%

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

      3. unpow2 9.4%

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

      4. unpow2 9.4%

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

      5. hypot-define 21.4%

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

   5. Simplified 21.4%

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

      1. associate-*l/ 21.4%

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

      2. pow1/2 21.4%

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

      3. pow1/2 21.5%

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

      4. pow-prod-down 21.6%

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

   7. Applied egg-rr 21.6%

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

   if 680 < F

   1. Initial program 20.8%

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

   3. Taylor expanded in C around 0 13.0%

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

      1. mul-1-neg 13.0%

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

      2. +-commutative 13.0%

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

      3. unpow2 13.0%

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

      4. unpow2 13.0%

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

      5. hypot-define 18.4%

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

   5. Simplified 18.4%

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

   6. Taylor expanded in A around 0 30.6%

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

4. Final simplification 25.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 680:\\ \;\;\;\;\frac{{\left(2 \cdot \left(F \cdot \left(A + \mathsf{hypot}\left(B, A\right)\right)\right)\right)}^{0.5}}{-B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 36.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;F \leq 450000000000:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(C + \mathsf{hypot}\left(B\_m, C\right)\right)\right)}}{-B\_m}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B\_m}}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= F 450000000000.0)
   (/ (sqrt (* 2.0 (* F (+ C (hypot B_m C))))) (- B_m))
   (* (sqrt 2.0) (- (sqrt (/ F B_m))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= 450000000000.0) {
		tmp = sqrt((2.0 * (F * (C + hypot(B_m, C))))) / -B_m;
	} else {
		tmp = sqrt(2.0) * -sqrt((F / B_m));
	}
	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 <= 450000000000.0) {
		tmp = Math.sqrt((2.0 * (F * (C + Math.hypot(B_m, C))))) / -B_m;
	} else {
		tmp = Math.sqrt(2.0) * -Math.sqrt((F / B_m));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if F <= 450000000000.0:
		tmp = math.sqrt((2.0 * (F * (C + math.hypot(B_m, C))))) / -B_m
	else:
		tmp = math.sqrt(2.0) * -math.sqrt((F / B_m))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (F <= 450000000000.0)
		tmp = Float64(sqrt(Float64(2.0 * Float64(F * Float64(C + hypot(B_m, C))))) / Float64(-B_m));
	else
		tmp = Float64(sqrt(2.0) * Float64(-sqrt(Float64(F / B_m))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (F <= 450000000000.0)
		tmp = sqrt((2.0 * (F * (C + hypot(B_m, C))))) / -B_m;
	else
		tmp = sqrt(2.0) * -sqrt((F / B_m));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[F, 450000000000.0], N[(N[Sqrt[N[(2.0 * N[(F * N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * (-N[Sqrt[N[(F / B$95$m), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < 4.5e11

   1. Initial program 20.6%

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

   3. Taylor expanded in A around 0 7.0%

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

      1. mul-1-neg 7.0%

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

      2. unpow2 7.0%

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

      3. unpow2 7.0%

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

      4. hypot-define 17.1%

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

   5. Simplified 17.1%

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

      1. associate-*l/ 17.2%

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

      2. pow1/2 17.2%

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

      3. pow1/2 17.3%

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

      4. pow-prod-down 17.3%

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

   7. Applied egg-rr 17.3%

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

      1. unpow1/2 17.2%

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

   9. Simplified 17.2%

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

   if 4.5e11 < F

   1. Initial program 21.3%

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

   3. Taylor expanded in C around 0 13.3%

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

      1. mul-1-neg 13.3%

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

      2. +-commutative 13.3%

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

      3. unpow2 13.3%

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

      4. unpow2 13.3%

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

      5. hypot-define 18.6%

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

   5. Simplified 18.6%

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

   6. Taylor expanded in A around 0 31.2%

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

4. Final simplification 23.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 450000000000:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)\right)}}{-B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 35.9% accurate, 2.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;F \leq 70000:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot F\right) \cdot \left(A + \mathsf{hypot}\left(B\_m, A\right)\right)}}{-B\_m}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B\_m}}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= F 70000.0)
   (/ (sqrt (* (* 2.0 F) (+ A (hypot B_m A)))) (- B_m))
   (* (sqrt 2.0) (- (sqrt (/ F B_m))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= 70000.0) {
		tmp = sqrt(((2.0 * F) * (A + hypot(B_m, A)))) / -B_m;
	} else {
		tmp = sqrt(2.0) * -sqrt((F / B_m));
	}
	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 <= 70000.0) {
		tmp = Math.sqrt(((2.0 * F) * (A + Math.hypot(B_m, A)))) / -B_m;
	} else {
		tmp = Math.sqrt(2.0) * -Math.sqrt((F / B_m));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if F <= 70000.0:
		tmp = math.sqrt(((2.0 * F) * (A + math.hypot(B_m, A)))) / -B_m
	else:
		tmp = math.sqrt(2.0) * -math.sqrt((F / B_m))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (F <= 70000.0)
		tmp = Float64(sqrt(Float64(Float64(2.0 * F) * Float64(A + hypot(B_m, A)))) / Float64(-B_m));
	else
		tmp = Float64(sqrt(2.0) * Float64(-sqrt(Float64(F / B_m))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (F <= 70000.0)
		tmp = sqrt(((2.0 * F) * (A + hypot(B_m, A)))) / -B_m;
	else
		tmp = sqrt(2.0) * -sqrt((F / B_m));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[F, 70000.0], N[(N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] * N[(A + N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * (-N[Sqrt[N[(F / B$95$m), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < 7e4

   1. Initial program 21.0%

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

   3. Taylor expanded in C around 0 9.4%

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

      1. mul-1-neg 9.4%

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

      2. +-commutative 9.4%

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

      3. unpow2 9.4%

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

      4. unpow2 9.4%

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

      5. hypot-define 21.4%

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

   5. Simplified 21.4%

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

      1. associate-*l/ 21.4%

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

      2. pow1/2 21.4%

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

      3. pow1/2 21.5%

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

      4. pow-prod-down 21.6%

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

   7. Applied egg-rr 21.6%

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

      1. unpow1/2 21.5%

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

      2. associate-*r* 21.5%

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

   9. Simplified 21.5%

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

   if 7e4 < F

   1. Initial program 20.8%

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

   3. Taylor expanded in C around 0 13.0%

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

      1. mul-1-neg 13.0%

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

      2. +-commutative 13.0%

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

      3. unpow2 13.0%

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

      4. unpow2 13.0%

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

      5. hypot-define 18.4%

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

   5. Simplified 18.4%

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

   6. Taylor expanded in A around 0 30.6%

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

4. Final simplification 25.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 70000:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot F\right) \cdot \left(A + \mathsf{hypot}\left(B, A\right)\right)}}{-B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 33.7% accurate, 3.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;F \leq 2.6 \cdot 10^{+55}:\\ \;\;\;\;\sqrt{B\_m \cdot F} \cdot \frac{\sqrt{2}}{-B\_m}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B\_m}}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= F 2.6e+55)
   (* (sqrt (* B_m F)) (/ (sqrt 2.0) (- B_m)))
   (* (sqrt 2.0) (- (sqrt (/ F B_m))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= 2.6e+55) {
		tmp = sqrt((B_m * F)) * (sqrt(2.0) / -B_m);
	} else {
		tmp = sqrt(2.0) * -sqrt((F / B_m));
	}
	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 <= 2.6d+55) then
        tmp = sqrt((b_m * f)) * (sqrt(2.0d0) / -b_m)
    else
        tmp = sqrt(2.0d0) * -sqrt((f / b_m))
    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 <= 2.6e+55) {
		tmp = Math.sqrt((B_m * F)) * (Math.sqrt(2.0) / -B_m);
	} else {
		tmp = Math.sqrt(2.0) * -Math.sqrt((F / B_m));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if F <= 2.6e+55:
		tmp = math.sqrt((B_m * F)) * (math.sqrt(2.0) / -B_m)
	else:
		tmp = math.sqrt(2.0) * -math.sqrt((F / B_m))
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (F <= 2.6e+55)
		tmp = Float64(sqrt(Float64(B_m * F)) * Float64(sqrt(2.0) / Float64(-B_m)));
	else
		tmp = Float64(sqrt(2.0) * Float64(-sqrt(Float64(F / B_m))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (F <= 2.6e+55)
		tmp = sqrt((B_m * F)) * (sqrt(2.0) / -B_m);
	else
		tmp = sqrt(2.0) * -sqrt((F / B_m));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[F, 2.6e+55], 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[Sqrt[N[(F / B$95$m), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < 2.6e55

   1. Initial program 22.5%

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

   3. Taylor expanded in C around 0 11.1%

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

      1. mul-1-neg 11.1%

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

      2. +-commutative 11.1%

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

      3. unpow2 11.1%

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

      4. unpow2 11.1%

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

      5. hypot-define 23.2%

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

   5. Simplified 23.2%

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

   6. Taylor expanded in A around 0 17.8%

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

   if 2.6e55 < 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 C around 0 10.7%

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

      1. mul-1-neg 10.7%

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

      2. +-commutative 10.7%

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

      3. unpow2 10.7%

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

      4. unpow2 10.7%

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

      5. hypot-define 14.7%

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

   5. Simplified 14.7%

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

   6. Taylor expanded in A around 0 29.0%

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

4. Final simplification 21.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 2.6 \cdot 10^{+55}:\\ \;\;\;\;\sqrt{B \cdot F} \cdot \frac{\sqrt{2}}{-B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 27.4% accurate, 3.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;A \leq 7.5 \cdot 10^{+124}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B\_m}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{A \cdot F} \cdot \left(-\frac{2}{B\_m}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= A 7.5e+124)
   (* (sqrt 2.0) (- (sqrt (/ F B_m))))
   (* (sqrt (* A F)) (- (/ 2.0 B_m)))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (A <= 7.5e+124) {
		tmp = sqrt(2.0) * -sqrt((F / B_m));
	} else {
		tmp = sqrt((A * F)) * -(2.0 / B_m);
	}
	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 (a <= 7.5d+124) then
        tmp = sqrt(2.0d0) * -sqrt((f / b_m))
    else
        tmp = sqrt((a * f)) * -(2.0d0 / b_m)
    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 (A <= 7.5e+124) {
		tmp = Math.sqrt(2.0) * -Math.sqrt((F / B_m));
	} else {
		tmp = Math.sqrt((A * F)) * -(2.0 / B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	tmp = 0
	if A <= 7.5e+124:
		tmp = math.sqrt(2.0) * -math.sqrt((F / B_m))
	else:
		tmp = math.sqrt((A * F)) * -(2.0 / B_m)
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	tmp = 0.0
	if (A <= 7.5e+124)
		tmp = Float64(sqrt(2.0) * Float64(-sqrt(Float64(F / B_m))));
	else
		tmp = Float64(sqrt(Float64(A * F)) * Float64(-Float64(2.0 / B_m)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (A <= 7.5e+124)
		tmp = sqrt(2.0) * -sqrt((F / B_m));
	else
		tmp = sqrt((A * F)) * -(2.0 / B_m);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := If[LessEqual[A, 7.5e+124], N[(N[Sqrt[2.0], $MachinePrecision] * (-N[Sqrt[N[(F / B$95$m), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[N[(A * F), $MachinePrecision]], $MachinePrecision] * (-N[(2.0 / B$95$m), $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if A < 7.50000000000000038e124

   1. Initial program 23.3%

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

   3. Taylor expanded in C around 0 12.2%

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

      1. mul-1-neg 12.2%

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

      2. +-commutative 12.2%

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

      3. unpow2 12.2%

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

      4. unpow2 12.2%

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

      5. hypot-define 20.5%

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

   5. Simplified 20.5%

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

   6. Taylor expanded in A around 0 20.6%

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

   if 7.50000000000000038e124 < A

   1. Initial program 10.1%

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

   3. Taylor expanded in C around 0 5.4%

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

      1. mul-1-neg 5.4%

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

      2. +-commutative 5.4%

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

      3. unpow2 5.4%

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

      4. unpow2 5.4%

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

      5. hypot-define 18.1%

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

   5. Simplified 18.1%

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

   6. Taylor expanded in B around 0 15.9%

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

      1. *-commutative 15.9%

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

      2. unpow2 15.9%

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

      3. rem-square-sqrt 16.0%

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

   8. Simplified 16.0%

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

4. Final simplification 19.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq 7.5 \cdot 10^{+124}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{A \cdot F} \cdot \left(-\frac{2}{B}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 8.2% accurate, 5.6× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := -\frac{2}{B\_m}\\ \mathbf{if}\;C \leq 1.3 \cdot 10^{-98}:\\ \;\;\;\;\sqrt{A \cdot F} \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{C \cdot F} \cdot t\_0\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (- (/ 2.0 B_m))))
   (if (<= C 1.3e-98) (* (sqrt (* A F)) t_0) (* (sqrt (* C F)) t_0))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double t_0 = -(2.0 / B_m);
	double tmp;
	if (C <= 1.3e-98) {
		tmp = sqrt((A * F)) * t_0;
	} else {
		tmp = sqrt((C * F)) * t_0;
	}
	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) :: t_0
    real(8) :: tmp
    t_0 = -(2.0d0 / b_m)
    if (c <= 1.3d-98) then
        tmp = sqrt((a * f)) * t_0
    else
        tmp = sqrt((c * f)) * t_0
    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 t_0 = -(2.0 / B_m);
	double tmp;
	if (C <= 1.3e-98) {
		tmp = Math.sqrt((A * F)) * t_0;
	} else {
		tmp = Math.sqrt((C * F)) * t_0;
	}
	return tmp;
}

Python

B_m = math.fabs(B)
def code(A, B_m, C, F):
	t_0 = -(2.0 / B_m)
	tmp = 0
	if C <= 1.3e-98:
		tmp = math.sqrt((A * F)) * t_0
	else:
		tmp = math.sqrt((C * F)) * t_0
	return tmp

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	t_0 = Float64(-Float64(2.0 / B_m))
	tmp = 0.0
	if (C <= 1.3e-98)
		tmp = Float64(sqrt(Float64(A * F)) * t_0);
	else
		tmp = Float64(sqrt(Float64(C * F)) * t_0);
	end
	return tmp
end

MATLAB

B_m = abs(B);
function tmp_2 = code(A, B_m, C, F)
	t_0 = -(2.0 / B_m);
	tmp = 0.0;
	if (C <= 1.3e-98)
		tmp = sqrt((A * F)) * t_0;
	else
		tmp = sqrt((C * F)) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = (-N[(2.0 / B$95$m), $MachinePrecision])}, If[LessEqual[C, 1.3e-98], N[(N[Sqrt[N[(A * F), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[Sqrt[N[(C * F), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if C < 1.30000000000000007e-98

   1. Initial program 19.6%

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

   3. Taylor expanded in C around 0 13.4%

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

      1. mul-1-neg 13.4%

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

      2. +-commutative 13.4%

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

      3. unpow2 13.4%

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

      4. unpow2 13.4%

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

      5. hypot-define 23.1%

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

   5. Simplified 23.1%

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

   6. Taylor expanded in B around 0 7.0%

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

      1. *-commutative 7.0%

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

      2. unpow2 7.0%

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

      3. rem-square-sqrt 7.0%

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

   8. Simplified 7.0%

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

   if 1.30000000000000007e-98 < C

   1. Initial program 23.3%

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

   3. Taylor expanded in A around 0 5.9%

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

      1. mul-1-neg 5.9%

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

      2. unpow2 5.9%

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

      3. unpow2 5.9%

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

      4. hypot-define 17.4%

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

   5. Simplified 17.4%

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

   6. Taylor expanded in B around 0 6.4%

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

      1. unpow2 6.4%

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

      2. rem-square-sqrt 6.4%

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

      3. *-commutative 6.4%

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

   8. Simplified 6.4%

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

4. Final simplification 6.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq 1.3 \cdot 10^{-98}:\\ \;\;\;\;\sqrt{A \cdot F} \cdot \left(-\frac{2}{B}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{C \cdot F} \cdot \left(-\frac{2}{B}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 4.9% accurate, 5.9× speedup

Math

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

FPCore

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

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	return sqrt((A * F)) * (2.0 / -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((a * f)) * (2.0d0 / -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((A * F)) * (2.0 / -B_m);
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 20.9%

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

3. Taylor expanded in C around 0 11.0%

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

   1. mul-1-neg 11.0%

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

   2. +-commutative 11.0%

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

   3. unpow2 11.0%

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

   4. unpow2 11.0%

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

   5. hypot-define 20.1%

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

5. Simplified 20.1%

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

6. Taylor expanded in B around 0 5.0%

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

   1. *-commutative 5.0%

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

   2. unpow2 5.0%

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

   3. rem-square-sqrt 5.0%

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

8. Simplified 5.0%

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

9. Final simplification 5.0%

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

Reproduce

herbie shell --seed 2024046 
(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))))