ABCF->ab-angle a

Percentage Accurate: 18.4% → 52.1%

Time: 25.6s

Alternatives: 13

Speedup: 3.1×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 18.4% 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: 52.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := A \cdot \left(C \cdot -4\right)\\ t_1 := \mathsf{fma}\left(B_m, B_m, t_0\right)\\ t_2 := \sqrt{A + \left(C + \mathsf{hypot}\left(B_m, A - C\right)\right)}\\ t_3 := \frac{t_2 \cdot \left(-\sqrt{t_1 \cdot \left(2 \cdot F\right)}\right)}{t_1}\\ \mathbf{if}\;{B_m}^{2} \leq 2 \cdot 10^{-167}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;{B_m}^{2} \leq 5 \cdot 10^{-93}:\\ \;\;\;\;\frac{{\left(\sqrt{\mathsf{hypot}\left(B_m, \sqrt{t_0}\right) \cdot \sqrt{2 \cdot F}}\right)}^{2} \cdot \left(-t_2\right)}{t_1}\\ \mathbf{elif}\;{B_m}^{2} \leq 10^{+130}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(C, B_m\right)} \cdot \sqrt{F}\right) \cdot \frac{-\sqrt{2}}{B_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (* A (* C -4.0)))
        (t_1 (fma B_m B_m t_0))
        (t_2 (sqrt (+ A (+ C (hypot B_m (- A C))))))
        (t_3 (/ (* t_2 (- (sqrt (* t_1 (* 2.0 F))))) t_1)))
   (if (<= (pow B_m 2.0) 2e-167)
     t_3
     (if (<= (pow B_m 2.0) 5e-93)
       (/
        (*
         (pow (sqrt (* (hypot B_m (sqrt t_0)) (sqrt (* 2.0 F)))) 2.0)
         (- t_2))
        t_1)
       (if (<= (pow B_m 2.0) 1e+130)
         t_3
         (*
          (* (sqrt (+ C (hypot C B_m))) (sqrt F))
          (/ (- (sqrt 2.0)) B_m)))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double t_0 = A * (C * -4.0);
	double t_1 = fma(B_m, B_m, t_0);
	double t_2 = sqrt((A + (C + hypot(B_m, (A - C)))));
	double t_3 = (t_2 * -sqrt((t_1 * (2.0 * F)))) / t_1;
	double tmp;
	if (pow(B_m, 2.0) <= 2e-167) {
		tmp = t_3;
	} else if (pow(B_m, 2.0) <= 5e-93) {
		tmp = (pow(sqrt((hypot(B_m, sqrt(t_0)) * sqrt((2.0 * F)))), 2.0) * -t_2) / t_1;
	} else if (pow(B_m, 2.0) <= 1e+130) {
		tmp = t_3;
	} else {
		tmp = (sqrt((C + hypot(C, B_m))) * sqrt(F)) * (-sqrt(2.0) / B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	t_0 = Float64(A * Float64(C * -4.0))
	t_1 = fma(B_m, B_m, t_0)
	t_2 = sqrt(Float64(A + Float64(C + hypot(B_m, Float64(A - C)))))
	t_3 = Float64(Float64(t_2 * Float64(-sqrt(Float64(t_1 * Float64(2.0 * F))))) / t_1)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 2e-167)
		tmp = t_3;
	elseif ((B_m ^ 2.0) <= 5e-93)
		tmp = Float64(Float64((sqrt(Float64(hypot(B_m, sqrt(t_0)) * sqrt(Float64(2.0 * F)))) ^ 2.0) * Float64(-t_2)) / t_1);
	elseif ((B_m ^ 2.0) <= 1e+130)
		tmp = t_3;
	else
		tmp = Float64(Float64(sqrt(Float64(C + hypot(C, B_m))) * sqrt(F)) * Float64(Float64(-sqrt(2.0)) / 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[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(B$95$m * B$95$m + t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(A + N[(C + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(t$95$2 * (-N[Sqrt[N[(t$95$1 * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$1), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e-167], t$95$3, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e-93], N[(N[(N[Power[N[Sqrt[N[(N[Sqrt[B$95$m ^ 2 + N[Sqrt[t$95$0], $MachinePrecision] ^ 2], $MachinePrecision] * N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision] * (-t$95$2)), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e+130], t$95$3, N[(N[(N[Sqrt[N[(C + N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $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) < 2e-167 or 4.99999999999999994e-93 < (pow.f64 B 2) < 1.0000000000000001e130

   1. Initial program 32.0%

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

   3. Simplified 39.6%

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

      1. pow1/2 39.6%

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

      2. *-commutative 39.6%

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

      3. unpow-prod-down 46.1%

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

      4. pow1/2 46.1%

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

      5. pow1/2 46.1%

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

      6. *-commutative 46.1%

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

   5. Applied egg-rr 46.1%

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

   if 2e-167 < (pow.f64 B 2) < 4.99999999999999994e-93

   1. Initial program 11.6%

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

   3. Simplified 14.7%

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

      1. pow1/2 14.7%

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

      2. *-commutative 14.7%

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

      3. unpow-prod-down 21.3%

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

      4. pow1/2 21.3%

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

      5. pow1/2 21.3%

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

      6. *-commutative 21.3%

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

   5. Applied egg-rr 21.3%

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

      1. add-sqr-sqrt 21.4%

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

      2. pow2 21.4%

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

      3. sqrt-prod 48.7%

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

      4. fma-udef 48.7%

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

      5. add-sqr-sqrt 48.7%

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

      6. hypot-def 48.7%

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

   7. Applied egg-rr 48.7%

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

   if 1.0000000000000001e130 < (pow.f64 B 2)

   1. Initial program 8.5%

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

   3. Simplified 11.8%

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

   4. Taylor expanded in A around 0 9.4%

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

      1. mul-1-neg 9.4%

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

      2. distribute-rgt-neg-in 9.4%

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

      3. unpow2 9.4%

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

      4. unpow2 9.4%

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

      5. hypot-def 28.2%

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

   6. Simplified 28.2%

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

      1. pow1/2 28.3%

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

      2. *-commutative 28.3%

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

      3. hypot-udef 9.5%

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

      4. unpow2 9.5%

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

      5. unpow2 9.5%

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

      6. unpow-prod-down 13.1%

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

   8. Applied egg-rr 35.9%

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

4. Final simplification 42.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 2 \cdot 10^{-167}:\\ \;\;\;\;\frac{\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(-\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(2 \cdot F\right)}\right)}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;{B}^{2} \leq 5 \cdot 10^{-93}:\\ \;\;\;\;\frac{{\left(\sqrt{\mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right) \cdot \sqrt{2 \cdot F}}\right)}^{2} \cdot \left(-\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)}\right)}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;{B}^{2} \leq 10^{+130}:\\ \;\;\;\;\frac{\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(-\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(2 \cdot F\right)}\right)}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \sqrt{F}\right) \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 2: 52.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := A \cdot \left(C \cdot -4\right)\\ t_1 := \mathsf{fma}\left(B_m, B_m, t_0\right)\\ t_2 := \sqrt{A + \left(C + \mathsf{hypot}\left(B_m, A - C\right)\right)}\\ t_3 := \frac{t_2 \cdot \left(-\sqrt{t_1 \cdot \left(2 \cdot F\right)}\right)}{t_1}\\ \mathbf{if}\;{B_m}^{2} \leq 2 \cdot 10^{-167}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;{B_m}^{2} \leq 5 \cdot 10^{-93}:\\ \;\;\;\;\frac{t_2 \cdot \left(\mathsf{hypot}\left(B_m, \sqrt{t_0}\right) \cdot \left(-\sqrt{2 \cdot F}\right)\right)}{t_1}\\ \mathbf{elif}\;{B_m}^{2} \leq 10^{+130}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(C, B_m\right)} \cdot \sqrt{F}\right) \cdot \frac{-\sqrt{2}}{B_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (* A (* C -4.0)))
        (t_1 (fma B_m B_m t_0))
        (t_2 (sqrt (+ A (+ C (hypot B_m (- A C))))))
        (t_3 (/ (* t_2 (- (sqrt (* t_1 (* 2.0 F))))) t_1)))
   (if (<= (pow B_m 2.0) 2e-167)
     t_3
     (if (<= (pow B_m 2.0) 5e-93)
       (/ (* t_2 (* (hypot B_m (sqrt t_0)) (- (sqrt (* 2.0 F))))) t_1)
       (if (<= (pow B_m 2.0) 1e+130)
         t_3
         (*
          (* (sqrt (+ C (hypot C B_m))) (sqrt F))
          (/ (- (sqrt 2.0)) B_m)))))))

C

B_m = fabs(B);
double code(double A, double B_m, double C, double F) {
	double t_0 = A * (C * -4.0);
	double t_1 = fma(B_m, B_m, t_0);
	double t_2 = sqrt((A + (C + hypot(B_m, (A - C)))));
	double t_3 = (t_2 * -sqrt((t_1 * (2.0 * F)))) / t_1;
	double tmp;
	if (pow(B_m, 2.0) <= 2e-167) {
		tmp = t_3;
	} else if (pow(B_m, 2.0) <= 5e-93) {
		tmp = (t_2 * (hypot(B_m, sqrt(t_0)) * -sqrt((2.0 * F)))) / t_1;
	} else if (pow(B_m, 2.0) <= 1e+130) {
		tmp = t_3;
	} else {
		tmp = (sqrt((C + hypot(C, B_m))) * sqrt(F)) * (-sqrt(2.0) / B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
function code(A, B_m, C, F)
	t_0 = Float64(A * Float64(C * -4.0))
	t_1 = fma(B_m, B_m, t_0)
	t_2 = sqrt(Float64(A + Float64(C + hypot(B_m, Float64(A - C)))))
	t_3 = Float64(Float64(t_2 * Float64(-sqrt(Float64(t_1 * Float64(2.0 * F))))) / t_1)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 2e-167)
		tmp = t_3;
	elseif ((B_m ^ 2.0) <= 5e-93)
		tmp = Float64(Float64(t_2 * Float64(hypot(B_m, sqrt(t_0)) * Float64(-sqrt(Float64(2.0 * F))))) / t_1);
	elseif ((B_m ^ 2.0) <= 1e+130)
		tmp = t_3;
	else
		tmp = Float64(Float64(sqrt(Float64(C + hypot(C, B_m))) * sqrt(F)) * Float64(Float64(-sqrt(2.0)) / 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[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(B$95$m * B$95$m + t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[N[(A + N[(C + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(t$95$2 * (-N[Sqrt[N[(t$95$1 * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$1), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e-167], t$95$3, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e-93], N[(N[(t$95$2 * N[(N[Sqrt[B$95$m ^ 2 + N[Sqrt[t$95$0], $MachinePrecision] ^ 2], $MachinePrecision] * (-N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e+130], t$95$3, N[(N[(N[Sqrt[N[(C + N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $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) < 2e-167 or 4.99999999999999994e-93 < (pow.f64 B 2) < 1.0000000000000001e130

   1. Initial program 32.0%

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

   3. Simplified 39.6%

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

      1. pow1/2 39.6%

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

      2. *-commutative 39.6%

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

      3. unpow-prod-down 46.1%

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

      4. pow1/2 46.1%

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

      5. pow1/2 46.1%

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

      6. *-commutative 46.1%

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

   5. Applied egg-rr 46.1%

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

   if 2e-167 < (pow.f64 B 2) < 4.99999999999999994e-93

   1. Initial program 11.6%

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

   3. Simplified 14.7%

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

      1. pow1/2 14.7%

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

      2. *-commutative 14.7%

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

      3. unpow-prod-down 21.3%

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

      4. pow1/2 21.3%

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

      5. pow1/2 21.3%

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

      6. *-commutative 21.3%

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

   5. Applied egg-rr 21.3%

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

      1. expm1-log1p-u 20.7%

         \[\leadsto \frac{-\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(2 \cdot F\right)}\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      2. expm1-udef 10.7%

         \[\leadsto \frac{-\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(2 \cdot F\right)}\right)} - 1\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      3. sqrt-prod 19.0%

         \[\leadsto \frac{-\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \cdot \sqrt{2 \cdot F}}\right)} - 1\right)}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      4. fma-udef 19.0%

         \[\leadsto \frac{-\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(e^{\mathsf{log1p}\left(\sqrt{\color{blue}{B \cdot B + A \cdot \left(C \cdot -4\right)}} \cdot \sqrt{2 \cdot F}\right)} - 1\right)}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      5. add-sqr-sqrt 19.0%

         \[\leadsto \frac{-\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(e^{\mathsf{log1p}\left(\sqrt{B \cdot B + \color{blue}{\sqrt{A \cdot \left(C \cdot -4\right)} \cdot \sqrt{A \cdot \left(C \cdot -4\right)}}} \cdot \sqrt{2 \cdot F}\right)} - 1\right)}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      6. hypot-def 19.0%

         \[\leadsto \frac{-\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(e^{\mathsf{log1p}\left(\color{blue}{\mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right)} \cdot \sqrt{2 \cdot F}\right)} - 1\right)}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   7. Applied egg-rr 19.0%

      \[\leadsto \frac{-\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \color{blue}{\left(e^{\mathsf{log1p}\left(\mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right) \cdot \sqrt{2 \cdot F}\right)} - 1\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
   8. Step-by-step derivation

      1. expm1-def 47.2%

         \[\leadsto \frac{-\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right) \cdot \sqrt{2 \cdot F}\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      2. expm1-log1p 48.6%

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

   9. Simplified 48.6%

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

   if 1.0000000000000001e130 < (pow.f64 B 2)

   1. Initial program 8.5%

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

   3. Simplified 11.8%

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

   4. Taylor expanded in A around 0 9.4%

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

      1. mul-1-neg 9.4%

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

      2. distribute-rgt-neg-in 9.4%

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

      3. unpow2 9.4%

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

      4. unpow2 9.4%

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

      5. hypot-def 28.2%

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

   6. Simplified 28.2%

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

      1. pow1/2 28.3%

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

      2. *-commutative 28.3%

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

      3. hypot-udef 9.5%

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

      4. unpow2 9.5%

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

      5. unpow2 9.5%

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

      6. unpow-prod-down 13.1%

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

   8. Applied egg-rr 35.9%

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

4. Final simplification 42.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 2 \cdot 10^{-167}:\\ \;\;\;\;\frac{\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(-\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(2 \cdot F\right)}\right)}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;{B}^{2} \leq 5 \cdot 10^{-93}:\\ \;\;\;\;\frac{\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(\mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right) \cdot \left(-\sqrt{2 \cdot F}\right)\right)}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;{B}^{2} \leq 10^{+130}:\\ \;\;\;\;\frac{\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(-\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(2 \cdot F\right)}\right)}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \sqrt{F}\right) \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 3: 52.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B_m, B_m, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;{B_m}^{2} \leq 10^{+130}:\\ \;\;\;\;\frac{\sqrt{A + \left(C + \mathsf{hypot}\left(B_m, A - C\right)\right)} \cdot \left(-\sqrt{t_0 \cdot \left(2 \cdot F\right)}\right)}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(C, B_m\right)} \cdot \sqrt{F}\right) \cdot \frac{-\sqrt{2}}{B_m}\\ \end{array} \end{array} \]

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) 1e+130)
     (/
      (* (sqrt (+ A (+ C (hypot B_m (- A C))))) (- (sqrt (* t_0 (* 2.0 F)))))
      t_0)
     (* (* (sqrt (+ C (hypot C B_m))) (sqrt F)) (/ (- (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) <= 1e+130) {
		tmp = (sqrt((A + (C + hypot(B_m, (A - C))))) * -sqrt((t_0 * (2.0 * F)))) / t_0;
	} else {
		tmp = (sqrt((C + hypot(C, B_m))) * sqrt(F)) * (-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) <= 1e+130)
		tmp = Float64(Float64(sqrt(Float64(A + Float64(C + hypot(B_m, Float64(A - C))))) * Float64(-sqrt(Float64(t_0 * Float64(2.0 * F))))) / t_0);
	else
		tmp = Float64(Float64(sqrt(Float64(C + hypot(C, B_m))) * sqrt(F)) * Float64(Float64(-sqrt(2.0)) / 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], 1e+130], N[(N[(N[Sqrt[N[(A + N[(C + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(t$95$0 * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[Sqrt[N[(C + N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $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) < 1.0000000000000001e130

   1. Initial program 29.2%

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

   3. Simplified 36.2%

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

      1. pow1/2 36.2%

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

      2. *-commutative 36.2%

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

      3. unpow-prod-down 42.7%

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

      4. pow1/2 42.7%

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

      5. pow1/2 42.7%

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

      6. *-commutative 42.7%

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

   5. Applied egg-rr 42.7%

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

   if 1.0000000000000001e130 < (pow.f64 B 2)

   1. Initial program 8.5%

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

   3. Simplified 11.8%

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

   4. Taylor expanded in A around 0 9.4%

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

      1. mul-1-neg 9.4%

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

      2. distribute-rgt-neg-in 9.4%

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

      3. unpow2 9.4%

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

      4. unpow2 9.4%

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

      5. hypot-def 28.2%

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

   6. Simplified 28.2%

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

      1. pow1/2 28.3%

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

      2. *-commutative 28.3%

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

      3. hypot-udef 9.5%

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

      4. unpow2 9.5%

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

      5. unpow2 9.5%

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

      6. unpow-prod-down 13.1%

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

   8. Applied egg-rr 35.9%

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

4. Final simplification 40.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{+130}:\\ \;\;\;\;\frac{\sqrt{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)} \cdot \left(-\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(2 \cdot F\right)}\right)}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \sqrt{F}\right) \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 4: 43.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B_m, B_m, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := t_0 \cdot \left(2 \cdot F\right)\\ \mathbf{if}\;{B_m}^{2} \leq 10^{-228}:\\ \;\;\;\;\frac{-\sqrt{t_1 \cdot \left(A + A\right)}}{t_0}\\ \mathbf{elif}\;{B_m}^{2} \leq 10^{-34}:\\ \;\;\;\;\frac{\sqrt{t_1} \cdot \left(-\sqrt{2 \cdot C}\right)}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(C, B_m\right)} \cdot \sqrt{F}\right) \cdot \frac{-\sqrt{2}}{B_m}\\ \end{array} \end{array} \]

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)))) (t_1 (* t_0 (* 2.0 F))))
   (if (<= (pow B_m 2.0) 1e-228)
     (/ (- (sqrt (* t_1 (+ A A)))) t_0)
     (if (<= (pow B_m 2.0) 1e-34)
       (/ (* (sqrt t_1) (- (sqrt (* 2.0 C)))) t_0)
       (* (* (sqrt (+ C (hypot C B_m))) (sqrt F)) (/ (- (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 t_1 = t_0 * (2.0 * F);
	double tmp;
	if (pow(B_m, 2.0) <= 1e-228) {
		tmp = -sqrt((t_1 * (A + A))) / t_0;
	} else if (pow(B_m, 2.0) <= 1e-34) {
		tmp = (sqrt(t_1) * -sqrt((2.0 * C))) / t_0;
	} else {
		tmp = (sqrt((C + hypot(C, B_m))) * sqrt(F)) * (-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)))
	t_1 = Float64(t_0 * Float64(2.0 * F))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e-228)
		tmp = Float64(Float64(-sqrt(Float64(t_1 * Float64(A + A)))) / t_0);
	elseif ((B_m ^ 2.0) <= 1e-34)
		tmp = Float64(Float64(sqrt(t_1) * Float64(-sqrt(Float64(2.0 * C)))) / t_0);
	else
		tmp = Float64(Float64(sqrt(Float64(C + hypot(C, B_m))) * sqrt(F)) * Float64(Float64(-sqrt(2.0)) / 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]}, Block[{t$95$1 = N[(t$95$0 * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-228], N[((-N[Sqrt[N[(t$95$1 * N[(A + A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-34], N[(N[(N[Sqrt[t$95$1], $MachinePrecision] * (-N[Sqrt[N[(2.0 * C), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[Sqrt[N[(C + N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $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) < 1.00000000000000003e-228

   1. Initial program 27.5%

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

   3. Simplified 34.8%

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

   4. Taylor expanded in C around -inf 32.9%

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

   if 1.00000000000000003e-228 < (pow.f64 B 2) < 9.99999999999999928e-35

   1. Initial program 16.5%

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

   3. Simplified 23.3%

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

      1. pow1/2 23.3%

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

      2. *-commutative 23.3%

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

      3. unpow-prod-down 32.8%

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

      4. pow1/2 32.8%

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

      5. pow1/2 32.8%

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

      6. *-commutative 32.8%

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

   5. Applied egg-rr 32.8%

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

   6. Taylor expanded in A around -inf 29.9%

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

   if 9.99999999999999928e-35 < (pow.f64 B 2)

   1. Initial program 18.5%

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

   3. Simplified 22.7%

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

   4. Taylor expanded in A around 0 12.4%

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

      1. mul-1-neg 12.4%

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

      2. distribute-rgt-neg-in 12.4%

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

      3. unpow2 12.4%

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

      4. unpow2 12.4%

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

      5. hypot-def 26.7%

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

   6. Simplified 26.7%

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

      1. pow1/2 26.7%

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

      2. *-commutative 26.7%

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

      3. hypot-udef 12.5%

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

      4. unpow2 12.5%

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

      5. unpow2 12.5%

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

      6. unpow-prod-down 15.9%

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

   8. Applied egg-rr 33.1%

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

4. Final simplification 32.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{-228}:\\ \;\;\;\;\frac{-\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(A + A\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;{B}^{2} \leq 10^{-34}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(2 \cdot F\right)} \cdot \left(-\sqrt{2 \cdot C}\right)}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \sqrt{F}\right) \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 5: 48.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B_m, B_m, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;{B_m}^{2} \leq 5 \cdot 10^{+133}:\\ \;\;\;\;\frac{-\sqrt{\left(A + \left(C + \mathsf{hypot}\left(B_m, A - C\right)\right)\right) \cdot \left(t_0 \cdot \left(2 \cdot F\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(C, B_m\right)} \cdot \sqrt{F}\right) \cdot \frac{-\sqrt{2}}{B_m}\\ \end{array} \end{array} \]

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) 5e+133)
     (/ (- (sqrt (* (+ A (+ C (hypot B_m (- A C)))) (* t_0 (* 2.0 F))))) t_0)
     (* (* (sqrt (+ C (hypot C B_m))) (sqrt F)) (/ (- (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) <= 5e+133) {
		tmp = -sqrt(((A + (C + hypot(B_m, (A - C)))) * (t_0 * (2.0 * F)))) / t_0;
	} else {
		tmp = (sqrt((C + hypot(C, B_m))) * sqrt(F)) * (-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) <= 5e+133)
		tmp = Float64(Float64(-sqrt(Float64(Float64(A + Float64(C + hypot(B_m, Float64(A - C)))) * Float64(t_0 * Float64(2.0 * F))))) / t_0);
	else
		tmp = Float64(Float64(sqrt(Float64(C + hypot(C, B_m))) * sqrt(F)) * Float64(Float64(-sqrt(2.0)) / 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], 5e+133], N[((-N[Sqrt[N[(N[(A + N[(C + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(t$95$0 * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], N[(N[(N[Sqrt[N[(C + N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $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) < 4.99999999999999961e133

   1. Initial program 29.3%

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

   3. Simplified 36.2%

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

   if 4.99999999999999961e133 < (pow.f64 B 2)

   1. Initial program 7.8%

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

   3. Simplified 11.1%

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

   4. Taylor expanded in A around 0 8.7%

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

      1. mul-1-neg 8.7%

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

      2. distribute-rgt-neg-in 8.7%

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

      3. unpow2 8.7%

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

      4. unpow2 8.7%

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

      5. hypot-def 28.1%

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

   6. Simplified 28.1%

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

      1. pow1/2 28.1%

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

      2. *-commutative 28.1%

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

      3. hypot-udef 8.7%

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

      4. unpow2 8.7%

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

      5. unpow2 8.7%

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

      6. unpow-prod-down 12.4%

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

   8. Applied egg-rr 35.9%

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

4. Final simplification 36.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 5 \cdot 10^{+133}:\\ \;\;\;\;\frac{-\sqrt{\left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right) \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(2 \cdot F\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \sqrt{F}\right) \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 6: 44.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B_m, B_m, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;{B_m}^{2} \leq 10^{-228}:\\ \;\;\;\;\frac{-\sqrt{\left(t_0 \cdot \left(2 \cdot F\right)\right) \cdot \left(A + A\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(C, B_m\right)} \cdot \sqrt{F}\right) \cdot \frac{-\sqrt{2}}{B_m}\\ \end{array} \end{array} \]

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) 1e-228)
     (/ (- (sqrt (* (* t_0 (* 2.0 F)) (+ A A)))) t_0)
     (* (* (sqrt (+ C (hypot C B_m))) (sqrt F)) (/ (- (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) <= 1e-228) {
		tmp = -sqrt(((t_0 * (2.0 * F)) * (A + A))) / t_0;
	} else {
		tmp = (sqrt((C + hypot(C, B_m))) * sqrt(F)) * (-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) <= 1e-228)
		tmp = Float64(Float64(-sqrt(Float64(Float64(t_0 * Float64(2.0 * F)) * Float64(A + A)))) / t_0);
	else
		tmp = Float64(Float64(sqrt(Float64(C + hypot(C, B_m))) * sqrt(F)) * Float64(Float64(-sqrt(2.0)) / 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], 1e-228], N[((-N[Sqrt[N[(N[(t$95$0 * N[(2.0 * F), $MachinePrecision]), $MachinePrecision] * N[(A + A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], N[(N[(N[Sqrt[N[(C + N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $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) < 1.00000000000000003e-228

   1. Initial program 27.5%

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

   3. Simplified 34.8%

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

   4. Taylor expanded in C around -inf 32.9%

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

   if 1.00000000000000003e-228 < (pow.f64 B 2)

   1. Initial program 18.0%

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

   3. Simplified 22.8%

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

   4. Taylor expanded in A around 0 11.6%

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

      1. mul-1-neg 11.6%

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

      2. distribute-rgt-neg-in 11.6%

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

      3. unpow2 11.6%

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

      4. unpow2 11.6%

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

      5. hypot-def 22.6%

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

   6. Simplified 22.6%

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

      1. pow1/2 22.6%

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

      2. *-commutative 22.6%

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

      3. hypot-udef 11.7%

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

      4. unpow2 11.7%

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

      5. unpow2 11.7%

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

      6. unpow-prod-down 14.9%

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

   8. Applied egg-rr 28.2%

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

4. Final simplification 29.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{-228}:\\ \;\;\;\;\frac{-\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(A + A\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \sqrt{F}\right) \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 7: 34.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B_m, B_m, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;B_m \leq 9 \cdot 10^{-18}:\\ \;\;\;\;-\frac{\sqrt{\left(t_0 \cdot \left(2 \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A + \mathsf{hypot}\left(B_m, A\right)\right)} \cdot \frac{-\sqrt{2}}{B_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* A (* C -4.0)))))
   (if (<= B_m 9e-18)
     (- (/ (sqrt (* (* t_0 (* 2.0 F)) (* 2.0 C))) t_0))
     (* (sqrt (* F (+ 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 (B_m <= 9e-18) {
		tmp = -(sqrt(((t_0 * (2.0 * F)) * (2.0 * C))) / t_0);
	} else {
		tmp = sqrt((F * (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 <= 9e-18)
		tmp = Float64(-Float64(sqrt(Float64(Float64(t_0 * Float64(2.0 * F)) * Float64(2.0 * C))) / t_0));
	else
		tmp = Float64(sqrt(Float64(F * Float64(A + hypot(B_m, A)))) * Float64(Float64(-sqrt(2.0)) / 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[B$95$m, 9e-18], (-N[(N[Sqrt[N[(N[(t$95$0 * N[(2.0 * F), $MachinePrecision]), $MachinePrecision] * N[(2.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$0), $MachinePrecision]), N[(N[Sqrt[N[(F * N[(A + N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[((-N[Sqrt[2.0], $MachinePrecision]) / B$95$m), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if B < 8.99999999999999987e-18

   1. Initial program 21.1%

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

   3. Simplified 27.5%

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

   4. Taylor expanded in A around -inf 16.1%

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

   if 8.99999999999999987e-18 < B

   1. Initial program 20.5%

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

   3. Simplified 24.0%

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

   4. Taylor expanded in C around 0 25.0%

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

      1. mul-1-neg 25.0%

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

      2. distribute-rgt-neg-in 25.0%

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

      3. +-commutative 25.0%

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

      4. unpow2 25.0%

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

      5. unpow2 25.0%

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

      6. hypot-def 48.7%

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

   6. Simplified 48.7%

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

4. Final simplification 25.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 9 \cdot 10^{-18}:\\ \;\;\;\;-\frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A + \mathsf{hypot}\left(B, A\right)\right)} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 8: 34.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B_m, B_m, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;B_m \leq 1.5 \cdot 10^{-73}:\\ \;\;\;\;\frac{-\sqrt{\left(t_0 \cdot \left(2 \cdot F\right)\right) \cdot \left(A + A\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A + \mathsf{hypot}\left(B_m, A\right)\right)} \cdot \frac{-\sqrt{2}}{B_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* A (* C -4.0)))))
   (if (<= B_m 1.5e-73)
     (/ (- (sqrt (* (* t_0 (* 2.0 F)) (+ A A)))) t_0)
     (* (sqrt (* F (+ 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 (B_m <= 1.5e-73) {
		tmp = -sqrt(((t_0 * (2.0 * F)) * (A + A))) / t_0;
	} else {
		tmp = sqrt((F * (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 <= 1.5e-73)
		tmp = Float64(Float64(-sqrt(Float64(Float64(t_0 * Float64(2.0 * F)) * Float64(A + A)))) / t_0);
	else
		tmp = Float64(sqrt(Float64(F * Float64(A + hypot(B_m, A)))) * Float64(Float64(-sqrt(2.0)) / 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[B$95$m, 1.5e-73], N[((-N[Sqrt[N[(N[(t$95$0 * N[(2.0 * F), $MachinePrecision]), $MachinePrecision] * N[(A + A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], N[(N[Sqrt[N[(F * N[(A + N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[((-N[Sqrt[2.0], $MachinePrecision]) / B$95$m), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if B < 1.5e-73

   1. Initial program 20.9%

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

   3. Simplified 26.4%

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

   4. Taylor expanded in C around -inf 19.1%

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

   if 1.5e-73 < B

   1. Initial program 21.0%

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

   3. Simplified 26.7%

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

   4. Taylor expanded in C around 0 23.8%

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

      1. mul-1-neg 23.8%

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

      2. distribute-rgt-neg-in 23.8%

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

      3. +-commutative 23.8%

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

      4. unpow2 23.8%

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

      5. unpow2 23.8%

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

      6. hypot-def 43.6%

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

   6. Simplified 43.6%

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

4. Final simplification 27.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.5 \cdot 10^{-73}:\\ \;\;\;\;\frac{-\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(A + A\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(A + \mathsf{hypot}\left(B, A\right)\right)} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 9: 35.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;F \leq 2.4 \cdot 10^{+54}:\\ \;\;\;\;\sqrt{F \cdot \left(A + \mathsf{hypot}\left(B_m, A\right)\right)} \cdot \frac{-\sqrt{2}}{B_m}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B_m}}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= F 2.4e+54)
   (* (sqrt (* F (+ A (hypot B_m A)))) (/ (- (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 <= 2.4e+54) {
		tmp = sqrt((F * (A + hypot(B_m, A)))) * (-sqrt(2.0) / 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 <= 2.4e+54) {
		tmp = Math.sqrt((F * (A + Math.hypot(B_m, A)))) * (-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 <= 2.4e+54:
		tmp = math.sqrt((F * (A + math.hypot(B_m, A)))) * (-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 <= 2.4e+54)
		tmp = Float64(sqrt(Float64(F * Float64(A + hypot(B_m, A)))) * Float64(Float64(-sqrt(2.0)) / 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 <= 2.4e+54)
		tmp = sqrt((F * (A + hypot(B_m, A)))) * (-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, 2.4e+54], N[(N[Sqrt[N[(F * N[(A + N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[((-N[Sqrt[2.0], $MachinePrecision]) / B$95$m), $MachinePrecision]), $MachinePrecision], 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 < 2.39999999999999998e54

   1. Initial program 22.2%

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

   3. Simplified 29.7%

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

   4. Taylor expanded in C around 0 9.0%

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

      1. mul-1-neg 9.0%

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

      2. distribute-rgt-neg-in 9.0%

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

      3. +-commutative 9.0%

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

      4. unpow2 9.0%

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

      5. unpow2 9.0%

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

      6. hypot-def 19.3%

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

   6. Simplified 19.3%

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

   if 2.39999999999999998e54 < F

   1. Initial program 18.2%

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

   3. Simplified 19.7%

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

   4. Taylor expanded in A around 0 9.5%

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

      1. mul-1-neg 9.5%

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

   6. Simplified 9.5%

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

   7. Taylor expanded in C around 0 21.0%

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

4. Final simplification 19.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 2.4 \cdot 10^{+54}:\\ \;\;\;\;\sqrt{F \cdot \left(A + \mathsf{hypot}\left(B, A\right)\right)} \cdot \frac{-\sqrt{2}}{B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \]

Alternative 10: 35.5% accurate, 3.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;F \leq 2.7 \cdot 10^{+28}:\\ \;\;\;\;-\frac{\sqrt{\left(2 \cdot F\right) \cdot \left(C + \mathsf{hypot}\left(C, B_m\right)\right)}}{B_m}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B_m}}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= F 2.7e+28)
   (- (/ (sqrt (* (* 2.0 F) (+ C (hypot C B_m)))) 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 <= 2.7e+28) {
		tmp = -(sqrt(((2.0 * F) * (C + hypot(C, B_m)))) / 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 <= 2.7e+28) {
		tmp = -(Math.sqrt(((2.0 * F) * (C + Math.hypot(C, B_m)))) / 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 <= 2.7e+28:
		tmp = -(math.sqrt(((2.0 * F) * (C + math.hypot(C, B_m)))) / 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 <= 2.7e+28)
		tmp = Float64(-Float64(sqrt(Float64(Float64(2.0 * F) * Float64(C + hypot(C, B_m)))) / 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 <= 2.7e+28)
		tmp = -(sqrt(((2.0 * F) * (C + hypot(C, B_m)))) / 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, 2.7e+28], (-N[(N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] * N[(C + N[Sqrt[C ^ 2 + B$95$m ^ 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 < 2.7000000000000002e28

   1. Initial program 22.0%

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

   3. Simplified 29.8%

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

   4. Taylor expanded in A around 0 7.7%

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

      1. mul-1-neg 7.7%

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

      2. distribute-rgt-neg-in 7.7%

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

      3. unpow2 7.7%

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

      4. unpow2 7.7%

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

      5. hypot-def 18.2%

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

   6. Simplified 18.2%

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

   8. Applied egg-rr 3.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(-\frac{{\left(2 \cdot \left(F \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)\right)\right)}^{0.5}}{B}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 17.5%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(-\frac{{\left(2 \cdot \left(F \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)\right)\right)}^{0.5}}{B}\right)\right)} \]

      2. expm1-log1p 18.3%

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

      3. distribute-neg-frac 18.3%

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

      4. unpow1/2 18.2%

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

      5. associate-*r* 18.2%

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

   10. Simplified 18.2%

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

   if 2.7000000000000002e28 < F

   1. Initial program 19.2%

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

   3. Simplified 20.9%

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

   4. Taylor expanded in A around 0 10.7%

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

      1. mul-1-neg 10.7%

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

   6. Simplified 10.7%

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

   7. Taylor expanded in C around 0 22.6%

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

4. Final simplification 19.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 2.7 \cdot 10^{+28}:\\ \;\;\;\;-\frac{\sqrt{\left(2 \cdot F\right) \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)}}{B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \]

Alternative 11: 33.6% accurate, 3.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ \begin{array}{l} \mathbf{if}\;F \leq 5.2 \cdot 10^{-25}:\\ \;\;\;\;\frac{\sqrt{2}}{B_m} \cdot \left(-\sqrt{B_m \cdot F}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B_m}}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
(FPCore (A B_m C F)
 :precision binary64
 (if (<= F 5.2e-25)
   (* (/ (sqrt 2.0) B_m) (- (sqrt (* B_m F))))
   (* (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 <= 5.2e-25) {
		tmp = (sqrt(2.0) / B_m) * -sqrt((B_m * F));
	} 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 <= 5.2d-25) then
        tmp = (sqrt(2.0d0) / b_m) * -sqrt((b_m * f))
    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 <= 5.2e-25) {
		tmp = (Math.sqrt(2.0) / B_m) * -Math.sqrt((B_m * F));
	} 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 <= 5.2e-25:
		tmp = (math.sqrt(2.0) / B_m) * -math.sqrt((B_m * F))
	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 <= 5.2e-25)
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(-sqrt(Float64(B_m * F))));
	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 <= 5.2e-25)
		tmp = (sqrt(2.0) / B_m) * -sqrt((B_m * F));
	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, 5.2e-25], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * (-N[Sqrt[N[(B$95$m * F), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * (-N[Sqrt[N[(F / B$95$m), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < 5.2e-25

   1. Initial program 21.6%

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

   3. Simplified 30.0%

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

   4. Taylor expanded in A around 0 7.1%

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

      1. mul-1-neg 7.1%

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

      2. distribute-rgt-neg-in 7.1%

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

      3. unpow2 7.1%

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

      4. unpow2 7.1%

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

      5. hypot-def 16.6%

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

   6. Simplified 16.6%

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

   7. Taylor expanded in C around 0 14.5%

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

   if 5.2e-25 < F

   1. Initial program 20.1%

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

   3. Simplified 21.9%

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

   4. Taylor expanded in A around 0 11.1%

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

      1. mul-1-neg 11.1%

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

   6. Simplified 11.1%

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

   7. Taylor expanded in C around 0 24.0%

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

4. Final simplification 18.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 5.2 \cdot 10^{-25}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{B \cdot F}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \]

Alternative 12: 26.6% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 21.0%

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

3. Simplified 26.5%

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

4. Taylor expanded in A around 0 8.8%

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

   1. mul-1-neg 8.8%

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

6. Simplified 8.8%

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

7. Taylor expanded in C around 0 16.9%

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

8. Final simplification 16.9%

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

Alternative 13: 5.1% accurate, 5.9× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 21.0%

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

3. Simplified 26.5%

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

4. Taylor expanded in A around 0 8.8%

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

   1. mul-1-neg 8.8%

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

   2. distribute-rgt-neg-in 8.8%

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

   3. unpow2 8.8%

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

   4. unpow2 8.8%

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

   5. hypot-def 17.0%

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

6. Simplified 17.0%

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

7. Taylor expanded in B around 0 2.8%

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

   1. mul-1-neg 2.8%

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

   2. *-commutative 2.8%

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

   3. *-commutative 2.8%

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

   4. unpow2 2.8%

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

   5. rem-square-sqrt 2.8%

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

9. Simplified 2.8%

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

10. Final simplification 2.8%

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

Reproduce

herbie shell --seed 2023337 
(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))))