ABCF->ab-angle a

Percentage Accurate: 19.0% → 46.5%

Time: 29.5s

Alternatives: 8

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: 19.0% 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: 46.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\\ t_2 := \mathsf{hypot}\left(B, A - C\right)\\ t_3 := {B}^{2} - 4 \cdot \left(A \cdot C\right)\\ t_4 := {B}^{2} - \left(4 \cdot A\right) \cdot C\\ t_5 := \frac{-\sqrt{\left(2 \cdot \left(t_4 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{t_4}\\ \mathbf{if}\;t_5 \leq -5 \cdot 10^{-210}:\\ \;\;\;\;\frac{\sqrt{2 \cdot t_0} \cdot \left(-\sqrt{F \cdot \left(\left(A + C\right) + t_2\right)}\right)}{t_3}\\ \mathbf{elif}\;t_5 \leq 2 \cdot 10^{-130}:\\ \;\;\;\;\frac{-\sqrt{\left(t_1 \cdot \left(2 \cdot F\right)\right) \cdot \left(-0.5 \cdot \left(B \cdot \left(B \cdot \frac{1}{C}\right)\right) + 2 \cdot A\right)}}{t_1}\\ \mathbf{elif}\;t_5 \leq \infty:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot t_0\right)} \cdot \left(-\sqrt{C + \left(A + t_2\right)}\right)}{t_3}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(B + C\right)} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (fma B B (* A (* C -4.0))))
        (t_1 (fma B B (* C (* A -4.0))))
        (t_2 (hypot B (- A C)))
        (t_3 (- (pow B 2.0) (* 4.0 (* A C))))
        (t_4 (- (pow B 2.0) (* (* 4.0 A) C)))
        (t_5
         (/
          (-
           (sqrt
            (*
             (* 2.0 (* t_4 F))
             (+ (+ A C) (sqrt (+ (pow B 2.0) (pow (- A C) 2.0)))))))
          t_4)))
   (if (<= t_5 -5e-210)
     (/ (* (sqrt (* 2.0 t_0)) (- (sqrt (* F (+ (+ A C) t_2))))) t_3)
     (if (<= t_5 2e-130)
       (/
        (-
         (sqrt
          (* (* t_1 (* 2.0 F)) (+ (* -0.5 (* B (* B (/ 1.0 C)))) (* 2.0 A)))))
        t_1)
       (if (<= t_5 INFINITY)
         (/ (* (sqrt (* 2.0 (* F t_0))) (- (sqrt (+ C (+ A t_2))))) t_3)
         (* (sqrt (* F (+ B C))) (/ (- (sqrt 2.0)) B)))))))

C

B = abs(B);
double code(double A, double B, double C, double F) {
	double t_0 = fma(B, B, (A * (C * -4.0)));
	double t_1 = fma(B, B, (C * (A * -4.0)));
	double t_2 = hypot(B, (A - C));
	double t_3 = pow(B, 2.0) - (4.0 * (A * C));
	double t_4 = pow(B, 2.0) - ((4.0 * A) * C);
	double t_5 = -sqrt(((2.0 * (t_4 * F)) * ((A + C) + sqrt((pow(B, 2.0) + pow((A - C), 2.0)))))) / t_4;
	double tmp;
	if (t_5 <= -5e-210) {
		tmp = (sqrt((2.0 * t_0)) * -sqrt((F * ((A + C) + t_2)))) / t_3;
	} else if (t_5 <= 2e-130) {
		tmp = -sqrt(((t_1 * (2.0 * F)) * ((-0.5 * (B * (B * (1.0 / C)))) + (2.0 * A)))) / t_1;
	} else if (t_5 <= ((double) INFINITY)) {
		tmp = (sqrt((2.0 * (F * t_0))) * -sqrt((C + (A + t_2)))) / t_3;
	} else {
		tmp = sqrt((F * (B + C))) * (-sqrt(2.0) / B);
	}
	return tmp;
}

Julia

B = abs(B)
function code(A, B, C, F)
	t_0 = fma(B, B, Float64(A * Float64(C * -4.0)))
	t_1 = fma(B, B, Float64(C * Float64(A * -4.0)))
	t_2 = hypot(B, Float64(A - C))
	t_3 = Float64((B ^ 2.0) - Float64(4.0 * Float64(A * C)))
	t_4 = Float64((B ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_5 = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(t_4 * F)) * Float64(Float64(A + C) + sqrt(Float64((B ^ 2.0) + (Float64(A - C) ^ 2.0))))))) / t_4)
	tmp = 0.0
	if (t_5 <= -5e-210)
		tmp = Float64(Float64(sqrt(Float64(2.0 * t_0)) * Float64(-sqrt(Float64(F * Float64(Float64(A + C) + t_2))))) / t_3);
	elseif (t_5 <= 2e-130)
		tmp = Float64(Float64(-sqrt(Float64(Float64(t_1 * Float64(2.0 * F)) * Float64(Float64(-0.5 * Float64(B * Float64(B * Float64(1.0 / C)))) + Float64(2.0 * A))))) / t_1);
	elseif (t_5 <= Inf)
		tmp = Float64(Float64(sqrt(Float64(2.0 * Float64(F * t_0))) * Float64(-sqrt(Float64(C + Float64(A + t_2))))) / t_3);
	else
		tmp = Float64(sqrt(Float64(F * Float64(B + C))) * Float64(Float64(-sqrt(2.0)) / B));
	end
	return tmp
end

Wolfram

NOTE: B should be positive before calling this function
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(B * B + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(B * B + N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]}, Block[{t$95$3 = N[(N[Power[B, 2.0], $MachinePrecision] - N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[Power[B, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[((-N[Sqrt[N[(N[(2.0 * N[(t$95$4 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[B, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$4), $MachinePrecision]}, If[LessEqual[t$95$5, -5e-210], N[(N[(N[Sqrt[N[(2.0 * t$95$0), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(F * N[(N[(A + C), $MachinePrecision] + t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$3), $MachinePrecision], If[LessEqual[t$95$5, 2e-130], N[((-N[Sqrt[N[(N[(t$95$1 * N[(2.0 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(-0.5 * N[(B * N[(B * N[(1.0 / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision], If[LessEqual[t$95$5, Infinity], N[(N[(N[Sqrt[N[(2.0 * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(C + N[(A + t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$3), $MachinePrecision], N[(N[Sqrt[N[(F * N[(B + C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[((-N[Sqrt[2.0], $MachinePrecision]) / B), $MachinePrecision]), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))) < -5.0000000000000002e-210

   1. Initial program 43.1%

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

   3. Simplified 43.1%

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

   5. Applied egg-rr 68.2%

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

   if -5.0000000000000002e-210 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))) < 2.0000000000000002e-130

   1. Initial program 6.0%

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

   3. Simplified 12.1%

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

   4. Taylor expanded in C around -inf 44.0%

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

      1. div-inv 44.0%

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

      2. unpow2 44.0%

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

      3. associate-*l* 44.0%

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

   6. Applied egg-rr 44.0%

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

   if 2.0000000000000002e-130 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))) < +inf.0

   1. Initial program 30.4%

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

   3. Simplified 30.4%

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

      1. unpow2 30.4%

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

      2. unpow2 30.4%

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

      3. hypot-udef 57.3%

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

      4. add-sqr-sqrt 57.3%

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

   5. Applied egg-rr 57.3%

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

      1. sqrt-prod 73.0%

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

      2. *-commutative 73.0%

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

      3. +-commutative 73.0%

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

      4. rem-square-sqrt 72.9%

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

      5. associate-+l+ 72.9%

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

      6. associate-*l* 72.9%

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

      7. *-commutative 72.9%

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

      8. cancel-sign-sub-inv 72.9%

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

      9. metadata-eval 72.9%

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

      10. *-commutative 72.9%

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

      11. *-commutative 72.9%

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

      12. associate-*r* 72.9%

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

   7. Applied egg-rr 72.9%

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

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

   1. Initial program 0.0%

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

   3. Simplified 1.5%

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

   4. Taylor expanded in B around inf 0.5%

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

      1. +-commutative 0.5%

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

   6. Simplified 0.5%

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

   7. Taylor expanded in A around 0 19.2%

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

      1. +-commutative 19.2%

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

   9. Simplified 19.2%

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

4. Final simplification 40.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \leq -5 \cdot 10^{-210}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \cdot \left(-\sqrt{F \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \leq 2 \cdot 10^{-130}:\\ \;\;\;\;\frac{-\sqrt{\left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(-0.5 \cdot \left(B \cdot \left(B \cdot \frac{1}{C}\right)\right) + 2 \cdot A\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\\ \mathbf{elif}\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \leq \infty:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)} \cdot \left(-\sqrt{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)}\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(B + C\right)} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 2: 38.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;{B}^{2} \leq 2 \cdot 10^{-208}:\\ \;\;\;\;\frac{-\sqrt{t_0 \cdot \left(F \cdot \left(2 \cdot \mathsf{fma}\left(-0.5, \frac{{B}^{2}}{C}, 2 \cdot A\right)\right)\right)}}{t_0}\\ \mathbf{elif}\;{B}^{2} \leq 10^{+292}:\\ \;\;\;\;\frac{\sqrt{2 \cdot t_0} \cdot \left(-\sqrt{F \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(B + C\right)} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (fma B B (* A (* C -4.0)))))
   (if (<= (pow B 2.0) 2e-208)
     (/
      (- (sqrt (* t_0 (* F (* 2.0 (fma -0.5 (/ (pow B 2.0) C) (* 2.0 A)))))))
      t_0)
     (if (<= (pow B 2.0) 1e+292)
       (/
        (* (sqrt (* 2.0 t_0)) (- (sqrt (* F (+ (+ A C) (hypot B (- A C)))))))
        (- (pow B 2.0) (* 4.0 (* A C))))
       (* (sqrt (* F (+ B C))) (/ (- (sqrt 2.0)) B))))))

C

B = abs(B);
double code(double A, double B, double C, double F) {
	double t_0 = fma(B, B, (A * (C * -4.0)));
	double tmp;
	if (pow(B, 2.0) <= 2e-208) {
		tmp = -sqrt((t_0 * (F * (2.0 * fma(-0.5, (pow(B, 2.0) / C), (2.0 * A)))))) / t_0;
	} else if (pow(B, 2.0) <= 1e+292) {
		tmp = (sqrt((2.0 * t_0)) * -sqrt((F * ((A + C) + hypot(B, (A - C)))))) / (pow(B, 2.0) - (4.0 * (A * C)));
	} else {
		tmp = sqrt((F * (B + C))) * (-sqrt(2.0) / B);
	}
	return tmp;
}

Julia

B = abs(B)
function code(A, B, C, F)
	t_0 = fma(B, B, Float64(A * Float64(C * -4.0)))
	tmp = 0.0
	if ((B ^ 2.0) <= 2e-208)
		tmp = Float64(Float64(-sqrt(Float64(t_0 * Float64(F * Float64(2.0 * fma(-0.5, Float64((B ^ 2.0) / C), Float64(2.0 * A))))))) / t_0);
	elseif ((B ^ 2.0) <= 1e+292)
		tmp = Float64(Float64(sqrt(Float64(2.0 * t_0)) * Float64(-sqrt(Float64(F * Float64(Float64(A + C) + hypot(B, Float64(A - C))))))) / Float64((B ^ 2.0) - Float64(4.0 * Float64(A * C))));
	else
		tmp = Float64(sqrt(Float64(F * Float64(B + C))) * Float64(Float64(-sqrt(2.0)) / B));
	end
	return tmp
end

Wolfram

NOTE: B should be positive before calling this function
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(B * B + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B, 2.0], $MachinePrecision], 2e-208], N[((-N[Sqrt[N[(t$95$0 * N[(F * N[(2.0 * N[(-0.5 * N[(N[Power[B, 2.0], $MachinePrecision] / C), $MachinePrecision] + N[(2.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], If[LessEqual[N[Power[B, 2.0], $MachinePrecision], 1e+292], N[(N[(N[Sqrt[N[(2.0 * t$95$0), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(F * N[(N[(A + C), $MachinePrecision] + N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / N[(N[Power[B, 2.0], $MachinePrecision] - N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(F * N[(B + C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[((-N[Sqrt[2.0], $MachinePrecision]) / B), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B 2) < 2.0000000000000002e-208

   1. Initial program 19.4%

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

   3. Simplified 28.5%

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

   4. Taylor expanded in C around -inf 30.0%

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

      1. distribute-frac-neg 30.0%

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

      2. associate-*r* 30.0%

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

      3. associate-*l* 30.0%

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

      4. associate-*r* 30.0%

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

      5. *-commutative 30.0%

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

      6. associate-*l* 30.0%

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

      7. fma-def 30.0%

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

      8. associate-*r* 30.0%

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

      9. *-commutative 30.0%

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

      10. associate-*l* 30.0%

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

   6. Applied egg-rr 30.0%

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

      1. associate-*l* 30.1%

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

   8. Simplified 30.1%

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

   if 2.0000000000000002e-208 < (pow.f64 B 2) < 1e292

   1. Initial program 26.1%

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

   3. Simplified 26.8%

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

   5. Applied egg-rr 45.5%

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

   if 1e292 < (pow.f64 B 2)

   1. Initial program 1.2%

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

   3. Simplified 1.2%

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

   4. Taylor expanded in B around inf 0.0%

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

      1. +-commutative 0.0%

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

   6. Simplified 0.0%

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

   7. Taylor expanded in A around 0 29.0%

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

      1. +-commutative 29.0%

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

   9. Simplified 29.0%

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

4. Final simplification 36.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 2 \cdot 10^{-208}:\\ \;\;\;\;\frac{-\sqrt{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(F \cdot \left(2 \cdot \mathsf{fma}\left(-0.5, \frac{{B}^{2}}{C}, 2 \cdot A\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;{B}^{2} \leq 10^{+292}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \cdot \left(-\sqrt{F \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)\right)}\right)}{{B}^{2} - 4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(B + C\right)} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 3: 39.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ \begin{array}{l} t_0 := -\sqrt{2}\\ t_1 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;F \leq 3.3 \cdot 10^{-307}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot t_1\right) \cdot \left(F \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{t_1}\\ \mathbf{elif}\;F \leq 44000000000:\\ \;\;\;\;\sqrt{F \cdot \left(B + C\right)} \cdot \frac{t_0}{B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot t_0\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (sqrt 2.0))) (t_1 (fma B B (* A (* C -4.0)))))
   (if (<= F 3.3e-307)
     (/ (- (sqrt (* (* 2.0 t_1) (* F (+ C (+ A (hypot B (- A C)))))))) t_1)
     (if (<= F 44000000000.0)
       (* (sqrt (* F (+ B C))) (/ t_0 B))
       (* (sqrt (/ F B)) t_0)))))

C

B = abs(B);
double code(double A, double B, double C, double F) {
	double t_0 = -sqrt(2.0);
	double t_1 = fma(B, B, (A * (C * -4.0)));
	double tmp;
	if (F <= 3.3e-307) {
		tmp = -sqrt(((2.0 * t_1) * (F * (C + (A + hypot(B, (A - C))))))) / t_1;
	} else if (F <= 44000000000.0) {
		tmp = sqrt((F * (B + C))) * (t_0 / B);
	} else {
		tmp = sqrt((F / B)) * t_0;
	}
	return tmp;
}

Julia

B = abs(B)
function code(A, B, C, F)
	t_0 = Float64(-sqrt(2.0))
	t_1 = fma(B, B, Float64(A * Float64(C * -4.0)))
	tmp = 0.0
	if (F <= 3.3e-307)
		tmp = Float64(Float64(-sqrt(Float64(Float64(2.0 * t_1) * Float64(F * Float64(C + Float64(A + hypot(B, Float64(A - C)))))))) / t_1);
	elseif (F <= 44000000000.0)
		tmp = Float64(sqrt(Float64(F * Float64(B + C))) * Float64(t_0 / B));
	else
		tmp = Float64(sqrt(Float64(F / B)) * t_0);
	end
	return tmp
end

Wolfram

NOTE: B should be positive before calling this function
code[A_, B_, C_, F_] := Block[{t$95$0 = (-N[Sqrt[2.0], $MachinePrecision])}, Block[{t$95$1 = N[(B * B + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, 3.3e-307], N[((-N[Sqrt[N[(N[(2.0 * t$95$1), $MachinePrecision] * N[(F * N[(C + N[(A + N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision], If[LessEqual[F, 44000000000.0], N[(N[Sqrt[N[(F * N[(B + C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(t$95$0 / B), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(F / B), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if F < 3.3e-307

   1. Initial program 17.8%

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

   3. Simplified 17.8%

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

      1. unpow2 17.8%

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

      2. unpow2 17.8%

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

      3. hypot-udef 36.6%

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

      4. add-sqr-sqrt 36.7%

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

   5. Applied egg-rr 36.7%

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

   7. Applied egg-rr 37.1%

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

   if 3.3e-307 < F < 4.4e10

   1. Initial program 15.2%

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

   3. Simplified 20.0%

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

   4. Taylor expanded in B around inf 4.4%

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

      1. +-commutative 4.4%

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

   6. Simplified 4.4%

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

   7. Taylor expanded in A around 0 20.4%

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

      1. +-commutative 20.4%

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

   9. Simplified 20.4%

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

   if 4.4e10 < F

   1. Initial program 16.0%

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

   3. Simplified 16.9%

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

   4. Taylor expanded in C around 0 8.7%

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

   5. Taylor expanded in A around 0 20.2%

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

4. Final simplification 22.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 3.3 \cdot 10^{-307}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot \left(F \cdot \left(C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;F \leq 44000000000:\\ \;\;\;\;\sqrt{F \cdot \left(B + C\right)} \cdot \frac{-\sqrt{2}}{B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]

Alternative 4: 33.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\\ \mathbf{if}\;B \leq 1.55 \cdot 10^{-48}:\\ \;\;\;\;\frac{-\sqrt{\left(t_0 \cdot \left(2 \cdot F\right)\right) \cdot \left(-0.5 \cdot \left(B \cdot \left(B \cdot \frac{1}{C}\right)\right) + 2 \cdot A\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(B + C\right)} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (fma B B (* C (* A -4.0)))))
   (if (<= B 1.55e-48)
     (/
      (-
       (sqrt
        (* (* t_0 (* 2.0 F)) (+ (* -0.5 (* B (* B (/ 1.0 C)))) (* 2.0 A)))))
      t_0)
     (* (sqrt (* F (+ B C))) (/ (- (sqrt 2.0)) B)))))

C

B = abs(B);
double code(double A, double B, double C, double F) {
	double t_0 = fma(B, B, (C * (A * -4.0)));
	double tmp;
	if (B <= 1.55e-48) {
		tmp = -sqrt(((t_0 * (2.0 * F)) * ((-0.5 * (B * (B * (1.0 / C)))) + (2.0 * A)))) / t_0;
	} else {
		tmp = sqrt((F * (B + C))) * (-sqrt(2.0) / B);
	}
	return tmp;
}

Julia

B = abs(B)
function code(A, B, C, F)
	t_0 = fma(B, B, Float64(C * Float64(A * -4.0)))
	tmp = 0.0
	if (B <= 1.55e-48)
		tmp = Float64(Float64(-sqrt(Float64(Float64(t_0 * Float64(2.0 * F)) * Float64(Float64(-0.5 * Float64(B * Float64(B * Float64(1.0 / C)))) + Float64(2.0 * A))))) / t_0);
	else
		tmp = Float64(sqrt(Float64(F * Float64(B + C))) * Float64(Float64(-sqrt(2.0)) / B));
	end
	return tmp
end

Wolfram

NOTE: B should be positive before calling this function
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(B * B + N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B, 1.55e-48], N[((-N[Sqrt[N[(N[(t$95$0 * N[(2.0 * F), $MachinePrecision]), $MachinePrecision] * N[(N[(-0.5 * N[(B * N[(B * N[(1.0 / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(2.0 * A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], N[(N[Sqrt[N[(F * N[(B + C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[((-N[Sqrt[2.0], $MachinePrecision]) / B), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if B < 1.55000000000000008e-48

   1. Initial program 18.3%

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

   3. Simplified 22.7%

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

   4. Taylor expanded in C around -inf 18.8%

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

      1. div-inv 18.8%

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

      2. unpow2 18.8%

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

      3. associate-*l* 18.8%

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

   6. Applied egg-rr 18.8%

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

   if 1.55000000000000008e-48 < B

   1. Initial program 10.6%

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

   3. Simplified 18.2%

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

   4. Taylor expanded in B around inf 11.9%

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

      1. +-commutative 11.9%

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

   6. Simplified 11.9%

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

   7. Taylor expanded in A around 0 46.5%

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

      1. +-commutative 46.5%

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

   9. Simplified 46.5%

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

4. Final simplification 27.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.55 \cdot 10^{-48}:\\ \;\;\;\;\frac{-\sqrt{\left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(-0.5 \cdot \left(B \cdot \left(B \cdot \frac{1}{C}\right)\right) + 2 \cdot A\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(B + C\right)} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 5: 37.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)\\ t_1 := -\sqrt{2}\\ \mathbf{if}\;F \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{-\sqrt{\left(t_0 \cdot \left(2 \cdot F\right)\right) \cdot \left(2 \cdot A\right)}}{t_0}\\ \mathbf{elif}\;F \leq 340000000:\\ \;\;\;\;\sqrt{F \cdot \left(B + C\right)} \cdot \frac{t_1}{B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot t_1\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (fma B B (* C (* A -4.0)))) (t_1 (- (sqrt 2.0))))
   (if (<= F -5e-310)
     (/ (- (sqrt (* (* t_0 (* 2.0 F)) (* 2.0 A)))) t_0)
     (if (<= F 340000000.0)
       (* (sqrt (* F (+ B C))) (/ t_1 B))
       (* (sqrt (/ F B)) t_1)))))

C

B = abs(B);
double code(double A, double B, double C, double F) {
	double t_0 = fma(B, B, (C * (A * -4.0)));
	double t_1 = -sqrt(2.0);
	double tmp;
	if (F <= -5e-310) {
		tmp = -sqrt(((t_0 * (2.0 * F)) * (2.0 * A))) / t_0;
	} else if (F <= 340000000.0) {
		tmp = sqrt((F * (B + C))) * (t_1 / B);
	} else {
		tmp = sqrt((F / B)) * t_1;
	}
	return tmp;
}

Julia

B = abs(B)
function code(A, B, C, F)
	t_0 = fma(B, B, Float64(C * Float64(A * -4.0)))
	t_1 = Float64(-sqrt(2.0))
	tmp = 0.0
	if (F <= -5e-310)
		tmp = Float64(Float64(-sqrt(Float64(Float64(t_0 * Float64(2.0 * F)) * Float64(2.0 * A)))) / t_0);
	elseif (F <= 340000000.0)
		tmp = Float64(sqrt(Float64(F * Float64(B + C))) * Float64(t_1 / B));
	else
		tmp = Float64(sqrt(Float64(F / B)) * t_1);
	end
	return tmp
end

Wolfram

NOTE: B should be positive before calling this function
code[A_, B_, C_, F_] := Block[{t$95$0 = N[(B * B + N[(C * N[(A * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = (-N[Sqrt[2.0], $MachinePrecision])}, If[LessEqual[F, -5e-310], N[((-N[Sqrt[N[(N[(t$95$0 * N[(2.0 * F), $MachinePrecision]), $MachinePrecision] * N[(2.0 * A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], If[LessEqual[F, 340000000.0], N[(N[Sqrt[N[(F * N[(B + C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(t$95$1 / B), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(F / B), $MachinePrecision]], $MachinePrecision] * t$95$1), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if F < -4.999999999999985e-310

   1. Initial program 18.9%

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

   3. Simplified 35.5%

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

   4. Taylor expanded in A around inf 22.8%

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

      1. *-commutative 22.8%

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

   6. Simplified 22.8%

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

   if -4.999999999999985e-310 < F < 3.4e8

   1. Initial program 15.0%

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

   3. Simplified 20.6%

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

   4. Taylor expanded in B around inf 4.5%

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

      1. +-commutative 4.5%

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

   6. Simplified 4.5%

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

   7. Taylor expanded in A around 0 20.2%

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

      1. +-commutative 20.2%

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

   9. Simplified 20.2%

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

   if 3.4e8 < F

   1. Initial program 16.0%

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

   3. Simplified 16.9%

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

   4. Taylor expanded in C around 0 8.7%

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

   5. Taylor expanded in A around 0 20.2%

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

4. Final simplification 20.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{-\sqrt{\left(\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right) \cdot \left(2 \cdot F\right)\right) \cdot \left(2 \cdot A\right)}}{\mathsf{fma}\left(B, B, C \cdot \left(A \cdot -4\right)\right)}\\ \mathbf{elif}\;F \leq 340000000:\\ \;\;\;\;\sqrt{F \cdot \left(B + C\right)} \cdot \frac{-\sqrt{2}}{B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]

Alternative 6: 34.3% accurate, 3.0× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ \begin{array}{l} t_0 := -\sqrt{2}\\ \mathbf{if}\;F \leq 4.6 \cdot 10^{-29}:\\ \;\;\;\;\sqrt{F \cdot \left(B + A\right)} \cdot \frac{t_0}{B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot t_0\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (sqrt 2.0))))
   (if (<= F 4.6e-29)
     (* (sqrt (* F (+ B A))) (/ t_0 B))
     (* (sqrt (/ F B)) t_0))))

C

B = abs(B);
double code(double A, double B, double C, double F) {
	double t_0 = -sqrt(2.0);
	double tmp;
	if (F <= 4.6e-29) {
		tmp = sqrt((F * (B + A))) * (t_0 / B);
	} else {
		tmp = sqrt((F / B)) * t_0;
	}
	return tmp;
}

Fortran

NOTE: B should be positive before calling this function
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
    real(8) :: tmp
    t_0 = -sqrt(2.0d0)
    if (f <= 4.6d-29) then
        tmp = sqrt((f * (b + a))) * (t_0 / b)
    else
        tmp = sqrt((f / b)) * t_0
    end if
    code = tmp
end function

Java

B = Math.abs(B);
public static double code(double A, double B, double C, double F) {
	double t_0 = -Math.sqrt(2.0);
	double tmp;
	if (F <= 4.6e-29) {
		tmp = Math.sqrt((F * (B + A))) * (t_0 / B);
	} else {
		tmp = Math.sqrt((F / B)) * t_0;
	}
	return tmp;
}

Python

B = abs(B)
def code(A, B, C, F):
	t_0 = -math.sqrt(2.0)
	tmp = 0
	if F <= 4.6e-29:
		tmp = math.sqrt((F * (B + A))) * (t_0 / B)
	else:
		tmp = math.sqrt((F / B)) * t_0
	return tmp

Julia

B = abs(B)
function code(A, B, C, F)
	t_0 = Float64(-sqrt(2.0))
	tmp = 0.0
	if (F <= 4.6e-29)
		tmp = Float64(sqrt(Float64(F * Float64(B + A))) * Float64(t_0 / B));
	else
		tmp = Float64(sqrt(Float64(F / B)) * t_0);
	end
	return tmp
end

MATLAB

B = abs(B)
function tmp_2 = code(A, B, C, F)
	t_0 = -sqrt(2.0);
	tmp = 0.0;
	if (F <= 4.6e-29)
		tmp = sqrt((F * (B + A))) * (t_0 / B);
	else
		tmp = sqrt((F / B)) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: B should be positive before calling this function
code[A_, B_, C_, F_] := Block[{t$95$0 = (-N[Sqrt[2.0], $MachinePrecision])}, If[LessEqual[F, 4.6e-29], N[(N[Sqrt[N[(F * N[(B + A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(t$95$0 / B), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(F / B), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if F < 4.59999999999999982e-29

   1. Initial program 15.4%

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

   3. Simplified 23.4%

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

   4. Taylor expanded in B around inf 4.9%

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

      1. +-commutative 4.9%

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

   6. Simplified 4.9%

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

   7. Taylor expanded in C around 0 15.2%

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

      1. +-commutative 15.2%

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

   9. Simplified 15.2%

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

   if 4.59999999999999982e-29 < F

   1. Initial program 16.4%

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

   3. Simplified 17.1%

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

   4. Taylor expanded in C around 0 9.1%

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

   5. Taylor expanded in A around 0 20.5%

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

4. Final simplification 17.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 4.6 \cdot 10^{-29}:\\ \;\;\;\;\sqrt{F \cdot \left(B + A\right)} \cdot \frac{-\sqrt{2}}{B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]

Alternative 7: 34.4% accurate, 3.0× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ \begin{array}{l} t_0 := -\sqrt{2}\\ \mathbf{if}\;F \leq 320000000:\\ \;\;\;\;\sqrt{F \cdot \left(B + C\right)} \cdot \frac{t_0}{B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot t_0\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (- (sqrt 2.0))))
   (if (<= F 320000000.0)
     (* (sqrt (* F (+ B C))) (/ t_0 B))
     (* (sqrt (/ F B)) t_0))))

C

B = abs(B);
double code(double A, double B, double C, double F) {
	double t_0 = -sqrt(2.0);
	double tmp;
	if (F <= 320000000.0) {
		tmp = sqrt((F * (B + C))) * (t_0 / B);
	} else {
		tmp = sqrt((F / B)) * t_0;
	}
	return tmp;
}

Fortran

NOTE: B should be positive before calling this function
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
    real(8) :: tmp
    t_0 = -sqrt(2.0d0)
    if (f <= 320000000.0d0) then
        tmp = sqrt((f * (b + c))) * (t_0 / b)
    else
        tmp = sqrt((f / b)) * t_0
    end if
    code = tmp
end function

Java

B = Math.abs(B);
public static double code(double A, double B, double C, double F) {
	double t_0 = -Math.sqrt(2.0);
	double tmp;
	if (F <= 320000000.0) {
		tmp = Math.sqrt((F * (B + C))) * (t_0 / B);
	} else {
		tmp = Math.sqrt((F / B)) * t_0;
	}
	return tmp;
}

Python

B = abs(B)
def code(A, B, C, F):
	t_0 = -math.sqrt(2.0)
	tmp = 0
	if F <= 320000000.0:
		tmp = math.sqrt((F * (B + C))) * (t_0 / B)
	else:
		tmp = math.sqrt((F / B)) * t_0
	return tmp

Julia

B = abs(B)
function code(A, B, C, F)
	t_0 = Float64(-sqrt(2.0))
	tmp = 0.0
	if (F <= 320000000.0)
		tmp = Float64(sqrt(Float64(F * Float64(B + C))) * Float64(t_0 / B));
	else
		tmp = Float64(sqrt(Float64(F / B)) * t_0);
	end
	return tmp
end

MATLAB

B = abs(B)
function tmp_2 = code(A, B, C, F)
	t_0 = -sqrt(2.0);
	tmp = 0.0;
	if (F <= 320000000.0)
		tmp = sqrt((F * (B + C))) * (t_0 / B);
	else
		tmp = sqrt((F / B)) * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: B should be positive before calling this function
code[A_, B_, C_, F_] := Block[{t$95$0 = (-N[Sqrt[2.0], $MachinePrecision])}, If[LessEqual[F, 320000000.0], N[(N[Sqrt[N[(F * N[(B + C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(t$95$0 / B), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(F / B), $MachinePrecision]], $MachinePrecision] * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if F < 3.2e8

   1. Initial program 15.7%

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

   3. Simplified 23.5%

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

   4. Taylor expanded in B around inf 5.2%

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

      1. +-commutative 5.2%

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

   6. Simplified 5.2%

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

   7. Taylor expanded in A around 0 16.5%

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

      1. +-commutative 16.5%

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

   9. Simplified 16.5%

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

   if 3.2e8 < F

   1. Initial program 16.0%

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

   3. Simplified 16.9%

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

   4. Taylor expanded in C around 0 8.7%

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

   5. Taylor expanded in A around 0 20.2%

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

4. Final simplification 18.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 320000000:\\ \;\;\;\;\sqrt{F \cdot \left(B + C\right)} \cdot \frac{-\sqrt{2}}{B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]

Alternative 8: 27.4% accurate, 3.1× speedup

Math

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

FPCore

NOTE: B should be positive before calling this function
(FPCore (A B C F) :precision binary64 (* (sqrt (/ F B)) (- (sqrt 2.0))))

C

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

Fortran

NOTE: B should be positive before calling this function
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
    code = sqrt((f / b)) * -sqrt(2.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: B should be positive before calling this function
code[A_, B_, C_, F_] := N[(N[Sqrt[N[(F / B), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision]

Derivation

1. Initial program 15.9%

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

3. Simplified 16.5%

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

4. Taylor expanded in C around 0 6.8%

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

5. Taylor expanded in A around 0 13.8%

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

6. Final simplification 13.8%

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

Reproduce

herbie shell --seed 2023305 
(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))))