ABCF->ab-angle a

Percentage Accurate: 18.4% → 58.2%

Time: 34.1s

Alternatives: 18

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: 58.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ \begin{array}{l} t_0 := \left(A + C\right) + \mathsf{hypot}\left(A - C, B\right)\\ t_1 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ t_2 := F \cdot t_1\\ t_3 := {B}^{2} - \left(4 \cdot A\right) \cdot C\\ t_4 := \frac{-\sqrt{\left(2 \cdot \left(t_3 \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{t_3}\\ \mathbf{if}\;t_4 \leq -5 \cdot 10^{-222}:\\ \;\;\;\;\frac{\sqrt{t_0} \cdot \left({\left(2 \cdot t_1\right)}^{0.5} \cdot \left(-\sqrt{F}\right)\right)}{t_1}\\ \mathbf{elif}\;t_4 \leq 10^{-56}:\\ \;\;\;\;\frac{-\sqrt{t_2 \cdot \left(2 \cdot \left(A + \left(A + -0.5 \cdot \frac{{B}^{2}}{C}\right)\right)\right)}}{t_1}\\ \mathbf{elif}\;t_4 \leq \infty:\\ \;\;\;\;\frac{\sqrt{2 \cdot t_0} \cdot \left(-\sqrt{t_2}\right)}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{F} \cdot \sqrt{A + \mathsf{hypot}\left(B, A\right)}\right) \cdot \left(\sqrt{2} \cdot \frac{-1}{B}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (+ (+ A C) (hypot (- A C) B)))
        (t_1 (fma B B (* A (* C -4.0))))
        (t_2 (* F t_1))
        (t_3 (- (pow B 2.0) (* (* 4.0 A) C)))
        (t_4
         (/
          (-
           (sqrt
            (*
             (* 2.0 (* t_3 F))
             (+ (+ A C) (sqrt (+ (pow B 2.0) (pow (- A C) 2.0)))))))
          t_3)))
   (if (<= t_4 -5e-222)
     (/ (* (sqrt t_0) (* (pow (* 2.0 t_1) 0.5) (- (sqrt F)))) t_1)
     (if (<= t_4 1e-56)
       (/
        (- (sqrt (* t_2 (* 2.0 (+ A (+ A (* -0.5 (/ (pow B 2.0) C))))))))
        t_1)
       (if (<= t_4 INFINITY)
         (/ (* (sqrt (* 2.0 t_0)) (- (sqrt t_2))) t_1)
         (*
          (* (sqrt F) (sqrt (+ A (hypot B A))))
          (* (sqrt 2.0) (/ -1.0 B))))))))

C

B = abs(B);
double code(double A, double B, double C, double F) {
	double t_0 = (A + C) + hypot((A - C), B);
	double t_1 = fma(B, B, (A * (C * -4.0)));
	double t_2 = F * t_1;
	double t_3 = pow(B, 2.0) - ((4.0 * A) * C);
	double t_4 = -sqrt(((2.0 * (t_3 * F)) * ((A + C) + sqrt((pow(B, 2.0) + pow((A - C), 2.0)))))) / t_3;
	double tmp;
	if (t_4 <= -5e-222) {
		tmp = (sqrt(t_0) * (pow((2.0 * t_1), 0.5) * -sqrt(F))) / t_1;
	} else if (t_4 <= 1e-56) {
		tmp = -sqrt((t_2 * (2.0 * (A + (A + (-0.5 * (pow(B, 2.0) / C))))))) / t_1;
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = (sqrt((2.0 * t_0)) * -sqrt(t_2)) / t_1;
	} else {
		tmp = (sqrt(F) * sqrt((A + hypot(B, A)))) * (sqrt(2.0) * (-1.0 / B));
	}
	return tmp;
}

Julia

B = abs(B)
function code(A, B, C, F)
	t_0 = Float64(Float64(A + C) + hypot(Float64(A - C), B))
	t_1 = fma(B, B, Float64(A * Float64(C * -4.0)))
	t_2 = Float64(F * t_1)
	t_3 = Float64((B ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_4 = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(t_3 * F)) * Float64(Float64(A + C) + sqrt(Float64((B ^ 2.0) + (Float64(A - C) ^ 2.0))))))) / t_3)
	tmp = 0.0
	if (t_4 <= -5e-222)
		tmp = Float64(Float64(sqrt(t_0) * Float64((Float64(2.0 * t_1) ^ 0.5) * Float64(-sqrt(F)))) / t_1);
	elseif (t_4 <= 1e-56)
		tmp = Float64(Float64(-sqrt(Float64(t_2 * Float64(2.0 * Float64(A + Float64(A + Float64(-0.5 * Float64((B ^ 2.0) / C)))))))) / t_1);
	elseif (t_4 <= Inf)
		tmp = Float64(Float64(sqrt(Float64(2.0 * t_0)) * Float64(-sqrt(t_2))) / t_1);
	else
		tmp = Float64(Float64(sqrt(F) * sqrt(Float64(A + hypot(B, A)))) * Float64(sqrt(2.0) * Float64(-1.0 / 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[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(B * B + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(F * t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(N[Power[B, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[((-N[Sqrt[N[(N[(2.0 * N[(t$95$3 * 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$3), $MachinePrecision]}, If[LessEqual[t$95$4, -5e-222], N[(N[(N[Sqrt[t$95$0], $MachinePrecision] * N[(N[Power[N[(2.0 * t$95$1), $MachinePrecision], 0.5], $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[t$95$4, 1e-56], N[((-N[Sqrt[N[(t$95$2 * N[(2.0 * N[(A + N[(A + N[(-0.5 * N[(N[Power[B, 2.0], $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[(N[(N[Sqrt[N[(2.0 * t$95$0), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[t$95$2], $MachinePrecision])), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(A + N[Sqrt[B ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(-1.0 / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))) < -5.00000000000000008e-222

   1. Initial program 42.0%

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

   3. Simplified 52.7%

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

      1. pow1/2 52.7%

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

      2. associate-*r* 52.7%

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

      3. unpow-prod-down 63.9%

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

      4. *-commutative 63.9%

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

      5. pow1/2 63.9%

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

      6. associate-+r+ 62.9%

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

      7. hypot-udef 49.6%

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

      8. unpow2 49.6%

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

      9. unpow2 49.6%

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

      10. +-commutative 49.6%

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

      11. +-commutative 49.6%

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

   5. Applied egg-rr 62.9%

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

      1. associate-*l* 62.9%

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

      2. unpow-prod-down 77.1%

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

      3. pow1/2 77.1%

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

   7. Applied egg-rr 77.1%

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

   if -5.00000000000000008e-222 < (/.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))) < 1e-56

   1. Initial program 10.0%

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

   3. Simplified 15.0%

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

   4. Taylor expanded in C around -inf 37.8%

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

   if 1e-56 < (/.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 50.1%

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

   3. Simplified 63.9%

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

      1. sqrt-prod 91.0%

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

      2. *-commutative 91.0%

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

      3. associate-+r+ 91.0%

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

      4. hypot-udef 56.9%

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

      5. unpow2 56.9%

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

      6. unpow2 56.9%

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

      7. +-commutative 56.9%

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

      8. +-commutative 56.9%

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

      9. unpow2 56.9%

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

      10. unpow2 56.9%

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

      11. hypot-def 91.0%

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

   5. Applied egg-rr 91.0%

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

      1. +-commutative 91.0%

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

   7. Simplified 91.0%

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

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

   1. Initial program 0.0%

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

   2. Taylor expanded in C around 0 1.7%

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

      1. mul-1-neg 1.7%

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

      2. distribute-rgt-neg-in 1.7%

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

      3. +-commutative 1.7%

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

      4. unpow2 1.7%

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

      5. unpow2 1.7%

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

      6. hypot-def 14.7%

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

   4. Simplified 14.7%

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

      1. pow1/2 14.7%

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

      2. *-commutative 14.7%

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

      3. unpow-prod-down 21.9%

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

      4. pow1/2 21.9%

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

      5. pow1/2 21.9%

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

   6. Applied egg-rr 21.9%

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

      1. div-inv 21.9%

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

   8. Applied egg-rr 21.9%

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

4. Final simplification 46.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \leq -5 \cdot 10^{-222}:\\ \;\;\;\;\frac{\sqrt{\left(A + C\right) + \mathsf{hypot}\left(A - C, B\right)} \cdot \left({\left(2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)}^{0.5} \cdot \left(-\sqrt{F}\right)\right)}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \leq 10^{-56}:\\ \;\;\;\;\frac{-\sqrt{\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot \left(2 \cdot \left(A + \left(A + -0.5 \cdot \frac{{B}^{2}}{C}\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C} \leq \infty:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(A - C, B\right)\right)} \cdot \left(-\sqrt{F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right)}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{F} \cdot \sqrt{A + \mathsf{hypot}\left(B, A\right)}\right) \cdot \left(\sqrt{2} \cdot \frac{-1}{B}\right)\\ \end{array} \]

Alternative 2: 45.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ \begin{array}{l} t_0 := A + \mathsf{hypot}\left(B, A\right)\\ t_1 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ t_2 := {B}^{2} - \left(4 \cdot A\right) \cdot C\\ t_3 := 2 \cdot \left(t_2 \cdot F\right)\\ \mathbf{if}\;{B}^{2} \leq 2 \cdot 10^{-280}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot t_1\right)\right) \cdot \left(2 \cdot A\right)}}{t_1}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{-167}:\\ \;\;\;\;\frac{-\sqrt{t_3 \cdot \left(2 \cdot C\right)}}{t_2}\\ \mathbf{elif}\;{B}^{2} \leq 10^{-62}:\\ \;\;\;\;\frac{-\sqrt{t_3 \cdot t_0}}{t_2}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{F} \cdot \sqrt{t_0}\right) \cdot \left(\sqrt{2} \cdot \frac{-1}{B}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
(FPCore (A B C F)
 :precision binary64
 (let* ((t_0 (+ A (hypot B A)))
        (t_1 (fma B B (* A (* C -4.0))))
        (t_2 (- (pow B 2.0) (* (* 4.0 A) C)))
        (t_3 (* 2.0 (* t_2 F))))
   (if (<= (pow B 2.0) 2e-280)
     (/ (- (sqrt (* (* 2.0 (* F t_1)) (* 2.0 A)))) t_1)
     (if (<= (pow B 2.0) 2e-167)
       (/ (- (sqrt (* t_3 (* 2.0 C)))) t_2)
       (if (<= (pow B 2.0) 1e-62)
         (/ (- (sqrt (* t_3 t_0))) t_2)
         (* (* (sqrt F) (sqrt t_0)) (* (sqrt 2.0) (/ -1.0 B))))))))

C

B = abs(B);
double code(double A, double B, double C, double F) {
	double t_0 = A + hypot(B, A);
	double t_1 = fma(B, B, (A * (C * -4.0)));
	double t_2 = pow(B, 2.0) - ((4.0 * A) * C);
	double t_3 = 2.0 * (t_2 * F);
	double tmp;
	if (pow(B, 2.0) <= 2e-280) {
		tmp = -sqrt(((2.0 * (F * t_1)) * (2.0 * A))) / t_1;
	} else if (pow(B, 2.0) <= 2e-167) {
		tmp = -sqrt((t_3 * (2.0 * C))) / t_2;
	} else if (pow(B, 2.0) <= 1e-62) {
		tmp = -sqrt((t_3 * t_0)) / t_2;
	} else {
		tmp = (sqrt(F) * sqrt(t_0)) * (sqrt(2.0) * (-1.0 / B));
	}
	return tmp;
}

Julia

B = abs(B)
function code(A, B, C, F)
	t_0 = Float64(A + hypot(B, A))
	t_1 = fma(B, B, Float64(A * Float64(C * -4.0)))
	t_2 = Float64((B ^ 2.0) - Float64(Float64(4.0 * A) * C))
	t_3 = Float64(2.0 * Float64(t_2 * F))
	tmp = 0.0
	if ((B ^ 2.0) <= 2e-280)
		tmp = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(F * t_1)) * Float64(2.0 * A)))) / t_1);
	elseif ((B ^ 2.0) <= 2e-167)
		tmp = Float64(Float64(-sqrt(Float64(t_3 * Float64(2.0 * C)))) / t_2);
	elseif ((B ^ 2.0) <= 1e-62)
		tmp = Float64(Float64(-sqrt(Float64(t_3 * t_0))) / t_2);
	else
		tmp = Float64(Float64(sqrt(F) * sqrt(t_0)) * Float64(sqrt(2.0) * Float64(-1.0 / 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[(A + N[Sqrt[B ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(B * B + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[B, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(2.0 * N[(t$95$2 * F), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B, 2.0], $MachinePrecision], 2e-280], N[((-N[Sqrt[N[(N[(2.0 * N[(F * t$95$1), $MachinePrecision]), $MachinePrecision] * N[(2.0 * A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision], If[LessEqual[N[Power[B, 2.0], $MachinePrecision], 2e-167], N[((-N[Sqrt[N[(t$95$3 * N[(2.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$2), $MachinePrecision], If[LessEqual[N[Power[B, 2.0], $MachinePrecision], 1e-62], N[((-N[Sqrt[N[(t$95$3 * t$95$0), $MachinePrecision]], $MachinePrecision]) / t$95$2), $MachinePrecision], N[(N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[t$95$0], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(-1.0 / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (pow.f64 B 2) < 1.9999999999999999e-280

   1. Initial program 17.3%

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

   3. Simplified 28.5%

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

      1. pow1/2 28.5%

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

      2. associate-*r* 28.5%

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

      3. unpow-prod-down 30.4%

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

      4. *-commutative 30.4%

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

      5. pow1/2 30.4%

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

      6. associate-+r+ 28.9%

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

      7. hypot-udef 20.0%

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

      8. unpow2 20.0%

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

      9. unpow2 20.0%

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

      10. +-commutative 20.0%

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

      11. +-commutative 20.0%

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

   5. Applied egg-rr 28.9%

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

   6. Taylor expanded in A around inf 22.5%

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

      1. expm1-log1p-u 21.9%

         \[\leadsto \frac{-\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot 2\right)}^{0.5} \cdot \sqrt{2 \cdot A}\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      2. expm1-udef 9.5%

         \[\leadsto \frac{-\color{blue}{\left(e^{\mathsf{log1p}\left({\left(\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot 2\right)}^{0.5} \cdot \sqrt{2 \cdot A}\right)} - 1\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      3. pow1/2 9.5%

         \[\leadsto \frac{-\left(e^{\mathsf{log1p}\left({\left(\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot 2\right)}^{0.5} \cdot \color{blue}{{\left(2 \cdot A\right)}^{0.5}}\right)} - 1\right)}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      4. pow-prod-down 11.4%

         \[\leadsto \frac{-\left(e^{\mathsf{log1p}\left(\color{blue}{{\left(\left(\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot 2\right) \cdot \left(2 \cdot A\right)\right)}^{0.5}}\right)} - 1\right)}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      5. *-commutative 11.4%

         \[\leadsto \frac{-\left(e^{\mathsf{log1p}\left({\left(\color{blue}{\left(2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)\right)} \cdot \left(2 \cdot A\right)\right)}^{0.5}\right)} - 1\right)}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   8. Applied egg-rr 11.4%

      \[\leadsto \frac{-\color{blue}{\left(e^{\mathsf{log1p}\left({\left(\left(2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)\right) \cdot \left(2 \cdot A\right)\right)}^{0.5}\right)} - 1\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
   9. Step-by-step derivation

      1. expm1-def 25.3%

         \[\leadsto \frac{-\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\left(2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)\right) \cdot \left(2 \cdot A\right)\right)}^{0.5}\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      2. expm1-log1p 26.1%

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

      3. unpow1/2 26.1%

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

   10. Simplified 26.1%

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

   if 1.9999999999999999e-280 < (pow.f64 B 2) < 2e-167

   1. Initial program 27.4%

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

   2. Taylor expanded in A around -inf 30.3%

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

   if 2e-167 < (pow.f64 B 2) < 1e-62

   1. Initial program 40.0%

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

   2. Taylor expanded in C around 0 36.5%

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

      1. +-commutative 36.5%

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

      2. unpow2 36.5%

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

      3. unpow2 36.5%

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

      4. hypot-def 41.3%

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

   4. Simplified 41.3%

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

   if 1e-62 < (pow.f64 B 2)

   1. Initial program 13.2%

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

   2. Taylor expanded in C around 0 10.2%

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

      1. mul-1-neg 10.2%

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

      2. distribute-rgt-neg-in 10.2%

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

      3. +-commutative 10.2%

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

      4. unpow2 10.2%

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

      5. unpow2 10.2%

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

      6. hypot-def 21.4%

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

   4. Simplified 21.4%

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

      1. pow1/2 21.4%

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

      2. *-commutative 21.4%

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

      3. unpow-prod-down 32.1%

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

      4. pow1/2 32.1%

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

      5. pow1/2 32.1%

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

   6. Applied egg-rr 32.1%

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

      1. div-inv 32.0%

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

   8. Applied egg-rr 32.0%

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

4. Final simplification 31.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 2 \cdot 10^{-280}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)\right) \cdot \left(2 \cdot A\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{-167}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\\ \mathbf{elif}\;{B}^{2} \leq 10^{-62}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \mathsf{hypot}\left(B, A\right)\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{F} \cdot \sqrt{A + \mathsf{hypot}\left(B, A\right)}\right) \cdot \left(\sqrt{2} \cdot \frac{-1}{B}\right)\\ \end{array} \]

Alternative 3: 44.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := {B}^{2} - \left(4 \cdot A\right) \cdot C\\ \mathbf{if}\;{B}^{2} \leq 2 \cdot 10^{-280}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot t_0\right)\right) \cdot \left(2 \cdot A\right)}}{t_0}\\ \mathbf{elif}\;{B}^{2} \leq 10^{-264}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(t_1 \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{t_1}\\ \mathbf{elif}\;{B}^{2} \leq 4 \cdot 10^{-221}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(2 \cdot t_0\right)} \cdot \left(-\sqrt{2 \cdot A}\right)}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{F} \cdot \sqrt{A + \mathsf{hypot}\left(B, A\right)}\right) \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \end{array} \]

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 (- (pow B 2.0) (* (* 4.0 A) C))))
   (if (<= (pow B 2.0) 2e-280)
     (/ (- (sqrt (* (* 2.0 (* F t_0)) (* 2.0 A)))) t_0)
     (if (<= (pow B 2.0) 1e-264)
       (/ (- (sqrt (* (* 2.0 (* t_1 F)) (* 2.0 C)))) t_1)
       (if (<= (pow B 2.0) 4e-221)
         (/ (* (sqrt (* F (* 2.0 t_0))) (- (sqrt (* 2.0 A)))) t_0)
         (* (* (sqrt F) (sqrt (+ A (hypot B A)))) (/ (- (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 = pow(B, 2.0) - ((4.0 * A) * C);
	double tmp;
	if (pow(B, 2.0) <= 2e-280) {
		tmp = -sqrt(((2.0 * (F * t_0)) * (2.0 * A))) / t_0;
	} else if (pow(B, 2.0) <= 1e-264) {
		tmp = -sqrt(((2.0 * (t_1 * F)) * (2.0 * C))) / t_1;
	} else if (pow(B, 2.0) <= 4e-221) {
		tmp = (sqrt((F * (2.0 * t_0))) * -sqrt((2.0 * A))) / t_0;
	} else {
		tmp = (sqrt(F) * sqrt((A + hypot(B, A)))) * (-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 = Float64((B ^ 2.0) - Float64(Float64(4.0 * A) * C))
	tmp = 0.0
	if ((B ^ 2.0) <= 2e-280)
		tmp = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(F * t_0)) * Float64(2.0 * A)))) / t_0);
	elseif ((B ^ 2.0) <= 1e-264)
		tmp = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(t_1 * F)) * Float64(2.0 * C)))) / t_1);
	elseif ((B ^ 2.0) <= 4e-221)
		tmp = Float64(Float64(sqrt(Float64(F * Float64(2.0 * t_0))) * Float64(-sqrt(Float64(2.0 * A)))) / t_0);
	else
		tmp = Float64(Float64(sqrt(F) * sqrt(Float64(A + hypot(B, A)))) * 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[(N[Power[B, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B, 2.0], $MachinePrecision], 2e-280], N[((-N[Sqrt[N[(N[(2.0 * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision] * N[(2.0 * A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], If[LessEqual[N[Power[B, 2.0], $MachinePrecision], 1e-264], N[((-N[Sqrt[N[(N[(2.0 * N[(t$95$1 * F), $MachinePrecision]), $MachinePrecision] * N[(2.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$1), $MachinePrecision], If[LessEqual[N[Power[B, 2.0], $MachinePrecision], 4e-221], N[(N[(N[Sqrt[N[(F * N[(2.0 * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(2.0 * A), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(A + N[Sqrt[B ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[((-N[Sqrt[2.0], $MachinePrecision]) / B), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (pow.f64 B 2) < 1.9999999999999999e-280

   1. Initial program 17.3%

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

   3. Simplified 28.5%

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

      1. pow1/2 28.5%

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

      2. associate-*r* 28.5%

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

      3. unpow-prod-down 30.4%

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

      4. *-commutative 30.4%

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

      5. pow1/2 30.4%

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

      6. associate-+r+ 28.9%

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

      7. hypot-udef 20.0%

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

      8. unpow2 20.0%

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

      9. unpow2 20.0%

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

      10. +-commutative 20.0%

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

      11. +-commutative 20.0%

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

   5. Applied egg-rr 28.9%

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

   6. Taylor expanded in A around inf 22.5%

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

      1. expm1-log1p-u 21.9%

         \[\leadsto \frac{-\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot 2\right)}^{0.5} \cdot \sqrt{2 \cdot A}\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      2. expm1-udef 9.5%

         \[\leadsto \frac{-\color{blue}{\left(e^{\mathsf{log1p}\left({\left(\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot 2\right)}^{0.5} \cdot \sqrt{2 \cdot A}\right)} - 1\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      3. pow1/2 9.5%

         \[\leadsto \frac{-\left(e^{\mathsf{log1p}\left({\left(\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot 2\right)}^{0.5} \cdot \color{blue}{{\left(2 \cdot A\right)}^{0.5}}\right)} - 1\right)}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      4. pow-prod-down 11.4%

         \[\leadsto \frac{-\left(e^{\mathsf{log1p}\left(\color{blue}{{\left(\left(\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot 2\right) \cdot \left(2 \cdot A\right)\right)}^{0.5}}\right)} - 1\right)}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      5. *-commutative 11.4%

         \[\leadsto \frac{-\left(e^{\mathsf{log1p}\left({\left(\color{blue}{\left(2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)\right)} \cdot \left(2 \cdot A\right)\right)}^{0.5}\right)} - 1\right)}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   8. Applied egg-rr 11.4%

      \[\leadsto \frac{-\color{blue}{\left(e^{\mathsf{log1p}\left({\left(\left(2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)\right) \cdot \left(2 \cdot A\right)\right)}^{0.5}\right)} - 1\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
   9. Step-by-step derivation

      1. expm1-def 25.3%

         \[\leadsto \frac{-\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\left(2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)\right) \cdot \left(2 \cdot A\right)\right)}^{0.5}\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      2. expm1-log1p 26.1%

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

      3. unpow1/2 26.1%

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

   10. Simplified 26.1%

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

   if 1.9999999999999999e-280 < (pow.f64 B 2) < 1e-264

   1. Initial program 30.3%

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

   2. Taylor expanded in A around -inf 58.4%

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

   if 1e-264 < (pow.f64 B 2) < 4.00000000000000007e-221

   1. Initial program 30.7%

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

   3. Simplified 34.3%

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

      1. pow1/2 34.3%

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

      2. associate-*r* 34.3%

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

      3. unpow-prod-down 39.9%

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

      4. *-commutative 39.9%

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

      5. pow1/2 39.9%

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

      6. associate-+r+ 37.8%

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

      7. hypot-udef 29.9%

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

      8. unpow2 29.9%

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

      9. unpow2 29.9%

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

      10. +-commutative 29.9%

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

      11. +-commutative 29.9%

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

   5. Applied egg-rr 37.8%

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

   6. Taylor expanded in A around inf 37.4%

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

      1. unpow1/2 37.4%

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

      2. associate-*l* 37.5%

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

   8. Applied egg-rr 37.5%

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

   if 4.00000000000000007e-221 < (pow.f64 B 2)

   1. Initial program 17.7%

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

   2. Taylor expanded in C around 0 10.8%

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

      1. mul-1-neg 10.8%

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

      2. distribute-rgt-neg-in 10.8%

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

      3. +-commutative 10.8%

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

      4. unpow2 10.8%

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

      5. unpow2 10.8%

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

      6. hypot-def 19.6%

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

   4. Simplified 19.6%

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

      1. pow1/2 19.6%

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

      2. *-commutative 19.6%

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

      3. unpow-prod-down 27.9%

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

      4. pow1/2 27.9%

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

      5. pow1/2 27.9%

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

   6. Applied egg-rr 27.9%

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

4. Final simplification 28.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 2 \cdot 10^{-280}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)\right) \cdot \left(2 \cdot A\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;{B}^{2} \leq 10^{-264}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\\ \mathbf{elif}\;{B}^{2} \leq 4 \cdot 10^{-221}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)} \cdot \left(-\sqrt{2 \cdot A}\right)}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{F} \cdot \sqrt{A + \mathsf{hypot}\left(B, A\right)}\right) \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 4: 45.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := \frac{-\sqrt{\left(2 \cdot \left(F \cdot t_0\right)\right) \cdot \left(2 \cdot A\right)}}{t_0}\\ t_2 := {B}^{2} - \left(4 \cdot A\right) \cdot C\\ \mathbf{if}\;{B}^{2} \leq 2 \cdot 10^{-280}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{-175}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(t_2 \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{t_2}\\ \mathbf{elif}\;{B}^{2} \leq 1.5 \cdot 10^{-137}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{F} \cdot \sqrt{A + \mathsf{hypot}\left(B, A\right)}\right) \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \end{array} \]

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 (/ (- (sqrt (* (* 2.0 (* F t_0)) (* 2.0 A)))) t_0))
        (t_2 (- (pow B 2.0) (* (* 4.0 A) C))))
   (if (<= (pow B 2.0) 2e-280)
     t_1
     (if (<= (pow B 2.0) 2e-175)
       (/ (- (sqrt (* (* 2.0 (* t_2 F)) (* 2.0 C)))) t_2)
       (if (<= (pow B 2.0) 1.5e-137)
         t_1
         (* (* (sqrt F) (sqrt (+ A (hypot B A)))) (/ (- (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 = -sqrt(((2.0 * (F * t_0)) * (2.0 * A))) / t_0;
	double t_2 = pow(B, 2.0) - ((4.0 * A) * C);
	double tmp;
	if (pow(B, 2.0) <= 2e-280) {
		tmp = t_1;
	} else if (pow(B, 2.0) <= 2e-175) {
		tmp = -sqrt(((2.0 * (t_2 * F)) * (2.0 * C))) / t_2;
	} else if (pow(B, 2.0) <= 1.5e-137) {
		tmp = t_1;
	} else {
		tmp = (sqrt(F) * sqrt((A + hypot(B, A)))) * (-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 = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(F * t_0)) * Float64(2.0 * A)))) / t_0)
	t_2 = Float64((B ^ 2.0) - Float64(Float64(4.0 * A) * C))
	tmp = 0.0
	if ((B ^ 2.0) <= 2e-280)
		tmp = t_1;
	elseif ((B ^ 2.0) <= 2e-175)
		tmp = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(t_2 * F)) * Float64(2.0 * C)))) / t_2);
	elseif ((B ^ 2.0) <= 1.5e-137)
		tmp = t_1;
	else
		tmp = Float64(Float64(sqrt(F) * sqrt(Float64(A + hypot(B, A)))) * 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[((-N[Sqrt[N[(N[(2.0 * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision] * N[(2.0 * A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[Power[B, 2.0], $MachinePrecision] - N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B, 2.0], $MachinePrecision], 2e-280], t$95$1, If[LessEqual[N[Power[B, 2.0], $MachinePrecision], 2e-175], N[((-N[Sqrt[N[(N[(2.0 * N[(t$95$2 * F), $MachinePrecision]), $MachinePrecision] * N[(2.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$2), $MachinePrecision], If[LessEqual[N[Power[B, 2.0], $MachinePrecision], 1.5e-137], t$95$1, N[(N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(A + N[Sqrt[B ^ 2 + A ^ 2], $MachinePrecision]), $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) < 1.9999999999999999e-280 or 2e-175 < (pow.f64 B 2) < 1.4999999999999999e-137

   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 27.8%

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

      1. pow1/2 27.8%

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

      2. associate-*r* 27.8%

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

      3. unpow-prod-down 30.3%

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

      4. *-commutative 30.3%

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

      5. pow1/2 30.3%

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

      6. associate-+r+ 29.0%

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

      7. hypot-udef 21.4%

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

      8. unpow2 21.4%

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

      9. unpow2 21.4%

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

      10. +-commutative 21.4%

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

      11. +-commutative 21.4%

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

   5. Applied egg-rr 29.0%

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

   6. Taylor expanded in A around inf 22.2%

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

      1. expm1-log1p-u 21.7%

         \[\leadsto \frac{-\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot 2\right)}^{0.5} \cdot \sqrt{2 \cdot A}\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      2. expm1-udef 8.3%

         \[\leadsto \frac{-\color{blue}{\left(e^{\mathsf{log1p}\left({\left(\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot 2\right)}^{0.5} \cdot \sqrt{2 \cdot A}\right)} - 1\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      3. pow1/2 8.3%

         \[\leadsto \frac{-\left(e^{\mathsf{log1p}\left({\left(\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot 2\right)}^{0.5} \cdot \color{blue}{{\left(2 \cdot A\right)}^{0.5}}\right)} - 1\right)}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      4. pow-prod-down 10.1%

         \[\leadsto \frac{-\left(e^{\mathsf{log1p}\left(\color{blue}{{\left(\left(\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot 2\right) \cdot \left(2 \cdot A\right)\right)}^{0.5}}\right)} - 1\right)}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      5. *-commutative 10.1%

         \[\leadsto \frac{-\left(e^{\mathsf{log1p}\left({\left(\color{blue}{\left(2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)\right)} \cdot \left(2 \cdot A\right)\right)}^{0.5}\right)} - 1\right)}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   8. Applied egg-rr 10.1%

      \[\leadsto \frac{-\color{blue}{\left(e^{\mathsf{log1p}\left({\left(\left(2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)\right) \cdot \left(2 \cdot A\right)\right)}^{0.5}\right)} - 1\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
   9. Step-by-step derivation

      1. expm1-def 26.1%

         \[\leadsto \frac{-\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\left(2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)\right) \cdot \left(2 \cdot A\right)\right)}^{0.5}\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      2. expm1-log1p 26.8%

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

      3. unpow1/2 26.8%

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

   10. Simplified 26.8%

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

   if 1.9999999999999999e-280 < (pow.f64 B 2) < 2e-175

   1. Initial program 28.6%

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

   2. Taylor expanded in A around -inf 31.9%

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

   if 1.4999999999999999e-137 < (pow.f64 B 2)

   1. Initial program 17.0%

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

   2. Taylor expanded in C around 0 10.4%

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

      1. mul-1-neg 10.4%

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

      2. distribute-rgt-neg-in 10.4%

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

      3. +-commutative 10.4%

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

      4. unpow2 10.4%

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

      5. unpow2 10.4%

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

      6. hypot-def 20.3%

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

   4. Simplified 20.3%

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

      1. pow1/2 20.4%

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

      2. *-commutative 20.4%

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

      3. unpow-prod-down 29.8%

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

      4. pow1/2 29.8%

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

      5. pow1/2 29.8%

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

   6. Applied egg-rr 29.8%

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

4. Final simplification 29.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 2 \cdot 10^{-280}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)\right) \cdot \left(2 \cdot A\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{-175}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{{B}^{2} - \left(4 \cdot A\right) \cdot C}\\ \mathbf{elif}\;{B}^{2} \leq 1.5 \cdot 10^{-137}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)\right) \cdot \left(2 \cdot A\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{F} \cdot \sqrt{A + \mathsf{hypot}\left(B, A\right)}\right) \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]

Alternative 5: 51.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;{B}^{2} \leq 6 \cdot 10^{+57}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(A - C, B\right)\right)} \cdot \left(-\sqrt{F \cdot t_0}\right)}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{F} \cdot \sqrt{A + \mathsf{hypot}\left(B, A\right)}\right) \cdot \left(\sqrt{2} \cdot \frac{-1}{B}\right)\\ \end{array} \end{array} \]

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) 6e+57)
     (/
      (* (sqrt (* 2.0 (+ (+ A C) (hypot (- A C) B)))) (- (sqrt (* F t_0))))
      t_0)
     (* (* (sqrt F) (sqrt (+ A (hypot B A)))) (* (sqrt 2.0) (/ -1.0 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) <= 6e+57) {
		tmp = (sqrt((2.0 * ((A + C) + hypot((A - C), B)))) * -sqrt((F * t_0))) / t_0;
	} else {
		tmp = (sqrt(F) * sqrt((A + hypot(B, A)))) * (sqrt(2.0) * (-1.0 / 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) <= 6e+57)
		tmp = Float64(Float64(sqrt(Float64(2.0 * Float64(Float64(A + C) + hypot(Float64(A - C), B)))) * Float64(-sqrt(Float64(F * t_0)))) / t_0);
	else
		tmp = Float64(Float64(sqrt(F) * sqrt(Float64(A + hypot(B, A)))) * Float64(sqrt(2.0) * Float64(-1.0 / 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], 6e+57], N[(N[(N[Sqrt[N[(2.0 * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[N[(F * t$95$0), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(A + N[Sqrt[B ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(-1.0 / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B 2) < 5.9999999999999999e57

   1. Initial program 24.9%

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

   3. Simplified 32.8%

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

      1. sqrt-prod 36.5%

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

      2. *-commutative 36.5%

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

      3. associate-+r+ 35.2%

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

      4. hypot-udef 26.2%

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

      5. unpow2 26.2%

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

      6. unpow2 26.2%

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

      7. +-commutative 26.2%

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

      8. +-commutative 26.2%

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

      9. unpow2 26.2%

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

      10. unpow2 26.2%

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

      11. hypot-def 35.2%

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

   5. Applied egg-rr 35.2%

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

      1. +-commutative 35.2%

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

   7. Simplified 35.2%

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

   if 5.9999999999999999e57 < (pow.f64 B 2)

   1. Initial program 9.5%

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

   2. Taylor expanded in C around 0 9.6%

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

      1. mul-1-neg 9.6%

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

      2. distribute-rgt-neg-in 9.6%

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

      3. +-commutative 9.6%

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

      4. unpow2 9.6%

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

      5. unpow2 9.6%

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

      6. hypot-def 23.7%

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

   4. Simplified 23.7%

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

      1. pow1/2 23.7%

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

      2. *-commutative 23.7%

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

      3. unpow-prod-down 37.4%

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

      4. pow1/2 37.4%

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

      5. pow1/2 37.4%

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

   6. Applied egg-rr 37.4%

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

      1. div-inv 37.4%

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

   8. Applied egg-rr 37.4%

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

4. Final simplification 36.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 6 \cdot 10^{+57}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(A - C, B\right)\right)} \cdot \left(-\sqrt{F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\right)}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{F} \cdot \sqrt{A + \mathsf{hypot}\left(B, A\right)}\right) \cdot \left(\sqrt{2} \cdot \frac{-1}{B}\right)\\ \end{array} \]

Alternative 6: 47.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;{B}^{2} \leq 6 \cdot 10^{+57}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(t_0 \cdot \left(F \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(A - C, B\right)\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{F} \cdot \sqrt{A + \mathsf{hypot}\left(B, A\right)}\right) \cdot \left(\sqrt{2} \cdot \frac{-1}{B}\right)\\ \end{array} \end{array} \]

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) 6e+57)
     (/ (- (sqrt (* 2.0 (* t_0 (* F (+ (+ A C) (hypot (- A C) B))))))) t_0)
     (* (* (sqrt F) (sqrt (+ A (hypot B A)))) (* (sqrt 2.0) (/ -1.0 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) <= 6e+57) {
		tmp = -sqrt((2.0 * (t_0 * (F * ((A + C) + hypot((A - C), B)))))) / t_0;
	} else {
		tmp = (sqrt(F) * sqrt((A + hypot(B, A)))) * (sqrt(2.0) * (-1.0 / 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) <= 6e+57)
		tmp = Float64(Float64(-sqrt(Float64(2.0 * Float64(t_0 * Float64(F * Float64(Float64(A + C) + hypot(Float64(A - C), B))))))) / t_0);
	else
		tmp = Float64(Float64(sqrt(F) * sqrt(Float64(A + hypot(B, A)))) * Float64(sqrt(2.0) * Float64(-1.0 / 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], 6e+57], N[((-N[Sqrt[N[(2.0 * N[(t$95$0 * N[(F * N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], N[(N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(A + N[Sqrt[B ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(-1.0 / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B 2) < 5.9999999999999999e57

   1. Initial program 24.9%

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

      1. expm1-log1p-u 11.3%

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

      2. expm1-udef 4.5%

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

   3. Applied egg-rr 5.6%

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

      1. expm1-def 16.5%

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

      2. expm1-log1p 31.0%

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

      3. distribute-neg-frac 31.0%

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

   5. Simplified 30.7%

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

   if 5.9999999999999999e57 < (pow.f64 B 2)

   1. Initial program 9.5%

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

   2. Taylor expanded in C around 0 9.6%

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

      1. mul-1-neg 9.6%

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

      2. distribute-rgt-neg-in 9.6%

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

      3. +-commutative 9.6%

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

      4. unpow2 9.6%

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

      5. unpow2 9.6%

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

      6. hypot-def 23.7%

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

   4. Simplified 23.7%

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

      1. pow1/2 23.7%

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

      2. *-commutative 23.7%

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

      3. unpow-prod-down 37.4%

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

      4. pow1/2 37.4%

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

      5. pow1/2 37.4%

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

   6. Applied egg-rr 37.4%

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

      1. div-inv 37.4%

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

   8. Applied egg-rr 37.4%

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

4. Final simplification 33.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 6 \cdot 10^{+57}:\\ \;\;\;\;\frac{-\sqrt{2 \cdot \left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot \left(F \cdot \left(\left(A + C\right) + \mathsf{hypot}\left(A - C, B\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{F} \cdot \sqrt{A + \mathsf{hypot}\left(B, A\right)}\right) \cdot \left(\sqrt{2} \cdot \frac{-1}{B}\right)\\ \end{array} \]

Alternative 7: 48.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;{B}^{2} \leq 6 \cdot 10^{+57}:\\ \;\;\;\;\frac{-\sqrt{\left(F \cdot t_0\right) \cdot \left(2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{t_0}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{F} \cdot \sqrt{A + \mathsf{hypot}\left(B, A\right)}\right) \cdot \left(\sqrt{2} \cdot \frac{-1}{B}\right)\\ \end{array} \end{array} \]

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) 6e+57)
     (/ (- (sqrt (* (* F t_0) (* 2.0 (+ A (+ C (hypot B (- A C)))))))) t_0)
     (* (* (sqrt F) (sqrt (+ A (hypot B A)))) (* (sqrt 2.0) (/ -1.0 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) <= 6e+57) {
		tmp = -sqrt(((F * t_0) * (2.0 * (A + (C + hypot(B, (A - C))))))) / t_0;
	} else {
		tmp = (sqrt(F) * sqrt((A + hypot(B, A)))) * (sqrt(2.0) * (-1.0 / 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) <= 6e+57)
		tmp = Float64(Float64(-sqrt(Float64(Float64(F * t_0) * Float64(2.0 * Float64(A + Float64(C + hypot(B, Float64(A - C)))))))) / t_0);
	else
		tmp = Float64(Float64(sqrt(F) * sqrt(Float64(A + hypot(B, A)))) * Float64(sqrt(2.0) * Float64(-1.0 / 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], 6e+57], N[((-N[Sqrt[N[(N[(F * t$95$0), $MachinePrecision] * N[(2.0 * N[(A + N[(C + N[Sqrt[B ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], N[(N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(A + N[Sqrt[B ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] * N[(-1.0 / B), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B 2) < 5.9999999999999999e57

   1. Initial program 24.9%

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

   3. Simplified 32.8%

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

   if 5.9999999999999999e57 < (pow.f64 B 2)

   1. Initial program 9.5%

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

   2. Taylor expanded in C around 0 9.6%

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

      1. mul-1-neg 9.6%

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

      2. distribute-rgt-neg-in 9.6%

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

      3. +-commutative 9.6%

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

      4. unpow2 9.6%

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

      5. unpow2 9.6%

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

      6. hypot-def 23.7%

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

   4. Simplified 23.7%

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

      1. pow1/2 23.7%

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

      2. *-commutative 23.7%

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

      3. unpow-prod-down 37.4%

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

      4. pow1/2 37.4%

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

      5. pow1/2 37.4%

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

   6. Applied egg-rr 37.4%

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

      1. div-inv 37.4%

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

   8. Applied egg-rr 37.4%

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

4. Final simplification 34.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 6 \cdot 10^{+57}:\\ \;\;\;\;\frac{-\sqrt{\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot \left(2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{F} \cdot \sqrt{A + \mathsf{hypot}\left(B, A\right)}\right) \cdot \left(\sqrt{2} \cdot \frac{-1}{B}\right)\\ \end{array} \]

Alternative 8: 39.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;F \leq 1.4 \cdot 10^{-301}:\\ \;\;\;\;\frac{1}{\frac{t_0}{-\sqrt{\left(2 \cdot \left(F \cdot t_0\right)\right) \cdot \left(2 \cdot A\right)}}}\\ \mathbf{elif}\;F \leq 1.7 \cdot 10^{+27}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \end{array} \]

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 (<= F 1.4e-301)
     (/ 1.0 (/ t_0 (- (sqrt (* (* 2.0 (* F t_0)) (* 2.0 A))))))
     (if (<= F 1.7e+27)
       (* (/ (sqrt 2.0) B) (- (sqrt (* F (+ C (hypot B C))))))
       (* (sqrt 2.0) (- (sqrt (/ F 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 (F <= 1.4e-301) {
		tmp = 1.0 / (t_0 / -sqrt(((2.0 * (F * t_0)) * (2.0 * A))));
	} else if (F <= 1.7e+27) {
		tmp = (sqrt(2.0) / B) * -sqrt((F * (C + hypot(B, C))));
	} else {
		tmp = sqrt(2.0) * -sqrt((F / 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 (F <= 1.4e-301)
		tmp = Float64(1.0 / Float64(t_0 / Float64(-sqrt(Float64(Float64(2.0 * Float64(F * t_0)) * Float64(2.0 * A))))));
	elseif (F <= 1.7e+27)
		tmp = Float64(Float64(sqrt(2.0) / B) * Float64(-sqrt(Float64(F * Float64(C + hypot(B, C))))));
	else
		tmp = Float64(sqrt(2.0) * Float64(-sqrt(Float64(F / 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[F, 1.4e-301], N[(1.0 / N[(t$95$0 / (-N[Sqrt[N[(N[(2.0 * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision] * N[(2.0 * A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 1.7e+27], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B), $MachinePrecision] * (-N[Sqrt[N[(F * N[(C + N[Sqrt[B ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * (-N[Sqrt[N[(F / B), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < 1.4000000000000001e-301

   1. Initial program 35.1%

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

   3. Simplified 46.2%

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

      1. pow1/2 46.2%

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

      2. associate-*r* 46.2%

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

      3. unpow-prod-down 60.7%

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

      4. *-commutative 60.7%

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

      5. pow1/2 60.7%

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

      6. associate-+r+ 60.7%

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

      7. hypot-udef 38.6%

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

      8. unpow2 38.6%

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

      9. unpow2 38.6%

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

      10. +-commutative 38.6%

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

      11. +-commutative 38.6%

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

   5. Applied egg-rr 60.7%

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

   6. Taylor expanded in A around inf 39.1%

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

      1. clear-num 39.3%

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

      2. inv-pow 39.3%

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

      3. pow1/2 39.3%

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

      4. pow-prod-down 32.1%

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

      5. *-commutative 32.1%

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

   8. Applied egg-rr 32.1%

      \[\leadsto \color{blue}{{\left(\frac{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}{-{\left(\left(2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)\right) \cdot \left(2 \cdot A\right)\right)}^{0.5}}\right)}^{-1}} \]
   9. Step-by-step derivation

      1. unpow-1 32.1%

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

      2. unpow1/2 32.1%

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

   10. Simplified 32.1%

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

   if 1.4000000000000001e-301 < F < 1.7e27

   1. Initial program 18.7%

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

   2. Taylor expanded in A around 0 9.0%

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

      1. mul-1-neg 9.0%

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

      2. distribute-rgt-neg-in 9.0%

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

      3. unpow2 9.0%

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

      4. unpow2 9.0%

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

      5. hypot-def 20.4%

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

   4. Simplified 20.4%

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

   if 1.7e27 < F

   1. Initial program 14.5%

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

   2. Taylor expanded in C around 0 8.5%

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

      1. mul-1-neg 8.5%

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

      2. distribute-rgt-neg-in 8.5%

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

      3. +-commutative 8.5%

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

      4. unpow2 8.5%

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

      5. unpow2 8.5%

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

      6. hypot-def 9.8%

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

   4. Simplified 9.8%

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

   5. Taylor expanded in A around 0 19.2%

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

      1. associate-*r* 19.2%

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

      2. mul-1-neg 19.2%

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

   7. Simplified 19.2%

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

4. Final simplification 21.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 1.4 \cdot 10^{-301}:\\ \;\;\;\;\frac{1}{\frac{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}{-\sqrt{\left(2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)\right) \cdot \left(2 \cdot A\right)}}}\\ \mathbf{elif}\;F \leq 1.7 \cdot 10^{+27}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \]

Alternative 9: 39.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;F \leq 1.46 \cdot 10^{-301}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot t_0\right)\right) \cdot \left(2 \cdot A\right)}}{t_0}\\ \mathbf{elif}\;F \leq 1.16 \cdot 10^{+27}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \end{array} \]

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 (<= F 1.46e-301)
     (/ (- (sqrt (* (* 2.0 (* F t_0)) (* 2.0 A)))) t_0)
     (if (<= F 1.16e+27)
       (* (/ (sqrt 2.0) B) (- (sqrt (* F (+ C (hypot B C))))))
       (* (sqrt 2.0) (- (sqrt (/ F 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 (F <= 1.46e-301) {
		tmp = -sqrt(((2.0 * (F * t_0)) * (2.0 * A))) / t_0;
	} else if (F <= 1.16e+27) {
		tmp = (sqrt(2.0) / B) * -sqrt((F * (C + hypot(B, C))));
	} else {
		tmp = sqrt(2.0) * -sqrt((F / 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 (F <= 1.46e-301)
		tmp = Float64(Float64(-sqrt(Float64(Float64(2.0 * Float64(F * t_0)) * Float64(2.0 * A)))) / t_0);
	elseif (F <= 1.16e+27)
		tmp = Float64(Float64(sqrt(2.0) / B) * Float64(-sqrt(Float64(F * Float64(C + hypot(B, C))))));
	else
		tmp = Float64(sqrt(2.0) * Float64(-sqrt(Float64(F / 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[F, 1.46e-301], N[((-N[Sqrt[N[(N[(2.0 * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision] * N[(2.0 * A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]) / t$95$0), $MachinePrecision], If[LessEqual[F, 1.16e+27], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B), $MachinePrecision] * (-N[Sqrt[N[(F * N[(C + N[Sqrt[B ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * (-N[Sqrt[N[(F / B), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if F < 1.46000000000000002e-301

   1. Initial program 35.1%

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

   3. Simplified 46.2%

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

      1. pow1/2 46.2%

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

      2. associate-*r* 46.2%

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

      3. unpow-prod-down 60.7%

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

      4. *-commutative 60.7%

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

      5. pow1/2 60.7%

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

      6. associate-+r+ 60.7%

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

      7. hypot-udef 38.6%

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

      8. unpow2 38.6%

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

      9. unpow2 38.6%

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

      10. +-commutative 38.6%

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

      11. +-commutative 38.6%

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

   5. Applied egg-rr 60.7%

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

   6. Taylor expanded in A around inf 39.1%

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

      1. expm1-log1p-u 37.6%

         \[\leadsto \frac{-\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot 2\right)}^{0.5} \cdot \sqrt{2 \cdot A}\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      2. expm1-udef 23.1%

         \[\leadsto \frac{-\color{blue}{\left(e^{\mathsf{log1p}\left({\left(\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot 2\right)}^{0.5} \cdot \sqrt{2 \cdot A}\right)} - 1\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      3. pow1/2 23.1%

         \[\leadsto \frac{-\left(e^{\mathsf{log1p}\left({\left(\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot 2\right)}^{0.5} \cdot \color{blue}{{\left(2 \cdot A\right)}^{0.5}}\right)} - 1\right)}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      4. pow-prod-down 16.4%

         \[\leadsto \frac{-\left(e^{\mathsf{log1p}\left(\color{blue}{{\left(\left(\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot 2\right) \cdot \left(2 \cdot A\right)\right)}^{0.5}}\right)} - 1\right)}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      5. *-commutative 16.4%

         \[\leadsto \frac{-\left(e^{\mathsf{log1p}\left({\left(\color{blue}{\left(2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)\right)} \cdot \left(2 \cdot A\right)\right)}^{0.5}\right)} - 1\right)}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   8. Applied egg-rr 16.4%

      \[\leadsto \frac{-\color{blue}{\left(e^{\mathsf{log1p}\left({\left(\left(2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)\right) \cdot \left(2 \cdot A\right)\right)}^{0.5}\right)} - 1\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]
   9. Step-by-step derivation

      1. expm1-def 31.0%

         \[\leadsto \frac{-\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\left(\left(2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)\right) \cdot \left(2 \cdot A\right)\right)}^{0.5}\right)\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      2. expm1-log1p 32.0%

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

      3. unpow1/2 32.0%

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

   10. Simplified 32.0%

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

   if 1.46000000000000002e-301 < F < 1.16e27

   1. Initial program 18.7%

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

   2. Taylor expanded in A around 0 9.0%

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

      1. mul-1-neg 9.0%

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

      2. distribute-rgt-neg-in 9.0%

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

      3. unpow2 9.0%

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

      4. unpow2 9.0%

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

      5. hypot-def 20.4%

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

   4. Simplified 20.4%

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

   if 1.16e27 < F

   1. Initial program 14.5%

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

   2. Taylor expanded in C around 0 8.5%

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

      1. mul-1-neg 8.5%

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

      2. distribute-rgt-neg-in 8.5%

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

      3. +-commutative 8.5%

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

      4. unpow2 8.5%

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

      5. unpow2 8.5%

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

      6. hypot-def 9.8%

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

   4. Simplified 9.8%

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

   5. Taylor expanded in A around 0 19.2%

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

      1. associate-*r* 19.2%

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

      2. mul-1-neg 19.2%

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

   7. Simplified 19.2%

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

4. Final simplification 21.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 1.46 \cdot 10^{-301}:\\ \;\;\;\;\frac{-\sqrt{\left(2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)\right) \cdot \left(2 \cdot A\right)}}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;F \leq 1.16 \cdot 10^{+27}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \]

Alternative 10: 39.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ \begin{array}{l} \mathbf{if}\;F \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\frac{\sqrt{2 \cdot A} \cdot \left(-{\left(2 \cdot \left(-4 \cdot \left(A \cdot \left(C \cdot F\right)\right)\right)\right)}^{0.5}\right)}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;F \leq 3.4 \cdot 10^{+24}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
(FPCore (A B C F)
 :precision binary64
 (if (<= F -5e-311)
   (/
    (* (sqrt (* 2.0 A)) (- (pow (* 2.0 (* -4.0 (* A (* C F)))) 0.5)))
    (fma B B (* A (* C -4.0))))
   (if (<= F 3.4e+24)
     (* (/ (sqrt 2.0) B) (- (sqrt (* F (+ C (hypot B C))))))
     (* (sqrt 2.0) (- (sqrt (/ F B)))))))

C

B = abs(B);
double code(double A, double B, double C, double F) {
	double tmp;
	if (F <= -5e-311) {
		tmp = (sqrt((2.0 * A)) * -pow((2.0 * (-4.0 * (A * (C * F)))), 0.5)) / fma(B, B, (A * (C * -4.0)));
	} else if (F <= 3.4e+24) {
		tmp = (sqrt(2.0) / B) * -sqrt((F * (C + hypot(B, C))));
	} else {
		tmp = sqrt(2.0) * -sqrt((F / B));
	}
	return tmp;
}

Julia

B = abs(B)
function code(A, B, C, F)
	tmp = 0.0
	if (F <= -5e-311)
		tmp = Float64(Float64(sqrt(Float64(2.0 * A)) * Float64(-(Float64(2.0 * Float64(-4.0 * Float64(A * Float64(C * F)))) ^ 0.5))) / fma(B, B, Float64(A * Float64(C * -4.0))));
	elseif (F <= 3.4e+24)
		tmp = Float64(Float64(sqrt(2.0) / B) * Float64(-sqrt(Float64(F * Float64(C + hypot(B, C))))));
	else
		tmp = Float64(sqrt(2.0) * Float64(-sqrt(Float64(F / B))));
	end
	return tmp
end

Wolfram

NOTE: B should be positive before calling this function
code[A_, B_, C_, F_] := If[LessEqual[F, -5e-311], N[(N[(N[Sqrt[N[(2.0 * A), $MachinePrecision]], $MachinePrecision] * (-N[Power[N[(2.0 * N[(-4.0 * N[(A * N[(C * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision])), $MachinePrecision] / N[(B * B + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, 3.4e+24], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B), $MachinePrecision] * (-N[Sqrt[N[(F * N[(C + N[Sqrt[B ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * (-N[Sqrt[N[(F / B), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -5.00000000000023e-311

   1. Initial program 37.7%

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

   3. Simplified 45.7%

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

      1. pow1/2 45.7%

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

      2. associate-*r* 45.7%

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

      3. unpow-prod-down 61.4%

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

      4. *-commutative 61.4%

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

      5. pow1/2 61.4%

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

      6. associate-+r+ 61.4%

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

      7. hypot-udef 41.5%

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

      8. unpow2 41.5%

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

      9. unpow2 41.5%

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

      10. +-commutative 41.5%

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

      11. +-commutative 41.5%

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

   5. Applied egg-rr 61.4%

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

   6. Taylor expanded in A around inf 38.0%

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

   7. Taylor expanded in B around 0 31.6%

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

      1. *-commutative 31.6%

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

   9. Simplified 31.6%

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

   if -5.00000000000023e-311 < F < 3.4000000000000001e24

   1. Initial program 18.5%

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

   2. Taylor expanded in A around 0 8.9%

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

      1. mul-1-neg 8.9%

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

      2. distribute-rgt-neg-in 8.9%

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

      3. unpow2 8.9%

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

      4. unpow2 8.9%

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

      5. hypot-def 20.1%

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

   4. Simplified 20.1%

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

   if 3.4000000000000001e24 < F

   1. Initial program 14.5%

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

   2. Taylor expanded in C around 0 8.5%

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

      1. mul-1-neg 8.5%

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

      2. distribute-rgt-neg-in 8.5%

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

      3. +-commutative 8.5%

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

      4. unpow2 8.5%

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

      5. unpow2 8.5%

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

      6. hypot-def 9.8%

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

   4. Simplified 9.8%

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

   5. Taylor expanded in A around 0 19.2%

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

      1. associate-*r* 19.2%

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

      2. mul-1-neg 19.2%

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

   7. Simplified 19.2%

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

4. Final simplification 20.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -5 \cdot 10^{-311}:\\ \;\;\;\;\frac{\sqrt{2 \cdot A} \cdot \left(-{\left(2 \cdot \left(-4 \cdot \left(A \cdot \left(C \cdot F\right)\right)\right)\right)}^{0.5}\right)}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;F \leq 3.4 \cdot 10^{+24}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \]

Alternative 11: 36.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ \begin{array}{l} \mathbf{if}\;F \leq 1.25 \cdot 10^{+73}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(A + \mathsf{hypot}\left(B, A\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
(FPCore (A B C F)
 :precision binary64
 (if (<= F 1.25e+73)
   (* (/ (sqrt 2.0) B) (- (sqrt (* F (+ A (hypot B A))))))
   (* (sqrt 2.0) (- (sqrt (/ F B))))))

C

B = abs(B);
double code(double A, double B, double C, double F) {
	double tmp;
	if (F <= 1.25e+73) {
		tmp = (sqrt(2.0) / B) * -sqrt((F * (A + hypot(B, A))));
	} else {
		tmp = sqrt(2.0) * -sqrt((F / B));
	}
	return tmp;
}

Java

B = Math.abs(B);
public static double code(double A, double B, double C, double F) {
	double tmp;
	if (F <= 1.25e+73) {
		tmp = (Math.sqrt(2.0) / B) * -Math.sqrt((F * (A + Math.hypot(B, A))));
	} else {
		tmp = Math.sqrt(2.0) * -Math.sqrt((F / B));
	}
	return tmp;
}

Python

B = abs(B)
def code(A, B, C, F):
	tmp = 0
	if F <= 1.25e+73:
		tmp = (math.sqrt(2.0) / B) * -math.sqrt((F * (A + math.hypot(B, A))))
	else:
		tmp = math.sqrt(2.0) * -math.sqrt((F / B))
	return tmp

Julia

B = abs(B)
function code(A, B, C, F)
	tmp = 0.0
	if (F <= 1.25e+73)
		tmp = Float64(Float64(sqrt(2.0) / B) * Float64(-sqrt(Float64(F * Float64(A + hypot(B, A))))));
	else
		tmp = Float64(sqrt(2.0) * Float64(-sqrt(Float64(F / B))));
	end
	return tmp
end

MATLAB

B = abs(B)
function tmp_2 = code(A, B, C, F)
	tmp = 0.0;
	if (F <= 1.25e+73)
		tmp = (sqrt(2.0) / B) * -sqrt((F * (A + hypot(B, A))));
	else
		tmp = sqrt(2.0) * -sqrt((F / B));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: B should be positive before calling this function
code[A_, B_, C_, F_] := If[LessEqual[F, 1.25e+73], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B), $MachinePrecision] * (-N[Sqrt[N[(F * N[(A + N[Sqrt[B ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * (-N[Sqrt[N[(F / B), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < 1.24999999999999994e73

   1. Initial program 20.2%

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

   2. Taylor expanded in C around 0 7.6%

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

      1. mul-1-neg 7.6%

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

      2. distribute-rgt-neg-in 7.6%

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

      3. +-commutative 7.6%

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

      4. unpow2 7.6%

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

      5. unpow2 7.6%

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

      6. hypot-def 16.8%

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

   4. Simplified 16.8%

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

   if 1.24999999999999994e73 < F

   1. Initial program 15.8%

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

   2. Taylor expanded in C around 0 8.4%

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

      1. mul-1-neg 8.4%

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

      2. distribute-rgt-neg-in 8.4%

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

      3. +-commutative 8.4%

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

      4. unpow2 8.4%

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

      5. unpow2 8.4%

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

      6. hypot-def 8.7%

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

   4. Simplified 8.7%

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

   5. Taylor expanded in A around 0 19.8%

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

      1. associate-*r* 19.8%

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

      2. mul-1-neg 19.8%

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

   7. Simplified 19.8%

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

4. Final simplification 17.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 1.25 \cdot 10^{+73}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(A + \mathsf{hypot}\left(B, A\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \]

Alternative 12: 36.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ \begin{array}{l} \mathbf{if}\;F \leq 1.4 \cdot 10^{+24}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
(FPCore (A B C F)
 :precision binary64
 (if (<= F 1.4e+24)
   (* (/ (sqrt 2.0) B) (- (sqrt (* F (+ C (hypot B C))))))
   (* (sqrt 2.0) (- (sqrt (/ F B))))))

C

B = abs(B);
double code(double A, double B, double C, double F) {
	double tmp;
	if (F <= 1.4e+24) {
		tmp = (sqrt(2.0) / B) * -sqrt((F * (C + hypot(B, C))));
	} else {
		tmp = sqrt(2.0) * -sqrt((F / B));
	}
	return tmp;
}

Java

B = Math.abs(B);
public static double code(double A, double B, double C, double F) {
	double tmp;
	if (F <= 1.4e+24) {
		tmp = (Math.sqrt(2.0) / B) * -Math.sqrt((F * (C + Math.hypot(B, C))));
	} else {
		tmp = Math.sqrt(2.0) * -Math.sqrt((F / B));
	}
	return tmp;
}

Python

B = abs(B)
def code(A, B, C, F):
	tmp = 0
	if F <= 1.4e+24:
		tmp = (math.sqrt(2.0) / B) * -math.sqrt((F * (C + math.hypot(B, C))))
	else:
		tmp = math.sqrt(2.0) * -math.sqrt((F / B))
	return tmp

Julia

B = abs(B)
function code(A, B, C, F)
	tmp = 0.0
	if (F <= 1.4e+24)
		tmp = Float64(Float64(sqrt(2.0) / B) * Float64(-sqrt(Float64(F * Float64(C + hypot(B, C))))));
	else
		tmp = Float64(sqrt(2.0) * Float64(-sqrt(Float64(F / B))));
	end
	return tmp
end

MATLAB

B = abs(B)
function tmp_2 = code(A, B, C, F)
	tmp = 0.0;
	if (F <= 1.4e+24)
		tmp = (sqrt(2.0) / B) * -sqrt((F * (C + hypot(B, C))));
	else
		tmp = sqrt(2.0) * -sqrt((F / B));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: B should be positive before calling this function
code[A_, B_, C_, F_] := If[LessEqual[F, 1.4e+24], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B), $MachinePrecision] * (-N[Sqrt[N[(F * N[(C + N[Sqrt[B ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * (-N[Sqrt[N[(F / B), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < 1.4000000000000001e24

   1. Initial program 21.6%

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

   2. Taylor expanded in A around 0 7.5%

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

      1. mul-1-neg 7.5%

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

      2. distribute-rgt-neg-in 7.5%

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

      3. unpow2 7.5%

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

      4. unpow2 7.5%

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

      5. hypot-def 16.9%

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

   4. Simplified 16.9%

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

   if 1.4000000000000001e24 < F

   1. Initial program 14.5%

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

   2. Taylor expanded in C around 0 8.5%

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

      1. mul-1-neg 8.5%

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

      2. distribute-rgt-neg-in 8.5%

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

      3. +-commutative 8.5%

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

      4. unpow2 8.5%

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

      5. unpow2 8.5%

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

      6. hypot-def 9.8%

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

   4. Simplified 9.8%

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

   5. Taylor expanded in A around 0 19.2%

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

      1. associate-*r* 19.2%

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

      2. mul-1-neg 19.2%

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

   7. Simplified 19.2%

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

4. Final simplification 17.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 1.4 \cdot 10^{+24}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \]

Alternative 13: 34.6% accurate, 3.0× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ \begin{array}{l} \mathbf{if}\;F \leq 1.6 \cdot 10^{-6}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{B \cdot F}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \end{array} \]

FPCore

NOTE: B should be positive before calling this function
(FPCore (A B C F)
 :precision binary64
 (if (<= F 1.6e-6)
   (* (/ (sqrt 2.0) B) (- (sqrt (* B F))))
   (* (sqrt 2.0) (- (sqrt (/ F B))))))

C

B = abs(B);
double code(double A, double B, double C, double F) {
	double tmp;
	if (F <= 1.6e-6) {
		tmp = (sqrt(2.0) / B) * -sqrt((B * F));
	} else {
		tmp = sqrt(2.0) * -sqrt((F / B));
	}
	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) :: tmp
    if (f <= 1.6d-6) then
        tmp = (sqrt(2.0d0) / b) * -sqrt((b * f))
    else
        tmp = sqrt(2.0d0) * -sqrt((f / b))
    end if
    code = tmp
end function

Java

B = Math.abs(B);
public static double code(double A, double B, double C, double F) {
	double tmp;
	if (F <= 1.6e-6) {
		tmp = (Math.sqrt(2.0) / B) * -Math.sqrt((B * F));
	} else {
		tmp = Math.sqrt(2.0) * -Math.sqrt((F / B));
	}
	return tmp;
}

Python

B = abs(B)
def code(A, B, C, F):
	tmp = 0
	if F <= 1.6e-6:
		tmp = (math.sqrt(2.0) / B) * -math.sqrt((B * F))
	else:
		tmp = math.sqrt(2.0) * -math.sqrt((F / B))
	return tmp

Julia

B = abs(B)
function code(A, B, C, F)
	tmp = 0.0
	if (F <= 1.6e-6)
		tmp = Float64(Float64(sqrt(2.0) / B) * Float64(-sqrt(Float64(B * F))));
	else
		tmp = Float64(sqrt(2.0) * Float64(-sqrt(Float64(F / B))));
	end
	return tmp
end

MATLAB

B = abs(B)
function tmp_2 = code(A, B, C, F)
	tmp = 0.0;
	if (F <= 1.6e-6)
		tmp = (sqrt(2.0) / B) * -sqrt((B * F));
	else
		tmp = sqrt(2.0) * -sqrt((F / B));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: B should be positive before calling this function
code[A_, B_, C_, F_] := If[LessEqual[F, 1.6e-6], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B), $MachinePrecision] * (-N[Sqrt[N[(B * F), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * (-N[Sqrt[N[(F / B), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < 1.5999999999999999e-6

   1. Initial program 23.0%

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

   2. Taylor expanded in C around 0 7.9%

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

      1. mul-1-neg 7.9%

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

      2. distribute-rgt-neg-in 7.9%

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

      3. +-commutative 7.9%

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

      4. unpow2 7.9%

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

      5. unpow2 7.9%

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

      6. hypot-def 17.7%

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

   4. Simplified 17.7%

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

   5. Taylor expanded in A around 0 15.8%

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

      1. *-commutative 15.8%

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

   7. Simplified 15.8%

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

   if 1.5999999999999999e-6 < F

   1. Initial program 13.9%

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

   2. Taylor expanded in C around 0 7.8%

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

      1. mul-1-neg 7.8%

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

      2. distribute-rgt-neg-in 7.8%

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

      3. +-commutative 7.8%

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

      4. unpow2 7.8%

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

      5. unpow2 7.8%

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

      6. hypot-def 9.9%

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

   4. Simplified 9.9%

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

   5. Taylor expanded in A around 0 17.8%

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

      1. associate-*r* 17.8%

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

      2. mul-1-neg 17.8%

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

   7. Simplified 17.8%

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

4. Final simplification 16.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 1.6 \cdot 10^{-6}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{B \cdot F}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{\frac{F}{B}}\right)\\ \end{array} \]

Alternative 14: 27.1% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

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(2.0d0) * -sqrt((f / b))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 18.6%

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

2. Taylor expanded in C around 0 7.9%

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

   1. mul-1-neg 7.9%

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

   2. distribute-rgt-neg-in 7.9%

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

   3. +-commutative 7.9%

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

   4. unpow2 7.9%

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

   5. unpow2 7.9%

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

   6. hypot-def 13.9%

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

4. Simplified 13.9%

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

5. Taylor expanded in A around 0 13.4%

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

   1. associate-*r* 13.4%

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

   2. mul-1-neg 13.4%

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

7. Simplified 13.4%

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

8. Final simplification 13.4%

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

Alternative 15: 5.4% accurate, 5.8× speedup

Math

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

FPCore

NOTE: B should be positive before calling this function
(FPCore (A B C F) :precision binary64 (* (pow (* A F) 0.5) (/ (- 2.0) B)))

C

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

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 = ((a * f) ** 0.5d0) * (-2.0d0 / b)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

NOTE: B should be positive before calling this function
code[A_, B_, C_, F_] := N[(N[Power[N[(A * F), $MachinePrecision], 0.5], $MachinePrecision] * N[((-2.0) / B), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 18.6%

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

2. Taylor expanded in C around 0 7.9%

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

   1. mul-1-neg 7.9%

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

   2. distribute-rgt-neg-in 7.9%

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

   3. +-commutative 7.9%

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

   4. unpow2 7.9%

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

   5. unpow2 7.9%

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

   6. hypot-def 13.9%

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

4. Simplified 13.9%

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

5. Taylor expanded in B around 0 2.5%

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

   1. associate-*r* 2.5%

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

   2. mul-1-neg 2.5%

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

   3. *-commutative 2.5%

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

   4. unpow2 2.5%

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

   5. rem-square-sqrt 2.5%

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

7. Simplified 2.5%

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

   1. pow1/2 2.6%

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

   2. *-commutative 2.6%

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

9. Applied egg-rr 2.6%

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

10. Final simplification 2.6%

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

Alternative 16: 5.3% accurate, 5.9× speedup

Math

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

FPCore

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

C

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

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((a * f)) * (-2.0d0 / b)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 18.6%

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

2. Taylor expanded in C around 0 7.9%

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

   1. mul-1-neg 7.9%

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

   2. distribute-rgt-neg-in 7.9%

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

   3. +-commutative 7.9%

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

   4. unpow2 7.9%

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

   5. unpow2 7.9%

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

   6. hypot-def 13.9%

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

4. Simplified 13.9%

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

5. Taylor expanded in B around 0 2.5%

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

   1. associate-*r* 2.5%

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

   2. mul-1-neg 2.5%

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

   3. *-commutative 2.5%

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

   4. unpow2 2.5%

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

   5. rem-square-sqrt 2.5%

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

7. Simplified 2.5%

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

8. Final simplification 2.5%

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

Alternative 17: 5.3% accurate, 5.9× speedup

Math

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

FPCore

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

C

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

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 = (2.0d0 * -sqrt((a * f))) / b
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 18.6%

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

2. Taylor expanded in C around 0 7.9%

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

   1. mul-1-neg 7.9%

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

   2. distribute-rgt-neg-in 7.9%

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

   3. +-commutative 7.9%

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

   4. unpow2 7.9%

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

   5. unpow2 7.9%

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

   6. hypot-def 13.9%

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

4. Simplified 13.9%

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

5. Taylor expanded in B around 0 2.5%

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

   1. associate-*r* 2.5%

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

   2. mul-1-neg 2.5%

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

   3. *-commutative 2.5%

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

   4. unpow2 2.5%

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

   5. rem-square-sqrt 2.5%

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

7. Simplified 2.5%

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

   1. associate-*r/ 2.5%

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

   2. *-commutative 2.5%

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

9. Applied egg-rr 2.5%

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

10. Final simplification 2.5%

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

Alternative 18: 1.2% accurate, 5.9× speedup

Math

\[\begin{array}{l} B = |B|\\ \\ \frac{2}{B} \cdot \sqrt{A \cdot F} \end{array} \]

FPCore

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

C

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

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 = (2.0d0 / b) * sqrt((a * f))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 18.6%

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

3. Simplified 24.1%

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

   1. pow1/2 24.1%

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

   2. associate-*r* 24.1%

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

   3. unpow-prod-down 28.5%

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

   4. *-commutative 28.5%

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

   5. pow1/2 28.5%

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

   6. associate-+r+ 27.7%

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

   7. hypot-udef 21.3%

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

   8. unpow2 21.3%

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

   9. unpow2 21.3%

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

   10. +-commutative 21.3%

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

   11. +-commutative 21.3%

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

5. Applied egg-rr 27.7%

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

6. Taylor expanded in A around inf 13.8%

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

7. Taylor expanded in B around -inf 3.4%

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

   1. unpow2 3.4%

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

   2. rem-square-sqrt 3.4%

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

9. Simplified 3.4%

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

10. Final simplification 3.4%

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

Reproduce

herbie shell --seed 2023318 
(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))))