ABCF->ab-angle b

Percentage Accurate: 18.7% → 55.7%

Time: 25.3s

Alternatives: 14

Speedup: 5.6×

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {B}^{2} - \left(4 \cdot A\right) \cdot C\\ \frac{-\sqrt{\left(2 \cdot \left(t\_0 \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{t\_0} \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 18.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {B}^{2} - \left(4 \cdot A\right) \cdot C\\ \frac{-\sqrt{\left(2 \cdot \left(t\_0 \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{\left(A - C\right)}^{2} + {B}^{2}}\right)}}{t\_0} \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 55.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := \left(4 \cdot A\right) \cdot C\\ t_2 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_1\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_1 - {B\_m}^{2}}\\ t_3 := \mathsf{fma}\left(C, A \cdot -4, {B\_m}^{2}\right)\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{-186}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B\_m, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot t\_3}}{-t\_3}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot t\_0\right) \cdot \left(2 \cdot \left(A + A\right) - \frac{{B\_m}^{2}}{C}\right)}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{2 \cdot \sqrt{2}}}{B\_m} \cdot \left(-e^{\left(\log \left(\mathsf{hypot}\left(B\_m, A\right) - A\right) - \log \left(\frac{-1}{F}\right)\right) \cdot 0.5}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* A (* C -4.0))))
        (t_1 (* (* 4.0 A) C))
        (t_2
         (/
          (sqrt
           (*
            (* 2.0 (* (- (pow B_m 2.0) t_1) F))
            (- (+ A C) (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))))))
          (- t_1 (pow B_m 2.0))))
        (t_3 (fma C (* A -4.0) (pow B_m 2.0))))
   (if (<= t_2 -5e-186)
     (/
      (* (sqrt (* F (+ A (- C (hypot B_m (- A C)))))) (sqrt (* 2.0 t_3)))
      (- t_3))
     (if (<= t_2 INFINITY)
       (/ (sqrt (* (* F t_0) (- (* 2.0 (+ A A)) (/ (pow B_m 2.0) C)))) (- t_0))
       (*
        (/ (cbrt (* 2.0 (sqrt 2.0))) B_m)
        (- (exp (* (- (log (- (hypot B_m A) A)) (log (/ -1.0 F))) 0.5))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (A * (C * -4.0)));
	double t_1 = (4.0 * A) * C;
	double t_2 = sqrt(((2.0 * ((pow(B_m, 2.0) - t_1) * F)) * ((A + C) - sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / (t_1 - pow(B_m, 2.0));
	double t_3 = fma(C, (A * -4.0), pow(B_m, 2.0));
	double tmp;
	if (t_2 <= -5e-186) {
		tmp = (sqrt((F * (A + (C - hypot(B_m, (A - C)))))) * sqrt((2.0 * t_3))) / -t_3;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = sqrt(((F * t_0) * ((2.0 * (A + A)) - (pow(B_m, 2.0) / C)))) / -t_0;
	} else {
		tmp = (cbrt((2.0 * sqrt(2.0))) / B_m) * -exp(((log((hypot(B_m, A) - A)) - log((-1.0 / F))) * 0.5));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	t_1 = Float64(Float64(4.0 * A) * C)
	t_2 = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_1) * F)) * Float64(Float64(A + C) - sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / Float64(t_1 - (B_m ^ 2.0)))
	t_3 = fma(C, Float64(A * -4.0), (B_m ^ 2.0))
	tmp = 0.0
	if (t_2 <= -5e-186)
		tmp = Float64(Float64(sqrt(Float64(F * Float64(A + Float64(C - hypot(B_m, Float64(A - C)))))) * sqrt(Float64(2.0 * t_3))) / Float64(-t_3));
	elseif (t_2 <= Inf)
		tmp = Float64(sqrt(Float64(Float64(F * t_0) * Float64(Float64(2.0 * Float64(A + A)) - Float64((B_m ^ 2.0) / C)))) / Float64(-t_0));
	else
		tmp = Float64(Float64(cbrt(Float64(2.0 * sqrt(2.0))) / B_m) * Float64(-exp(Float64(Float64(log(Float64(hypot(B_m, A) - A)) - log(Float64(-1.0 / F))) * 0.5))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$1), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$1 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(C * N[(A * -4.0), $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e-186], N[(N[(N[Sqrt[N[(F * N[(A + N[(C - N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(2.0 * t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / (-t$95$3)), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(N[Sqrt[N[(N[(F * t$95$0), $MachinePrecision] * N[(N[(2.0 * N[(A + A), $MachinePrecision]), $MachinePrecision] - N[(N[Power[B$95$m, 2.0], $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], N[(N[(N[Power[N[(2.0 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / B$95$m), $MachinePrecision] * (-N[Exp[N[(N[(N[Log[N[(N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision] - A), $MachinePrecision]], $MachinePrecision] - N[Log[N[(-1.0 / F), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -5e-186

   1. Initial program 43.0%

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

   3. Simplified 43.5%

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

      1. pow1/2 43.5%

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

      2. associate-*r* 51.3%

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

      3. unpow-prod-down 69.0%

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

      4. associate-+r- 68.1%

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

      5. hypot-undefine 49.9%

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

      6. unpow2 49.9%

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

      7. unpow2 49.9%

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

      8. +-commutative 49.9%

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

      9. unpow2 49.9%

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

      10. unpow2 49.9%

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

      11. hypot-define 68.1%

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

      12. pow1/2 68.1%

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

   6. Applied egg-rr 68.1%

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

      1. unpow1/2 68.1%

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

      2. associate-+r- 69.0%

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

      3. hypot-undefine 49.9%

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

      4. unpow2 49.9%

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

      5. unpow2 49.9%

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

      6. +-commutative 49.9%

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

      7. unpow2 49.9%

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

      8. unpow2 49.9%

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

      9. hypot-undefine 69.0%

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

   8. Simplified 69.0%

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

   if -5e-186 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < +inf.0

   1. Initial program 11.3%

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

   3. Simplified 20.5%

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

   5. Taylor expanded in C around inf 32.6%

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

   if +inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)))

   1. Initial program 0.0%

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

   3. Taylor expanded in C around 0 1.6%

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

      1. mul-1-neg 1.6%

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

      2. +-commutative 1.6%

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

      3. unpow2 1.6%

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

      4. unpow2 1.6%

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

      5. hypot-define 16.4%

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

   5. Simplified 16.4%

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

      1. pow1/2 16.4%

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

      2. pow-to-exp 15.4%

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

   7. Applied egg-rr 15.4%

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

   8. Taylor expanded in F around -inf 1.7%

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

      1. mul-1-neg 1.7%

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

      2. +-commutative 1.7%

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

      3. unpow2 1.7%

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

      4. unpow2 1.7%

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

      5. hypot-undefine 24.1%

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

      6. mul-1-neg 24.1%

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

   10. Simplified 24.1%

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

       1. add-cbrt-cube 24.1%

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

       2. pow1/3 24.1%

          \[\leadsto -\frac{\color{blue}{{\left(\left(\sqrt{2} \cdot \sqrt{2}\right) \cdot \sqrt{2}\right)}^{0.3333333333333333}}}{B} \cdot e^{\left(\log \left(-\left(A - \mathsf{hypot}\left(B, A\right)\right)\right) + \left(-\log \left(\frac{-1}{F}\right)\right)\right) \cdot 0.5} \]

       3. pow1/2 24.1%

          \[\leadsto -\frac{{\left(\left(\color{blue}{{2}^{0.5}} \cdot \sqrt{2}\right) \cdot \sqrt{2}\right)}^{0.3333333333333333}}{B} \cdot e^{\left(\log \left(-\left(A - \mathsf{hypot}\left(B, A\right)\right)\right) + \left(-\log \left(\frac{-1}{F}\right)\right)\right) \cdot 0.5} \]

       4. pow1/2 24.1%

          \[\leadsto -\frac{{\left(\left({2}^{0.5} \cdot \color{blue}{{2}^{0.5}}\right) \cdot \sqrt{2}\right)}^{0.3333333333333333}}{B} \cdot e^{\left(\log \left(-\left(A - \mathsf{hypot}\left(B, A\right)\right)\right) + \left(-\log \left(\frac{-1}{F}\right)\right)\right) \cdot 0.5} \]

       5. pow-prod-up 24.1%

          \[\leadsto -\frac{{\left(\color{blue}{{2}^{\left(0.5 + 0.5\right)}} \cdot \sqrt{2}\right)}^{0.3333333333333333}}{B} \cdot e^{\left(\log \left(-\left(A - \mathsf{hypot}\left(B, A\right)\right)\right) + \left(-\log \left(\frac{-1}{F}\right)\right)\right) \cdot 0.5} \]

       6. metadata-eval 24.1%

          \[\leadsto -\frac{{\left({2}^{\color{blue}{1}} \cdot \sqrt{2}\right)}^{0.3333333333333333}}{B} \cdot e^{\left(\log \left(-\left(A - \mathsf{hypot}\left(B, A\right)\right)\right) + \left(-\log \left(\frac{-1}{F}\right)\right)\right) \cdot 0.5} \]

       7. metadata-eval 24.1%

          \[\leadsto -\frac{{\left(\color{blue}{2} \cdot \sqrt{2}\right)}^{0.3333333333333333}}{B} \cdot e^{\left(\log \left(-\left(A - \mathsf{hypot}\left(B, A\right)\right)\right) + \left(-\log \left(\frac{-1}{F}\right)\right)\right) \cdot 0.5} \]

   12. Applied egg-rr 24.1%

       \[\leadsto -\frac{\color{blue}{{\left(2 \cdot \sqrt{2}\right)}^{0.3333333333333333}}}{B} \cdot e^{\left(\log \left(-\left(A - \mathsf{hypot}\left(B, A\right)\right)\right) + \left(-\log \left(\frac{-1}{F}\right)\right)\right) \cdot 0.5} \]
   13. Step-by-step derivation

       1. unpow1/3 24.1%

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

   14. Simplified 24.1%

       \[\leadsto -\frac{\color{blue}{\sqrt[3]{2 \cdot \sqrt{2}}}}{B} \cdot e^{\left(\log \left(-\left(A - \mathsf{hypot}\left(B, A\right)\right)\right) + \left(-\log \left(\frac{-1}{F}\right)\right)\right) \cdot 0.5} \]
3. Recombined 3 regimes into one program.

4. Final simplification 43.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -5 \cdot 10^{-186}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(C, A \cdot -4, {B}^{2}\right)}}{-\mathsf{fma}\left(C, A \cdot -4, {B}^{2}\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot \left(2 \cdot \left(A + A\right) - \frac{{B}^{2}}{C}\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{2 \cdot \sqrt{2}}}{B} \cdot \left(-e^{\left(\log \left(\mathsf{hypot}\left(B, A\right) - A\right) - \log \left(\frac{-1}{F}\right)\right) \cdot 0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 54.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := F \cdot t\_0\\ t_2 := -t\_0\\ \mathbf{if}\;{B\_m}^{2} \leq 10^{-223}:\\ \;\;\;\;\frac{\sqrt{\left(4 \cdot A\right) \cdot t\_1}}{t\_2}\\ \mathbf{elif}\;{B\_m}^{2} \leq 4 \cdot 10^{+91}:\\ \;\;\;\;\frac{\sqrt{t\_1 \cdot \left(2 \cdot \left(A + \left(C - \mathsf{hypot}\left(B\_m, A - C\right)\right)\right)\right)}}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{2 \cdot \sqrt{2}}}{B\_m} \cdot \left(-e^{\left(\log \left(\mathsf{hypot}\left(B\_m, A\right) - A\right) - \log \left(\frac{-1}{F}\right)\right) \cdot 0.5}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* A (* C -4.0)))) (t_1 (* F t_0)) (t_2 (- t_0)))
   (if (<= (pow B_m 2.0) 1e-223)
     (/ (sqrt (* (* 4.0 A) t_1)) t_2)
     (if (<= (pow B_m 2.0) 4e+91)
       (/ (sqrt (* t_1 (* 2.0 (+ A (- C (hypot B_m (- A C))))))) t_2)
       (*
        (/ (cbrt (* 2.0 (sqrt 2.0))) B_m)
        (- (exp (* (- (log (- (hypot B_m A) A)) (log (/ -1.0 F))) 0.5))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (A * (C * -4.0)));
	double t_1 = F * t_0;
	double t_2 = -t_0;
	double tmp;
	if (pow(B_m, 2.0) <= 1e-223) {
		tmp = sqrt(((4.0 * A) * t_1)) / t_2;
	} else if (pow(B_m, 2.0) <= 4e+91) {
		tmp = sqrt((t_1 * (2.0 * (A + (C - hypot(B_m, (A - C))))))) / t_2;
	} else {
		tmp = (cbrt((2.0 * sqrt(2.0))) / B_m) * -exp(((log((hypot(B_m, A) - A)) - log((-1.0 / F))) * 0.5));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	t_1 = Float64(F * t_0)
	t_2 = Float64(-t_0)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e-223)
		tmp = Float64(sqrt(Float64(Float64(4.0 * A) * t_1)) / t_2);
	elseif ((B_m ^ 2.0) <= 4e+91)
		tmp = Float64(sqrt(Float64(t_1 * Float64(2.0 * Float64(A + Float64(C - hypot(B_m, Float64(A - C))))))) / t_2);
	else
		tmp = Float64(Float64(cbrt(Float64(2.0 * sqrt(2.0))) / B_m) * Float64(-exp(Float64(Float64(log(Float64(hypot(B_m, A) - A)) - log(Float64(-1.0 / F))) * 0.5))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(F * t$95$0), $MachinePrecision]}, Block[{t$95$2 = (-t$95$0)}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-223], N[(N[Sqrt[N[(N[(4.0 * A), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 4e+91], N[(N[Sqrt[N[(t$95$1 * N[(2.0 * N[(A + N[(C - N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision], N[(N[(N[Power[N[(2.0 * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / B$95$m), $MachinePrecision] * (-N[Exp[N[(N[(N[Log[N[(N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision] - A), $MachinePrecision]], $MachinePrecision] - N[Log[N[(-1.0 / F), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 9.9999999999999997e-224

   1. Initial program 11.2%

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

   3. Simplified 24.2%

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

   5. Taylor expanded in A around -inf 30.7%

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

   if 9.9999999999999997e-224 < (pow.f64 B #s(literal 2 binary64)) < 4.00000000000000032e91

   1. Initial program 46.4%

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

   3. Simplified 51.6%

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

   if 4.00000000000000032e91 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 9.3%

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

   3. Taylor expanded in C around 0 10.1%

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

      1. mul-1-neg 10.1%

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

      2. +-commutative 10.1%

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

      3. unpow2 10.1%

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

      4. unpow2 10.1%

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

      5. hypot-define 25.7%

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

   5. Simplified 25.7%

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

      1. pow1/2 25.7%

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

      2. pow-to-exp 24.3%

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

   7. Applied egg-rr 24.3%

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

   8. Taylor expanded in F around -inf 10.4%

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

      1. mul-1-neg 10.4%

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

      2. +-commutative 10.4%

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

      3. unpow2 10.4%

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

      4. unpow2 10.4%

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

      5. hypot-undefine 33.1%

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

      6. mul-1-neg 33.1%

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

   10. Simplified 33.1%

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

       1. add-cbrt-cube 33.2%

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

       2. pow1/3 33.1%

          \[\leadsto -\frac{\color{blue}{{\left(\left(\sqrt{2} \cdot \sqrt{2}\right) \cdot \sqrt{2}\right)}^{0.3333333333333333}}}{B} \cdot e^{\left(\log \left(-\left(A - \mathsf{hypot}\left(B, A\right)\right)\right) + \left(-\log \left(\frac{-1}{F}\right)\right)\right) \cdot 0.5} \]

       3. pow1/2 33.1%

          \[\leadsto -\frac{{\left(\left(\color{blue}{{2}^{0.5}} \cdot \sqrt{2}\right) \cdot \sqrt{2}\right)}^{0.3333333333333333}}{B} \cdot e^{\left(\log \left(-\left(A - \mathsf{hypot}\left(B, A\right)\right)\right) + \left(-\log \left(\frac{-1}{F}\right)\right)\right) \cdot 0.5} \]

       4. pow1/2 33.1%

          \[\leadsto -\frac{{\left(\left({2}^{0.5} \cdot \color{blue}{{2}^{0.5}}\right) \cdot \sqrt{2}\right)}^{0.3333333333333333}}{B} \cdot e^{\left(\log \left(-\left(A - \mathsf{hypot}\left(B, A\right)\right)\right) + \left(-\log \left(\frac{-1}{F}\right)\right)\right) \cdot 0.5} \]

       5. pow-prod-up 33.1%

          \[\leadsto -\frac{{\left(\color{blue}{{2}^{\left(0.5 + 0.5\right)}} \cdot \sqrt{2}\right)}^{0.3333333333333333}}{B} \cdot e^{\left(\log \left(-\left(A - \mathsf{hypot}\left(B, A\right)\right)\right) + \left(-\log \left(\frac{-1}{F}\right)\right)\right) \cdot 0.5} \]

       6. metadata-eval 33.1%

          \[\leadsto -\frac{{\left({2}^{\color{blue}{1}} \cdot \sqrt{2}\right)}^{0.3333333333333333}}{B} \cdot e^{\left(\log \left(-\left(A - \mathsf{hypot}\left(B, A\right)\right)\right) + \left(-\log \left(\frac{-1}{F}\right)\right)\right) \cdot 0.5} \]

       7. metadata-eval 33.1%

          \[\leadsto -\frac{{\left(\color{blue}{2} \cdot \sqrt{2}\right)}^{0.3333333333333333}}{B} \cdot e^{\left(\log \left(-\left(A - \mathsf{hypot}\left(B, A\right)\right)\right) + \left(-\log \left(\frac{-1}{F}\right)\right)\right) \cdot 0.5} \]

   12. Applied egg-rr 33.1%

       \[\leadsto -\frac{\color{blue}{{\left(2 \cdot \sqrt{2}\right)}^{0.3333333333333333}}}{B} \cdot e^{\left(\log \left(-\left(A - \mathsf{hypot}\left(B, A\right)\right)\right) + \left(-\log \left(\frac{-1}{F}\right)\right)\right) \cdot 0.5} \]
   13. Step-by-step derivation

       1. unpow1/3 33.2%

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

   14. Simplified 33.2%

       \[\leadsto -\frac{\color{blue}{\sqrt[3]{2 \cdot \sqrt{2}}}}{B} \cdot e^{\left(\log \left(-\left(A - \mathsf{hypot}\left(B, A\right)\right)\right) + \left(-\log \left(\frac{-1}{F}\right)\right)\right) \cdot 0.5} \]
3. Recombined 3 regimes into one program.

4. Final simplification 36.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{-223}:\\ \;\;\;\;\frac{\sqrt{\left(4 \cdot A\right) \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;{B}^{2} \leq 4 \cdot 10^{+91}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot \left(2 \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt[3]{2 \cdot \sqrt{2}}}{B} \cdot \left(-e^{\left(\log \left(\mathsf{hypot}\left(B, A\right) - A\right) - \log \left(\frac{-1}{F}\right)\right) \cdot 0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 54.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := F \cdot t\_0\\ t_2 := -t\_0\\ \mathbf{if}\;{B\_m}^{2} \leq 10^{-223}:\\ \;\;\;\;\frac{\sqrt{\left(4 \cdot A\right) \cdot t\_1}}{t\_2}\\ \mathbf{elif}\;{B\_m}^{2} \leq 4 \cdot 10^{+91}:\\ \;\;\;\;\frac{\sqrt{t\_1 \cdot \left(2 \cdot \left(A + \left(C - \mathsf{hypot}\left(B\_m, A - C\right)\right)\right)\right)}}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;e^{0.5 \cdot \left(\log \left(\mathsf{hypot}\left(B\_m, A\right) - A\right) + \log \left(-F\right)\right)} \cdot \frac{-\sqrt{2}}{B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* A (* C -4.0)))) (t_1 (* F t_0)) (t_2 (- t_0)))
   (if (<= (pow B_m 2.0) 1e-223)
     (/ (sqrt (* (* 4.0 A) t_1)) t_2)
     (if (<= (pow B_m 2.0) 4e+91)
       (/ (sqrt (* t_1 (* 2.0 (+ A (- C (hypot B_m (- A C))))))) t_2)
       (*
        (exp (* 0.5 (+ (log (- (hypot B_m A) A)) (log (- F)))))
        (/ (- (sqrt 2.0)) B_m))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (A * (C * -4.0)));
	double t_1 = F * t_0;
	double t_2 = -t_0;
	double tmp;
	if (pow(B_m, 2.0) <= 1e-223) {
		tmp = sqrt(((4.0 * A) * t_1)) / t_2;
	} else if (pow(B_m, 2.0) <= 4e+91) {
		tmp = sqrt((t_1 * (2.0 * (A + (C - hypot(B_m, (A - C))))))) / t_2;
	} else {
		tmp = exp((0.5 * (log((hypot(B_m, A) - A)) + log(-F)))) * (-sqrt(2.0) / B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	t_1 = Float64(F * t_0)
	t_2 = Float64(-t_0)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e-223)
		tmp = Float64(sqrt(Float64(Float64(4.0 * A) * t_1)) / t_2);
	elseif ((B_m ^ 2.0) <= 4e+91)
		tmp = Float64(sqrt(Float64(t_1 * Float64(2.0 * Float64(A + Float64(C - hypot(B_m, Float64(A - C))))))) / t_2);
	else
		tmp = Float64(exp(Float64(0.5 * Float64(log(Float64(hypot(B_m, A) - A)) + log(Float64(-F))))) * Float64(Float64(-sqrt(2.0)) / B_m));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(F * t$95$0), $MachinePrecision]}, Block[{t$95$2 = (-t$95$0)}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-223], N[(N[Sqrt[N[(N[(4.0 * A), $MachinePrecision] * t$95$1), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 4e+91], N[(N[Sqrt[N[(t$95$1 * N[(2.0 * N[(A + N[(C - N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision], N[(N[Exp[N[(0.5 * N[(N[Log[N[(N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision] - A), $MachinePrecision]], $MachinePrecision] + N[Log[(-F)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[((-N[Sqrt[2.0], $MachinePrecision]) / B$95$m), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 9.9999999999999997e-224

   1. Initial program 11.2%

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

   3. Simplified 24.2%

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

   5. Taylor expanded in A around -inf 30.7%

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

   if 9.9999999999999997e-224 < (pow.f64 B #s(literal 2 binary64)) < 4.00000000000000032e91

   1. Initial program 46.4%

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

   3. Simplified 51.6%

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

   if 4.00000000000000032e91 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 9.3%

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

   3. Taylor expanded in C around 0 10.1%

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

      1. mul-1-neg 10.1%

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

      2. +-commutative 10.1%

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

      3. unpow2 10.1%

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

      4. unpow2 10.1%

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

      5. hypot-define 25.7%

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

   5. Simplified 25.7%

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

      1. pow1/2 25.7%

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

      2. pow-to-exp 24.3%

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

   7. Applied egg-rr 24.3%

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

   8. Taylor expanded in F around -inf 10.4%

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

      1. mul-1-neg 10.4%

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

      2. +-commutative 10.4%

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

      3. unpow2 10.4%

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

      4. unpow2 10.4%

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

      5. hypot-undefine 33.1%

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

      6. mul-1-neg 33.1%

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

   10. Simplified 33.1%

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

   11. Taylor expanded in F around -inf 10.4%

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

       1. sub-neg 10.4%

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

       2. +-commutative 10.4%

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

       3. unpow2 10.4%

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

       4. unpow2 10.4%

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

       5. hypot-undefine 33.1%

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

       6. log-rec 33.1%

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

       7. associate-/r/ 33.1%

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

       8. metadata-eval 33.1%

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

       9. neg-mul-1 33.1%

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

   13. Simplified 33.1%

       \[\leadsto -\frac{\sqrt{2}}{B} \cdot e^{\color{blue}{0.5 \cdot \left(\log \left(\mathsf{hypot}\left(B, A\right) - A\right) + \log \left(-F\right)\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 36.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{-223}:\\ \;\;\;\;\frac{\sqrt{\left(4 \cdot A\right) \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;{B}^{2} \leq 4 \cdot 10^{+91}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot \left(2 \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{0.5 \cdot \left(\log \left(\mathsf{hypot}\left(B, A\right) - A\right) + \log \left(-F\right)\right)} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 51.0% accurate, 1.4× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := -t\_0\\ t_2 := F \cdot t\_0\\ \mathbf{if}\;B\_m \leq 1.8 \cdot 10^{-109}:\\ \;\;\;\;\frac{\sqrt{\left(4 \cdot A\right) \cdot t\_2}}{t\_1}\\ \mathbf{elif}\;B\_m \leq 1.25 \cdot 10^{+45}:\\ \;\;\;\;\frac{\sqrt{t\_2 \cdot \left(2 \cdot \left(A + \left(C - \mathsf{hypot}\left(B\_m, A - C\right)\right)\right)\right)}}{t\_1}\\ \mathbf{elif}\;B\_m \leq 2.5 \cdot 10^{+174}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \frac{\mathsf{hypot}\left(A, B\_m\right) - A}{\frac{-1}{F}}}}{-B\_m}\\ \mathbf{else}:\\ \;\;\;\;e^{0.5 \cdot \left(\log \left(-F\right) + \log B\_m\right)} \cdot \frac{-\sqrt{2}}{B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* A (* C -4.0)))) (t_1 (- t_0)) (t_2 (* F t_0)))
   (if (<= B_m 1.8e-109)
     (/ (sqrt (* (* 4.0 A) t_2)) t_1)
     (if (<= B_m 1.25e+45)
       (/ (sqrt (* t_2 (* 2.0 (+ A (- C (hypot B_m (- A C))))))) t_1)
       (if (<= B_m 2.5e+174)
         (/ (sqrt (* 2.0 (/ (- (hypot A B_m) A) (/ -1.0 F)))) (- B_m))
         (*
          (exp (* 0.5 (+ (log (- F)) (log B_m))))
          (/ (- (sqrt 2.0)) B_m)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (A * (C * -4.0)));
	double t_1 = -t_0;
	double t_2 = F * t_0;
	double tmp;
	if (B_m <= 1.8e-109) {
		tmp = sqrt(((4.0 * A) * t_2)) / t_1;
	} else if (B_m <= 1.25e+45) {
		tmp = sqrt((t_2 * (2.0 * (A + (C - hypot(B_m, (A - C))))))) / t_1;
	} else if (B_m <= 2.5e+174) {
		tmp = sqrt((2.0 * ((hypot(A, B_m) - A) / (-1.0 / F)))) / -B_m;
	} else {
		tmp = exp((0.5 * (log(-F) + log(B_m)))) * (-sqrt(2.0) / B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	t_1 = Float64(-t_0)
	t_2 = Float64(F * t_0)
	tmp = 0.0
	if (B_m <= 1.8e-109)
		tmp = Float64(sqrt(Float64(Float64(4.0 * A) * t_2)) / t_1);
	elseif (B_m <= 1.25e+45)
		tmp = Float64(sqrt(Float64(t_2 * Float64(2.0 * Float64(A + Float64(C - hypot(B_m, Float64(A - C))))))) / t_1);
	elseif (B_m <= 2.5e+174)
		tmp = Float64(sqrt(Float64(2.0 * Float64(Float64(hypot(A, B_m) - A) / Float64(-1.0 / F)))) / Float64(-B_m));
	else
		tmp = Float64(exp(Float64(0.5 * Float64(log(Float64(-F)) + log(B_m)))) * Float64(Float64(-sqrt(2.0)) / B_m));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = (-t$95$0)}, Block[{t$95$2 = N[(F * t$95$0), $MachinePrecision]}, If[LessEqual[B$95$m, 1.8e-109], N[(N[Sqrt[N[(N[(4.0 * A), $MachinePrecision] * t$95$2), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[B$95$m, 1.25e+45], N[(N[Sqrt[N[(t$95$2 * N[(2.0 * N[(A + N[(C - N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[B$95$m, 2.5e+174], N[(N[Sqrt[N[(2.0 * N[(N[(N[Sqrt[A ^ 2 + B$95$m ^ 2], $MachinePrecision] - A), $MachinePrecision] / N[(-1.0 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision], N[(N[Exp[N[(0.5 * N[(N[Log[(-F)], $MachinePrecision] + N[Log[B$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[((-N[Sqrt[2.0], $MachinePrecision]) / B$95$m), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if B < 1.8e-109

   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 24.9%

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

   5. Taylor expanded in A around -inf 20.7%

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

   if 1.8e-109 < B < 1.25e45

   1. Initial program 38.6%

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

   3. Simplified 44.1%

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

   if 1.25e45 < B < 2.4999999999999998e174

   1. Initial program 23.2%

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

   3. Taylor expanded in C around 0 45.0%

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

      1. mul-1-neg 45.0%

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

      2. +-commutative 45.0%

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

      3. unpow2 45.0%

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

      4. unpow2 45.0%

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

      5. hypot-define 58.3%

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

   5. Simplified 58.3%

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

      1. pow1/2 58.3%

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

      2. pow-to-exp 55.2%

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

   7. Applied egg-rr 55.2%

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

   8. Taylor expanded in F around -inf 45.9%

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

      1. mul-1-neg 45.9%

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

      2. +-commutative 45.9%

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

      3. unpow2 45.9%

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

      4. unpow2 45.9%

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

      5. hypot-undefine 62.1%

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

      6. mul-1-neg 62.1%

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

   10. Simplified 62.1%

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

       1. neg-sub0 62.1%

          \[\leadsto \color{blue}{0 - \frac{\sqrt{2}}{B} \cdot e^{\left(\log \left(-\left(A - \mathsf{hypot}\left(B, A\right)\right)\right) + \left(-\log \left(\frac{-1}{F}\right)\right)\right) \cdot 0.5}} \]

       2. associate-*l/ 62.1%

          \[\leadsto 0 - \color{blue}{\frac{\sqrt{2} \cdot e^{\left(\log \left(-\left(A - \mathsf{hypot}\left(B, A\right)\right)\right) + \left(-\log \left(\frac{-1}{F}\right)\right)\right) \cdot 0.5}}{B}} \]

   12. Applied egg-rr 58.8%

       \[\leadsto \color{blue}{0 - \frac{{\left(2 \cdot \frac{-\left(A - \mathsf{hypot}\left(B, A\right)\right)}{\frac{-1}{F}}\right)}^{0.5}}{B}} \]
   13. Step-by-step derivation

       1. neg-sub0 58.8%

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

       2. distribute-neg-frac2 58.8%

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

       3. unpow1/2 58.8%

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

       4. distribute-frac-neg 58.8%

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

       5. distribute-neg-frac2 58.8%

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

       6. hypot-undefine 45.5%

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

       7. unpow2 45.5%

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

       8. unpow2 45.5%

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

       9. +-commutative 45.5%

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

       10. unpow2 45.5%

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

       11. unpow2 45.5%

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

       12. hypot-define 58.8%

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

   14. Simplified 58.8%

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

   if 2.4999999999999998e174 < B

   1. Initial program 0.0%

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

   3. Taylor expanded in C around 0 2.3%

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

      1. mul-1-neg 2.3%

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

      2. +-commutative 2.3%

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

      3. unpow2 2.3%

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

      4. unpow2 2.3%

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

      5. hypot-define 49.6%

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

   5. Simplified 49.6%

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

      1. pow1/2 49.6%

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

      2. pow-to-exp 46.4%

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

   7. Applied egg-rr 46.4%

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

   8. Taylor expanded in F around -inf 2.3%

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

      1. mul-1-neg 2.3%

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

      2. +-commutative 2.3%

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

      3. unpow2 2.3%

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

      4. unpow2 2.3%

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

      5. hypot-undefine 74.2%

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

      6. mul-1-neg 74.2%

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

   10. Simplified 74.2%

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

   11. Taylor expanded in A around 0 66.5%

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

       1. sub-neg 66.5%

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

       2. log-rec 66.5%

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

       3. associate-/r/ 66.5%

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

       4. metadata-eval 66.5%

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

       5. neg-mul-1 66.5%

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

   13. Simplified 66.5%

       \[\leadsto -\frac{\sqrt{2}}{B} \cdot e^{\color{blue}{0.5 \cdot \left(\log B + \log \left(-F\right)\right)}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 32.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.8 \cdot 10^{-109}:\\ \;\;\;\;\frac{\sqrt{\left(4 \cdot A\right) \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 1.25 \cdot 10^{+45}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot \left(2 \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 2.5 \cdot 10^{+174}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \frac{\mathsf{hypot}\left(A, B\right) - A}{\frac{-1}{F}}}}{-B}\\ \mathbf{else}:\\ \;\;\;\;e^{0.5 \cdot \left(\log \left(-F\right) + \log B\right)} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 46.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ \mathbf{if}\;{B\_m}^{2} \leq 2 \cdot 10^{-41}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_0\right) \cdot F\right)\right) \cdot \left(2 \cdot A\right)}}{t\_0 - {B\_m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \frac{\mathsf{hypot}\left(A, B\_m\right) - A}{\frac{-1}{F}}}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (* (* 4.0 A) C)))
   (if (<= (pow B_m 2.0) 2e-41)
     (/
      (sqrt (* (* 2.0 (* (- (pow B_m 2.0) t_0) F)) (* 2.0 A)))
      (- t_0 (pow B_m 2.0)))
     (/ (sqrt (* 2.0 (/ (- (hypot A B_m) A) (/ -1.0 F)))) (- B_m)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = (4.0 * A) * C;
	double tmp;
	if (pow(B_m, 2.0) <= 2e-41) {
		tmp = sqrt(((2.0 * ((pow(B_m, 2.0) - t_0) * F)) * (2.0 * A))) / (t_0 - pow(B_m, 2.0));
	} else {
		tmp = sqrt((2.0 * ((hypot(A, B_m) - A) / (-1.0 / F)))) / -B_m;
	}
	return tmp;
}

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double t_0 = (4.0 * A) * C;
	double tmp;
	if (Math.pow(B_m, 2.0) <= 2e-41) {
		tmp = Math.sqrt(((2.0 * ((Math.pow(B_m, 2.0) - t_0) * F)) * (2.0 * A))) / (t_0 - Math.pow(B_m, 2.0));
	} else {
		tmp = Math.sqrt((2.0 * ((Math.hypot(A, B_m) - A) / (-1.0 / F)))) / -B_m;
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	t_0 = (4.0 * A) * C
	tmp = 0
	if math.pow(B_m, 2.0) <= 2e-41:
		tmp = math.sqrt(((2.0 * ((math.pow(B_m, 2.0) - t_0) * F)) * (2.0 * A))) / (t_0 - math.pow(B_m, 2.0))
	else:
		tmp = math.sqrt((2.0 * ((math.hypot(A, B_m) - A) / (-1.0 / F)))) / -B_m
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(Float64(4.0 * A) * C)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 2e-41)
		tmp = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_0) * F)) * Float64(2.0 * A))) / Float64(t_0 - (B_m ^ 2.0)));
	else
		tmp = Float64(sqrt(Float64(2.0 * Float64(Float64(hypot(A, B_m) - A) / Float64(-1.0 / F)))) / Float64(-B_m));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	t_0 = (4.0 * A) * C;
	tmp = 0.0;
	if ((B_m ^ 2.0) <= 2e-41)
		tmp = sqrt(((2.0 * (((B_m ^ 2.0) - t_0) * F)) * (2.0 * A))) / (t_0 - (B_m ^ 2.0));
	else
		tmp = sqrt((2.0 * ((hypot(A, B_m) - A) / (-1.0 / F)))) / -B_m;
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e-41], N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(2.0 * A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(2.0 * N[(N[(N[Sqrt[A ^ 2 + B$95$m ^ 2], $MachinePrecision] - A), $MachinePrecision] / N[(-1.0 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 2.00000000000000001e-41

   1. Initial program 20.7%

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

   3. Taylor expanded in A around -inf 28.3%

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

   if 2.00000000000000001e-41 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 17.3%

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

   3. Taylor expanded in C around 0 12.9%

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

      1. mul-1-neg 12.9%

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

      2. +-commutative 12.9%

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

      3. unpow2 12.9%

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

      4. unpow2 12.9%

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

      5. hypot-define 25.3%

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

   5. Simplified 25.3%

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

      1. pow1/2 25.3%

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

      2. pow-to-exp 23.9%

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

   7. Applied egg-rr 23.9%

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

   8. Taylor expanded in F around -inf 12.8%

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

      1. mul-1-neg 12.8%

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

      2. +-commutative 12.8%

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

      3. unpow2 12.8%

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

      4. unpow2 12.8%

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

      5. hypot-undefine 30.9%

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

      6. mul-1-neg 30.9%

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

   10. Simplified 30.9%

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

       1. neg-sub0 30.9%

          \[\leadsto \color{blue}{0 - \frac{\sqrt{2}}{B} \cdot e^{\left(\log \left(-\left(A - \mathsf{hypot}\left(B, A\right)\right)\right) + \left(-\log \left(\frac{-1}{F}\right)\right)\right) \cdot 0.5}} \]

       2. associate-*l/ 30.9%

          \[\leadsto 0 - \color{blue}{\frac{\sqrt{2} \cdot e^{\left(\log \left(-\left(A - \mathsf{hypot}\left(B, A\right)\right)\right) + \left(-\log \left(\frac{-1}{F}\right)\right)\right) \cdot 0.5}}{B}} \]

   12. Applied egg-rr 25.5%

       \[\leadsto \color{blue}{0 - \frac{{\left(2 \cdot \frac{-\left(A - \mathsf{hypot}\left(B, A\right)\right)}{\frac{-1}{F}}\right)}^{0.5}}{B}} \]
   13. Step-by-step derivation

       1. neg-sub0 25.5%

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

       2. distribute-neg-frac2 25.5%

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

       3. unpow1/2 25.5%

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

       4. distribute-frac-neg 25.5%

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

       5. distribute-neg-frac2 25.5%

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

       6. hypot-undefine 13.0%

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

       7. unpow2 13.0%

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

       8. unpow2 13.0%

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

       9. +-commutative 13.0%

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

       10. unpow2 13.0%

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

       11. unpow2 13.0%

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

       12. hypot-define 25.5%

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

   14. Simplified 25.5%

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

4. Final simplification 26.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 2 \cdot 10^{-41}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot A\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \frac{\mathsf{hypot}\left(A, B\right) - A}{\frac{-1}{F}}}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 46.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 2 \cdot 10^{-41}:\\ \;\;\;\;\frac{\sqrt{\left(4 \cdot A\right) \cdot \left(F \cdot t\_0\right)}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \frac{\mathsf{hypot}\left(A, B\_m\right) - A}{\frac{-1}{F}}}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* A (* C -4.0)))))
   (if (<= (pow B_m 2.0) 2e-41)
     (/ (sqrt (* (* 4.0 A) (* F t_0))) (- t_0))
     (/ (sqrt (* 2.0 (/ (- (hypot A B_m) A) (/ -1.0 F)))) (- B_m)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (A * (C * -4.0)));
	double tmp;
	if (pow(B_m, 2.0) <= 2e-41) {
		tmp = sqrt(((4.0 * A) * (F * t_0))) / -t_0;
	} else {
		tmp = sqrt((2.0 * ((hypot(A, B_m) - A) / (-1.0 / F)))) / -B_m;
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 2e-41)
		tmp = Float64(sqrt(Float64(Float64(4.0 * A) * Float64(F * t_0))) / Float64(-t_0));
	else
		tmp = Float64(sqrt(Float64(2.0 * Float64(Float64(hypot(A, B_m) - A) / Float64(-1.0 / F)))) / Float64(-B_m));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e-41], N[(N[Sqrt[N[(N[(4.0 * A), $MachinePrecision] * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], N[(N[Sqrt[N[(2.0 * N[(N[(N[Sqrt[A ^ 2 + B$95$m ^ 2], $MachinePrecision] - A), $MachinePrecision] / N[(-1.0 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 2.00000000000000001e-41

   1. Initial program 20.7%

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

   3. Simplified 31.3%

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

   5. Taylor expanded in A around -inf 28.3%

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

   if 2.00000000000000001e-41 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 17.3%

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

   3. Taylor expanded in C around 0 12.9%

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

      1. mul-1-neg 12.9%

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

      2. +-commutative 12.9%

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

      3. unpow2 12.9%

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

      4. unpow2 12.9%

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

      5. hypot-define 25.3%

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

   5. Simplified 25.3%

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

      1. pow1/2 25.3%

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

      2. pow-to-exp 23.9%

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

   7. Applied egg-rr 23.9%

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

   8. Taylor expanded in F around -inf 12.8%

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

      1. mul-1-neg 12.8%

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

      2. +-commutative 12.8%

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

      3. unpow2 12.8%

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

      4. unpow2 12.8%

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

      5. hypot-undefine 30.9%

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

      6. mul-1-neg 30.9%

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

   10. Simplified 30.9%

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

       1. neg-sub0 30.9%

          \[\leadsto \color{blue}{0 - \frac{\sqrt{2}}{B} \cdot e^{\left(\log \left(-\left(A - \mathsf{hypot}\left(B, A\right)\right)\right) + \left(-\log \left(\frac{-1}{F}\right)\right)\right) \cdot 0.5}} \]

       2. associate-*l/ 30.9%

          \[\leadsto 0 - \color{blue}{\frac{\sqrt{2} \cdot e^{\left(\log \left(-\left(A - \mathsf{hypot}\left(B, A\right)\right)\right) + \left(-\log \left(\frac{-1}{F}\right)\right)\right) \cdot 0.5}}{B}} \]

   12. Applied egg-rr 25.5%

       \[\leadsto \color{blue}{0 - \frac{{\left(2 \cdot \frac{-\left(A - \mathsf{hypot}\left(B, A\right)\right)}{\frac{-1}{F}}\right)}^{0.5}}{B}} \]
   13. Step-by-step derivation

       1. neg-sub0 25.5%

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

       2. distribute-neg-frac2 25.5%

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

       3. unpow1/2 25.5%

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

       4. distribute-frac-neg 25.5%

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

       5. distribute-neg-frac2 25.5%

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

       6. hypot-undefine 13.0%

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

       7. unpow2 13.0%

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

       8. unpow2 13.0%

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

       9. +-commutative 13.0%

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

       10. unpow2 13.0%

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

       11. unpow2 13.0%

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

       12. hypot-define 25.5%

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

   14. Simplified 25.5%

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

4. Final simplification 26.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 2 \cdot 10^{-41}:\\ \;\;\;\;\frac{\sqrt{\left(4 \cdot A\right) \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \frac{\mathsf{hypot}\left(A, B\right) - A}{\frac{-1}{F}}}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 42.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 10^{-223}:\\ \;\;\;\;\frac{\sqrt{\left(A \cdot -8\right) \cdot \left(C \cdot \left(F \cdot \left(A + A\right)\right)\right)}}{-\mathsf{fma}\left(C, A \cdot -4, {B\_m}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \frac{\mathsf{hypot}\left(A, B\_m\right) - A}{\frac{-1}{F}}}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 1e-223)
   (/
    (sqrt (* (* A -8.0) (* C (* F (+ A A)))))
    (- (fma C (* A -4.0) (pow B_m 2.0))))
   (/ (sqrt (* 2.0 (/ (- (hypot A B_m) A) (/ -1.0 F)))) (- B_m))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (pow(B_m, 2.0) <= 1e-223) {
		tmp = sqrt(((A * -8.0) * (C * (F * (A + A))))) / -fma(C, (A * -4.0), pow(B_m, 2.0));
	} else {
		tmp = sqrt((2.0 * ((hypot(A, B_m) - A) / (-1.0 / F)))) / -B_m;
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e-223)
		tmp = Float64(sqrt(Float64(Float64(A * -8.0) * Float64(C * Float64(F * Float64(A + A))))) / Float64(-fma(C, Float64(A * -4.0), (B_m ^ 2.0))));
	else
		tmp = Float64(sqrt(Float64(2.0 * Float64(Float64(hypot(A, B_m) - A) / Float64(-1.0 / F)))) / Float64(-B_m));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e-223], N[(N[Sqrt[N[(N[(A * -8.0), $MachinePrecision] * N[(C * N[(F * N[(A + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-N[(C * N[(A * -4.0), $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[N[(2.0 * N[(N[(N[Sqrt[A ^ 2 + B$95$m ^ 2], $MachinePrecision] - A), $MachinePrecision] / N[(-1.0 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 9.9999999999999997e-224

   1. Initial program 11.2%

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

   3. Simplified 21.8%

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

   5. Taylor expanded in C around inf 28.6%

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

      1. associate-*r* 28.6%

         \[\leadsto \frac{\sqrt{\color{blue}{\left(-8 \cdot A\right) \cdot \left(C \cdot \left(F \cdot \left(A - -1 \cdot A\right)\right)\right)}}}{-\mathsf{fma}\left(C, A \cdot -4, {B}^{2}\right)} \]

      2. mul-1-neg 28.6%

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

   7. Simplified 28.6%

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

   if 9.9999999999999997e-224 < (pow.f64 B #s(literal 2 binary64))

   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. Add Preprocessing

   3. Taylor expanded in C around 0 13.4%

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

      1. mul-1-neg 13.4%

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

      2. +-commutative 13.4%

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

      3. unpow2 13.4%

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

      4. unpow2 13.4%

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

      5. hypot-define 23.4%

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

   5. Simplified 23.4%

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

      1. pow1/2 23.4%

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

      2. pow-to-exp 22.2%

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

   7. Applied egg-rr 22.2%

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

   8. Taylor expanded in F around -inf 13.2%

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

      1. mul-1-neg 13.2%

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

      2. +-commutative 13.2%

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

      3. unpow2 13.2%

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

      4. unpow2 13.2%

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

      5. hypot-undefine 27.7%

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

      6. mul-1-neg 27.7%

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

   10. Simplified 27.7%

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

       1. neg-sub0 27.7%

          \[\leadsto \color{blue}{0 - \frac{\sqrt{2}}{B} \cdot e^{\left(\log \left(-\left(A - \mathsf{hypot}\left(B, A\right)\right)\right) + \left(-\log \left(\frac{-1}{F}\right)\right)\right) \cdot 0.5}} \]

       2. associate-*l/ 27.7%

          \[\leadsto 0 - \color{blue}{\frac{\sqrt{2} \cdot e^{\left(\log \left(-\left(A - \mathsf{hypot}\left(B, A\right)\right)\right) + \left(-\log \left(\frac{-1}{F}\right)\right)\right) \cdot 0.5}}{B}} \]

   12. Applied egg-rr 23.5%

       \[\leadsto \color{blue}{0 - \frac{{\left(2 \cdot \frac{-\left(A - \mathsf{hypot}\left(B, A\right)\right)}{\frac{-1}{F}}\right)}^{0.5}}{B}} \]
   13. Step-by-step derivation

       1. neg-sub0 23.5%

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

       2. distribute-neg-frac2 23.5%

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

       3. unpow1/2 23.5%

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

       4. distribute-frac-neg 23.5%

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

       5. distribute-neg-frac2 23.5%

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

       6. hypot-undefine 13.5%

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

       7. unpow2 13.5%

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

       8. unpow2 13.5%

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

       9. +-commutative 13.5%

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

       10. unpow2 13.5%

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

       11. unpow2 13.5%

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

       12. hypot-define 23.5%

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

   14. Simplified 23.5%

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

4. Final simplification 25.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{-223}:\\ \;\;\;\;\frac{\sqrt{\left(A \cdot -8\right) \cdot \left(C \cdot \left(F \cdot \left(A + A\right)\right)\right)}}{-\mathsf{fma}\left(C, A \cdot -4, {B}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \frac{\mathsf{hypot}\left(A, B\right) - A}{\frac{-1}{F}}}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 50.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ \mathbf{if}\;B\_m \leq 9 \cdot 10^{-21}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_0\right) \cdot F\right)\right) \cdot \left(2 \cdot A\right)}}{t\_0 - {B\_m}^{2}}\\ \mathbf{elif}\;B\_m \leq 2.7 \cdot 10^{+175}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \frac{\mathsf{hypot}\left(A, B\_m\right) - A}{\frac{-1}{F}}}}{-B\_m}\\ \mathbf{else}:\\ \;\;\;\;e^{0.5 \cdot \left(\log \left(-F\right) + \log B\_m\right)} \cdot \frac{-\sqrt{2}}{B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (* (* 4.0 A) C)))
   (if (<= B_m 9e-21)
     (/
      (sqrt (* (* 2.0 (* (- (pow B_m 2.0) t_0) F)) (* 2.0 A)))
      (- t_0 (pow B_m 2.0)))
     (if (<= B_m 2.7e+175)
       (/ (sqrt (* 2.0 (/ (- (hypot A B_m) A) (/ -1.0 F)))) (- B_m))
       (* (exp (* 0.5 (+ (log (- F)) (log B_m)))) (/ (- (sqrt 2.0)) B_m))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = (4.0 * A) * C;
	double tmp;
	if (B_m <= 9e-21) {
		tmp = sqrt(((2.0 * ((pow(B_m, 2.0) - t_0) * F)) * (2.0 * A))) / (t_0 - pow(B_m, 2.0));
	} else if (B_m <= 2.7e+175) {
		tmp = sqrt((2.0 * ((hypot(A, B_m) - A) / (-1.0 / F)))) / -B_m;
	} else {
		tmp = exp((0.5 * (log(-F) + log(B_m)))) * (-sqrt(2.0) / B_m);
	}
	return tmp;
}

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double t_0 = (4.0 * A) * C;
	double tmp;
	if (B_m <= 9e-21) {
		tmp = Math.sqrt(((2.0 * ((Math.pow(B_m, 2.0) - t_0) * F)) * (2.0 * A))) / (t_0 - Math.pow(B_m, 2.0));
	} else if (B_m <= 2.7e+175) {
		tmp = Math.sqrt((2.0 * ((Math.hypot(A, B_m) - A) / (-1.0 / F)))) / -B_m;
	} else {
		tmp = Math.exp((0.5 * (Math.log(-F) + Math.log(B_m)))) * (-Math.sqrt(2.0) / B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	t_0 = (4.0 * A) * C
	tmp = 0
	if B_m <= 9e-21:
		tmp = math.sqrt(((2.0 * ((math.pow(B_m, 2.0) - t_0) * F)) * (2.0 * A))) / (t_0 - math.pow(B_m, 2.0))
	elif B_m <= 2.7e+175:
		tmp = math.sqrt((2.0 * ((math.hypot(A, B_m) - A) / (-1.0 / F)))) / -B_m
	else:
		tmp = math.exp((0.5 * (math.log(-F) + math.log(B_m)))) * (-math.sqrt(2.0) / B_m)
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(Float64(4.0 * A) * C)
	tmp = 0.0
	if (B_m <= 9e-21)
		tmp = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_0) * F)) * Float64(2.0 * A))) / Float64(t_0 - (B_m ^ 2.0)));
	elseif (B_m <= 2.7e+175)
		tmp = Float64(sqrt(Float64(2.0 * Float64(Float64(hypot(A, B_m) - A) / Float64(-1.0 / F)))) / Float64(-B_m));
	else
		tmp = Float64(exp(Float64(0.5 * Float64(log(Float64(-F)) + log(B_m)))) * Float64(Float64(-sqrt(2.0)) / B_m));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	t_0 = (4.0 * A) * C;
	tmp = 0.0;
	if (B_m <= 9e-21)
		tmp = sqrt(((2.0 * (((B_m ^ 2.0) - t_0) * F)) * (2.0 * A))) / (t_0 - (B_m ^ 2.0));
	elseif (B_m <= 2.7e+175)
		tmp = sqrt((2.0 * ((hypot(A, B_m) - A) / (-1.0 / F)))) / -B_m;
	else
		tmp = exp((0.5 * (log(-F) + log(B_m)))) * (-sqrt(2.0) / B_m);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, If[LessEqual[B$95$m, 9e-21], N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(2.0 * A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 2.7e+175], N[(N[Sqrt[N[(2.0 * N[(N[(N[Sqrt[A ^ 2 + B$95$m ^ 2], $MachinePrecision] - A), $MachinePrecision] / N[(-1.0 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision], N[(N[Exp[N[(0.5 * N[(N[Log[(-F)], $MachinePrecision] + N[Log[B$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[((-N[Sqrt[2.0], $MachinePrecision]) / B$95$m), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if B < 8.99999999999999936e-21

   1. Initial program 19.4%

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

   3. Taylor expanded in A around -inf 19.6%

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

   if 8.99999999999999936e-21 < B < 2.7000000000000001e175

   1. Initial program 30.2%

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

   3. Taylor expanded in C around 0 41.1%

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

      1. mul-1-neg 41.1%

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

      2. +-commutative 41.1%

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

      3. unpow2 41.1%

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

      4. unpow2 41.1%

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

      5. hypot-define 49.3%

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

   5. Simplified 49.3%

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

      1. pow1/2 49.3%

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

      2. pow-to-exp 46.6%

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

   7. Applied egg-rr 46.6%

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

   8. Taylor expanded in F around -inf 40.8%

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

      1. mul-1-neg 40.8%

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

      2. +-commutative 40.8%

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

      3. unpow2 40.8%

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

      4. unpow2 40.8%

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

      5. hypot-undefine 50.8%

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

      6. mul-1-neg 50.8%

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

   10. Simplified 50.8%

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

       1. neg-sub0 50.8%

          \[\leadsto \color{blue}{0 - \frac{\sqrt{2}}{B} \cdot e^{\left(\log \left(-\left(A - \mathsf{hypot}\left(B, A\right)\right)\right) + \left(-\log \left(\frac{-1}{F}\right)\right)\right) \cdot 0.5}} \]

       2. associate-*l/ 50.8%

          \[\leadsto 0 - \color{blue}{\frac{\sqrt{2} \cdot e^{\left(\log \left(-\left(A - \mathsf{hypot}\left(B, A\right)\right)\right) + \left(-\log \left(\frac{-1}{F}\right)\right)\right) \cdot 0.5}}{B}} \]

   12. Applied egg-rr 49.7%

       \[\leadsto \color{blue}{0 - \frac{{\left(2 \cdot \frac{-\left(A - \mathsf{hypot}\left(B, A\right)\right)}{\frac{-1}{F}}\right)}^{0.5}}{B}} \]
   13. Step-by-step derivation

       1. neg-sub0 49.7%

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

       2. distribute-neg-frac2 49.7%

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

       3. unpow1/2 49.7%

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

       4. distribute-frac-neg 49.7%

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

       5. distribute-neg-frac2 49.7%

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

       6. hypot-undefine 41.5%

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

       7. unpow2 41.5%

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

       8. unpow2 41.5%

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

       9. +-commutative 41.5%

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

       10. unpow2 41.5%

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

       11. unpow2 41.5%

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

       12. hypot-define 49.7%

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

   14. Simplified 49.7%

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

   if 2.7000000000000001e175 < B

   1. Initial program 0.0%

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

   3. Taylor expanded in C around 0 2.3%

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

      1. mul-1-neg 2.3%

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

      2. +-commutative 2.3%

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

      3. unpow2 2.3%

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

      4. unpow2 2.3%

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

      5. hypot-define 49.6%

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

   5. Simplified 49.6%

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

      1. pow1/2 49.6%

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

      2. pow-to-exp 46.4%

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

   7. Applied egg-rr 46.4%

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

   8. Taylor expanded in F around -inf 2.3%

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

      1. mul-1-neg 2.3%

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

      2. +-commutative 2.3%

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

      3. unpow2 2.3%

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

      4. unpow2 2.3%

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

      5. hypot-undefine 74.2%

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

      6. mul-1-neg 74.2%

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

   10. Simplified 74.2%

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

   11. Taylor expanded in A around 0 66.5%

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

       1. sub-neg 66.5%

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

       2. log-rec 66.5%

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

       3. associate-/r/ 66.5%

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

       4. metadata-eval 66.5%

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

       5. neg-mul-1 66.5%

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

   13. Simplified 66.5%

       \[\leadsto -\frac{\sqrt{2}}{B} \cdot e^{\color{blue}{0.5 \cdot \left(\log B + \log \left(-F\right)\right)}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 29.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 9 \cdot 10^{-21}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot A\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;B \leq 2.7 \cdot 10^{+175}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \frac{\mathsf{hypot}\left(A, B\right) - A}{\frac{-1}{F}}}}{-B}\\ \mathbf{else}:\\ \;\;\;\;e^{0.5 \cdot \left(\log \left(-F\right) + \log B\right)} \cdot \frac{-\sqrt{2}}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 30.8% accurate, 3.0× speedup

Math

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

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (/ (sqrt (* 2.0 (/ (- (hypot A B_m) A) (/ -1.0 F)))) (- B_m)))

C

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

Java

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

Python

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

Julia

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

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp = code(A, B_m, C, F)
	tmp = sqrt((2.0 * ((hypot(A, B_m) - A) / (-1.0 / F)))) / -B_m;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := N[(N[Sqrt[N[(2.0 * N[(N[(N[Sqrt[A ^ 2 + B$95$m ^ 2], $MachinePrecision] - A), $MachinePrecision] / N[(-1.0 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision]

Derivation

1. Initial program 18.9%

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

3. Taylor expanded in C around 0 10.4%

   \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}\right)} \]
4. 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. +-commutative 10.4%

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

   3. unpow2 10.4%

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

   4. unpow2 10.4%

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

   5. hypot-define 17.0%

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

5. Simplified 17.0%

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

   1. pow1/2 17.0%

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

   2. pow-to-exp 16.1%

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

7. Applied egg-rr 16.1%

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

8. Taylor expanded in F around -inf 10.4%

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

   1. mul-1-neg 10.4%

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

   2. +-commutative 10.4%

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

   3. unpow2 10.4%

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

   4. unpow2 10.4%

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

   5. hypot-undefine 20.0%

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

   6. mul-1-neg 20.0%

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

10. Simplified 20.0%

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

    1. neg-sub0 20.0%

       \[\leadsto \color{blue}{0 - \frac{\sqrt{2}}{B} \cdot e^{\left(\log \left(-\left(A - \mathsf{hypot}\left(B, A\right)\right)\right) + \left(-\log \left(\frac{-1}{F}\right)\right)\right) \cdot 0.5}} \]

    2. associate-*l/ 20.0%

       \[\leadsto 0 - \color{blue}{\frac{\sqrt{2} \cdot e^{\left(\log \left(-\left(A - \mathsf{hypot}\left(B, A\right)\right)\right) + \left(-\log \left(\frac{-1}{F}\right)\right)\right) \cdot 0.5}}{B}} \]

12. Applied egg-rr 17.1%

    \[\leadsto \color{blue}{0 - \frac{{\left(2 \cdot \frac{-\left(A - \mathsf{hypot}\left(B, A\right)\right)}{\frac{-1}{F}}\right)}^{0.5}}{B}} \]
13. Step-by-step derivation

    1. neg-sub0 17.1%

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

    2. distribute-neg-frac2 17.1%

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

    3. unpow1/2 17.0%

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

    4. distribute-frac-neg 17.0%

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

    5. distribute-neg-frac2 17.0%

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

    6. hypot-undefine 10.5%

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

    7. unpow2 10.5%

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

    8. unpow2 10.5%

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

    9. +-commutative 10.5%

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

    10. unpow2 10.5%

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

    11. unpow2 10.5%

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

    12. hypot-define 17.0%

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

14. Simplified 17.0%

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

15. Final simplification 17.0%

    \[\leadsto \frac{\sqrt{2 \cdot \frac{\mathsf{hypot}\left(A, B\right) - A}{\frac{-1}{F}}}}{-B} \]
16. Add Preprocessing

Alternative 10: 30.8% accurate, 3.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \frac{\sqrt{2 \cdot \left(F \cdot \left(A - \mathsf{hypot}\left(B\_m, A\right)\right)\right)}}{-B\_m} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (/ (sqrt (* 2.0 (* F (- A (hypot B_m A))))) (- B_m)))

C

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

Java

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

Python

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

Julia

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

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp = code(A, B_m, C, F)
	tmp = sqrt((2.0 * (F * (A - hypot(B_m, A))))) / -B_m;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := N[(N[Sqrt[N[(2.0 * N[(F * N[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision]

Derivation

1. Initial program 18.9%

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

3. Taylor expanded in C around 0 10.4%

   \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}\right)} \]
4. 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. +-commutative 10.4%

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

   3. unpow2 10.4%

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

   4. unpow2 10.4%

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

   5. hypot-define 17.0%

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

5. Simplified 17.0%

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

   1. neg-sub0 17.0%

      \[\leadsto \color{blue}{0 - \frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}} \]

   2. associate-*l/ 17.0%

      \[\leadsto 0 - \color{blue}{\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}}{B}} \]

   3. pow1/2 17.0%

      \[\leadsto 0 - \frac{\color{blue}{{2}^{0.5}} \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}}{B} \]

   4. pow1/2 17.0%

      \[\leadsto 0 - \frac{{2}^{0.5} \cdot \color{blue}{{\left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}^{0.5}}}{B} \]

   5. pow-prod-down 17.1%

      \[\leadsto 0 - \frac{\color{blue}{{\left(2 \cdot \left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)\right)}^{0.5}}}{B} \]

7. Applied egg-rr 17.1%

   \[\leadsto \color{blue}{0 - \frac{{\left(2 \cdot \left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)\right)}^{0.5}}{B}} \]
8. Step-by-step derivation

   1. neg-sub0 17.1%

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

   2. distribute-neg-frac2 17.1%

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

   3. unpow1/2 17.1%

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

9. Simplified 17.1%

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

Alternative 11: 26.6% accurate, 5.6× speedup

Math

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

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= A -5e+220)
   (* (sqrt (* A F)) (/ -2.0 B_m))
   (/ (sqrt (* -2.0 (* B_m F))) (- B_m))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (A <= -5e+220) {
		tmp = sqrt((A * F)) * (-2.0 / B_m);
	} else {
		tmp = sqrt((-2.0 * (B_m * F))) / -B_m;
	}
	return tmp;
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (a <= (-5d+220)) then
        tmp = sqrt((a * f)) * ((-2.0d0) / b_m)
    else
        tmp = sqrt(((-2.0d0) * (b_m * f))) / -b_m
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (A <= -5e+220) {
		tmp = Math.sqrt((A * F)) * (-2.0 / B_m);
	} else {
		tmp = Math.sqrt((-2.0 * (B_m * F))) / -B_m;
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if A <= -5e+220:
		tmp = math.sqrt((A * F)) * (-2.0 / B_m)
	else:
		tmp = math.sqrt((-2.0 * (B_m * F))) / -B_m
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (A <= -5e+220)
		tmp = Float64(sqrt(Float64(A * F)) * Float64(-2.0 / B_m));
	else
		tmp = Float64(sqrt(Float64(-2.0 * Float64(B_m * F))) / Float64(-B_m));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (A <= -5e+220)
		tmp = sqrt((A * F)) * (-2.0 / B_m);
	else
		tmp = sqrt((-2.0 * (B_m * F))) / -B_m;
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[A, -5e+220], N[(N[Sqrt[N[(A * F), $MachinePrecision]], $MachinePrecision] * N[(-2.0 / B$95$m), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(-2.0 * N[(B$95$m * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if A < -5.0000000000000002e220

   1. Initial program 1.9%

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

   3. Taylor expanded in C around 0 1.2%

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

      1. mul-1-neg 1.2%

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

      2. +-commutative 1.2%

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

      3. unpow2 1.2%

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

      4. unpow2 1.2%

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

      5. hypot-define 17.7%

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

   5. Simplified 17.7%

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

      1. pow1/2 18.2%

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

      2. pow-to-exp 17.3%

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

   7. Applied egg-rr 17.3%

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

   8. Taylor expanded in A around -inf 0.0%

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

      1. *-commutative 0.0%

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

      2. unpow2 0.0%

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

      3. rem-square-sqrt 17.5%

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

      4. unpow2 17.5%

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

      5. rem-square-sqrt 17.8%

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

      6. metadata-eval 17.8%

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

   10. Simplified 17.8%

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

   if -5.0000000000000002e220 < A

   1. Initial program 20.2%

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

   3. Taylor expanded in C around 0 11.1%

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

      1. mul-1-neg 11.1%

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

      2. +-commutative 11.1%

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

      3. unpow2 11.1%

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

      4. unpow2 11.1%

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

      5. hypot-define 16.9%

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

   5. Simplified 16.9%

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

      1. neg-sub0 16.9%

         \[\leadsto \color{blue}{0 - \frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}} \]

      2. associate-*l/ 16.9%

         \[\leadsto 0 - \color{blue}{\frac{\sqrt{2} \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}}{B}} \]

      3. pow1/2 16.9%

         \[\leadsto 0 - \frac{\color{blue}{{2}^{0.5}} \cdot \sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}}{B} \]

      4. pow1/2 16.9%

         \[\leadsto 0 - \frac{{2}^{0.5} \cdot \color{blue}{{\left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}^{0.5}}}{B} \]

      5. pow-prod-down 17.0%

         \[\leadsto 0 - \frac{\color{blue}{{\left(2 \cdot \left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)\right)}^{0.5}}}{B} \]

   7. Applied egg-rr 17.0%

      \[\leadsto \color{blue}{0 - \frac{{\left(2 \cdot \left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)\right)}^{0.5}}{B}} \]
   8. Step-by-step derivation

      1. neg-sub0 17.0%

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

      2. distribute-neg-frac2 17.0%

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

      3. unpow1/2 17.0%

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

   9. Simplified 17.0%

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

   10. Taylor expanded in A around 0 16.2%

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

4. Final simplification 16.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -5 \cdot 10^{+220}:\\ \;\;\;\;\sqrt{A \cdot F} \cdot \frac{-2}{B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{-2 \cdot \left(B \cdot F\right)}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 8.8% accurate, 5.9× speedup

Math

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

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F) :precision binary64 (* (sqrt (* A F)) (/ -2.0 B_m)))

C

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

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    code = sqrt((a * f)) * ((-2.0d0) / b_m)
end function

Java

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

Python

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

Julia

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

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp = code(A, B_m, C, F)
	tmp = sqrt((A * F)) * (-2.0 / B_m);
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := N[(N[Sqrt[N[(A * F), $MachinePrecision]], $MachinePrecision] * N[(-2.0 / B$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 18.9%

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

3. Taylor expanded in C around 0 10.4%

   \[\leadsto \color{blue}{-1 \cdot \left(\frac{\sqrt{2}}{B} \cdot \sqrt{F \cdot \left(A - \sqrt{{A}^{2} + {B}^{2}}\right)}\right)} \]
4. 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. +-commutative 10.4%

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

   3. unpow2 10.4%

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

   4. unpow2 10.4%

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

   5. hypot-define 17.0%

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

5. Simplified 17.0%

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

   1. pow1/2 17.0%

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

   2. pow-to-exp 16.1%

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

7. Applied egg-rr 16.1%

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

8. Taylor expanded in A around -inf 0.0%

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

   1. *-commutative 0.0%

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

   2. unpow2 0.0%

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

   3. rem-square-sqrt 2.9%

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

   4. unpow2 2.9%

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

   5. rem-square-sqrt 2.9%

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

   6. metadata-eval 2.9%

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

10. Simplified 2.9%

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

11. Final simplification 2.9%

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

Alternative 13: 1.7% accurate, 6.0× speedup

Math

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

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F) :precision binary64 (pow (* 2.0 (/ F B_m)) 0.5))

C

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

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    code = (2.0d0 * (f / b_m)) ** 0.5d0
end function

Java

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

Python

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

Julia

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

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp = code(A, B_m, C, F)
	tmp = (2.0 * (F / B_m)) ^ 0.5;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := N[Power[N[(2.0 * N[(F / B$95$m), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]

Derivation

1. Initial program 18.9%

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

3. Taylor expanded in B around -inf 0.0%

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

   1. mul-1-neg 0.0%

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

   2. unpow2 0.0%

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

   3. rem-square-sqrt 1.6%

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

5. Simplified 1.6%

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

6. Taylor expanded in F around 0 1.6%

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

   1. sqrt-unprod 1.6%

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

   2. pow1/2 1.9%

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

8. Applied egg-rr 1.9%

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

9. Final simplification 1.9%

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

Alternative 14: 1.5% accurate, 6.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \sqrt{\frac{2 \cdot F}{B\_m}} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F) :precision binary64 (sqrt (/ (* 2.0 F) B_m)))

C

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

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    code = sqrt(((2.0d0 * f) / b_m))
end function

Java

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

Python

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

Julia

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

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp = code(A, B_m, C, F)
	tmp = sqrt(((2.0 * F) / B_m));
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] / B$95$m), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 18.9%

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

3. Taylor expanded in B around -inf 0.0%

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

   1. mul-1-neg 0.0%

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

   2. unpow2 0.0%

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

   3. rem-square-sqrt 1.6%

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

5. Simplified 1.6%

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

6. Taylor expanded in F around 0 1.6%

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

   1. pow1 1.6%

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

   2. sqrt-unprod 1.6%

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

8. Applied egg-rr 1.6%

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

   1. unpow1 1.6%

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

   2. *-commutative 1.6%

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

   3. associate-*r/ 1.6%

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

10. Simplified 1.6%

    \[\leadsto \color{blue}{\sqrt{\frac{2 \cdot F}{B}}} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024165 
(FPCore (A B C F)
  :name "ABCF->ab-angle b"
  :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))))