ABCF->ab-angle b

Percentage Accurate: 19.3% → 56.9%

Time: 26.1s

Alternatives: 18

Speedup: 5.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 19.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 56.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := \sqrt[3]{\frac{-1}{F}}\\ t_2 := \left(4 \cdot A\right) \cdot C\\ t_3 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_2\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_2 - {B\_m}^{2}}\\ t_4 := \mathsf{fma}\left(C, A \cdot -4, {B\_m}^{2}\right)\\ \mathbf{if}\;t\_3 \leq -2 \cdot 10^{-204}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B\_m, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot t\_4}}{-t\_4}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot t\_0\right) \cdot \left(2 \cdot \left(A + \left(A + -0.5 \cdot \frac{{B\_m}^{2}}{C}\right)\right)\right)}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(\log \left(\mathsf{hypot}\left(B\_m, A\right) - A\right) - \left(\log \left({t\_1}^{2}\right) + \log t\_1\right)\right) \cdot 0.5} \cdot \frac{\sqrt{2}}{-B\_m}\\ \end{array} \end{array} \]

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 (cbrt (/ -1.0 F)))
        (t_2 (* (* 4.0 A) C))
        (t_3
         (/
          (sqrt
           (*
            (* 2.0 (* (- (pow B_m 2.0) t_2) F))
            (- (+ A C) (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))))))
          (- t_2 (pow B_m 2.0))))
        (t_4 (fma C (* A -4.0) (pow B_m 2.0))))
   (if (<= t_3 -2e-204)
     (/
      (* (sqrt (* F (+ A (- C (hypot B_m (- A C)))))) (sqrt (* 2.0 t_4)))
      (- t_4))
     (if (<= t_3 INFINITY)
       (/
        (sqrt (* (* F t_0) (* 2.0 (+ A (+ A (* -0.5 (/ (pow B_m 2.0) C)))))))
        (- t_0))
       (*
        (exp
         (*
          (- (log (- (hypot B_m A) A)) (+ (log (pow t_1 2.0)) (log t_1)))
          0.5))
        (/ (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 = cbrt((-1.0 / F));
	double t_2 = (4.0 * A) * C;
	double t_3 = sqrt(((2.0 * ((pow(B_m, 2.0) - t_2) * F)) * ((A + C) - sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / (t_2 - pow(B_m, 2.0));
	double t_4 = fma(C, (A * -4.0), pow(B_m, 2.0));
	double tmp;
	if (t_3 <= -2e-204) {
		tmp = (sqrt((F * (A + (C - hypot(B_m, (A - C)))))) * sqrt((2.0 * t_4))) / -t_4;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = sqrt(((F * t_0) * (2.0 * (A + (A + (-0.5 * (pow(B_m, 2.0) / C))))))) / -t_0;
	} else {
		tmp = exp(((log((hypot(B_m, A) - A)) - (log(pow(t_1, 2.0)) + log(t_1))) * 0.5)) * (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 = cbrt(Float64(-1.0 / F))
	t_2 = Float64(Float64(4.0 * A) * C)
	t_3 = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_2) * F)) * Float64(Float64(A + C) - sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / Float64(t_2 - (B_m ^ 2.0)))
	t_4 = fma(C, Float64(A * -4.0), (B_m ^ 2.0))
	tmp = 0.0
	if (t_3 <= -2e-204)
		tmp = Float64(Float64(sqrt(Float64(F * Float64(A + Float64(C - hypot(B_m, Float64(A - C)))))) * sqrt(Float64(2.0 * t_4))) / Float64(-t_4));
	elseif (t_3 <= Inf)
		tmp = Float64(sqrt(Float64(Float64(F * t_0) * Float64(2.0 * Float64(A + Float64(A + Float64(-0.5 * Float64((B_m ^ 2.0) / C))))))) / Float64(-t_0));
	else
		tmp = Float64(exp(Float64(Float64(log(Float64(hypot(B_m, A) - A)) - Float64(log((t_1 ^ 2.0)) + log(t_1))) * 0.5)) * Float64(sqrt(2.0) / 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]}, Block[{t$95$1 = N[Power[N[(-1.0 / F), $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$2 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$2), $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$2 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(C * N[(A * -4.0), $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -2e-204], 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$4), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / (-t$95$4)), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(N[Sqrt[N[(N[(F * t$95$0), $MachinePrecision] * N[(2.0 * N[(A + N[(A + N[(-0.5 * N[(N[Power[B$95$m, 2.0], $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], N[(N[Exp[N[(N[(N[Log[N[(N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision] - A), $MachinePrecision]], $MachinePrecision] - N[(N[Log[N[Power[t$95$1, 2.0], $MachinePrecision]], $MachinePrecision] + N[Log[t$95$1], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $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))) < -2e-204

   1. Initial program 51.1%

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

   3. Simplified 46.8%

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

      1. pow1/2 46.8%

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

      2. associate-*r* 62.0%

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

      3. unpow-prod-down 77.2%

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

      4. associate-+r- 76.4%

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

      5. hypot-undefine 61.8%

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

      6. unpow2 61.8%

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

      7. unpow2 61.8%

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

      8. +-commutative 61.8%

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

      9. unpow2 61.8%

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

      10. unpow2 61.8%

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

      11. hypot-define 76.4%

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

      12. pow1/2 76.4%

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

   6. Applied egg-rr 76.4%

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

      1. unpow1/2 76.4%

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

      2. sub-neg 76.4%

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

      3. associate-+l+ 77.2%

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

      4. sub-neg 77.2%

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

      5. hypot-undefine 61.8%

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

      6. unpow2 61.8%

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

      7. unpow2 61.8%

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

      8. +-commutative 61.8%

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

      9. unpow2 61.8%

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

      10. unpow2 61.8%

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

      11. hypot-undefine 77.2%

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

   8. Simplified 77.2%

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

   if -2e-204 < (/.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 23.5%

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

   3. Simplified 32.2%

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

   5. Taylor expanded in C around inf 34.0%

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

      1. mul-1-neg 34.0%

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

   7. Simplified 34.0%

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

   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 2.0%

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

      1. mul-1-neg 2.0%

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

      2. +-commutative 2.0%

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

   5. Simplified 2.0%

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

      1. pow1/2 2.0%

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

      2. pow-to-exp 2.0%

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

      3. unpow2 2.0%

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

      4. unpow2 2.0%

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

      5. hypot-undefine 15.8%

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

   7. Applied egg-rr 15.8%

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

   8. Taylor expanded in F around -inf 1.9%

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

      1. mul-1-neg 1.9%

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

      2. unsub-neg 1.9%

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

      3. mul-1-neg 1.9%

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

      4. +-commutative 1.9%

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

      5. unpow2 1.9%

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

      6. unpow2 1.9%

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

      7. hypot-undefine 23.9%

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

   10. Simplified 23.9%

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

       1. add-cube-cbrt 23.9%

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

       2. log-prod 23.9%

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

       3. pow2 23.9%

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

   12. Applied egg-rr 23.9%

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

4. Final simplification 44.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -2 \cdot 10^{-204}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(C, A \cdot -4, {B}^{2}\right)}}{-\mathsf{fma}\left(C, A \cdot -4, {B}^{2}\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot \left(2 \cdot \left(A + \left(A + -0.5 \cdot \frac{{B}^{2}}{C}\right)\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{\left(\log \left(\mathsf{hypot}\left(B, A\right) - A\right) - \left(\log \left({\left(\sqrt[3]{\frac{-1}{F}}\right)}^{2}\right) + \log \left(\sqrt[3]{\frac{-1}{F}}\right)\right)\right) \cdot 0.5} \cdot \frac{\sqrt{2}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 57.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := \left(4 \cdot A\right) \cdot C\\ t_2 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_1\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_1 - {B\_m}^{2}}\\ t_3 := \mathsf{fma}\left(C, A \cdot -4, {B\_m}^{2}\right)\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{-204}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B\_m, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot t\_3}}{-t\_3}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot t\_0\right) \cdot \left(2 \cdot \left(A + \left(A + -0.5 \cdot \frac{{B\_m}^{2}}{C}\right)\right)\right)}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B\_m} \cdot \left(-e^{0.5 \cdot \left(\log \left(\mathsf{hypot}\left(B\_m, A\right) - A\right) - \log \left(\frac{-1}{F}\right)\right)}\right)\\ \end{array} \end{array} \]

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 -2e-204)
     (/
      (* (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 (* -0.5 (/ (pow B_m 2.0) C)))))))
        (- t_0))
       (*
        (/ (sqrt 2.0) B_m)
        (- (exp (* 0.5 (- (log (- (hypot B_m A) A)) (log (/ -1.0 F)))))))))))

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 <= -2e-204) {
		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 + (-0.5 * (pow(B_m, 2.0) / C))))))) / -t_0;
	} else {
		tmp = (sqrt(2.0) / B_m) * -exp((0.5 * (log((hypot(B_m, A) - A)) - log((-1.0 / F)))));
	}
	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 <= -2e-204)
		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(2.0 * Float64(A + Float64(A + Float64(-0.5 * Float64((B_m ^ 2.0) / C))))))) / Float64(-t_0));
	else
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(-exp(Float64(0.5 * Float64(log(Float64(hypot(B_m, A) - A)) - log(Float64(-1.0 / F)))))));
	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, -2e-204], 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[(2.0 * N[(A + N[(A + N[(-0.5 * N[(N[Power[B$95$m, 2.0], $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * (-N[Exp[N[(0.5 * 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]), $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))) < -2e-204

   1. Initial program 51.1%

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

   3. Simplified 46.8%

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

      1. pow1/2 46.8%

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

      2. associate-*r* 62.0%

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

      3. unpow-prod-down 77.2%

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

      4. associate-+r- 76.4%

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

      5. hypot-undefine 61.8%

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

      6. unpow2 61.8%

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

      7. unpow2 61.8%

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

      8. +-commutative 61.8%

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

      9. unpow2 61.8%

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

      10. unpow2 61.8%

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

      11. hypot-define 76.4%

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

      12. pow1/2 76.4%

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

   6. Applied egg-rr 76.4%

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

      1. unpow1/2 76.4%

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

      2. sub-neg 76.4%

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

      3. associate-+l+ 77.2%

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

      4. sub-neg 77.2%

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

      5. hypot-undefine 61.8%

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

      6. unpow2 61.8%

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

      7. unpow2 61.8%

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

      8. +-commutative 61.8%

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

      9. unpow2 61.8%

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

      10. unpow2 61.8%

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

      11. hypot-undefine 77.2%

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

   8. Simplified 77.2%

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

   if -2e-204 < (/.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 23.5%

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

   3. Simplified 32.2%

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

   5. Taylor expanded in C around inf 34.0%

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

      1. mul-1-neg 34.0%

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

   7. Simplified 34.0%

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

   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 2.0%

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

      1. mul-1-neg 2.0%

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

      2. +-commutative 2.0%

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

   5. Simplified 2.0%

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

      1. pow1/2 2.0%

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

      2. pow-to-exp 2.0%

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

      3. unpow2 2.0%

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

      4. unpow2 2.0%

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

      5. hypot-undefine 15.8%

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

   7. Applied egg-rr 15.8%

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

   8. Taylor expanded in F around -inf 1.9%

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

      1. mul-1-neg 1.9%

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

      2. unsub-neg 1.9%

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

      3. mul-1-neg 1.9%

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

      4. +-commutative 1.9%

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

      5. unpow2 1.9%

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

      6. unpow2 1.9%

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

      7. hypot-undefine 23.9%

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

   10. Simplified 23.9%

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

   11. Taylor expanded in F around -inf 1.9%

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

       1. +-commutative 1.9%

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

       2. unpow2 1.9%

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

       3. unpow2 1.9%

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

       4. hypot-undefine 23.9%

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

   13. Simplified 23.9%

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

4. Final simplification 45.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -2 \cdot 10^{-204}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(C, A \cdot -4, {B}^{2}\right)}}{-\mathsf{fma}\left(C, A \cdot -4, {B}^{2}\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot \left(2 \cdot \left(A + \left(A + -0.5 \cdot \frac{{B}^{2}}{C}\right)\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-e^{0.5 \cdot \left(\log \left(\mathsf{hypot}\left(B, A\right) - A\right) - \log \left(\frac{-1}{F}\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 54.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 2 \cdot 10^{+39}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot t\_0\right) \cdot \left(2 \cdot \left(A + \left(A + -0.5 \cdot \frac{{B\_m}^{2}}{C}\right)\right)\right)}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B\_m} \cdot \left(-{\left(e^{0.5}\right)}^{\left(\log \left(\mathsf{hypot}\left(A, B\_m\right) - A\right) - \log \left(\frac{-1}{F}\right)\right)}\right)\\ \end{array} \end{array} \]

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)))))
   (if (<= (pow B_m 2.0) 2e+39)
     (/
      (sqrt (* (* F t_0) (* 2.0 (+ A (+ A (* -0.5 (/ (pow B_m 2.0) C)))))))
      (- t_0))
     (*
      (/ (sqrt 2.0) B_m)
      (- (pow (exp 0.5) (- (log (- (hypot A B_m) A)) (log (/ -1.0 F)))))))))

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+39) {
		tmp = sqrt(((F * t_0) * (2.0 * (A + (A + (-0.5 * (pow(B_m, 2.0) / C))))))) / -t_0;
	} else {
		tmp = (sqrt(2.0) / B_m) * -pow(exp(0.5), (log((hypot(A, B_m) - A)) - log((-1.0 / F))));
	}
	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+39)
		tmp = Float64(sqrt(Float64(Float64(F * t_0) * Float64(2.0 * Float64(A + Float64(A + Float64(-0.5 * Float64((B_m ^ 2.0) / C))))))) / Float64(-t_0));
	else
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(-(exp(0.5) ^ Float64(log(Float64(hypot(A, B_m) - A)) - log(Float64(-1.0 / F))))));
	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+39], N[(N[Sqrt[N[(N[(F * t$95$0), $MachinePrecision] * N[(2.0 * N[(A + N[(A + N[(-0.5 * N[(N[Power[B$95$m, 2.0], $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * (-N[Power[N[Exp[0.5], $MachinePrecision], N[(N[Log[N[(N[Sqrt[A ^ 2 + B$95$m ^ 2], $MachinePrecision] - A), $MachinePrecision]], $MachinePrecision] - N[Log[N[(-1.0 / F), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 1.99999999999999988e39

   1. Initial program 25.3%

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

   3. Simplified 35.0%

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

   5. Taylor expanded in C around inf 24.4%

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

      1. mul-1-neg 24.4%

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

   7. Simplified 24.4%

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

   if 1.99999999999999988e39 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 20.4%

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

   3. Taylor expanded in C around 0 15.8%

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

      1. mul-1-neg 15.8%

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

      2. +-commutative 15.8%

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

   5. Simplified 15.8%

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

      1. pow1/2 15.8%

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

      2. pow-to-exp 14.9%

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

      3. unpow2 14.9%

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

      4. unpow2 14.9%

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

      5. hypot-undefine 30.3%

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

   7. Applied egg-rr 30.3%

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

   8. Taylor expanded in F around -inf 14.7%

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

      1. mul-1-neg 14.7%

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

      2. unsub-neg 14.7%

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

      3. mul-1-neg 14.7%

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

      4. +-commutative 14.7%

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

      5. unpow2 14.7%

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

      6. unpow2 14.7%

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

      7. hypot-undefine 38.3%

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

   10. Simplified 38.3%

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

   11. Taylor expanded in F around -inf 14.7%

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

       1. mul-1-neg 14.7%

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

       2. associate-/l* 14.7%

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

       3. distribute-lft-neg-in 14.7%

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

   13. Simplified 38.3%

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

4. Final simplification 30.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 2 \cdot 10^{+39}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot \left(2 \cdot \left(A + \left(A + -0.5 \cdot \frac{{B}^{2}}{C}\right)\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-{\left(e^{0.5}\right)}^{\left(\log \left(\mathsf{hypot}\left(A, B\right) - A\right) - \log \left(\frac{-1}{F}\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 54.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 2 \cdot 10^{+39}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot t\_0\right) \cdot \left(2 \cdot \left(A + \left(A + -0.5 \cdot \frac{{B\_m}^{2}}{C}\right)\right)\right)}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B\_m} \cdot \left(-e^{0.5 \cdot \left(\log \left(\mathsf{hypot}\left(B\_m, A\right) - A\right) - \log \left(\frac{-1}{F}\right)\right)}\right)\\ \end{array} \end{array} \]

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)))))
   (if (<= (pow B_m 2.0) 2e+39)
     (/
      (sqrt (* (* F t_0) (* 2.0 (+ A (+ A (* -0.5 (/ (pow B_m 2.0) C)))))))
      (- t_0))
     (*
      (/ (sqrt 2.0) B_m)
      (- (exp (* 0.5 (- (log (- (hypot B_m A) A)) (log (/ -1.0 F))))))))))

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+39) {
		tmp = sqrt(((F * t_0) * (2.0 * (A + (A + (-0.5 * (pow(B_m, 2.0) / C))))))) / -t_0;
	} else {
		tmp = (sqrt(2.0) / B_m) * -exp((0.5 * (log((hypot(B_m, A) - A)) - log((-1.0 / F)))));
	}
	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+39)
		tmp = Float64(sqrt(Float64(Float64(F * t_0) * Float64(2.0 * Float64(A + Float64(A + Float64(-0.5 * Float64((B_m ^ 2.0) / C))))))) / Float64(-t_0));
	else
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(-exp(Float64(0.5 * Float64(log(Float64(hypot(B_m, A) - A)) - log(Float64(-1.0 / F)))))));
	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+39], N[(N[Sqrt[N[(N[(F * t$95$0), $MachinePrecision] * N[(2.0 * N[(A + N[(A + N[(-0.5 * N[(N[Power[B$95$m, 2.0], $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * (-N[Exp[N[(0.5 * 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]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 1.99999999999999988e39

   1. Initial program 25.3%

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

   3. Simplified 35.0%

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

   5. Taylor expanded in C around inf 24.4%

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

      1. mul-1-neg 24.4%

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

   7. Simplified 24.4%

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

   if 1.99999999999999988e39 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 20.4%

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

   3. Taylor expanded in C around 0 15.8%

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

      1. mul-1-neg 15.8%

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

      2. +-commutative 15.8%

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

   5. Simplified 15.8%

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

      1. pow1/2 15.8%

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

      2. pow-to-exp 14.9%

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

      3. unpow2 14.9%

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

      4. unpow2 14.9%

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

      5. hypot-undefine 30.3%

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

   7. Applied egg-rr 30.3%

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

   8. Taylor expanded in F around -inf 14.7%

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

      1. mul-1-neg 14.7%

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

      2. unsub-neg 14.7%

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

      3. mul-1-neg 14.7%

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

      4. +-commutative 14.7%

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

      5. unpow2 14.7%

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

      6. unpow2 14.7%

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

      7. hypot-undefine 38.3%

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

   10. Simplified 38.3%

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

   11. Taylor expanded in F around -inf 14.7%

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

       1. +-commutative 14.7%

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

       2. unpow2 14.7%

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

       3. unpow2 14.7%

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

       4. hypot-undefine 38.3%

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

   13. Simplified 38.3%

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

4. Final simplification 30.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 2 \cdot 10^{+39}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot \left(2 \cdot \left(A + \left(A + -0.5 \cdot \frac{{B}^{2}}{C}\right)\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-e^{0.5 \cdot \left(\log \left(\mathsf{hypot}\left(B, A\right) - A\right) - \log \left(\frac{-1}{F}\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 50.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;B\_m \leq 3.5 \cdot 10^{+21}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot t\_0\right) \cdot \left(2 \cdot \left(A + \left(A + -0.5 \cdot \frac{{B\_m}^{2}}{C}\right)\right)\right)}}{-t\_0}\\ \mathbf{elif}\;B\_m \leq 1.65 \cdot 10^{+205}:\\ \;\;\;\;\frac{-1}{B\_m} \cdot \sqrt{2 \cdot \left(F \cdot \left(A - \mathsf{hypot}\left(B\_m, A\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{0.5 \cdot \left(\log \left(-F\right) + \left(\log B\_m - \frac{A}{B\_m}\right)\right)} \cdot \frac{\sqrt{2}}{-B\_m}\\ \end{array} \end{array} \]

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)))))
   (if (<= B_m 3.5e+21)
     (/
      (sqrt (* (* F t_0) (* 2.0 (+ A (+ A (* -0.5 (/ (pow B_m 2.0) C)))))))
      (- t_0))
     (if (<= B_m 1.65e+205)
       (* (/ -1.0 B_m) (sqrt (* 2.0 (* F (- A (hypot B_m A))))))
       (*
        (exp (* 0.5 (+ (log (- F)) (- (log B_m) (/ A 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 tmp;
	if (B_m <= 3.5e+21) {
		tmp = sqrt(((F * t_0) * (2.0 * (A + (A + (-0.5 * (pow(B_m, 2.0) / C))))))) / -t_0;
	} else if (B_m <= 1.65e+205) {
		tmp = (-1.0 / B_m) * sqrt((2.0 * (F * (A - hypot(B_m, A)))));
	} else {
		tmp = exp((0.5 * (log(-F) + (log(B_m) - (A / 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)))
	tmp = 0.0
	if (B_m <= 3.5e+21)
		tmp = Float64(sqrt(Float64(Float64(F * t_0) * Float64(2.0 * Float64(A + Float64(A + Float64(-0.5 * Float64((B_m ^ 2.0) / C))))))) / Float64(-t_0));
	elseif (B_m <= 1.65e+205)
		tmp = Float64(Float64(-1.0 / B_m) * sqrt(Float64(2.0 * Float64(F * Float64(A - hypot(B_m, A))))));
	else
		tmp = Float64(exp(Float64(0.5 * Float64(log(Float64(-F)) + Float64(log(B_m) - Float64(A / B_m))))) * Float64(sqrt(2.0) / 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[B$95$m, 3.5e+21], N[(N[Sqrt[N[(N[(F * t$95$0), $MachinePrecision] * N[(2.0 * N[(A + N[(A + N[(-0.5 * N[(N[Power[B$95$m, 2.0], $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], If[LessEqual[B$95$m, 1.65e+205], N[(N[(-1.0 / B$95$m), $MachinePrecision] * N[Sqrt[N[(2.0 * N[(F * N[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Exp[N[(0.5 * N[(N[Log[(-F)], $MachinePrecision] + N[(N[Log[B$95$m], $MachinePrecision] - N[(A / B$95$m), $MachinePrecision]), $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 < 3.5e21

   1. Initial program 26.1%

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

   3. Simplified 33.8%

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

   5. Taylor expanded in C around inf 20.3%

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

      1. mul-1-neg 20.3%

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

   7. Simplified 20.3%

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

   if 3.5e21 < B < 1.6500000000000001e205

   1. Initial program 22.9%

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

   3. Taylor expanded in C around 0 38.0%

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

      1. mul-1-neg 38.0%

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

      2. +-commutative 38.0%

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

      3. unpow2 38.0%

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

      4. unpow2 38.0%

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

      5. hypot-define 48.5%

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

   5. Simplified 48.5%

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

      1. div-inv 48.4%

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

   7. Applied egg-rr 48.4%

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

      1. pow1/2 48.4%

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

      2. exp-to-pow 45.3%

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

      3. div-inv 45.3%

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

      4. associate-*l/ 45.3%

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

      5. exp-to-pow 48.4%

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

      6. pow1/2 48.4%

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

      7. sqrt-prod 48.6%

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

      8. distribute-frac-neg2 48.6%

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

      9. clear-num 48.6%

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

      10. frac-2neg 48.6%

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

      11. metadata-eval 48.6%

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

   9. Applied egg-rr 48.6%

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

       1. associate-/r/ 48.7%

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

       2. associate-*r* 48.7%

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

       3. hypot-undefine 38.1%

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

       4. unpow2 38.1%

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

       5. unpow2 38.1%

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

       6. +-commutative 38.1%

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

       7. *-commutative 38.1%

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

       8. +-commutative 38.1%

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

       9. unpow2 38.1%

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

       10. unpow2 38.1%

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

       11. hypot-undefine 48.7%

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

   11. Simplified 48.7%

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

   if 1.6500000000000001e205 < 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 2.4%

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

      1. mul-1-neg 2.4%

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

      2. +-commutative 2.4%

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

   5. Simplified 2.4%

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

      1. pow1/2 2.4%

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

      2. pow-to-exp 2.4%

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

      3. unpow2 2.4%

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

      4. unpow2 2.4%

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

      5. hypot-undefine 54.1%

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

   7. Applied egg-rr 54.1%

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

   8. Taylor expanded in B around inf 84.6%

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

      1. neg-mul-1 84.6%

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

      2. mul-1-neg 84.6%

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

      3. unsub-neg 84.6%

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

      4. mul-1-neg 84.6%

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

      5. log-rec 84.6%

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

      6. remove-double-div 84.6%

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

   10. Simplified 84.6%

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

4. Final simplification 30.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 3.5 \cdot 10^{+21}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot \left(2 \cdot \left(A + \left(A + -0.5 \cdot \frac{{B}^{2}}{C}\right)\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 1.65 \cdot 10^{+205}:\\ \;\;\;\;\frac{-1}{B} \cdot \sqrt{2 \cdot \left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{0.5 \cdot \left(\log \left(-F\right) + \left(\log B - \frac{A}{B}\right)\right)} \cdot \frac{\sqrt{2}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 51.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;B\_m \leq 4800:\\ \;\;\;\;\frac{\sqrt{\left(4 \cdot A\right) \cdot \left(F \cdot t\_0\right)}}{-t\_0}\\ \mathbf{elif}\;B\_m \leq 7.2 \cdot 10^{+204}:\\ \;\;\;\;\frac{{\left(2 \cdot \frac{\mathsf{hypot}\left(B\_m, A\right) - A}{\frac{-1}{F}}\right)}^{0.5}}{-B\_m}\\ \mathbf{else}:\\ \;\;\;\;e^{0.5 \cdot \left(\log \left(-F\right) + \left(\log B\_m - \frac{A}{B\_m}\right)\right)} \cdot \frac{\sqrt{2}}{-B\_m}\\ \end{array} \end{array} \]

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)))))
   (if (<= B_m 4800.0)
     (/ (sqrt (* (* 4.0 A) (* F t_0))) (- t_0))
     (if (<= B_m 7.2e+204)
       (/ (pow (* 2.0 (/ (- (hypot B_m A) A) (/ -1.0 F))) 0.5) (- B_m))
       (*
        (exp (* 0.5 (+ (log (- F)) (- (log B_m) (/ A 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 tmp;
	if (B_m <= 4800.0) {
		tmp = sqrt(((4.0 * A) * (F * t_0))) / -t_0;
	} else if (B_m <= 7.2e+204) {
		tmp = pow((2.0 * ((hypot(B_m, A) - A) / (-1.0 / F))), 0.5) / -B_m;
	} else {
		tmp = exp((0.5 * (log(-F) + (log(B_m) - (A / 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)))
	tmp = 0.0
	if (B_m <= 4800.0)
		tmp = Float64(sqrt(Float64(Float64(4.0 * A) * Float64(F * t_0))) / Float64(-t_0));
	elseif (B_m <= 7.2e+204)
		tmp = Float64((Float64(2.0 * Float64(Float64(hypot(B_m, A) - A) / Float64(-1.0 / F))) ^ 0.5) / Float64(-B_m));
	else
		tmp = Float64(exp(Float64(0.5 * Float64(log(Float64(-F)) + Float64(log(B_m) - Float64(A / B_m))))) * Float64(sqrt(2.0) / 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[B$95$m, 4800.0], N[(N[Sqrt[N[(N[(4.0 * A), $MachinePrecision] * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], If[LessEqual[B$95$m, 7.2e+204], N[(N[Power[N[(2.0 * N[(N[(N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision] - A), $MachinePrecision] / N[(-1.0 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision] / (-B$95$m)), $MachinePrecision], N[(N[Exp[N[(0.5 * N[(N[Log[(-F)], $MachinePrecision] + N[(N[Log[B$95$m], $MachinePrecision] - N[(A / B$95$m), $MachinePrecision]), $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 < 4800

   1. Initial program 26.7%

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

   3. Simplified 34.4%

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

   5. Taylor expanded in A around -inf 20.6%

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

   if 4800 < B < 7.2000000000000005e204

   1. Initial program 21.0%

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

   3. Taylor expanded in C around 0 34.9%

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

      1. mul-1-neg 34.9%

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

      2. +-commutative 34.9%

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

   5. Simplified 34.9%

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

      1. pow1/2 35.0%

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

      2. pow-to-exp 32.7%

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

      3. unpow2 32.7%

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

      4. unpow2 32.7%

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

      5. hypot-undefine 41.6%

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

   7. Applied egg-rr 41.6%

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

   8. Taylor expanded in F around -inf 32.2%

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

      1. mul-1-neg 32.2%

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

      2. unsub-neg 32.2%

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

      3. mul-1-neg 32.2%

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

      4. +-commutative 32.2%

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

      5. unpow2 32.2%

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

      6. unpow2 32.2%

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

      7. hypot-undefine 44.7%

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

   10. Simplified 44.7%

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

       1. associate-*l/ 44.7%

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

       2. pow1/2 44.7%

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

       3. exp-prod 41.0%

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

       4. pow-prod-down 40.9%

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

       5. diff-log 41.6%

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

       6. add-exp-log 44.7%

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

   12. Applied egg-rr 44.7%

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

   if 7.2000000000000005e204 < 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 2.4%

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

      1. mul-1-neg 2.4%

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

      2. +-commutative 2.4%

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

   5. Simplified 2.4%

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

      1. pow1/2 2.4%

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

      2. pow-to-exp 2.4%

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

      3. unpow2 2.4%

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

      4. unpow2 2.4%

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

      5. hypot-undefine 54.1%

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

   7. Applied egg-rr 54.1%

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

   8. Taylor expanded in B around inf 84.6%

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

      1. neg-mul-1 84.6%

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

      2. mul-1-neg 84.6%

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

      3. unsub-neg 84.6%

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

      4. mul-1-neg 84.6%

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

      5. log-rec 84.6%

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

      6. remove-double-div 84.6%

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

   10. Simplified 84.6%

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

4. Final simplification 30.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 4800:\\ \;\;\;\;\frac{\sqrt{\left(4 \cdot A\right) \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 7.2 \cdot 10^{+204}:\\ \;\;\;\;\frac{{\left(2 \cdot \frac{\mathsf{hypot}\left(B, A\right) - A}{\frac{-1}{F}}\right)}^{0.5}}{-B}\\ \mathbf{else}:\\ \;\;\;\;e^{0.5 \cdot \left(\log \left(-F\right) + \left(\log B - \frac{A}{B}\right)\right)} \cdot \frac{\sqrt{2}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 40.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 4 \cdot 10^{-174}:\\ \;\;\;\;\frac{\sqrt{-16 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right)}}{\left(4 \cdot A\right) \cdot C - {B\_m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(2 \cdot \frac{\mathsf{hypot}\left(B\_m, A\right) - A}{\frac{-1}{F}}\right)}^{0.5}}{-B\_m}\\ \end{array} \end{array} \]

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) 4e-174)
   (/
    (sqrt (* -16.0 (* (pow A 2.0) (* C F))))
    (- (* (* 4.0 A) C) (pow B_m 2.0)))
   (/ (pow (* 2.0 (/ (- (hypot B_m A) A) (/ -1.0 F))) 0.5) (- 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) <= 4e-174) {
		tmp = sqrt((-16.0 * (pow(A, 2.0) * (C * F)))) / (((4.0 * A) * C) - pow(B_m, 2.0));
	} else {
		tmp = pow((2.0 * ((hypot(B_m, A) - A) / (-1.0 / F))), 0.5) / -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 tmp;
	if (Math.pow(B_m, 2.0) <= 4e-174) {
		tmp = Math.sqrt((-16.0 * (Math.pow(A, 2.0) * (C * F)))) / (((4.0 * A) * C) - Math.pow(B_m, 2.0));
	} else {
		tmp = Math.pow((2.0 * ((Math.hypot(B_m, A) - A) / (-1.0 / F))), 0.5) / -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 math.pow(B_m, 2.0) <= 4e-174:
		tmp = math.sqrt((-16.0 * (math.pow(A, 2.0) * (C * F)))) / (((4.0 * A) * C) - math.pow(B_m, 2.0))
	else:
		tmp = math.pow((2.0 * ((math.hypot(B_m, A) - A) / (-1.0 / F))), 0.5) / -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) <= 4e-174)
		tmp = Float64(sqrt(Float64(-16.0 * Float64((A ^ 2.0) * Float64(C * F)))) / Float64(Float64(Float64(4.0 * A) * C) - (B_m ^ 2.0)));
	else
		tmp = Float64((Float64(2.0 * Float64(Float64(hypot(B_m, A) - A) / Float64(-1.0 / F))) ^ 0.5) / 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 ((B_m ^ 2.0) <= 4e-174)
		tmp = sqrt((-16.0 * ((A ^ 2.0) * (C * F)))) / (((4.0 * A) * C) - (B_m ^ 2.0));
	else
		tmp = ((2.0 * ((hypot(B_m, A) - A) / (-1.0 / F))) ^ 0.5) / -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[N[Power[B$95$m, 2.0], $MachinePrecision], 4e-174], N[(N[Sqrt[N[(-16.0 * N[(N[Power[A, 2.0], $MachinePrecision] * N[(C * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision] - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(2.0 * N[(N[(N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision] - A), $MachinePrecision] / N[(-1.0 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision] / (-B$95$m)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 4e-174

   1. Initial program 23.5%

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

   3. Simplified 29.6%

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

   5. Taylor expanded in A around -inf 20.7%

      \[\leadsto \frac{\sqrt{\color{blue}{-16 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right)}}}{-\mathsf{fma}\left(C, A \cdot -4, {B}^{2}\right)} \]

   6. Taylor expanded in C around 0 20.7%

      \[\leadsto \frac{\sqrt{-16 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right)}}{\color{blue}{4 \cdot \left(A \cdot C\right) - {B}^{2}}} \]
   7. Step-by-step derivation

      1. associate-*r* 20.7%

         \[\leadsto \frac{\sqrt{-16 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right)}}{\color{blue}{\left(4 \cdot A\right) \cdot C} - {B}^{2}} \]

   8. Simplified 20.7%

      \[\leadsto \frac{\sqrt{-16 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right)}}{\color{blue}{\left(4 \cdot A\right) \cdot C - {B}^{2}}} \]

   if 4e-174 < (pow.f64 B #s(literal 2 binary64))

   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 13.9%

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

      1. mul-1-neg 13.9%

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

      2. +-commutative 13.9%

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

   5. Simplified 13.9%

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

      1. pow1/2 13.9%

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

      2. pow-to-exp 13.1%

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

      3. unpow2 13.1%

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

      4. unpow2 13.1%

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

      5. hypot-undefine 23.5%

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

   7. Applied egg-rr 23.5%

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

   8. Taylor expanded in F around -inf 12.9%

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

      1. mul-1-neg 12.9%

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

      2. unsub-neg 12.9%

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

      3. mul-1-neg 12.9%

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

      4. +-commutative 12.9%

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

      5. unpow2 12.9%

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

      6. unpow2 12.9%

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

      7. hypot-undefine 28.7%

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

   10. Simplified 28.7%

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

       1. associate-*l/ 28.7%

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

       2. pow1/2 28.7%

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

       3. exp-prod 23.0%

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

       4. pow-prod-down 23.0%

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

       5. diff-log 23.5%

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

       6. add-exp-log 25.0%

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

   12. Applied egg-rr 25.0%

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

4. Final simplification 23.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 4 \cdot 10^{-174}:\\ \;\;\;\;\frac{\sqrt{-16 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(2 \cdot \frac{\mathsf{hypot}\left(B, A\right) - A}{\frac{-1}{F}}\right)}^{0.5}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 51.2% accurate, 1.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;B\_m \leq 5000:\\ \;\;\;\;\frac{\sqrt{\left(4 \cdot A\right) \cdot \left(F \cdot t\_0\right)}}{-t\_0}\\ \mathbf{elif}\;B\_m \leq 1.65 \cdot 10^{+205}:\\ \;\;\;\;\frac{{\left(2 \cdot \frac{\mathsf{hypot}\left(B\_m, A\right) - A}{\frac{-1}{F}}\right)}^{0.5}}{-B\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B\_m} \cdot \left(-e^{0.5 \cdot \left(\log \left(-F\right) + \log B\_m\right)}\right)\\ \end{array} \end{array} \]

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)))))
   (if (<= B_m 5000.0)
     (/ (sqrt (* (* 4.0 A) (* F t_0))) (- t_0))
     (if (<= B_m 1.65e+205)
       (/ (pow (* 2.0 (/ (- (hypot B_m A) A) (/ -1.0 F))) 0.5) (- B_m))
       (* (/ (sqrt 2.0) B_m) (- (exp (* 0.5 (+ (log (- F)) (log 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 (B_m <= 5000.0) {
		tmp = sqrt(((4.0 * A) * (F * t_0))) / -t_0;
	} else if (B_m <= 1.65e+205) {
		tmp = pow((2.0 * ((hypot(B_m, A) - A) / (-1.0 / F))), 0.5) / -B_m;
	} else {
		tmp = (sqrt(2.0) / B_m) * -exp((0.5 * (log(-F) + log(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 <= 5000.0)
		tmp = Float64(sqrt(Float64(Float64(4.0 * A) * Float64(F * t_0))) / Float64(-t_0));
	elseif (B_m <= 1.65e+205)
		tmp = Float64((Float64(2.0 * Float64(Float64(hypot(B_m, A) - A) / Float64(-1.0 / F))) ^ 0.5) / Float64(-B_m));
	else
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(-exp(Float64(0.5 * Float64(log(Float64(-F)) + log(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[B$95$m, 5000.0], N[(N[Sqrt[N[(N[(4.0 * A), $MachinePrecision] * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], If[LessEqual[B$95$m, 1.65e+205], N[(N[Power[N[(2.0 * N[(N[(N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision] - A), $MachinePrecision] / N[(-1.0 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision] / (-B$95$m)), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * (-N[Exp[N[(0.5 * N[(N[Log[(-F)], $MachinePrecision] + N[Log[B$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if B < 5e3

   1. Initial program 26.7%

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

   3. Simplified 34.4%

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

   5. Taylor expanded in A around -inf 20.6%

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

   if 5e3 < B < 1.6500000000000001e205

   1. Initial program 21.0%

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

   3. Taylor expanded in C around 0 34.9%

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

      1. mul-1-neg 34.9%

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

      2. +-commutative 34.9%

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

   5. Simplified 34.9%

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

      1. pow1/2 35.0%

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

      2. pow-to-exp 32.7%

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

      3. unpow2 32.7%

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

      4. unpow2 32.7%

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

      5. hypot-undefine 41.6%

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

   7. Applied egg-rr 41.6%

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

   8. Taylor expanded in F around -inf 32.2%

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

      1. mul-1-neg 32.2%

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

      2. unsub-neg 32.2%

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

      3. mul-1-neg 32.2%

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

      4. +-commutative 32.2%

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

      5. unpow2 32.2%

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

      6. unpow2 32.2%

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

      7. hypot-undefine 44.7%

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

   10. Simplified 44.7%

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

       1. associate-*l/ 44.7%

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

       2. pow1/2 44.7%

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

       3. exp-prod 41.0%

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

       4. pow-prod-down 40.9%

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

       5. diff-log 41.6%

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

       6. add-exp-log 44.7%

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

   12. Applied egg-rr 44.7%

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

   if 1.6500000000000001e205 < 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 2.4%

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

      1. mul-1-neg 2.4%

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

      2. +-commutative 2.4%

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

   5. Simplified 2.4%

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

      1. pow1/2 2.4%

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

      2. pow-to-exp 2.4%

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

      3. unpow2 2.4%

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

      4. unpow2 2.4%

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

      5. hypot-undefine 54.1%

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

   7. Applied egg-rr 54.1%

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

   8. Taylor expanded in B around inf 83.8%

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

      1. neg-mul-1 83.8%

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

      2. mul-1-neg 83.8%

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

      3. log-rec 83.8%

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

      4. remove-double-div 83.8%

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

   10. Simplified 83.8%

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

4. Final simplification 30.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 5000:\\ \;\;\;\;\frac{\sqrt{\left(4 \cdot A\right) \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 1.65 \cdot 10^{+205}:\\ \;\;\;\;\frac{{\left(2 \cdot \frac{\mathsf{hypot}\left(B, A\right) - A}{\frac{-1}{F}}\right)}^{0.5}}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-e^{0.5 \cdot \left(\log \left(-F\right) + \log B\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 47.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;B\_m \leq 1000:\\ \;\;\;\;\frac{\sqrt{\left(4 \cdot A\right) \cdot \left(F \cdot t\_0\right)}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(2 \cdot \frac{\mathsf{hypot}\left(B\_m, A\right) - A}{\frac{-1}{F}}\right)}^{0.5}}{-B\_m}\\ \end{array} \end{array} \]

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 (<= B_m 1000.0)
     (/ (sqrt (* (* 4.0 A) (* F t_0))) (- t_0))
     (/ (pow (* 2.0 (/ (- (hypot B_m A) A) (/ -1.0 F))) 0.5) (- 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 (B_m <= 1000.0) {
		tmp = sqrt(((4.0 * A) * (F * t_0))) / -t_0;
	} else {
		tmp = pow((2.0 * ((hypot(B_m, A) - A) / (-1.0 / F))), 0.5) / -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 <= 1000.0)
		tmp = Float64(sqrt(Float64(Float64(4.0 * A) * Float64(F * t_0))) / Float64(-t_0));
	else
		tmp = Float64((Float64(2.0 * Float64(Float64(hypot(B_m, A) - A) / Float64(-1.0 / F))) ^ 0.5) / 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[B$95$m, 1000.0], N[(N[Sqrt[N[(N[(4.0 * A), $MachinePrecision] * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], N[(N[Power[N[(2.0 * N[(N[(N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision] - A), $MachinePrecision] / N[(-1.0 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision] / (-B$95$m)), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if B < 1e3

   1. Initial program 26.7%

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

   3. Simplified 34.4%

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

   5. Taylor expanded in A around -inf 20.6%

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

   if 1e3 < B

   1. Initial program 13.9%

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

   3. Taylor expanded in C around 0 23.9%

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

      1. mul-1-neg 23.9%

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

      2. +-commutative 23.9%

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

   5. Simplified 23.9%

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

      1. pow1/2 24.0%

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

      2. pow-to-exp 22.4%

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

      3. unpow2 22.4%

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

      4. unpow2 22.4%

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

      5. hypot-undefine 45.9%

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

   7. Applied egg-rr 45.9%

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

   8. Taylor expanded in F around -inf 22.1%

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

      1. mul-1-neg 22.1%

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

      2. unsub-neg 22.1%

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

      3. mul-1-neg 22.1%

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

      4. +-commutative 22.1%

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

      5. unpow2 22.1%

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

      6. unpow2 22.1%

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

      7. hypot-undefine 58.2%

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

   10. Simplified 58.2%

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

       1. associate-*l/ 58.2%

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

       2. pow1/2 58.2%

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

       3. exp-prod 44.8%

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

       4. pow-prod-down 44.8%

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

       5. diff-log 45.8%

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

       6. add-exp-log 49.0%

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

   12. Applied egg-rr 49.0%

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

4. Final simplification 28.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1000:\\ \;\;\;\;\frac{\sqrt{\left(4 \cdot A\right) \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(2 \cdot \frac{\mathsf{hypot}\left(B, A\right) - A}{\frac{-1}{F}}\right)}^{0.5}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 45.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 6.5 \cdot 10^{-84}:\\ \;\;\;\;\frac{\sqrt{\left(A \cdot -8\right) \cdot \left(C \cdot \left(F \cdot \left(A + A\right)\right)\right)}}{-\mathsf{fma}\left(C, A \cdot -4, {B\_m}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(2 \cdot \frac{\mathsf{hypot}\left(B\_m, A\right) - A}{\frac{-1}{F}}\right)}^{0.5}}{-B\_m}\\ \end{array} \end{array} \]

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 (<= B_m 6.5e-84)
   (/
    (sqrt (* (* A -8.0) (* C (* F (+ A A)))))
    (- (fma C (* A -4.0) (pow B_m 2.0))))
   (/ (pow (* 2.0 (/ (- (hypot B_m A) A) (/ -1.0 F))) 0.5) (- 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 (B_m <= 6.5e-84) {
		tmp = sqrt(((A * -8.0) * (C * (F * (A + A))))) / -fma(C, (A * -4.0), pow(B_m, 2.0));
	} else {
		tmp = pow((2.0 * ((hypot(B_m, A) - A) / (-1.0 / F))), 0.5) / -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 <= 6.5e-84)
		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((Float64(2.0 * Float64(Float64(hypot(B_m, A) - A) / Float64(-1.0 / F))) ^ 0.5) / 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[B$95$m, 6.5e-84], 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[Power[N[(2.0 * N[(N[(N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision] - A), $MachinePrecision] / N[(-1.0 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision] / (-B$95$m)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 6.50000000000000022e-84

   1. Initial program 27.0%

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

   3. Simplified 29.2%

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

   5. Taylor expanded in C around inf 18.6%

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

      1. associate-*r* 18.6%

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

      2. mul-1-neg 18.6%

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

   7. Simplified 18.6%

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

   if 6.50000000000000022e-84 < B

   1. Initial program 15.9%

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

   3. Taylor expanded in C around 0 24.4%

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

      1. mul-1-neg 24.4%

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

      2. +-commutative 24.4%

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

   5. Simplified 24.4%

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

      1. pow1/2 24.5%

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

      2. pow-to-exp 23.1%

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

      3. unpow2 23.1%

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

      4. unpow2 23.1%

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

      5. hypot-undefine 41.7%

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

   7. Applied egg-rr 41.7%

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

   8. Taylor expanded in F around -inf 22.7%

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

      1. mul-1-neg 22.7%

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

      2. unsub-neg 22.7%

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

      3. mul-1-neg 22.7%

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

      4. +-commutative 22.7%

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

      5. unpow2 22.7%

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

      6. unpow2 22.7%

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

      7. hypot-undefine 51.4%

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

   10. Simplified 51.4%

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

       1. associate-*l/ 51.4%

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

       2. pow1/2 51.4%

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

       3. exp-prod 40.8%

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

       4. pow-prod-down 40.8%

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

       5. diff-log 41.7%

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

       6. add-exp-log 44.4%

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

   12. Applied egg-rr 44.4%

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

4. Final simplification 27.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 6.5 \cdot 10^{-84}:\\ \;\;\;\;\frac{\sqrt{\left(A \cdot -8\right) \cdot \left(C \cdot \left(F \cdot \left(A + A\right)\right)\right)}}{-\mathsf{fma}\left(C, A \cdot -4, {B}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(2 \cdot \frac{\mathsf{hypot}\left(B, A\right) - A}{\frac{-1}{F}}\right)}^{0.5}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 40.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 4 \cdot 10^{-174}:\\ \;\;\;\;\frac{\sqrt{-16 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right)}}{\left(4 \cdot A\right) \cdot C}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(2 \cdot \frac{\mathsf{hypot}\left(B\_m, A\right) - A}{\frac{-1}{F}}\right)}^{0.5}}{-B\_m}\\ \end{array} \end{array} \]

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) 4e-174)
   (/ (sqrt (* -16.0 (* (pow A 2.0) (* C F)))) (* (* 4.0 A) C))
   (/ (pow (* 2.0 (/ (- (hypot B_m A) A) (/ -1.0 F))) 0.5) (- 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) <= 4e-174) {
		tmp = sqrt((-16.0 * (pow(A, 2.0) * (C * F)))) / ((4.0 * A) * C);
	} else {
		tmp = pow((2.0 * ((hypot(B_m, A) - A) / (-1.0 / F))), 0.5) / -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 tmp;
	if (Math.pow(B_m, 2.0) <= 4e-174) {
		tmp = Math.sqrt((-16.0 * (Math.pow(A, 2.0) * (C * F)))) / ((4.0 * A) * C);
	} else {
		tmp = Math.pow((2.0 * ((Math.hypot(B_m, A) - A) / (-1.0 / F))), 0.5) / -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 math.pow(B_m, 2.0) <= 4e-174:
		tmp = math.sqrt((-16.0 * (math.pow(A, 2.0) * (C * F)))) / ((4.0 * A) * C)
	else:
		tmp = math.pow((2.0 * ((math.hypot(B_m, A) - A) / (-1.0 / F))), 0.5) / -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) <= 4e-174)
		tmp = Float64(sqrt(Float64(-16.0 * Float64((A ^ 2.0) * Float64(C * F)))) / Float64(Float64(4.0 * A) * C));
	else
		tmp = Float64((Float64(2.0 * Float64(Float64(hypot(B_m, A) - A) / Float64(-1.0 / F))) ^ 0.5) / 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 ((B_m ^ 2.0) <= 4e-174)
		tmp = sqrt((-16.0 * ((A ^ 2.0) * (C * F)))) / ((4.0 * A) * C);
	else
		tmp = ((2.0 * ((hypot(B_m, A) - A) / (-1.0 / F))) ^ 0.5) / -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[N[Power[B$95$m, 2.0], $MachinePrecision], 4e-174], N[(N[Sqrt[N[(-16.0 * N[(N[Power[A, 2.0], $MachinePrecision] * N[(C * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(2.0 * N[(N[(N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision] - A), $MachinePrecision] / N[(-1.0 / F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision] / (-B$95$m)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 4e-174

   1. Initial program 23.5%

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

   3. Simplified 29.6%

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

   5. Taylor expanded in A around -inf 20.7%

      \[\leadsto \frac{\sqrt{\color{blue}{-16 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right)}}}{-\mathsf{fma}\left(C, A \cdot -4, {B}^{2}\right)} \]

   6. Taylor expanded in C around inf 20.6%

      \[\leadsto \frac{\sqrt{-16 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right)}}{\color{blue}{4 \cdot \left(A \cdot C\right)}} \]
   7. Step-by-step derivation

      1. associate-*r* 20.6%

         \[\leadsto \frac{\sqrt{-16 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right)}}{\color{blue}{\left(4 \cdot A\right) \cdot C}} \]

   8. Simplified 20.6%

      \[\leadsto \frac{\sqrt{-16 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right)}}{\color{blue}{\left(4 \cdot A\right) \cdot C}} \]

   if 4e-174 < (pow.f64 B #s(literal 2 binary64))

   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 13.9%

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

      1. mul-1-neg 13.9%

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

      2. +-commutative 13.9%

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

   5. Simplified 13.9%

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

      1. pow1/2 13.9%

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

      2. pow-to-exp 13.1%

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

      3. unpow2 13.1%

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

      4. unpow2 13.1%

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

      5. hypot-undefine 23.5%

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

   7. Applied egg-rr 23.5%

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

   8. Taylor expanded in F around -inf 12.9%

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

      1. mul-1-neg 12.9%

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

      2. unsub-neg 12.9%

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

      3. mul-1-neg 12.9%

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

      4. +-commutative 12.9%

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

      5. unpow2 12.9%

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

      6. unpow2 12.9%

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

      7. hypot-undefine 28.7%

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

   10. Simplified 28.7%

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

       1. associate-*l/ 28.7%

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

       2. pow1/2 28.7%

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

       3. exp-prod 23.0%

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

       4. pow-prod-down 23.0%

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

       5. diff-log 23.5%

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

       6. add-exp-log 25.0%

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

   12. Applied egg-rr 25.0%

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

4. Final simplification 23.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 4 \cdot 10^{-174}:\\ \;\;\;\;\frac{\sqrt{-16 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right)}}{\left(4 \cdot A\right) \cdot C}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(2 \cdot \frac{\mathsf{hypot}\left(B, A\right) - A}{\frac{-1}{F}}\right)}^{0.5}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 40.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 3.1 \cdot 10^{-87}:\\ \;\;\;\;\frac{\sqrt{-16 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right)}}{\left(4 \cdot A\right) \cdot C}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(A - \mathsf{hypot}\left(B\_m, A\right)\right)\right)}}{-B\_m}\\ \end{array} \end{array} \]

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 (<= B_m 3.1e-87)
   (/ (sqrt (* -16.0 (* (pow A 2.0) (* C F)))) (* (* 4.0 A) C))
   (/ (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) {
	double tmp;
	if (B_m <= 3.1e-87) {
		tmp = sqrt((-16.0 * (pow(A, 2.0) * (C * F)))) / ((4.0 * A) * C);
	} else {
		tmp = sqrt((2.0 * (F * (A - hypot(B_m, A))))) / -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 tmp;
	if (B_m <= 3.1e-87) {
		tmp = Math.sqrt((-16.0 * (Math.pow(A, 2.0) * (C * F)))) / ((4.0 * A) * C);
	} else {
		tmp = Math.sqrt((2.0 * (F * (A - Math.hypot(B_m, A))))) / -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 B_m <= 3.1e-87:
		tmp = math.sqrt((-16.0 * (math.pow(A, 2.0) * (C * F)))) / ((4.0 * A) * C)
	else:
		tmp = math.sqrt((2.0 * (F * (A - math.hypot(B_m, A))))) / -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 <= 3.1e-87)
		tmp = Float64(sqrt(Float64(-16.0 * Float64((A ^ 2.0) * Float64(C * F)))) / Float64(Float64(4.0 * A) * C));
	else
		tmp = Float64(sqrt(Float64(2.0 * Float64(F * Float64(A - hypot(B_m, A))))) / 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 (B_m <= 3.1e-87)
		tmp = sqrt((-16.0 * ((A ^ 2.0) * (C * F)))) / ((4.0 * A) * C);
	else
		tmp = sqrt((2.0 * (F * (A - hypot(B_m, A))))) / -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[B$95$m, 3.1e-87], N[(N[Sqrt[N[(-16.0 * N[(N[Power[A, 2.0], $MachinePrecision] * N[(C * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]), $MachinePrecision], 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. Split input into 2 regimes

2. if B < 3.09999999999999998e-87

   1. Initial program 27.1%

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

   3. Simplified 29.4%

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

   5. Taylor expanded in A around -inf 13.5%

      \[\leadsto \frac{\sqrt{\color{blue}{-16 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right)}}}{-\mathsf{fma}\left(C, A \cdot -4, {B}^{2}\right)} \]

   6. Taylor expanded in C around inf 14.6%

      \[\leadsto \frac{\sqrt{-16 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right)}}{\color{blue}{4 \cdot \left(A \cdot C\right)}} \]
   7. Step-by-step derivation

      1. associate-*r* 14.6%

         \[\leadsto \frac{\sqrt{-16 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right)}}{\color{blue}{\left(4 \cdot A\right) \cdot C}} \]

   8. Simplified 14.6%

      \[\leadsto \frac{\sqrt{-16 \cdot \left({A}^{2} \cdot \left(C \cdot F\right)\right)}}{\color{blue}{\left(4 \cdot A\right) \cdot C}} \]

   if 3.09999999999999998e-87 < B

   1. Initial program 15.8%

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

   3. Taylor expanded in C around 0 24.2%

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

      1. mul-1-neg 24.2%

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

      2. +-commutative 24.2%

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

      3. unpow2 24.2%

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

      4. unpow2 24.2%

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

      5. hypot-define 43.8%

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

   5. Simplified 43.8%

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

      1. neg-sub0 43.8%

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

      2. associate-*l/ 43.8%

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

      3. pow1/2 43.8%

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

      4. pow1/2 43.8%

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

      5. hypot-undefine 24.2%

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

      6. unpow2 24.2%

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

      7. unpow2 24.2%

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

      8. pow-prod-down 24.2%

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

      9. unpow2 24.2%

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

      10. unpow2 24.2%

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

      11. hypot-undefine 43.9%

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

   7. Applied egg-rr 43.9%

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

      1. neg-sub0 43.9%

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

      2. distribute-neg-frac2 43.9%

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

      3. unpow1/2 43.9%

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

   9. Simplified 43.9%

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

Alternative 13: 32.0% 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 23.3%

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

3. Taylor expanded in C around 0 9.9%

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

   1. mul-1-neg 9.9%

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

   2. +-commutative 9.9%

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

   3. unpow2 9.9%

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

   4. unpow2 9.9%

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

   5. hypot-define 17.1%

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

5. Simplified 17.1%

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

   1. neg-sub0 17.1%

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

   2. associate-*l/ 17.1%

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

   3. pow1/2 17.1%

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

   4. pow1/2 17.1%

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

   5. hypot-undefine 10.0%

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

   6. unpow2 10.0%

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

   7. unpow2 10.0%

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

   8. pow-prod-down 10.0%

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

   9. unpow2 10.0%

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

   10. unpow2 10.0%

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

   11. hypot-undefine 17.1%

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

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 14: 27.1% accurate, 5.7× speedup

Math

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

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 (- (* A F) (* 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) {
	return sqrt((2.0 * ((A * F) - (B_m * 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 * ((a * f) - (b_m * 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 * ((A * F) - (B_m * 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 * ((A * F) - (B_m * 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(A * F) - Float64(B_m * 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 * ((A * F) - (B_m * 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[(A * F), $MachinePrecision] - N[(B$95$m * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision]

Derivation

1. Initial program 23.3%

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

3. Taylor expanded in C around 0 9.9%

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

   1. mul-1-neg 9.9%

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

   2. +-commutative 9.9%

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

   3. unpow2 9.9%

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

   4. unpow2 9.9%

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

   5. hypot-define 17.1%

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

5. Simplified 17.1%

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

   1. neg-sub0 17.1%

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

   2. associate-*l/ 17.1%

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

   3. pow1/2 17.1%

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

   4. pow1/2 17.1%

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

   5. hypot-undefine 10.0%

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

   6. unpow2 10.0%

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

   7. unpow2 10.0%

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

   8. pow-prod-down 10.0%

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

   9. unpow2 10.0%

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

   10. unpow2 10.0%

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

   11. hypot-undefine 17.1%

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

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. Taylor expanded in A around 0 15.1%

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

    1. +-commutative 15.1%

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

    2. mul-1-neg 15.1%

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

    3. unsub-neg 15.1%

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

    4. *-commutative 15.1%

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

12. Simplified 15.1%

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

13. Final simplification 15.1%

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

Alternative 15: 26.8% accurate, 5.9× speedup

Math

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

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 (* (* B_m 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(((B_m * 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(((b_m * 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(((B_m * 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(((B_m * 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(Float64(B_m * F) * -2.0)) / 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(((B_m * 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[(N[(B$95$m * F), $MachinePrecision] * -2.0), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision]

Derivation

1. Initial program 23.3%

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

3. Taylor expanded in C around 0 9.9%

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

   1. mul-1-neg 9.9%

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

   2. +-commutative 9.9%

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

   3. unpow2 9.9%

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

   4. unpow2 9.9%

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

   5. hypot-define 17.1%

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

5. Simplified 17.1%

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

   1. neg-sub0 17.1%

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

   2. associate-*l/ 17.1%

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

   3. pow1/2 17.1%

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

   4. pow1/2 17.1%

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

   5. hypot-undefine 10.0%

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

   6. unpow2 10.0%

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

   7. unpow2 10.0%

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

   8. pow-prod-down 10.0%

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

   9. unpow2 10.0%

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

   10. unpow2 10.0%

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

   11. hypot-undefine 17.1%

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

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. Taylor expanded in A around 0 15.4%

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

11. Final simplification 15.4%

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

Alternative 16: 9.0% 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 23.3%

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

3. Taylor expanded in C around 0 9.9%

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

   1. mul-1-neg 9.9%

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

   2. +-commutative 9.9%

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

   3. unpow2 9.9%

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

   4. unpow2 9.9%

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

   5. hypot-define 17.1%

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

5. Simplified 17.1%

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

   1. div-inv 17.0%

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

7. Applied egg-rr 17.0%

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

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. unpow2 0.0%

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

   4. rem-square-sqrt 2.9%

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

   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 17: 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 23.3%

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

3. Taylor expanded in 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 2.3%

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

5. Simplified 2.3%

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

6. Taylor expanded in F around 0 2.3%

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

   1. *-commutative 2.3%

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

   2. pow1/2 2.3%

      \[\leadsto \color{blue}{{2}^{0.5}} \cdot \sqrt{\frac{F}{B}} \]

   3. pow1/2 2.4%

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

   4. pow-prod-down 2.4%

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

8. Applied egg-rr 2.4%

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

Alternative 18: 1.6% 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 23.3%

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

3. Taylor expanded in 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 2.3%

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

5. Simplified 2.3%

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

6. Taylor expanded in F around 0 2.3%

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

   1. pow1 2.3%

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

   2. *-commutative 2.3%

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

   3. sqrt-unprod 2.3%

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

8. Applied egg-rr 2.3%

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

   1. unpow1 2.3%

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

   2. associate-*r/ 2.3%

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

10. Simplified 2.3%

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

Reproduce

herbie shell --seed 2024170 
(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))))