ABCF->ab-angle b

Percentage Accurate: 19.4% → 52.3%

Time: 25.9s

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: 19.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 52.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ t_1 := 2 \cdot \left(\left({B\_m}^{2} - t\_0\right) \cdot F\right)\\ t_2 := t\_0 - {B\_m}^{2}\\ t_3 := \frac{\sqrt{t\_1 \cdot \left(\left(A + C\right) - \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_2}\\ t_4 := \mathsf{fma}\left(C, A \cdot -4, {B\_m}^{2}\right)\\ \mathbf{if}\;t\_3 \leq -1 \cdot 10^{-210}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B\_m, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot t\_4}}{-t\_4}\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{+187}:\\ \;\;\;\;\frac{\sqrt{t\_1 \cdot \left(A + \left(A + -0.5 \cdot \frac{{B\_m}^{2}}{C}\right)\right)}}{t\_2}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;\frac{{\left(e^{0.25 \cdot \left(\log \left(-8 \cdot \left(C \cdot F\right)\right) + -2 \cdot \log \left(\frac{-1}{A}\right)\right)} \cdot \sqrt{\sqrt{2}}\right)}^{2}}{-\mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{B\_m}{\sqrt{2}} \cdot \sqrt{\frac{\frac{1}{F}}{A - \mathsf{hypot}\left(A, B\_m\right)}}}\\ \end{array} \end{array} \]

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 (* (* 4.0 A) C))
        (t_1 (* 2.0 (* (- (pow B_m 2.0) t_0) F)))
        (t_2 (- t_0 (pow B_m 2.0)))
        (t_3
         (/
          (sqrt (* t_1 (- (+ A C) (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))))))
          t_2))
        (t_4 (fma C (* A -4.0) (pow B_m 2.0))))
   (if (<= t_3 -1e-210)
     (/
      (* (sqrt (* F (+ A (- C (hypot B_m (- A C)))))) (sqrt (* 2.0 t_4)))
      (- t_4))
     (if (<= t_3 2e+187)
       (/ (sqrt (* t_1 (+ A (+ A (* -0.5 (/ (pow B_m 2.0) C)))))) t_2)
       (if (<= t_3 INFINITY)
         (/
          (pow
           (*
            (exp (* 0.25 (+ (log (* -8.0 (* C F))) (* -2.0 (log (/ -1.0 A))))))
            (sqrt (sqrt 2.0)))
           2.0)
          (- (fma B_m B_m (* A (* C -4.0)))))
         (/
          -1.0
          (* (/ B_m (sqrt 2.0)) (sqrt (/ (/ 1.0 F) (- A (hypot 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 t_0 = (4.0 * A) * C;
	double t_1 = 2.0 * ((pow(B_m, 2.0) - t_0) * F);
	double t_2 = t_0 - pow(B_m, 2.0);
	double t_3 = sqrt((t_1 * ((A + C) - sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / t_2;
	double t_4 = fma(C, (A * -4.0), pow(B_m, 2.0));
	double tmp;
	if (t_3 <= -1e-210) {
		tmp = (sqrt((F * (A + (C - hypot(B_m, (A - C)))))) * sqrt((2.0 * t_4))) / -t_4;
	} else if (t_3 <= 2e+187) {
		tmp = sqrt((t_1 * (A + (A + (-0.5 * (pow(B_m, 2.0) / C)))))) / t_2;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = pow((exp((0.25 * (log((-8.0 * (C * F))) + (-2.0 * log((-1.0 / A)))))) * sqrt(sqrt(2.0))), 2.0) / -fma(B_m, B_m, (A * (C * -4.0)));
	} else {
		tmp = -1.0 / ((B_m / sqrt(2.0)) * sqrt(((1.0 / F) / (A - hypot(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)
	t_0 = Float64(Float64(4.0 * A) * C)
	t_1 = Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_0) * F))
	t_2 = Float64(t_0 - (B_m ^ 2.0))
	t_3 = Float64(sqrt(Float64(t_1 * Float64(Float64(A + C) - sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / t_2)
	t_4 = fma(C, Float64(A * -4.0), (B_m ^ 2.0))
	tmp = 0.0
	if (t_3 <= -1e-210)
		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 <= 2e+187)
		tmp = Float64(sqrt(Float64(t_1 * Float64(A + Float64(A + Float64(-0.5 * Float64((B_m ^ 2.0) / C)))))) / t_2);
	elseif (t_3 <= Inf)
		tmp = Float64((Float64(exp(Float64(0.25 * Float64(log(Float64(-8.0 * Float64(C * F))) + Float64(-2.0 * log(Float64(-1.0 / A)))))) * sqrt(sqrt(2.0))) ^ 2.0) / Float64(-fma(B_m, B_m, Float64(A * Float64(C * -4.0)))));
	else
		tmp = Float64(-1.0 / Float64(Float64(B_m / sqrt(2.0)) * sqrt(Float64(Float64(1.0 / F) / Float64(A - hypot(A, 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[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$1 = N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(t$95$1 * 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] / t$95$2), $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, -1e-210], 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, 2e+187], N[(N[Sqrt[N[(t$95$1 * N[(A + N[(A + N[(-0.5 * N[(N[Power[B$95$m, 2.0], $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[t$95$3, Infinity], N[(N[Power[N[(N[Exp[N[(0.25 * N[(N[Log[N[(-8.0 * N[(C * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(-2.0 * N[Log[N[(-1.0 / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[Sqrt[2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / (-N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], N[(-1.0 / N[(N[(B$95$m / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(1.0 / F), $MachinePrecision] / N[(A - N[Sqrt[A ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 4 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))) < -1e-210

   1. Initial program 42.5%

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

   3. Simplified 38.8%

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

      1. pow1/2 38.8%

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

      2. associate-*r* 49.0%

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

      3. unpow-prod-down 63.7%

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

      4. associate-+r- 62.7%

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

      5. hypot-undefine 50.1%

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

      6. unpow2 50.1%

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

      7. unpow2 50.1%

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

      8. +-commutative 50.1%

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

      9. unpow2 50.1%

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

      10. unpow2 50.1%

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

      11. hypot-define 62.7%

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

      12. pow1/2 62.7%

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

   6. Applied egg-rr 62.7%

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

      1. unpow1/2 62.7%

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

      2. sub-neg 62.7%

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

      3. associate-+l+ 63.7%

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

      4. sub-neg 63.7%

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

      5. hypot-undefine 50.1%

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

      6. unpow2 50.1%

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

      7. unpow2 50.1%

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

      8. +-commutative 50.1%

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

      9. unpow2 50.1%

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

      10. unpow2 50.1%

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

      11. hypot-undefine 63.7%

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

   8. Simplified 63.7%

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

   if -1e-210 < (/.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))) < 1.99999999999999981e187

   1. Initial program 11.0%

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

   3. Taylor expanded in C around inf 25.9%

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

      1. mul-1-neg 25.9%

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

      2. associate--l+ 25.9%

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

   5. Simplified 25.9%

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

   if 1.99999999999999981e187 < (/.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 4.7%

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

   3. Simplified 27.0%

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

      1. add-sqr-sqrt 26.8%

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

      2. pow2 26.8%

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

   6. Applied egg-rr 26.8%

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

   7. Taylor expanded in A around -inf 57.3%

      \[\leadsto \frac{{\color{blue}{\left(e^{0.25 \cdot \left(\log \left(-8 \cdot \left(C \cdot F\right)\right) + -2 \cdot \log \left(\frac{-1}{A}\right)\right)} \cdot \sqrt{\sqrt{2}}\right)}}^{2}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

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

   1. Initial program 0.0%

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

   3. Taylor expanded in C around 0 1.6%

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

      1. mul-1-neg 1.6%

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

      2. +-commutative 1.6%

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

   5. Simplified 1.6%

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

      1. add-cbrt-cube 1.6%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot \color{blue}{\sqrt[3]{\left(\sqrt{F \cdot \left(A - \sqrt{{B}^{2} + {A}^{2}}\right)} \cdot \sqrt{F \cdot \left(A - \sqrt{{B}^{2} + {A}^{2}}\right)}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{{B}^{2} + {A}^{2}}\right)}}} \]

      2. pow1/3 1.6%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot \color{blue}{{\left(\left(\sqrt{F \cdot \left(A - \sqrt{{B}^{2} + {A}^{2}}\right)} \cdot \sqrt{F \cdot \left(A - \sqrt{{B}^{2} + {A}^{2}}\right)}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{{B}^{2} + {A}^{2}}\right)}\right)}^{0.3333333333333333}} \]

   7. Applied egg-rr 8.6%

      \[\leadsto -\frac{\sqrt{2}}{B} \cdot \color{blue}{{\left({\left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}^{1.5}\right)}^{0.3333333333333333}} \]
   8. Step-by-step derivation

      1. associate-*l/ 8.6%

         \[\leadsto -\color{blue}{\frac{\sqrt{2} \cdot {\left({\left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}^{1.5}\right)}^{0.3333333333333333}}{B}} \]

      2. pow-pow 13.4%

         \[\leadsto -\frac{\sqrt{2} \cdot \color{blue}{{\left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}}}{B} \]

      3. metadata-eval 13.4%

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

      4. pow1/2 13.4%

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

      5. sqrt-prod 13.4%

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

      6. distribute-frac-neg2 13.4%

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

      7. clear-num 13.4%

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

      8. associate-*r* 13.4%

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

   9. Applied egg-rr 13.4%

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

   10. Taylor expanded in F around 0 1.5%

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

       1. mul-1-neg 1.5%

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

       2. distribute-rgt-neg-in 1.5%

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

       3. associate-/r* 1.5%

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

       4. unpow2 1.5%

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

       5. unpow2 1.5%

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

       6. hypot-define 12.2%

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

   12. Simplified 12.2%

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

4. Final simplification 35.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -1 \cdot 10^{-210}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(C, A \cdot -4, {B}^{2}\right)}}{-\mathsf{fma}\left(C, A \cdot -4, {B}^{2}\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq 2 \cdot 10^{+187}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \left(A + -0.5 \cdot \frac{{B}^{2}}{C}\right)\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{{\left(e^{0.25 \cdot \left(\log \left(-8 \cdot \left(C \cdot F\right)\right) + -2 \cdot \log \left(\frac{-1}{A}\right)\right)} \cdot \sqrt{\sqrt{2}}\right)}^{2}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{B}{\sqrt{2}} \cdot \sqrt{\frac{\frac{1}{F}}{A - \mathsf{hypot}\left(A, B\right)}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 45.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ \mathbf{if}\;B\_m \leq 3.1 \cdot 10^{-174}:\\ \;\;\;\;\frac{{\left(e^{0.25 \cdot \left(\log \left(-8 \cdot \left(C \cdot F\right)\right) + -2 \cdot \log \left(\frac{-1}{A}\right)\right)} \cdot \sqrt{\sqrt{2}}\right)}^{2}}{-\mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;B\_m \leq 2.45 \cdot 10^{+20}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_0\right) \cdot F\right)\right) \cdot \left(A + \left(A + -0.5 \cdot \frac{{B\_m}^{2}}{C}\right)\right)}}{t\_0 - {B\_m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{B\_m}{\sqrt{2}} \cdot \sqrt{\frac{\frac{1}{F}}{A - \mathsf{hypot}\left(A, B\_m\right)}}}\\ \end{array} \end{array} \]

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 (* (* 4.0 A) C)))
   (if (<= B_m 3.1e-174)
     (/
      (pow
       (*
        (exp (* 0.25 (+ (log (* -8.0 (* C F))) (* -2.0 (log (/ -1.0 A))))))
        (sqrt (sqrt 2.0)))
       2.0)
      (- (fma B_m B_m (* A (* C -4.0)))))
     (if (<= B_m 2.45e+20)
       (/
        (sqrt
         (*
          (* 2.0 (* (- (pow B_m 2.0) t_0) F))
          (+ A (+ A (* -0.5 (/ (pow B_m 2.0) C))))))
        (- t_0 (pow B_m 2.0)))
       (/
        -1.0
        (* (/ B_m (sqrt 2.0)) (sqrt (/ (/ 1.0 F) (- A (hypot 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 t_0 = (4.0 * A) * C;
	double tmp;
	if (B_m <= 3.1e-174) {
		tmp = pow((exp((0.25 * (log((-8.0 * (C * F))) + (-2.0 * log((-1.0 / A)))))) * sqrt(sqrt(2.0))), 2.0) / -fma(B_m, B_m, (A * (C * -4.0)));
	} else if (B_m <= 2.45e+20) {
		tmp = sqrt(((2.0 * ((pow(B_m, 2.0) - t_0) * F)) * (A + (A + (-0.5 * (pow(B_m, 2.0) / C)))))) / (t_0 - pow(B_m, 2.0));
	} else {
		tmp = -1.0 / ((B_m / sqrt(2.0)) * sqrt(((1.0 / F) / (A - hypot(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)
	t_0 = Float64(Float64(4.0 * A) * C)
	tmp = 0.0
	if (B_m <= 3.1e-174)
		tmp = Float64((Float64(exp(Float64(0.25 * Float64(log(Float64(-8.0 * Float64(C * F))) + Float64(-2.0 * log(Float64(-1.0 / A)))))) * sqrt(sqrt(2.0))) ^ 2.0) / Float64(-fma(B_m, B_m, Float64(A * Float64(C * -4.0)))));
	elseif (B_m <= 2.45e+20)
		tmp = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_0) * F)) * Float64(A + Float64(A + Float64(-0.5 * Float64((B_m ^ 2.0) / C)))))) / Float64(t_0 - (B_m ^ 2.0)));
	else
		tmp = Float64(-1.0 / Float64(Float64(B_m / sqrt(2.0)) * sqrt(Float64(Float64(1.0 / F) / Float64(A - hypot(A, 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[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, If[LessEqual[B$95$m, 3.1e-174], N[(N[Power[N[(N[Exp[N[(0.25 * N[(N[Log[N[(-8.0 * N[(C * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + N[(-2.0 * N[Log[N[(-1.0 / A), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[Sqrt[2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / (-N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], If[LessEqual[B$95$m, 2.45e+20], N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(A + N[(A + N[(-0.5 * N[(N[Power[B$95$m, 2.0], $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(N[(B$95$m / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(1.0 / F), $MachinePrecision] / N[(A - N[Sqrt[A ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if B < 3.0999999999999999e-174

   1. Initial program 16.4%

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

   3. Simplified 21.8%

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

      1. add-sqr-sqrt 21.7%

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

      2. pow2 21.7%

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

   6. Applied egg-rr 20.5%

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

   7. Taylor expanded in A around -inf 14.4%

      \[\leadsto \frac{{\color{blue}{\left(e^{0.25 \cdot \left(\log \left(-8 \cdot \left(C \cdot F\right)\right) + -2 \cdot \log \left(\frac{-1}{A}\right)\right)} \cdot \sqrt{\sqrt{2}}\right)}}^{2}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   if 3.0999999999999999e-174 < B < 2.45e20

   1. Initial program 25.5%

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

   3. Taylor expanded in C around inf 21.3%

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

      1. mul-1-neg 21.3%

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

      2. associate--l+ 21.3%

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

   5. Simplified 21.3%

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

   if 2.45e20 < B

   1. Initial program 16.2%

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

   3. Taylor expanded in C around 0 23.4%

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

      1. mul-1-neg 23.4%

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

      2. +-commutative 23.4%

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

   5. Simplified 23.4%

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

      1. add-cbrt-cube 19.0%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot \color{blue}{\sqrt[3]{\left(\sqrt{F \cdot \left(A - \sqrt{{B}^{2} + {A}^{2}}\right)} \cdot \sqrt{F \cdot \left(A - \sqrt{{B}^{2} + {A}^{2}}\right)}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{{B}^{2} + {A}^{2}}\right)}}} \]

      2. pow1/3 17.9%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot \color{blue}{{\left(\left(\sqrt{F \cdot \left(A - \sqrt{{B}^{2} + {A}^{2}}\right)} \cdot \sqrt{F \cdot \left(A - \sqrt{{B}^{2} + {A}^{2}}\right)}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{{B}^{2} + {A}^{2}}\right)}\right)}^{0.3333333333333333}} \]

   7. Applied egg-rr 29.1%

      \[\leadsto -\frac{\sqrt{2}}{B} \cdot \color{blue}{{\left({\left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}^{1.5}\right)}^{0.3333333333333333}} \]
   8. Step-by-step derivation

      1. associate-*l/ 29.1%

         \[\leadsto -\color{blue}{\frac{\sqrt{2} \cdot {\left({\left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}^{1.5}\right)}^{0.3333333333333333}}{B}} \]

      2. pow-pow 42.4%

         \[\leadsto -\frac{\sqrt{2} \cdot \color{blue}{{\left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}}}{B} \]

      3. metadata-eval 42.4%

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

      4. pow1/2 42.4%

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

      5. sqrt-prod 42.6%

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

      6. distribute-frac-neg2 42.6%

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

      7. clear-num 42.6%

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

      8. associate-*r* 42.6%

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

   9. Applied egg-rr 42.6%

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

   10. Taylor expanded in F around 0 23.4%

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

       1. mul-1-neg 23.4%

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

       2. distribute-rgt-neg-in 23.4%

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

       3. associate-/r* 23.4%

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

       4. unpow2 23.4%

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

       5. unpow2 23.4%

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

       6. hypot-define 41.9%

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

   12. Simplified 41.9%

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

4. Final simplification 22.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 3.1 \cdot 10^{-174}:\\ \;\;\;\;\frac{{\left(e^{0.25 \cdot \left(\log \left(-8 \cdot \left(C \cdot F\right)\right) + -2 \cdot \log \left(\frac{-1}{A}\right)\right)} \cdot \sqrt{\sqrt{2}}\right)}^{2}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 2.45 \cdot 10^{+20}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \left(A + -0.5 \cdot \frac{{B}^{2}}{C}\right)\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{B}{\sqrt{2}} \cdot \sqrt{\frac{\frac{1}{F}}{A - \mathsf{hypot}\left(A, B\right)}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 47.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ \mathbf{if}\;{B\_m}^{2} \leq 10^{+41}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_0\right) \cdot F\right)\right) \cdot \left(A + \left(A + -0.5 \cdot \frac{{B\_m}^{2}}{C}\right)\right)}}{t\_0 - {B\_m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{B\_m}{\sqrt{2}} \cdot \sqrt{\frac{\frac{1}{F}}{A - \mathsf{hypot}\left(A, B\_m\right)}}}\\ \end{array} \end{array} \]

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 (* (* 4.0 A) C)))
   (if (<= (pow B_m 2.0) 1e+41)
     (/
      (sqrt
       (*
        (* 2.0 (* (- (pow B_m 2.0) t_0) F))
        (+ A (+ A (* -0.5 (/ (pow B_m 2.0) C))))))
      (- t_0 (pow B_m 2.0)))
     (/
      -1.0
      (* (/ B_m (sqrt 2.0)) (sqrt (/ (/ 1.0 F) (- A (hypot 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 t_0 = (4.0 * A) * C;
	double tmp;
	if (pow(B_m, 2.0) <= 1e+41) {
		tmp = sqrt(((2.0 * ((pow(B_m, 2.0) - t_0) * F)) * (A + (A + (-0.5 * (pow(B_m, 2.0) / C)))))) / (t_0 - pow(B_m, 2.0));
	} else {
		tmp = -1.0 / ((B_m / sqrt(2.0)) * sqrt(((1.0 / F) / (A - hypot(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 t_0 = (4.0 * A) * C;
	double tmp;
	if (Math.pow(B_m, 2.0) <= 1e+41) {
		tmp = Math.sqrt(((2.0 * ((Math.pow(B_m, 2.0) - t_0) * F)) * (A + (A + (-0.5 * (Math.pow(B_m, 2.0) / C)))))) / (t_0 - Math.pow(B_m, 2.0));
	} else {
		tmp = -1.0 / ((B_m / Math.sqrt(2.0)) * Math.sqrt(((1.0 / F) / (A - Math.hypot(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):
	t_0 = (4.0 * A) * C
	tmp = 0
	if math.pow(B_m, 2.0) <= 1e+41:
		tmp = math.sqrt(((2.0 * ((math.pow(B_m, 2.0) - t_0) * F)) * (A + (A + (-0.5 * (math.pow(B_m, 2.0) / C)))))) / (t_0 - math.pow(B_m, 2.0))
	else:
		tmp = -1.0 / ((B_m / math.sqrt(2.0)) * math.sqrt(((1.0 / F) / (A - math.hypot(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)
	t_0 = Float64(Float64(4.0 * A) * C)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e+41)
		tmp = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_0) * F)) * Float64(A + Float64(A + Float64(-0.5 * Float64((B_m ^ 2.0) / C)))))) / Float64(t_0 - (B_m ^ 2.0)));
	else
		tmp = Float64(-1.0 / Float64(Float64(B_m / sqrt(2.0)) * sqrt(Float64(Float64(1.0 / F) / Float64(A - hypot(A, B_m))))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	t_0 = (4.0 * A) * C;
	tmp = 0.0;
	if ((B_m ^ 2.0) <= 1e+41)
		tmp = sqrt(((2.0 * (((B_m ^ 2.0) - t_0) * F)) * (A + (A + (-0.5 * ((B_m ^ 2.0) / C)))))) / (t_0 - (B_m ^ 2.0));
	else
		tmp = -1.0 / ((B_m / sqrt(2.0)) * sqrt(((1.0 / F) / (A - hypot(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_] := Block[{t$95$0 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e+41], N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(A + N[(A + N[(-0.5 * N[(N[Power[B$95$m, 2.0], $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(N[(B$95$m / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(1.0 / F), $MachinePrecision] / N[(A - N[Sqrt[A ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 1.00000000000000001e41

   1. Initial program 19.8%

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

   3. Taylor expanded in C around inf 23.4%

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

      1. mul-1-neg 23.4%

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

      2. associate--l+ 23.4%

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

   5. Simplified 23.4%

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

   if 1.00000000000000001e41 < (pow.f64 B #s(literal 2 binary64))

   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 12.8%

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

      1. mul-1-neg 12.8%

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

      2. +-commutative 12.8%

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

   5. Simplified 12.8%

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

      1. add-cbrt-cube 10.5%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot \color{blue}{\sqrt[3]{\left(\sqrt{F \cdot \left(A - \sqrt{{B}^{2} + {A}^{2}}\right)} \cdot \sqrt{F \cdot \left(A - \sqrt{{B}^{2} + {A}^{2}}\right)}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{{B}^{2} + {A}^{2}}\right)}}} \]

      2. pow1/3 9.9%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot \color{blue}{{\left(\left(\sqrt{F \cdot \left(A - \sqrt{{B}^{2} + {A}^{2}}\right)} \cdot \sqrt{F \cdot \left(A - \sqrt{{B}^{2} + {A}^{2}}\right)}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{{B}^{2} + {A}^{2}}\right)}\right)}^{0.3333333333333333}} \]

   7. Applied egg-rr 16.3%

      \[\leadsto -\frac{\sqrt{2}}{B} \cdot \color{blue}{{\left({\left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}^{1.5}\right)}^{0.3333333333333333}} \]
   8. Step-by-step derivation

      1. associate-*l/ 16.3%

         \[\leadsto -\color{blue}{\frac{\sqrt{2} \cdot {\left({\left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}^{1.5}\right)}^{0.3333333333333333}}{B}} \]

      2. pow-pow 23.4%

         \[\leadsto -\frac{\sqrt{2} \cdot \color{blue}{{\left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}}}{B} \]

      3. metadata-eval 23.4%

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

      4. pow1/2 23.4%

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

      5. sqrt-prod 23.4%

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

      6. distribute-frac-neg2 23.4%

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

      7. clear-num 23.5%

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

      8. associate-*r* 23.5%

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

   9. Applied egg-rr 23.5%

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

   10. Taylor expanded in F around 0 12.8%

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

       1. mul-1-neg 12.8%

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

       2. distribute-rgt-neg-in 12.8%

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

       3. associate-/r* 12.8%

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

       4. unpow2 12.8%

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

       5. unpow2 12.8%

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

       6. hypot-define 22.7%

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

   12. Simplified 22.7%

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

4. Final simplification 23.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{+41}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(A + \left(A + -0.5 \cdot \frac{{B}^{2}}{C}\right)\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{B}{\sqrt{2}} \cdot \sqrt{\frac{\frac{1}{F}}{A - \mathsf{hypot}\left(A, B\right)}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 47.7% accurate, 1.5× speedup

Math

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

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 (* (* 4.0 A) C)))
   (if (<= (pow B_m 2.0) 1e+41)
     (/
      (sqrt (* (* 2.0 (* (- (pow B_m 2.0) t_0) F)) (* 2.0 A)))
      (- t_0 (pow B_m 2.0)))
     (/
      -1.0
      (* (/ B_m (sqrt 2.0)) (sqrt (/ (/ 1.0 F) (- A (hypot 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 t_0 = (4.0 * A) * C;
	double tmp;
	if (pow(B_m, 2.0) <= 1e+41) {
		tmp = sqrt(((2.0 * ((pow(B_m, 2.0) - t_0) * F)) * (2.0 * A))) / (t_0 - pow(B_m, 2.0));
	} else {
		tmp = -1.0 / ((B_m / sqrt(2.0)) * sqrt(((1.0 / F) / (A - hypot(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 t_0 = (4.0 * A) * C;
	double tmp;
	if (Math.pow(B_m, 2.0) <= 1e+41) {
		tmp = Math.sqrt(((2.0 * ((Math.pow(B_m, 2.0) - t_0) * F)) * (2.0 * A))) / (t_0 - Math.pow(B_m, 2.0));
	} else {
		tmp = -1.0 / ((B_m / Math.sqrt(2.0)) * Math.sqrt(((1.0 / F) / (A - Math.hypot(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):
	t_0 = (4.0 * A) * C
	tmp = 0
	if math.pow(B_m, 2.0) <= 1e+41:
		tmp = math.sqrt(((2.0 * ((math.pow(B_m, 2.0) - t_0) * F)) * (2.0 * A))) / (t_0 - math.pow(B_m, 2.0))
	else:
		tmp = -1.0 / ((B_m / math.sqrt(2.0)) * math.sqrt(((1.0 / F) / (A - math.hypot(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)
	t_0 = Float64(Float64(4.0 * A) * C)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e+41)
		tmp = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_0) * F)) * Float64(2.0 * A))) / Float64(t_0 - (B_m ^ 2.0)));
	else
		tmp = Float64(-1.0 / Float64(Float64(B_m / sqrt(2.0)) * sqrt(Float64(Float64(1.0 / F) / Float64(A - hypot(A, B_m))))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	t_0 = (4.0 * A) * C;
	tmp = 0.0;
	if ((B_m ^ 2.0) <= 1e+41)
		tmp = sqrt(((2.0 * (((B_m ^ 2.0) - t_0) * F)) * (2.0 * A))) / (t_0 - (B_m ^ 2.0));
	else
		tmp = -1.0 / ((B_m / sqrt(2.0)) * sqrt(((1.0 / F) / (A - hypot(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_] := Block[{t$95$0 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e+41], N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(2.0 * A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(-1.0 / N[(N[(B$95$m / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(1.0 / F), $MachinePrecision] / N[(A - N[Sqrt[A ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 1.00000000000000001e41

   1. Initial program 19.8%

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

   3. Taylor expanded in A around -inf 22.1%

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

   if 1.00000000000000001e41 < (pow.f64 B #s(literal 2 binary64))

   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 12.8%

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

      1. mul-1-neg 12.8%

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

      2. +-commutative 12.8%

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

   5. Simplified 12.8%

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

      1. add-cbrt-cube 10.5%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot \color{blue}{\sqrt[3]{\left(\sqrt{F \cdot \left(A - \sqrt{{B}^{2} + {A}^{2}}\right)} \cdot \sqrt{F \cdot \left(A - \sqrt{{B}^{2} + {A}^{2}}\right)}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{{B}^{2} + {A}^{2}}\right)}}} \]

      2. pow1/3 9.9%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot \color{blue}{{\left(\left(\sqrt{F \cdot \left(A - \sqrt{{B}^{2} + {A}^{2}}\right)} \cdot \sqrt{F \cdot \left(A - \sqrt{{B}^{2} + {A}^{2}}\right)}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{{B}^{2} + {A}^{2}}\right)}\right)}^{0.3333333333333333}} \]

   7. Applied egg-rr 16.3%

      \[\leadsto -\frac{\sqrt{2}}{B} \cdot \color{blue}{{\left({\left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}^{1.5}\right)}^{0.3333333333333333}} \]
   8. Step-by-step derivation

      1. associate-*l/ 16.3%

         \[\leadsto -\color{blue}{\frac{\sqrt{2} \cdot {\left({\left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}^{1.5}\right)}^{0.3333333333333333}}{B}} \]

      2. pow-pow 23.4%

         \[\leadsto -\frac{\sqrt{2} \cdot \color{blue}{{\left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}}}{B} \]

      3. metadata-eval 23.4%

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

      4. pow1/2 23.4%

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

      5. sqrt-prod 23.4%

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

      6. distribute-frac-neg2 23.4%

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

      7. clear-num 23.5%

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

      8. associate-*r* 23.5%

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

   9. Applied egg-rr 23.5%

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

   10. Taylor expanded in F around 0 12.8%

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

       1. mul-1-neg 12.8%

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

       2. distribute-rgt-neg-in 12.8%

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

       3. associate-/r* 12.8%

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

       4. unpow2 12.8%

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

       5. unpow2 12.8%

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

       6. hypot-define 22.7%

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

   12. Simplified 22.7%

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

4. Final simplification 22.4%

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

Alternative 5: 47.7% accurate, 1.5× speedup

Math

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

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) 1e+41)
     (/ (sqrt (* (* 4.0 A) (* F t_0))) (- t_0))
     (/
      -1.0
      (* (/ B_m (sqrt 2.0)) (sqrt (/ (/ 1.0 F) (- A (hypot 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 t_0 = fma(B_m, B_m, (A * (C * -4.0)));
	double tmp;
	if (pow(B_m, 2.0) <= 1e+41) {
		tmp = sqrt(((4.0 * A) * (F * t_0))) / -t_0;
	} else {
		tmp = -1.0 / ((B_m / sqrt(2.0)) * sqrt(((1.0 / F) / (A - hypot(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)
	t_0 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 1e+41)
		tmp = Float64(sqrt(Float64(Float64(4.0 * A) * Float64(F * t_0))) / Float64(-t_0));
	else
		tmp = Float64(-1.0 / Float64(Float64(B_m / sqrt(2.0)) * sqrt(Float64(Float64(1.0 / F) / Float64(A - hypot(A, B_m))))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e+41], N[(N[Sqrt[N[(N[(4.0 * A), $MachinePrecision] * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], N[(-1.0 / N[(N[(B$95$m / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(1.0 / F), $MachinePrecision] / N[(A - N[Sqrt[A ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 1.00000000000000001e41

   1. Initial program 19.8%

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

   3. Simplified 27.8%

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

   5. Taylor expanded in A around -inf 22.1%

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

   if 1.00000000000000001e41 < (pow.f64 B #s(literal 2 binary64))

   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 12.8%

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

      1. mul-1-neg 12.8%

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

      2. +-commutative 12.8%

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

   5. Simplified 12.8%

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

      1. add-cbrt-cube 10.5%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot \color{blue}{\sqrt[3]{\left(\sqrt{F \cdot \left(A - \sqrt{{B}^{2} + {A}^{2}}\right)} \cdot \sqrt{F \cdot \left(A - \sqrt{{B}^{2} + {A}^{2}}\right)}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{{B}^{2} + {A}^{2}}\right)}}} \]

      2. pow1/3 9.9%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot \color{blue}{{\left(\left(\sqrt{F \cdot \left(A - \sqrt{{B}^{2} + {A}^{2}}\right)} \cdot \sqrt{F \cdot \left(A - \sqrt{{B}^{2} + {A}^{2}}\right)}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{{B}^{2} + {A}^{2}}\right)}\right)}^{0.3333333333333333}} \]

   7. Applied egg-rr 16.3%

      \[\leadsto -\frac{\sqrt{2}}{B} \cdot \color{blue}{{\left({\left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}^{1.5}\right)}^{0.3333333333333333}} \]
   8. Step-by-step derivation

      1. associate-*l/ 16.3%

         \[\leadsto -\color{blue}{\frac{\sqrt{2} \cdot {\left({\left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}^{1.5}\right)}^{0.3333333333333333}}{B}} \]

      2. pow-pow 23.4%

         \[\leadsto -\frac{\sqrt{2} \cdot \color{blue}{{\left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}}}{B} \]

      3. metadata-eval 23.4%

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

      4. pow1/2 23.4%

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

      5. sqrt-prod 23.4%

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

      6. distribute-frac-neg2 23.4%

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

      7. clear-num 23.5%

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

      8. associate-*r* 23.5%

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

   9. Applied egg-rr 23.5%

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

   10. Taylor expanded in F around 0 12.8%

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

       1. mul-1-neg 12.8%

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

       2. distribute-rgt-neg-in 12.8%

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

       3. associate-/r* 12.8%

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

       4. unpow2 12.8%

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

       5. unpow2 12.8%

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

       6. hypot-define 22.7%

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

   12. Simplified 22.7%

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

4. Final simplification 22.4%

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

Alternative 6: 46.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 2 \cdot 10^{-53}:\\ \;\;\;\;\frac{\sqrt{\left(A \cdot -8\right) \cdot \left(C \cdot \left(F \cdot \left(A + A\right)\right)\right)}}{-\mathsf{fma}\left(C, A \cdot -4, {B\_m}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{B\_m}{\sqrt{2}} \cdot \sqrt{\frac{\frac{1}{F}}{A - \mathsf{hypot}\left(A, B\_m\right)}}}\\ \end{array} \end{array} \]

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
 (if (<= (pow B_m 2.0) 2e-53)
   (/
    (sqrt (* (* A -8.0) (* C (* F (+ A A)))))
    (- (fma C (* A -4.0) (pow B_m 2.0))))
   (/ -1.0 (* (/ B_m (sqrt 2.0)) (sqrt (/ (/ 1.0 F) (- A (hypot 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 (pow(B_m, 2.0) <= 2e-53) {
		tmp = sqrt(((A * -8.0) * (C * (F * (A + A))))) / -fma(C, (A * -4.0), pow(B_m, 2.0));
	} else {
		tmp = -1.0 / ((B_m / sqrt(2.0)) * sqrt(((1.0 / F) / (A - hypot(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 ^ 2.0) <= 2e-53)
		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(-1.0 / Float64(Float64(B_m / sqrt(2.0)) * sqrt(Float64(Float64(1.0 / F) / Float64(A - hypot(A, B_m))))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e-53], 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[(-1.0 / N[(N[(B$95$m / N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(1.0 / F), $MachinePrecision] / N[(A - N[Sqrt[A ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 2.00000000000000006e-53

   1. Initial program 19.6%

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

   3. Simplified 20.5%

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

   5. Taylor expanded in C around inf 21.9%

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

      1. associate-*r* 21.9%

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

      2. mul-1-neg 21.9%

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

   7. Simplified 21.9%

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

   if 2.00000000000000006e-53 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 16.4%

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

   3. Taylor expanded in C around 0 12.9%

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

      1. mul-1-neg 12.9%

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

      2. +-commutative 12.9%

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

   5. Simplified 12.9%

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

      1. add-cbrt-cube 10.3%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot \color{blue}{\sqrt[3]{\left(\sqrt{F \cdot \left(A - \sqrt{{B}^{2} + {A}^{2}}\right)} \cdot \sqrt{F \cdot \left(A - \sqrt{{B}^{2} + {A}^{2}}\right)}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{{B}^{2} + {A}^{2}}\right)}}} \]

      2. pow1/3 9.8%

         \[\leadsto -\frac{\sqrt{2}}{B} \cdot \color{blue}{{\left(\left(\sqrt{F \cdot \left(A - \sqrt{{B}^{2} + {A}^{2}}\right)} \cdot \sqrt{F \cdot \left(A - \sqrt{{B}^{2} + {A}^{2}}\right)}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{{B}^{2} + {A}^{2}}\right)}\right)}^{0.3333333333333333}} \]

   7. Applied egg-rr 15.5%

      \[\leadsto -\frac{\sqrt{2}}{B} \cdot \color{blue}{{\left({\left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}^{1.5}\right)}^{0.3333333333333333}} \]
   8. Step-by-step derivation

      1. associate-*l/ 15.5%

         \[\leadsto -\color{blue}{\frac{\sqrt{2} \cdot {\left({\left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}^{1.5}\right)}^{0.3333333333333333}}{B}} \]

      2. pow-pow 22.4%

         \[\leadsto -\frac{\sqrt{2} \cdot \color{blue}{{\left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}}}{B} \]

      3. metadata-eval 22.4%

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

      4. pow1/2 22.3%

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

      5. sqrt-prod 22.4%

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

      6. distribute-frac-neg2 22.4%

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

      7. clear-num 22.4%

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

      8. associate-*r* 22.4%

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

   9. Applied egg-rr 22.4%

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

   10. Taylor expanded in F around 0 12.4%

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

       1. mul-1-neg 12.4%

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

       2. distribute-rgt-neg-in 12.4%

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

       3. associate-/r* 12.4%

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

       4. unpow2 12.4%

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

       5. unpow2 12.4%

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

       6. hypot-define 21.6%

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

   12. Simplified 21.6%

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

4. Final simplification 21.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 2 \cdot 10^{-53}:\\ \;\;\;\;\frac{\sqrt{\left(A \cdot -8\right) \cdot \left(C \cdot \left(F \cdot \left(A + A\right)\right)\right)}}{-\mathsf{fma}\left(C, A \cdot -4, {B}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{B}{\sqrt{2}} \cdot \sqrt{\frac{\frac{1}{F}}{A - \mathsf{hypot}\left(A, B\right)}}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 32.0% accurate, 3.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \frac{-1}{B\_m \cdot {\left(\left(2 \cdot F\right) \cdot \left(A - \mathsf{hypot}\left(B\_m, A\right)\right)\right)}^{-0.5}} \end{array} \]

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
 (/ -1.0 (* B_m (pow (* (* 2.0 F) (- A (hypot B_m A))) -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 -1.0 / (B_m * pow(((2.0 * F) * (A - hypot(B_m, A))), -0.5));
}

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 -1.0 / (B_m * Math.pow(((2.0 * F) * (A - Math.hypot(B_m, A))), -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 -1.0 / (B_m * math.pow(((2.0 * F) * (A - math.hypot(B_m, A))), -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(-1.0 / Float64(B_m * (Float64(Float64(2.0 * F) * Float64(A - hypot(B_m, A))) ^ -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 = -1.0 / (B_m * (((2.0 * F) * (A - hypot(B_m, A))) ^ -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[(-1.0 / N[(B$95$m * N[Power[N[(N[(2.0 * F), $MachinePrecision] * N[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 17.8%

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

3. Taylor expanded in C around 0 9.1%

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

   1. mul-1-neg 9.1%

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

   2. +-commutative 9.1%

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

5. Simplified 9.1%

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

   1. add-cbrt-cube 7.3%

      \[\leadsto -\frac{\sqrt{2}}{B} \cdot \color{blue}{\sqrt[3]{\left(\sqrt{F \cdot \left(A - \sqrt{{B}^{2} + {A}^{2}}\right)} \cdot \sqrt{F \cdot \left(A - \sqrt{{B}^{2} + {A}^{2}}\right)}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{{B}^{2} + {A}^{2}}\right)}}} \]

   2. pow1/3 6.9%

      \[\leadsto -\frac{\sqrt{2}}{B} \cdot \color{blue}{{\left(\left(\sqrt{F \cdot \left(A - \sqrt{{B}^{2} + {A}^{2}}\right)} \cdot \sqrt{F \cdot \left(A - \sqrt{{B}^{2} + {A}^{2}}\right)}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{{B}^{2} + {A}^{2}}\right)}\right)}^{0.3333333333333333}} \]

7. Applied egg-rr 10.5%

   \[\leadsto -\frac{\sqrt{2}}{B} \cdot \color{blue}{{\left({\left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}^{1.5}\right)}^{0.3333333333333333}} \]
8. Step-by-step derivation

   1. associate-*l/ 10.5%

      \[\leadsto -\color{blue}{\frac{\sqrt{2} \cdot {\left({\left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}^{1.5}\right)}^{0.3333333333333333}}{B}} \]

   2. pow-pow 14.7%

      \[\leadsto -\frac{\sqrt{2} \cdot \color{blue}{{\left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}}}{B} \]

   3. metadata-eval 14.7%

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

   4. pow1/2 14.6%

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

   5. sqrt-prod 14.7%

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

   6. distribute-frac-neg2 14.7%

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

   7. clear-num 14.7%

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

   8. associate-*r* 14.7%

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

9. Applied egg-rr 14.7%

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

    1. add-sqr-sqrt 1.2%

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

    2. sqrt-unprod 2.2%

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

    3. sqr-neg 2.2%

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

    4. sqrt-prod 1.3%

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

    5. add-sqr-sqrt 15.4%

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

    6. remove-double-neg 15.4%

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

    7. distribute-frac-neg 15.4%

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

    8. div-inv 15.3%

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

    9. distribute-rgt-neg-in 15.3%

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

    10. add-sqr-sqrt 14.0%

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

    11. sqrt-unprod 15.8%

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

    12. sqr-neg 15.8%

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

    13. sqrt-prod 13.4%

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

    14. add-sqr-sqrt 14.7%

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

    15. pow1/2 14.7%

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

    16. pow-flip 14.8%

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

    17. associate-*l* 14.7%

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

    18. metadata-eval 14.7%

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

11. Applied egg-rr 14.7%

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

    1. associate-*r* 14.8%

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

    2. *-commutative 14.8%

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

13. Simplified 14.8%

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

14. Final simplification 14.8%

    \[\leadsto \frac{-1}{B \cdot {\left(\left(2 \cdot F\right) \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}^{-0.5}} \]
15. Add Preprocessing

Alternative 8: 32.0% accurate, 3.0× speedup

Math

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

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
 (/ -1.0 (/ B_m (sqrt (* (* 2.0 F) (- A (hypot B_m A)))))))

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 -1.0 / (B_m / sqrt(((2.0 * F) * (A - hypot(B_m, A)))));
}

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 -1.0 / (B_m / Math.sqrt(((2.0 * F) * (A - Math.hypot(B_m, A)))));
}

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 -1.0 / (B_m / math.sqrt(((2.0 * F) * (A - math.hypot(B_m, A)))))

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(-1.0 / Float64(B_m / sqrt(Float64(Float64(2.0 * F) * Float64(A - hypot(B_m, A))))))
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 = -1.0 / (B_m / sqrt(((2.0 * F) * (A - hypot(B_m, A)))));
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[(-1.0 / N[(B$95$m / N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] * N[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 17.8%

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

3. Taylor expanded in C around 0 9.1%

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

   1. mul-1-neg 9.1%

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

   2. +-commutative 9.1%

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

5. Simplified 9.1%

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

   1. add-cbrt-cube 7.3%

      \[\leadsto -\frac{\sqrt{2}}{B} \cdot \color{blue}{\sqrt[3]{\left(\sqrt{F \cdot \left(A - \sqrt{{B}^{2} + {A}^{2}}\right)} \cdot \sqrt{F \cdot \left(A - \sqrt{{B}^{2} + {A}^{2}}\right)}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{{B}^{2} + {A}^{2}}\right)}}} \]

   2. pow1/3 6.9%

      \[\leadsto -\frac{\sqrt{2}}{B} \cdot \color{blue}{{\left(\left(\sqrt{F \cdot \left(A - \sqrt{{B}^{2} + {A}^{2}}\right)} \cdot \sqrt{F \cdot \left(A - \sqrt{{B}^{2} + {A}^{2}}\right)}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{{B}^{2} + {A}^{2}}\right)}\right)}^{0.3333333333333333}} \]

7. Applied egg-rr 10.5%

   \[\leadsto -\frac{\sqrt{2}}{B} \cdot \color{blue}{{\left({\left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}^{1.5}\right)}^{0.3333333333333333}} \]
8. Step-by-step derivation

   1. associate-*l/ 10.5%

      \[\leadsto -\color{blue}{\frac{\sqrt{2} \cdot {\left({\left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}^{1.5}\right)}^{0.3333333333333333}}{B}} \]

   2. pow-pow 14.7%

      \[\leadsto -\frac{\sqrt{2} \cdot \color{blue}{{\left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}}}{B} \]

   3. metadata-eval 14.7%

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

   4. pow1/2 14.6%

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

   5. sqrt-prod 14.7%

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

   6. distribute-frac-neg2 14.7%

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

   7. clear-num 14.7%

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

   8. associate-*r* 14.7%

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

9. Applied egg-rr 14.7%

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

10. Final simplification 14.7%

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

Alternative 9: 32.0% accurate, 3.0× speedup

Math

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

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)))) (/ -1.0 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(((2.0 * F) * (A - hypot(B_m, A)))) * (-1.0 / 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)))) * (-1.0 / 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(((2.0 * F) * (A - math.hypot(B_m, A)))) * (-1.0 / 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(2.0 * F) * Float64(A - hypot(B_m, A)))) * Float64(-1.0 / 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(((2.0 * F) * (A - hypot(B_m, A)))) * (-1.0 / 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[(2.0 * F), $MachinePrecision] * N[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / B$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 17.8%

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

3. Taylor expanded in C around 0 9.1%

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

   1. mul-1-neg 9.1%

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

   2. +-commutative 9.1%

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

   3. unpow2 9.1%

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

   4. unpow2 9.1%

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

   5. hypot-define 14.6%

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

5. Simplified 14.6%

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

   1. neg-sub0 14.6%

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

   2. associate-*l/ 14.6%

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

   3. pow1/2 14.6%

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

   4. pow1/2 14.7%

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

   5. hypot-undefine 9.2%

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

   6. unpow2 9.2%

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

   7. unpow2 9.2%

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

   8. pow-prod-down 9.2%

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

   9. unpow2 9.2%

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

   10. unpow2 9.2%

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

   11. hypot-undefine 14.7%

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

7. Applied egg-rr 14.7%

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

   1. neg-sub0 14.7%

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

   2. distribute-neg-frac2 14.7%

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

   3. unpow1/2 14.7%

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

9. Simplified 14.7%

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

    1. clear-num 14.7%

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

    2. inv-pow 14.7%

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

    3. associate-*r* 14.7%

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

11. Applied egg-rr 14.7%

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

    1. unpow-1 14.7%

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

    2. associate-/r/ 14.7%

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

13. Simplified 14.7%

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

14. Final simplification 14.7%

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

Alternative 10: 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 17.8%

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

3. Taylor expanded in C around 0 9.1%

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

   1. mul-1-neg 9.1%

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

   2. +-commutative 9.1%

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

   3. unpow2 9.1%

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

   4. unpow2 9.1%

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

   5. hypot-define 14.6%

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

5. Simplified 14.6%

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

   1. neg-sub0 14.6%

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

   2. associate-*l/ 14.6%

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

   3. pow1/2 14.6%

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

   4. pow1/2 14.7%

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

   5. hypot-undefine 9.2%

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

   6. unpow2 9.2%

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

   7. unpow2 9.2%

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

   8. pow-prod-down 9.2%

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

   9. unpow2 9.2%

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

   10. unpow2 9.2%

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

   11. hypot-undefine 14.7%

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

7. Applied egg-rr 14.7%

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

   1. neg-sub0 14.7%

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

   2. distribute-neg-frac2 14.7%

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

   3. unpow1/2 14.7%

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

9. Simplified 14.7%

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

Alternative 11: 27.0% accurate, 5.4× speedup

Math

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

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
 (/ -1.0 (/ B_m (sqrt (* B_m (+ (* F -2.0) (* 2.0 (/ (* A 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 -1.0 / (B_m / sqrt((B_m * ((F * -2.0) + (2.0 * ((A * 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 = (-1.0d0) / (b_m / sqrt((b_m * ((f * (-2.0d0)) + (2.0d0 * ((a * 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 -1.0 / (B_m / Math.sqrt((B_m * ((F * -2.0) + (2.0 * ((A * 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 -1.0 / (B_m / math.sqrt((B_m * ((F * -2.0) + (2.0 * ((A * 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(-1.0 / Float64(B_m / sqrt(Float64(B_m * Float64(Float64(F * -2.0) + Float64(2.0 * Float64(Float64(A * 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 = -1.0 / (B_m / sqrt((B_m * ((F * -2.0) + (2.0 * ((A * 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[(-1.0 / N[(B$95$m / N[Sqrt[N[(B$95$m * N[(N[(F * -2.0), $MachinePrecision] + N[(2.0 * N[(N[(A * F), $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 17.8%

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

3. Taylor expanded in C around 0 9.1%

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

   1. mul-1-neg 9.1%

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

   2. +-commutative 9.1%

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

5. Simplified 9.1%

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

   1. add-cbrt-cube 7.3%

      \[\leadsto -\frac{\sqrt{2}}{B} \cdot \color{blue}{\sqrt[3]{\left(\sqrt{F \cdot \left(A - \sqrt{{B}^{2} + {A}^{2}}\right)} \cdot \sqrt{F \cdot \left(A - \sqrt{{B}^{2} + {A}^{2}}\right)}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{{B}^{2} + {A}^{2}}\right)}}} \]

   2. pow1/3 6.9%

      \[\leadsto -\frac{\sqrt{2}}{B} \cdot \color{blue}{{\left(\left(\sqrt{F \cdot \left(A - \sqrt{{B}^{2} + {A}^{2}}\right)} \cdot \sqrt{F \cdot \left(A - \sqrt{{B}^{2} + {A}^{2}}\right)}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{{B}^{2} + {A}^{2}}\right)}\right)}^{0.3333333333333333}} \]

7. Applied egg-rr 10.5%

   \[\leadsto -\frac{\sqrt{2}}{B} \cdot \color{blue}{{\left({\left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}^{1.5}\right)}^{0.3333333333333333}} \]
8. Step-by-step derivation

   1. associate-*l/ 10.5%

      \[\leadsto -\color{blue}{\frac{\sqrt{2} \cdot {\left({\left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}^{1.5}\right)}^{0.3333333333333333}}{B}} \]

   2. pow-pow 14.7%

      \[\leadsto -\frac{\sqrt{2} \cdot \color{blue}{{\left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}}}{B} \]

   3. metadata-eval 14.7%

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

   4. pow1/2 14.6%

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

   5. sqrt-prod 14.7%

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

   6. distribute-frac-neg2 14.7%

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

   7. clear-num 14.7%

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

   8. associate-*r* 14.7%

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

9. Applied egg-rr 14.7%

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

10. Taylor expanded in B around inf 12.7%

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

11. Final simplification 12.7%

    \[\leadsto \frac{-1}{\frac{B}{\sqrt{B \cdot \left(F \cdot -2 + 2 \cdot \frac{A \cdot F}{B}\right)}}} \]
12. Add Preprocessing

Alternative 12: 27.1% accurate, 5.5× speedup

Math

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

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
 (/ -1.0 (/ B_m (sqrt (+ (* -2.0 (* B_m F)) (* 2.0 (* A 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) {
	return -1.0 / (B_m / sqrt(((-2.0 * (B_m * F)) + (2.0 * (A * F)))));
}

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 = (-1.0d0) / (b_m / sqrt((((-2.0d0) * (b_m * f)) + (2.0d0 * (a * f)))))
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 -1.0 / (B_m / Math.sqrt(((-2.0 * (B_m * F)) + (2.0 * (A * F)))));
}

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 -1.0 / (B_m / math.sqrt(((-2.0 * (B_m * F)) + (2.0 * (A * F)))))

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(-1.0 / Float64(B_m / sqrt(Float64(Float64(-2.0 * Float64(B_m * F)) + Float64(2.0 * Float64(A * F))))))
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 = -1.0 / (B_m / sqrt(((-2.0 * (B_m * F)) + (2.0 * (A * F)))));
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[(-1.0 / N[(B$95$m / N[Sqrt[N[(N[(-2.0 * N[(B$95$m * F), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(A * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 17.8%

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

3. Taylor expanded in C around 0 9.1%

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

   1. mul-1-neg 9.1%

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

   2. +-commutative 9.1%

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

5. Simplified 9.1%

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

   1. add-cbrt-cube 7.3%

      \[\leadsto -\frac{\sqrt{2}}{B} \cdot \color{blue}{\sqrt[3]{\left(\sqrt{F \cdot \left(A - \sqrt{{B}^{2} + {A}^{2}}\right)} \cdot \sqrt{F \cdot \left(A - \sqrt{{B}^{2} + {A}^{2}}\right)}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{{B}^{2} + {A}^{2}}\right)}}} \]

   2. pow1/3 6.9%

      \[\leadsto -\frac{\sqrt{2}}{B} \cdot \color{blue}{{\left(\left(\sqrt{F \cdot \left(A - \sqrt{{B}^{2} + {A}^{2}}\right)} \cdot \sqrt{F \cdot \left(A - \sqrt{{B}^{2} + {A}^{2}}\right)}\right) \cdot \sqrt{F \cdot \left(A - \sqrt{{B}^{2} + {A}^{2}}\right)}\right)}^{0.3333333333333333}} \]

7. Applied egg-rr 10.5%

   \[\leadsto -\frac{\sqrt{2}}{B} \cdot \color{blue}{{\left({\left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}^{1.5}\right)}^{0.3333333333333333}} \]
8. Step-by-step derivation

   1. associate-*l/ 10.5%

      \[\leadsto -\color{blue}{\frac{\sqrt{2} \cdot {\left({\left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}^{1.5}\right)}^{0.3333333333333333}}{B}} \]

   2. pow-pow 14.7%

      \[\leadsto -\frac{\sqrt{2} \cdot \color{blue}{{\left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}}}{B} \]

   3. metadata-eval 14.7%

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

   4. pow1/2 14.6%

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

   5. sqrt-prod 14.7%

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

   6. distribute-frac-neg2 14.7%

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

   7. clear-num 14.7%

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

   8. associate-*r* 14.7%

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

9. Applied egg-rr 14.7%

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

10. Taylor expanded in A around 0 12.6%

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

11. Final simplification 12.6%

    \[\leadsto \frac{-1}{\frac{B}{\sqrt{-2 \cdot \left(B \cdot F\right) + 2 \cdot \left(A \cdot F\right)}}} \]
12. Add Preprocessing

Alternative 13: 27.1% accurate, 5.6× speedup

Math

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

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 (* B_m F)) (* 2.0 (* A 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 * (B_m * F)) + (2.0 * (A * 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) * (b_m * f)) + (2.0d0 * (a * 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 * (B_m * F)) + (2.0 * (A * 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 * (B_m * F)) + (2.0 * (A * 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(Float64(-2.0 * Float64(B_m * F)) + Float64(2.0 * Float64(A * 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 * (B_m * F)) + (2.0 * (A * 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[(N[(-2.0 * N[(B$95$m * F), $MachinePrecision]), $MachinePrecision] + N[(2.0 * N[(A * F), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision]

Derivation

1. Initial program 17.8%

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

3. Taylor expanded in C around 0 9.1%

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

   1. mul-1-neg 9.1%

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

   2. +-commutative 9.1%

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

   3. unpow2 9.1%

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

   4. unpow2 9.1%

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

   5. hypot-define 14.6%

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

5. Simplified 14.6%

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

   1. neg-sub0 14.6%

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

   2. associate-*l/ 14.6%

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

   3. pow1/2 14.6%

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

   4. pow1/2 14.7%

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

   5. hypot-undefine 9.2%

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

   6. unpow2 9.2%

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

   7. unpow2 9.2%

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

   8. pow-prod-down 9.2%

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

   9. unpow2 9.2%

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

   10. unpow2 9.2%

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

   11. hypot-undefine 14.7%

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

7. Applied egg-rr 14.7%

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

   1. neg-sub0 14.7%

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

   2. distribute-neg-frac2 14.7%

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

   3. unpow1/2 14.7%

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

9. Simplified 14.7%

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

10. Taylor expanded in A around 0 12.6%

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

Alternative 14: 26.8% accurate, 5.9× speedup

Math

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

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

Derivation

1. Initial program 17.8%

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

3. Taylor expanded in C around 0 9.1%

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

   1. mul-1-neg 9.1%

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

   2. +-commutative 9.1%

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

   3. unpow2 9.1%

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

   4. unpow2 9.1%

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

   5. hypot-define 14.6%

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

5. Simplified 14.6%

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

   1. neg-sub0 14.6%

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

   2. associate-*l/ 14.6%

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

   3. pow1/2 14.6%

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

   4. pow1/2 14.7%

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

   5. hypot-undefine 9.2%

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

   6. unpow2 9.2%

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

   7. unpow2 9.2%

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

   8. pow-prod-down 9.2%

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

   9. unpow2 9.2%

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

   10. unpow2 9.2%

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

   11. hypot-undefine 14.7%

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

7. Applied egg-rr 14.7%

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

   1. neg-sub0 14.7%

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

   2. distribute-neg-frac2 14.7%

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

   3. unpow1/2 14.7%

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

9. Simplified 14.7%

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

10. Taylor expanded in A around 0 13.4%

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

    1. associate-*r* 13.4%

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

12. Simplified 13.4%

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

13. Final simplification 13.4%

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

Alternative 15: 26.8% accurate, 5.9× speedup

Math

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

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 (* 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 * (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) * (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 * (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 * (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(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 * (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[(B$95$m * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision]

Derivation

1. Initial program 17.8%

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

3. Taylor expanded in C around 0 9.1%

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

   1. mul-1-neg 9.1%

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

   2. +-commutative 9.1%

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

   3. unpow2 9.1%

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

   4. unpow2 9.1%

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

   5. hypot-define 14.6%

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

5. Simplified 14.6%

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

   1. neg-sub0 14.6%

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

   2. associate-*l/ 14.6%

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

   3. pow1/2 14.6%

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

   4. pow1/2 14.7%

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

   5. hypot-undefine 9.2%

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

   6. unpow2 9.2%

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

   7. unpow2 9.2%

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

   8. pow-prod-down 9.2%

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

   9. unpow2 9.2%

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

   10. unpow2 9.2%

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

   11. hypot-undefine 14.7%

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

7. Applied egg-rr 14.7%

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

   1. neg-sub0 14.7%

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

   2. distribute-neg-frac2 14.7%

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

   3. unpow1/2 14.7%

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

9. Simplified 14.7%

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

10. Taylor expanded in A around 0 13.4%

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

Alternative 16: 2.4% accurate, 5.9× speedup

Math

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

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 (/ 1.0 (sqrt (/ B_m (* F -2.0)))))

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 1.0 / sqrt((B_m / (F * -2.0)));
}

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 = 1.0d0 / sqrt((b_m / (f * (-2.0d0))))
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 1.0 / Math.sqrt((B_m / (F * -2.0)));
}

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 1.0 / math.sqrt((B_m / (F * -2.0)))

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(1.0 / sqrt(Float64(B_m / Float64(F * -2.0))))
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 = 1.0 / sqrt((B_m / (F * -2.0)));
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[(1.0 / N[Sqrt[N[(B$95$m / N[(F * -2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 17.8%

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

3. Taylor expanded in B around -inf 0.0%

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

   1. mul-1-neg 0.0%

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

   2. unpow2 0.0%

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

   3. rem-square-sqrt 1.8%

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

5. Simplified 1.8%

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

6. Taylor expanded in F around 0 1.8%

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

   1. *-commutative 1.8%

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

   2. pow1/2 1.8%

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

   3. metadata-eval 1.8%

      \[\leadsto {2}^{\color{blue}{\left(1.5 \cdot 0.3333333333333333\right)}} \cdot \sqrt{\frac{F}{B}} \]

   4. pow1/2 1.9%

      \[\leadsto {2}^{\left(1.5 \cdot 0.3333333333333333\right)} \cdot \color{blue}{{\left(\frac{F}{B}\right)}^{0.5}} \]

   5. metadata-eval 1.9%

      \[\leadsto {2}^{\left(1.5 \cdot 0.3333333333333333\right)} \cdot {\left(\frac{F}{B}\right)}^{\color{blue}{\left(1.5 \cdot 0.3333333333333333\right)}} \]

   6. pow-prod-down 1.9%

      \[\leadsto \color{blue}{{\left(2 \cdot \frac{F}{B}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} \]

   7. metadata-eval 1.9%

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

8. Applied egg-rr 1.9%

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

   1. unpow1/2 1.8%

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

   2. associate-*r/ 1.8%

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

10. Simplified 1.8%

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

    1. clear-num 1.9%

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

    2. sqrt-div 1.6%

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

    3. metadata-eval 1.6%

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

    4. frac-2neg 1.6%

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

    5. add-sqr-sqrt 1.2%

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

    6. sqrt-unprod 2.6%

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

    7. sqr-neg 2.6%

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

    8. sqrt-prod 1.3%

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

    9. add-sqr-sqrt 1.9%

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

    10. *-commutative 1.9%

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

    11. distribute-rgt-neg-in 1.9%

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

    12. metadata-eval 1.9%

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

12. Applied egg-rr 1.9%

    \[\leadsto \color{blue}{\frac{1}{\sqrt{\frac{B}{F \cdot -2}}}} \]
13. Add Preprocessing

Alternative 17: 2.4% accurate, 5.9× speedup

Math

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

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 (/ 1.0 (/ (/ B_m F) -2.0))))

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((1.0 / ((B_m / F) / -2.0)));
}

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((1.0d0 / ((b_m / f) / (-2.0d0))))
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((1.0 / ((B_m / F) / -2.0)));
}

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((1.0 / ((B_m / F) / -2.0)))

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

Derivation

1. Initial program 17.8%

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

3. Taylor expanded in B around -inf 0.0%

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

   1. mul-1-neg 0.0%

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

   2. unpow2 0.0%

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

   3. rem-square-sqrt 1.8%

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

5. Simplified 1.8%

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

6. Taylor expanded in F around 0 1.8%

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

   1. *-commutative 1.8%

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

   2. pow1/2 1.8%

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

   3. metadata-eval 1.8%

      \[\leadsto {2}^{\color{blue}{\left(1.5 \cdot 0.3333333333333333\right)}} \cdot \sqrt{\frac{F}{B}} \]

   4. pow1/2 1.9%

      \[\leadsto {2}^{\left(1.5 \cdot 0.3333333333333333\right)} \cdot \color{blue}{{\left(\frac{F}{B}\right)}^{0.5}} \]

   5. metadata-eval 1.9%

      \[\leadsto {2}^{\left(1.5 \cdot 0.3333333333333333\right)} \cdot {\left(\frac{F}{B}\right)}^{\color{blue}{\left(1.5 \cdot 0.3333333333333333\right)}} \]

   6. pow-prod-down 1.9%

      \[\leadsto \color{blue}{{\left(2 \cdot \frac{F}{B}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} \]

   7. metadata-eval 1.9%

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

8. Applied egg-rr 1.9%

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

   1. unpow1/2 1.8%

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

   2. associate-*r/ 1.8%

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

10. Simplified 1.8%

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

    1. clear-num 1.9%

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

    2. inv-pow 1.9%

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

    3. frac-2neg 1.9%

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

    4. add-sqr-sqrt 1.2%

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

    5. sqrt-unprod 2.5%

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

    6. sqr-neg 2.5%

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

    7. sqrt-prod 1.3%

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

    8. add-sqr-sqrt 2.1%

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

    9. *-commutative 2.1%

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

    10. distribute-rgt-neg-in 2.1%

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

    11. metadata-eval 2.1%

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

12. Applied egg-rr 2.1%

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

    1. unpow-1 2.1%

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

    2. associate-/r* 2.1%

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

14. Simplified 2.1%

    \[\leadsto \sqrt{\color{blue}{\frac{1}{\frac{\frac{B}{F}}{-2}}}} \]
15. Add Preprocessing

Alternative 18: 2.4% accurate, 6.0× speedup

Math

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

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

Derivation

1. Initial program 17.8%

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

3. Taylor expanded in B around -inf 0.0%

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

   1. mul-1-neg 0.0%

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

   2. unpow2 0.0%

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

   3. rem-square-sqrt 1.8%

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

5. Simplified 1.8%

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

6. Taylor expanded in F around 0 1.8%

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

   1. *-commutative 1.8%

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

   2. pow1/2 1.8%

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

   3. metadata-eval 1.8%

      \[\leadsto {2}^{\color{blue}{\left(1.5 \cdot 0.3333333333333333\right)}} \cdot \sqrt{\frac{F}{B}} \]

   4. pow1/2 1.9%

      \[\leadsto {2}^{\left(1.5 \cdot 0.3333333333333333\right)} \cdot \color{blue}{{\left(\frac{F}{B}\right)}^{0.5}} \]

   5. metadata-eval 1.9%

      \[\leadsto {2}^{\left(1.5 \cdot 0.3333333333333333\right)} \cdot {\left(\frac{F}{B}\right)}^{\color{blue}{\left(1.5 \cdot 0.3333333333333333\right)}} \]

   6. pow-prod-down 1.9%

      \[\leadsto \color{blue}{{\left(2 \cdot \frac{F}{B}\right)}^{\left(1.5 \cdot 0.3333333333333333\right)}} \]

   7. metadata-eval 1.9%

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

8. Applied egg-rr 1.9%

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

   1. unpow1/2 1.8%

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

   2. associate-*r/ 1.8%

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

10. Simplified 1.8%

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

    1. associate-/l* 1.8%

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

    2. sqrt-prod 1.8%

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

    3. pow1 1.8%

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

    4. metadata-eval 1.8%

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

    5. sqrt-pow1 3.2%

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

    6. *-commutative 3.2%

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

    7. *-un-lft-identity 3.2%

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

    8. *-commutative 3.2%

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

    9. sqrt-pow1 1.8%

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

    10. metadata-eval 1.8%

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

    11. pow1 1.8%

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

    12. sqrt-prod 1.8%

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

    13. associate-/l* 1.8%

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

    14. frac-2neg 1.8%

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

    15. *-commutative 1.8%

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

    16. distribute-rgt-neg-in 1.8%

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

    17. metadata-eval 1.8%

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

    18. add-sqr-sqrt 1.2%

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

    19. sqrt-unprod 2.5%

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

    20. sqr-neg 2.5%

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

    21. sqrt-prod 1.3%

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

    22. add-sqr-sqrt 2.1%

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

12. Applied egg-rr 2.1%

    \[\leadsto \color{blue}{1 \cdot \sqrt{\frac{F \cdot -2}{B}}} \]
13. Step-by-step derivation

    1. *-lft-identity 2.1%

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

    2. associate-/l* 2.1%

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

14. Simplified 2.1%

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

Reproduce

herbie shell --seed 2024163 
(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))))