ABCF->ab-angle b

Percentage Accurate: 19.6% → 51.4%

Time: 22.5s

Alternatives: 13

Speedup: 5.6×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 19.6% 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: 51.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := A \cdot \left(C \cdot -4\right)\\ t_1 := \mathsf{fma}\left(B\_m, B\_m, t\_0\right)\\ t_2 := -t\_1\\ t_3 := \left(4 \cdot A\right) \cdot C\\ t_4 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_3\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_3 - {B\_m}^{2}}\\ t_5 := A + \left(A + -0.5 \cdot \frac{{B\_m}^{2}}{C}\right)\\ t_6 := F \cdot t\_1\\ \mathbf{if}\;t\_4 \leq -\infty:\\ \;\;\;\;\frac{\mathsf{hypot}\left(B\_m, \sqrt{t\_0}\right) \cdot \sqrt{\left(2 \cdot F\right) \cdot t\_5}}{t\_2}\\ \mathbf{elif}\;t\_4 \leq -5 \cdot 10^{-216}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(A - C, B\_m\right)\right)\right) \cdot t\_6} \cdot \frac{-1}{t\_1}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot t\_5\right) \cdot t\_6}}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot F\right) \cdot \left(C - \mathsf{hypot}\left(C, B\_m\right)\right)}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (* A (* C -4.0)))
        (t_1 (fma B_m B_m t_0))
        (t_2 (- t_1))
        (t_3 (* (* 4.0 A) C))
        (t_4
         (/
          (sqrt
           (*
            (* 2.0 (* (- (pow B_m 2.0) t_3) F))
            (- (+ A C) (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))))))
          (- t_3 (pow B_m 2.0))))
        (t_5 (+ A (+ A (* -0.5 (/ (pow B_m 2.0) C)))))
        (t_6 (* F t_1)))
   (if (<= t_4 (- INFINITY))
     (/ (* (hypot B_m (sqrt t_0)) (sqrt (* (* 2.0 F) t_5))) t_2)
     (if (<= t_4 -5e-216)
       (* (sqrt (* (* 2.0 (- (+ A C) (hypot (- A C) B_m))) t_6)) (/ -1.0 t_1))
       (if (<= t_4 INFINITY)
         (/ (sqrt (* (* 2.0 t_5) t_6)) t_2)
         (/ (sqrt (* (* 2.0 F) (- C (hypot C B_m)))) (- 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 = A * (C * -4.0);
	double t_1 = fma(B_m, B_m, t_0);
	double t_2 = -t_1;
	double t_3 = (4.0 * A) * C;
	double t_4 = sqrt(((2.0 * ((pow(B_m, 2.0) - t_3) * F)) * ((A + C) - sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / (t_3 - pow(B_m, 2.0));
	double t_5 = A + (A + (-0.5 * (pow(B_m, 2.0) / C)));
	double t_6 = F * t_1;
	double tmp;
	if (t_4 <= -((double) INFINITY)) {
		tmp = (hypot(B_m, sqrt(t_0)) * sqrt(((2.0 * F) * t_5))) / t_2;
	} else if (t_4 <= -5e-216) {
		tmp = sqrt(((2.0 * ((A + C) - hypot((A - C), B_m))) * t_6)) * (-1.0 / t_1);
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = sqrt(((2.0 * t_5) * t_6)) / t_2;
	} else {
		tmp = sqrt(((2.0 * F) * (C - hypot(C, B_m)))) / -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(A * Float64(C * -4.0))
	t_1 = fma(B_m, B_m, t_0)
	t_2 = Float64(-t_1)
	t_3 = Float64(Float64(4.0 * A) * C)
	t_4 = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_3) * F)) * Float64(Float64(A + C) - sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / Float64(t_3 - (B_m ^ 2.0)))
	t_5 = Float64(A + Float64(A + Float64(-0.5 * Float64((B_m ^ 2.0) / C))))
	t_6 = Float64(F * t_1)
	tmp = 0.0
	if (t_4 <= Float64(-Inf))
		tmp = Float64(Float64(hypot(B_m, sqrt(t_0)) * sqrt(Float64(Float64(2.0 * F) * t_5))) / t_2);
	elseif (t_4 <= -5e-216)
		tmp = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64(A + C) - hypot(Float64(A - C), B_m))) * t_6)) * Float64(-1.0 / t_1));
	elseif (t_4 <= Inf)
		tmp = Float64(sqrt(Float64(Float64(2.0 * t_5) * t_6)) / t_2);
	else
		tmp = Float64(sqrt(Float64(Float64(2.0 * F) * Float64(C - hypot(C, B_m)))) / Float64(-B_m));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(B$95$m * B$95$m + t$95$0), $MachinePrecision]}, Block[{t$95$2 = (-t$95$1)}, Block[{t$95$3 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$3), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$3 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(A + N[(A + N[(-0.5 * N[(N[Power[B$95$m, 2.0], $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(F * t$95$1), $MachinePrecision]}, If[LessEqual[t$95$4, (-Infinity)], N[(N[(N[Sqrt[B$95$m ^ 2 + N[Sqrt[t$95$0], $MachinePrecision] ^ 2], $MachinePrecision] * N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] * t$95$5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[t$95$4, -5e-216], N[(N[Sqrt[N[(N[(2.0 * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$6), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[(N[Sqrt[N[(N[(2.0 * t$95$5), $MachinePrecision] * t$95$6), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision], N[(N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] * N[(C - N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $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))) < -inf.0

   1. Initial program 3.1%

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

   3. Simplified 15.6%

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

   5. Taylor expanded in C around inf 10.4%

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

      1. mul-1-neg 10.4%

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

   7. Simplified 10.4%

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

      1. pow1/2 11.4%

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

      2. associate-*l* 11.4%

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

      3. unpow-prod-down 22.1%

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

      4. pow1/2 22.1%

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

      5. fma-undefine 22.1%

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

      6. add-sqr-sqrt 22.0%

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

      7. hypot-define 22.0%

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

      8. associate-*r* 22.0%

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

      9. associate-*r/ 22.0%

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

   9. Applied egg-rr 22.0%

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

       1. associate-*l* 22.0%

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

       2. unpow1/2 21.4%

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

       3. associate-*r* 21.4%

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

       4. associate-+r- 21.4%

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

       5. associate-*r/ 21.4%

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

   11. Simplified 21.4%

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

   if -inf.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -5.00000000000000021e-216

   1. Initial program 99.5%

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

   3. Simplified 99.4%

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

      1. div-inv 99.5%

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

   6. Applied egg-rr 99.5%

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

   if -5.00000000000000021e-216 < (/.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 30.1%

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

   3. Simplified 34.4%

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

   5. Taylor expanded in C around inf 36.1%

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

      1. mul-1-neg 36.1%

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

   7. Simplified 36.1%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in A around 0 4.2%

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

      1. mul-1-neg 4.2%

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

      2. +-commutative 4.2%

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

      3. unpow2 4.2%

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

      4. unpow2 4.2%

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

      5. hypot-define 29.0%

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

   5. Simplified 29.0%

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

      1. neg-sub0 29.0%

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

      2. associate-*l/ 29.0%

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

      3. pow1/2 29.0%

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

      4. pow1/2 29.0%

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

      5. pow-prod-down 29.1%

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

   7. Applied egg-rr 29.1%

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

      1. neg-sub0 29.1%

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

      2. distribute-neg-frac2 29.1%

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

      3. unpow1/2 29.0%

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

      4. associate-*r* 29.0%

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

   9. Simplified 29.0%

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

4. Final simplification 36.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -\infty:\\ \;\;\;\;\frac{\mathsf{hypot}\left(B, \sqrt{A \cdot \left(C \cdot -4\right)}\right) \cdot \sqrt{\left(2 \cdot F\right) \cdot \left(A + \left(A + -0.5 \cdot \frac{{B}^{2}}{C}\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -5 \cdot 10^{-216}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(A - C, B\right)\right)\right) \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(A + \left(A + -0.5 \cdot \frac{{B}^{2}}{C}\right)\right)\right) \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot F\right) \cdot \left(C - \mathsf{hypot}\left(C, B\right)\right)}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 52.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := \left(4 \cdot A\right) \cdot C\\ t_2 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_1\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_1 - {B\_m}^{2}}\\ t_3 := \mathsf{fma}\left(C, A \cdot -4, {B\_m}^{2}\right)\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{-216}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B\_m, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot t\_3}}{-t\_3}\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(A + \left(A + -0.5 \cdot \frac{{B\_m}^{2}}{C}\right)\right)\right) \cdot \left(F \cdot t\_0\right)}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot F\right) \cdot \left(C - \mathsf{hypot}\left(C, B\_m\right)\right)}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* A (* C -4.0))))
        (t_1 (* (* 4.0 A) C))
        (t_2
         (/
          (sqrt
           (*
            (* 2.0 (* (- (pow B_m 2.0) t_1) F))
            (- (+ A C) (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))))))
          (- t_1 (pow B_m 2.0))))
        (t_3 (fma C (* A -4.0) (pow B_m 2.0))))
   (if (<= t_2 -5e-216)
     (/
      (* (sqrt (* F (+ A (- C (hypot B_m (- A C)))))) (sqrt (* 2.0 t_3)))
      (- t_3))
     (if (<= t_2 INFINITY)
       (/
        (sqrt (* (* 2.0 (+ A (+ A (* -0.5 (/ (pow B_m 2.0) C))))) (* F t_0)))
        (- t_0))
       (/ (sqrt (* (* 2.0 F) (- C (hypot C B_m)))) (- B_m))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (A * (C * -4.0)));
	double t_1 = (4.0 * A) * C;
	double t_2 = sqrt(((2.0 * ((pow(B_m, 2.0) - t_1) * F)) * ((A + C) - sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / (t_1 - pow(B_m, 2.0));
	double t_3 = fma(C, (A * -4.0), pow(B_m, 2.0));
	double tmp;
	if (t_2 <= -5e-216) {
		tmp = (sqrt((F * (A + (C - hypot(B_m, (A - C)))))) * sqrt((2.0 * t_3))) / -t_3;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = sqrt(((2.0 * (A + (A + (-0.5 * (pow(B_m, 2.0) / C))))) * (F * t_0))) / -t_0;
	} else {
		tmp = sqrt(((2.0 * F) * (C - hypot(C, B_m)))) / -B_m;
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	t_1 = Float64(Float64(4.0 * A) * C)
	t_2 = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_1) * F)) * Float64(Float64(A + C) - sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / Float64(t_1 - (B_m ^ 2.0)))
	t_3 = fma(C, Float64(A * -4.0), (B_m ^ 2.0))
	tmp = 0.0
	if (t_2 <= -5e-216)
		tmp = Float64(Float64(sqrt(Float64(F * Float64(A + Float64(C - hypot(B_m, Float64(A - C)))))) * sqrt(Float64(2.0 * t_3))) / Float64(-t_3));
	elseif (t_2 <= Inf)
		tmp = Float64(sqrt(Float64(Float64(2.0 * Float64(A + Float64(A + Float64(-0.5 * Float64((B_m ^ 2.0) / C))))) * Float64(F * t_0))) / Float64(-t_0));
	else
		tmp = Float64(sqrt(Float64(Float64(2.0 * F) * Float64(C - hypot(C, B_m)))) / Float64(-B_m));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$1), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$1 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(C * N[(A * -4.0), $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e-216], N[(N[(N[Sqrt[N[(F * N[(A + N[(C - N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(2.0 * t$95$3), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / (-t$95$3)), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(N[Sqrt[N[(N[(2.0 * N[(A + N[(A + N[(-0.5 * N[(N[Power[B$95$m, 2.0], $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], N[(N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] * N[(C - N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (-.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -5.00000000000000021e-216

   1. Initial program 37.1%

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

   3. Simplified 35.3%

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

      1. pow1/2 35.3%

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

      2. associate-*r* 45.2%

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

      3. unpow-prod-down 60.0%

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

      4. associate-+r- 58.8%

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

      5. hypot-undefine 47.0%

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

      6. unpow2 47.0%

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

      7. unpow2 47.0%

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

      8. +-commutative 47.0%

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

      9. unpow2 47.0%

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

      10. unpow2 47.0%

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

      11. hypot-define 58.8%

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

      12. pow1/2 58.8%

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

   6. Applied egg-rr 58.8%

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

      1. unpow1/2 58.8%

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

      2. associate-+r- 60.0%

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

      3. hypot-undefine 47.0%

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

      4. unpow2 47.0%

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

      5. unpow2 47.0%

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

      6. +-commutative 47.0%

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

      7. unpow2 47.0%

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

      8. unpow2 47.0%

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

      9. hypot-undefine 60.0%

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

   8. Simplified 60.0%

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

   if -5.00000000000000021e-216 < (/.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 30.1%

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

   3. Simplified 34.4%

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

   5. Taylor expanded in C around inf 36.1%

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

      1. mul-1-neg 36.1%

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

   7. Simplified 36.1%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in A around 0 4.2%

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

      1. mul-1-neg 4.2%

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

      2. +-commutative 4.2%

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

      3. unpow2 4.2%

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

      4. unpow2 4.2%

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

      5. hypot-define 29.0%

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

   5. Simplified 29.0%

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

      1. neg-sub0 29.0%

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

      2. associate-*l/ 29.0%

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

      3. pow1/2 29.0%

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

      4. pow1/2 29.0%

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

      5. pow-prod-down 29.1%

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

   7. Applied egg-rr 29.1%

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

      1. neg-sub0 29.1%

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

      2. distribute-neg-frac2 29.1%

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

      3. unpow1/2 29.0%

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

      4. associate-*r* 29.0%

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

   9. Simplified 29.0%

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

4. Final simplification 39.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -5 \cdot 10^{-216}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt{2 \cdot \mathsf{fma}\left(C, A \cdot -4, {B}^{2}\right)}}{-\mathsf{fma}\left(C, A \cdot -4, {B}^{2}\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(A + \left(A + -0.5 \cdot \frac{{B}^{2}}{C}\right)\right)\right) \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot F\right) \cdot \left(C - \mathsf{hypot}\left(C, B\right)\right)}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 41.7% accurate, 1.4× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;F \leq -2.5 \cdot 10^{+145}:\\ \;\;\;\;\frac{-1}{\frac{B\_m}{\sqrt{{B\_m}^{2} \cdot \frac{F}{-C}}}}\\ \mathbf{elif}\;F \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sqrt{2}}{B\_m} \cdot \left(-\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B\_m, A\right)\right)}\right)\\ \mathbf{elif}\;F \leq 8.5 \cdot 10^{+156}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(A - C, B\_m\right)\right)\right) \cdot \left(F \cdot t\_0\right)} \cdot \frac{-1}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(C - \mathsf{hypot}\left(C, B\_m\right)\right) \cdot \left(F \cdot -2\right)}}{B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* A (* C -4.0)))))
   (if (<= F -2.5e+145)
     (/ -1.0 (/ B_m (sqrt (* (pow B_m 2.0) (/ F (- C))))))
     (if (<= F -1e-310)
       (* (/ (sqrt 2.0) B_m) (- (sqrt (* F (- A (hypot B_m A))))))
       (if (<= F 8.5e+156)
         (*
          (sqrt (* (* 2.0 (- (+ A C) (hypot (- A C) B_m))) (* F t_0)))
          (/ -1.0 t_0))
         (/ (sqrt (* (- C (hypot C B_m)) (* F -2.0))) B_m))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (A * (C * -4.0)));
	double tmp;
	if (F <= -2.5e+145) {
		tmp = -1.0 / (B_m / sqrt((pow(B_m, 2.0) * (F / -C))));
	} else if (F <= -1e-310) {
		tmp = (sqrt(2.0) / B_m) * -sqrt((F * (A - hypot(B_m, A))));
	} else if (F <= 8.5e+156) {
		tmp = sqrt(((2.0 * ((A + C) - hypot((A - C), B_m))) * (F * t_0))) * (-1.0 / t_0);
	} else {
		tmp = sqrt(((C - hypot(C, B_m)) * (F * -2.0))) / B_m;
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	tmp = 0.0
	if (F <= -2.5e+145)
		tmp = Float64(-1.0 / Float64(B_m / sqrt(Float64((B_m ^ 2.0) * Float64(F / Float64(-C))))));
	elseif (F <= -1e-310)
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(-sqrt(Float64(F * Float64(A - hypot(B_m, A))))));
	elseif (F <= 8.5e+156)
		tmp = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64(A + C) - hypot(Float64(A - C), B_m))) * Float64(F * t_0))) * Float64(-1.0 / t_0));
	else
		tmp = Float64(sqrt(Float64(Float64(C - hypot(C, B_m)) * Float64(F * -2.0))) / B_m);
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, -2.5e+145], N[(-1.0 / N[(B$95$m / N[Sqrt[N[(N[Power[B$95$m, 2.0], $MachinePrecision] * N[(F / (-C)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -1e-310], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * (-N[Sqrt[N[(F * N[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], If[LessEqual[F, 8.5e+156], N[(N[Sqrt[N[(N[(2.0 * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(N[(C - N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision] * N[(F * -2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / B$95$m), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if F < -2.49999999999999983e145

   1. Initial program 10.1%

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

   3. Taylor expanded in A around 0 2.4%

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

      1. mul-1-neg 2.4%

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

      2. +-commutative 2.4%

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

      3. unpow2 2.4%

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

      4. unpow2 2.4%

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

      5. hypot-define 2.5%

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

   5. Simplified 2.5%

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

      1. neg-sub0 2.5%

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

      2. associate-*l/ 2.5%

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

      3. pow1/2 2.5%

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

      4. pow1/2 2.5%

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

      5. pow-prod-down 2.5%

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

   7. Applied egg-rr 2.5%

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

      1. neg-sub0 2.5%

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

      2. distribute-neg-frac2 2.5%

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

      3. unpow1/2 2.5%

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

      4. associate-*r* 2.5%

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

   9. Simplified 2.5%

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

   10. Taylor expanded in C around inf 6.7%

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

       1. clear-num 6.7%

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

       2. inv-pow 6.7%

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

       3. mul-1-neg 6.7%

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

       4. associate-/l* 6.5%

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

   12. Applied egg-rr 6.5%

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

       1. unpow-1 6.5%

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

       2. distribute-rgt-neg-in 6.5%

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

       3. distribute-neg-frac2 6.5%

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

   14. Simplified 6.5%

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

   if -2.49999999999999983e145 < F < -9.999999999999969e-311

   1. Initial program 15.1%

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

   3. Taylor expanded in C around 0 8.8%

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

      1. mul-1-neg 8.8%

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

      2. +-commutative 8.8%

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

      3. unpow2 8.8%

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

      4. unpow2 8.8%

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

      5. hypot-define 26.4%

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

   5. Simplified 26.4%

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

   if -9.999999999999969e-311 < F < 8.49999999999999948e156

   1. Initial program 35.7%

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

   3. Simplified 56.4%

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

      1. div-inv 56.4%

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

   6. Applied egg-rr 65.7%

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

   if 8.49999999999999948e156 < F

   1. Initial program 19.1%

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

   3. Taylor expanded in A around 0 11.8%

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

      1. mul-1-neg 11.8%

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

      2. +-commutative 11.8%

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

      3. unpow2 11.8%

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

      4. unpow2 11.8%

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

      5. hypot-define 23.5%

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

   5. Simplified 23.5%

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

      1. neg-sub0 23.5%

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

      2. associate-*l/ 23.5%

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

      3. pow1/2 23.5%

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

      4. pow1/2 23.9%

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

      5. pow-prod-down 23.9%

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

   7. Applied egg-rr 23.9%

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

      1. neg-sub0 23.9%

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

      2. distribute-neg-frac2 23.9%

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

      3. unpow1/2 23.5%

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

      4. associate-*r* 23.5%

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

   9. Simplified 23.5%

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

   11. Applied egg-rr 25.4%

       \[\leadsto \color{blue}{0 + \frac{\sqrt{-2 \cdot \left(F \cdot \left(C - \mathsf{hypot}\left(C, B\right)\right)\right)}}{B}} \]
   12. Step-by-step derivation

       1. +-lft-identity 25.4%

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

       2. associate-*r* 25.4%

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

       3. *-commutative 25.4%

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

   13. Simplified 25.4%

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

4. Final simplification 28.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -2.5 \cdot 10^{+145}:\\ \;\;\;\;\frac{-1}{\frac{B}{\sqrt{{B}^{2} \cdot \frac{F}{-C}}}}\\ \mathbf{elif}\;F \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}\right)\\ \mathbf{elif}\;F \leq 8.5 \cdot 10^{+156}:\\ \;\;\;\;\sqrt{\left(2 \cdot \left(\left(A + C\right) - \mathsf{hypot}\left(A - C, B\right)\right)\right) \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)} \cdot \frac{-1}{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(C - \mathsf{hypot}\left(C, B\right)\right) \cdot \left(F \cdot -2\right)}}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 39.0% accurate, 2.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;F \leq -1.35 \cdot 10^{+145}:\\ \;\;\;\;\frac{-1}{\frac{B\_m}{\sqrt{{B\_m}^{2} \cdot \frac{F}{-C}}}}\\ \mathbf{elif}\;F \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sqrt{2}}{B\_m} \cdot \left(-\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B\_m, A\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(C - \mathsf{hypot}\left(C, B\_m\right)\right) \cdot \left(F \cdot -2\right)}}{B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= F -1.35e+145)
   (/ -1.0 (/ B_m (sqrt (* (pow B_m 2.0) (/ F (- C))))))
   (if (<= F -1e-310)
     (* (/ (sqrt 2.0) B_m) (- (sqrt (* F (- A (hypot B_m A))))))
     (/ (sqrt (* (- C (hypot C B_m)) (* F -2.0))) B_m))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= -1.35e+145) {
		tmp = -1.0 / (B_m / sqrt((pow(B_m, 2.0) * (F / -C))));
	} else if (F <= -1e-310) {
		tmp = (sqrt(2.0) / B_m) * -sqrt((F * (A - hypot(B_m, A))));
	} else {
		tmp = sqrt(((C - hypot(C, B_m)) * (F * -2.0))) / B_m;
	}
	return tmp;
}

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= -1.35e+145) {
		tmp = -1.0 / (B_m / Math.sqrt((Math.pow(B_m, 2.0) * (F / -C))));
	} else if (F <= -1e-310) {
		tmp = (Math.sqrt(2.0) / B_m) * -Math.sqrt((F * (A - Math.hypot(B_m, A))));
	} else {
		tmp = Math.sqrt(((C - Math.hypot(C, B_m)) * (F * -2.0))) / B_m;
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if F <= -1.35e+145:
		tmp = -1.0 / (B_m / math.sqrt((math.pow(B_m, 2.0) * (F / -C))))
	elif F <= -1e-310:
		tmp = (math.sqrt(2.0) / B_m) * -math.sqrt((F * (A - math.hypot(B_m, A))))
	else:
		tmp = math.sqrt(((C - math.hypot(C, B_m)) * (F * -2.0))) / B_m
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (F <= -1.35e+145)
		tmp = Float64(-1.0 / Float64(B_m / sqrt(Float64((B_m ^ 2.0) * Float64(F / Float64(-C))))));
	elseif (F <= -1e-310)
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(-sqrt(Float64(F * Float64(A - hypot(B_m, A))))));
	else
		tmp = Float64(sqrt(Float64(Float64(C - hypot(C, B_m)) * Float64(F * -2.0))) / B_m);
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (F <= -1.35e+145)
		tmp = -1.0 / (B_m / sqrt(((B_m ^ 2.0) * (F / -C))));
	elseif (F <= -1e-310)
		tmp = (sqrt(2.0) / B_m) * -sqrt((F * (A - hypot(B_m, A))));
	else
		tmp = sqrt(((C - hypot(C, B_m)) * (F * -2.0))) / B_m;
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[F, -1.35e+145], N[(-1.0 / N[(B$95$m / N[Sqrt[N[(N[Power[B$95$m, 2.0], $MachinePrecision] * N[(F / (-C)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[F, -1e-310], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * (-N[Sqrt[N[(F * N[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[N[(N[(C - N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision] * N[(F * -2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / B$95$m), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.35000000000000011e145

   1. Initial program 10.1%

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

   3. Taylor expanded in A around 0 2.4%

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

      1. mul-1-neg 2.4%

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

      2. +-commutative 2.4%

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

      3. unpow2 2.4%

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

      4. unpow2 2.4%

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

      5. hypot-define 2.5%

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

   5. Simplified 2.5%

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

      1. neg-sub0 2.5%

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

      2. associate-*l/ 2.5%

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

      3. pow1/2 2.5%

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

      4. pow1/2 2.5%

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

      5. pow-prod-down 2.5%

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

   7. Applied egg-rr 2.5%

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

      1. neg-sub0 2.5%

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

      2. distribute-neg-frac2 2.5%

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

      3. unpow1/2 2.5%

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

      4. associate-*r* 2.5%

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

   9. Simplified 2.5%

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

   10. Taylor expanded in C around inf 6.7%

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

       1. clear-num 6.7%

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

       2. inv-pow 6.7%

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

       3. mul-1-neg 6.7%

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

       4. associate-/l* 6.5%

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

   12. Applied egg-rr 6.5%

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

       1. unpow-1 6.5%

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

       2. distribute-rgt-neg-in 6.5%

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

       3. distribute-neg-frac2 6.5%

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

   14. Simplified 6.5%

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

   if -1.35000000000000011e145 < F < -9.999999999999969e-311

   1. Initial program 15.1%

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

   3. Taylor expanded in C around 0 8.8%

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

      1. mul-1-neg 8.8%

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

      2. +-commutative 8.8%

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

      3. unpow2 8.8%

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

      4. unpow2 8.8%

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

      5. hypot-define 26.4%

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

   5. Simplified 26.4%

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

   if -9.999999999999969e-311 < F

   1. Initial program 31.0%

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

   3. Taylor expanded in A around 0 11.8%

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

      1. mul-1-neg 11.8%

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

      2. +-commutative 11.8%

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

      3. unpow2 11.8%

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

      4. unpow2 11.8%

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

      5. hypot-define 23.4%

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

   5. Simplified 23.4%

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

      1. neg-sub0 23.4%

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

      2. associate-*l/ 23.4%

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

      3. pow1/2 23.4%

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

      4. pow1/2 23.6%

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

      5. pow-prod-down 23.6%

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

   7. Applied egg-rr 23.6%

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

      1. neg-sub0 23.6%

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

      2. distribute-neg-frac2 23.6%

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

      3. unpow1/2 23.4%

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

      4. associate-*r* 23.4%

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

   9. Simplified 23.4%

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

   11. Applied egg-rr 25.6%

       \[\leadsto \color{blue}{0 + \frac{\sqrt{-2 \cdot \left(F \cdot \left(C - \mathsf{hypot}\left(C, B\right)\right)\right)}}{B}} \]
   12. Step-by-step derivation

       1. +-lft-identity 25.6%

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

       2. associate-*r* 25.6%

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

       3. *-commutative 25.6%

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

   13. Simplified 25.6%

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

4. Final simplification 21.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.35 \cdot 10^{+145}:\\ \;\;\;\;\frac{-1}{\frac{B}{\sqrt{{B}^{2} \cdot \frac{F}{-C}}}}\\ \mathbf{elif}\;F \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(-\sqrt{F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(C - \mathsf{hypot}\left(C, B\right)\right) \cdot \left(F \cdot -2\right)}}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 34.4% accurate, 2.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;F \leq -1.3 \cdot 10^{+145}:\\ \;\;\;\;\frac{\sqrt{{B\_m}^{2} \cdot \frac{F}{-C}}}{-B\_m}\\ \mathbf{elif}\;F \leq -3.8 \cdot 10^{-307}:\\ \;\;\;\;\frac{\sqrt{-2 \cdot \left(B\_m \cdot F\right)}}{-B\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(C - \mathsf{hypot}\left(C, B\_m\right)\right) \cdot \left(F \cdot -2\right)}}{B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= F -1.3e+145)
   (/ (sqrt (* (pow B_m 2.0) (/ F (- C)))) (- B_m))
   (if (<= F -3.8e-307)
     (/ (sqrt (* -2.0 (* B_m F))) (- B_m))
     (/ (sqrt (* (- C (hypot C B_m)) (* F -2.0))) B_m))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= -1.3e+145) {
		tmp = sqrt((pow(B_m, 2.0) * (F / -C))) / -B_m;
	} else if (F <= -3.8e-307) {
		tmp = sqrt((-2.0 * (B_m * F))) / -B_m;
	} else {
		tmp = sqrt(((C - hypot(C, B_m)) * (F * -2.0))) / B_m;
	}
	return tmp;
}

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= -1.3e+145) {
		tmp = Math.sqrt((Math.pow(B_m, 2.0) * (F / -C))) / -B_m;
	} else if (F <= -3.8e-307) {
		tmp = Math.sqrt((-2.0 * (B_m * F))) / -B_m;
	} else {
		tmp = Math.sqrt(((C - Math.hypot(C, B_m)) * (F * -2.0))) / B_m;
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if F <= -1.3e+145:
		tmp = math.sqrt((math.pow(B_m, 2.0) * (F / -C))) / -B_m
	elif F <= -3.8e-307:
		tmp = math.sqrt((-2.0 * (B_m * F))) / -B_m
	else:
		tmp = math.sqrt(((C - math.hypot(C, B_m)) * (F * -2.0))) / B_m
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (F <= -1.3e+145)
		tmp = Float64(sqrt(Float64((B_m ^ 2.0) * Float64(F / Float64(-C)))) / Float64(-B_m));
	elseif (F <= -3.8e-307)
		tmp = Float64(sqrt(Float64(-2.0 * Float64(B_m * F))) / Float64(-B_m));
	else
		tmp = Float64(sqrt(Float64(Float64(C - hypot(C, B_m)) * Float64(F * -2.0))) / B_m);
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (F <= -1.3e+145)
		tmp = sqrt(((B_m ^ 2.0) * (F / -C))) / -B_m;
	elseif (F <= -3.8e-307)
		tmp = sqrt((-2.0 * (B_m * F))) / -B_m;
	else
		tmp = sqrt(((C - hypot(C, B_m)) * (F * -2.0))) / B_m;
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[F, -1.3e+145], N[(N[Sqrt[N[(N[Power[B$95$m, 2.0], $MachinePrecision] * N[(F / (-C)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision], If[LessEqual[F, -3.8e-307], N[(N[Sqrt[N[(-2.0 * N[(B$95$m * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision], N[(N[Sqrt[N[(N[(C - N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision] * N[(F * -2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / B$95$m), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if F < -1.30000000000000001e145

   1. Initial program 10.1%

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

   3. Taylor expanded in A around 0 2.4%

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

      1. mul-1-neg 2.4%

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

      2. +-commutative 2.4%

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

      3. unpow2 2.4%

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

      4. unpow2 2.4%

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

      5. hypot-define 2.5%

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

   5. Simplified 2.5%

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

      1. neg-sub0 2.5%

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

      2. associate-*l/ 2.5%

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

      3. pow1/2 2.5%

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

      4. pow1/2 2.5%

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

      5. pow-prod-down 2.5%

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

   7. Applied egg-rr 2.5%

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

      1. neg-sub0 2.5%

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

      2. distribute-neg-frac2 2.5%

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

      3. unpow1/2 2.5%

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

      4. associate-*r* 2.5%

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

   9. Simplified 2.5%

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

   10. Taylor expanded in C around inf 6.7%

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

       1. mul-1-neg 6.7%

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

       2. associate-/l* 6.5%

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

       3. distribute-rgt-neg-in 6.5%

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

       4. distribute-neg-frac2 6.5%

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

   12. Simplified 6.5%

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

   if -1.30000000000000001e145 < F < -3.79999999999999985e-307

   1. Initial program 15.2%

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

   3. Taylor expanded in A around 0 9.0%

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

      1. mul-1-neg 9.0%

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

      2. +-commutative 9.0%

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

      3. unpow2 9.0%

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

      4. unpow2 9.0%

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

      5. hypot-define 27.8%

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

   5. Simplified 27.8%

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

      1. neg-sub0 27.8%

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

      2. associate-*l/ 27.8%

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

      3. pow1/2 27.8%

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

      4. pow1/2 27.8%

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

      5. pow-prod-down 27.9%

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

   7. Applied egg-rr 27.9%

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

      1. neg-sub0 27.9%

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

      2. distribute-neg-frac2 27.9%

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

      3. unpow1/2 27.9%

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

      4. associate-*r* 27.9%

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

   9. Simplified 27.9%

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

   10. Taylor expanded in C around 0 23.3%

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

   if -3.79999999999999985e-307 < F

   1. Initial program 30.6%

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

   3. Taylor expanded in A around 0 11.7%

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

      1. mul-1-neg 11.7%

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

      2. +-commutative 11.7%

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

      3. unpow2 11.7%

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

      4. unpow2 11.7%

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

      5. hypot-define 23.1%

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

   5. Simplified 23.1%

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

      1. neg-sub0 23.1%

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

      2. associate-*l/ 23.1%

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

      3. pow1/2 23.1%

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

      4. pow1/2 23.3%

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

      5. pow-prod-down 23.3%

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

   7. Applied egg-rr 23.3%

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

      1. neg-sub0 23.3%

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

      2. distribute-neg-frac2 23.3%

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

      3. unpow1/2 23.1%

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

      4. associate-*r* 23.1%

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

   9. Simplified 23.1%

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

   11. Applied egg-rr 25.2%

       \[\leadsto \color{blue}{0 + \frac{\sqrt{-2 \cdot \left(F \cdot \left(C - \mathsf{hypot}\left(C, B\right)\right)\right)}}{B}} \]
   12. Step-by-step derivation

       1. +-lft-identity 25.2%

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

       2. associate-*r* 25.2%

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

       3. *-commutative 25.2%

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

   13. Simplified 25.2%

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

4. Final simplification 20.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1.3 \cdot 10^{+145}:\\ \;\;\;\;\frac{\sqrt{{B}^{2} \cdot \frac{F}{-C}}}{-B}\\ \mathbf{elif}\;F \leq -3.8 \cdot 10^{-307}:\\ \;\;\;\;\frac{\sqrt{-2 \cdot \left(B \cdot F\right)}}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(C - \mathsf{hypot}\left(C, B\right)\right) \cdot \left(F \cdot -2\right)}}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 34.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;F \leq -5.1 \cdot 10^{+144}:\\ \;\;\;\;\frac{-1}{\frac{B\_m}{\sqrt{{B\_m}^{2} \cdot \frac{F}{-C}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot F\right) \cdot \left(C - \mathsf{hypot}\left(C, B\_m\right)\right)}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= F -5.1e+144)
   (/ -1.0 (/ B_m (sqrt (* (pow B_m 2.0) (/ F (- C))))))
   (/ (sqrt (* (* 2.0 F) (- C (hypot C B_m)))) (- 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 (F <= -5.1e+144) {
		tmp = -1.0 / (B_m / sqrt((pow(B_m, 2.0) * (F / -C))));
	} else {
		tmp = sqrt(((2.0 * F) * (C - hypot(C, B_m)))) / -B_m;
	}
	return tmp;
}

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= -5.1e+144) {
		tmp = -1.0 / (B_m / Math.sqrt((Math.pow(B_m, 2.0) * (F / -C))));
	} else {
		tmp = Math.sqrt(((2.0 * F) * (C - Math.hypot(C, B_m)))) / -B_m;
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if F <= -5.1e+144:
		tmp = -1.0 / (B_m / math.sqrt((math.pow(B_m, 2.0) * (F / -C))))
	else:
		tmp = math.sqrt(((2.0 * F) * (C - math.hypot(C, B_m)))) / -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 (F <= -5.1e+144)
		tmp = Float64(-1.0 / Float64(B_m / sqrt(Float64((B_m ^ 2.0) * Float64(F / Float64(-C))))));
	else
		tmp = Float64(sqrt(Float64(Float64(2.0 * F) * Float64(C - hypot(C, B_m)))) / Float64(-B_m));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (F <= -5.1e+144)
		tmp = -1.0 / (B_m / sqrt(((B_m ^ 2.0) * (F / -C))));
	else
		tmp = sqrt(((2.0 * F) * (C - hypot(C, B_m)))) / -B_m;
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[F, -5.1e+144], N[(-1.0 / N[(B$95$m / N[Sqrt[N[(N[Power[B$95$m, 2.0], $MachinePrecision] * N[(F / (-C)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] * N[(C - N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -5.0999999999999999e144

   1. Initial program 10.1%

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

   3. Taylor expanded in A around 0 2.4%

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

      1. mul-1-neg 2.4%

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

      2. +-commutative 2.4%

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

      3. unpow2 2.4%

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

      4. unpow2 2.4%

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

      5. hypot-define 2.5%

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

   5. Simplified 2.5%

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

      1. neg-sub0 2.5%

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

      2. associate-*l/ 2.5%

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

      3. pow1/2 2.5%

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

      4. pow1/2 2.5%

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

      5. pow-prod-down 2.5%

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

   7. Applied egg-rr 2.5%

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

      1. neg-sub0 2.5%

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

      2. distribute-neg-frac2 2.5%

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

      3. unpow1/2 2.5%

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

      4. associate-*r* 2.5%

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

   9. Simplified 2.5%

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

   10. Taylor expanded in C around inf 6.7%

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

       1. clear-num 6.7%

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

       2. inv-pow 6.7%

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

       3. mul-1-neg 6.7%

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

       4. associate-/l* 6.5%

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

   12. Applied egg-rr 6.5%

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

       1. unpow-1 6.5%

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

       2. distribute-rgt-neg-in 6.5%

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

       3. distribute-neg-frac2 6.5%

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

   14. Simplified 6.5%

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

   if -5.0999999999999999e144 < F

   1. Initial program 19.9%

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

   3. Taylor expanded in A around 0 9.8%

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

      1. mul-1-neg 9.8%

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

      2. +-commutative 9.8%

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

      3. unpow2 9.8%

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

      4. unpow2 9.8%

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

      5. hypot-define 26.3%

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

   5. Simplified 26.3%

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

      1. neg-sub0 26.3%

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

      2. associate-*l/ 26.3%

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

      3. pow1/2 26.3%

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

      4. pow1/2 26.4%

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

      5. pow-prod-down 26.5%

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

   7. Applied egg-rr 26.5%

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

      1. neg-sub0 26.5%

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

      2. distribute-neg-frac2 26.5%

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

      3. unpow1/2 26.4%

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

      4. associate-*r* 26.4%

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

   9. Simplified 26.4%

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

4. Final simplification 22.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -5.1 \cdot 10^{+144}:\\ \;\;\;\;\frac{-1}{\frac{B}{\sqrt{{B}^{2} \cdot \frac{F}{-C}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot F\right) \cdot \left(C - \mathsf{hypot}\left(C, B\right)\right)}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 34.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;F \leq -4.4 \cdot 10^{+144}:\\ \;\;\;\;\frac{\sqrt{{B\_m}^{2} \cdot \frac{F}{-C}}}{-B\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot F\right) \cdot \left(C - \mathsf{hypot}\left(C, B\_m\right)\right)}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= F -4.4e+144)
   (/ (sqrt (* (pow B_m 2.0) (/ F (- C)))) (- B_m))
   (/ (sqrt (* (* 2.0 F) (- C (hypot C B_m)))) (- 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 (F <= -4.4e+144) {
		tmp = sqrt((pow(B_m, 2.0) * (F / -C))) / -B_m;
	} else {
		tmp = sqrt(((2.0 * F) * (C - hypot(C, B_m)))) / -B_m;
	}
	return tmp;
}

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= -4.4e+144) {
		tmp = Math.sqrt((Math.pow(B_m, 2.0) * (F / -C))) / -B_m;
	} else {
		tmp = Math.sqrt(((2.0 * F) * (C - Math.hypot(C, B_m)))) / -B_m;
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if F <= -4.4e+144:
		tmp = math.sqrt((math.pow(B_m, 2.0) * (F / -C))) / -B_m
	else:
		tmp = math.sqrt(((2.0 * F) * (C - math.hypot(C, B_m)))) / -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 (F <= -4.4e+144)
		tmp = Float64(sqrt(Float64((B_m ^ 2.0) * Float64(F / Float64(-C)))) / Float64(-B_m));
	else
		tmp = Float64(sqrt(Float64(Float64(2.0 * F) * Float64(C - hypot(C, B_m)))) / Float64(-B_m));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (F <= -4.4e+144)
		tmp = sqrt(((B_m ^ 2.0) * (F / -C))) / -B_m;
	else
		tmp = sqrt(((2.0 * F) * (C - hypot(C, B_m)))) / -B_m;
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[F, -4.4e+144], N[(N[Sqrt[N[(N[Power[B$95$m, 2.0], $MachinePrecision] * N[(F / (-C)), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision], N[(N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] * N[(C - N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -4.39999999999999976e144

   1. Initial program 10.1%

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

   3. Taylor expanded in A around 0 2.4%

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

      1. mul-1-neg 2.4%

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

      2. +-commutative 2.4%

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

      3. unpow2 2.4%

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

      4. unpow2 2.4%

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

      5. hypot-define 2.5%

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

   5. Simplified 2.5%

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

      1. neg-sub0 2.5%

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

      2. associate-*l/ 2.5%

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

      3. pow1/2 2.5%

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

      4. pow1/2 2.5%

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

      5. pow-prod-down 2.5%

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

   7. Applied egg-rr 2.5%

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

      1. neg-sub0 2.5%

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

      2. distribute-neg-frac2 2.5%

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

      3. unpow1/2 2.5%

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

      4. associate-*r* 2.5%

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

   9. Simplified 2.5%

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

   10. Taylor expanded in C around inf 6.7%

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

       1. mul-1-neg 6.7%

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

       2. associate-/l* 6.5%

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

       3. distribute-rgt-neg-in 6.5%

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

       4. distribute-neg-frac2 6.5%

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

   12. Simplified 6.5%

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

   if -4.39999999999999976e144 < F

   1. Initial program 19.9%

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

   3. Taylor expanded in A around 0 9.8%

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

      1. mul-1-neg 9.8%

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

      2. +-commutative 9.8%

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

      3. unpow2 9.8%

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

      4. unpow2 9.8%

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

      5. hypot-define 26.3%

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

   5. Simplified 26.3%

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

      1. neg-sub0 26.3%

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

      2. associate-*l/ 26.3%

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

      3. pow1/2 26.3%

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

      4. pow1/2 26.4%

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

      5. pow-prod-down 26.5%

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

   7. Applied egg-rr 26.5%

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

      1. neg-sub0 26.5%

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

      2. distribute-neg-frac2 26.5%

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

      3. unpow1/2 26.4%

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

      4. associate-*r* 26.4%

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

   9. Simplified 26.4%

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

Alternative 8: 33.4% accurate, 2.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;F \leq -3.8 \cdot 10^{-307}:\\ \;\;\;\;\frac{\sqrt{-2 \cdot \left(B\_m \cdot F\right)}}{-B\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(C - \mathsf{hypot}\left(C, B\_m\right)\right) \cdot \left(F \cdot -2\right)}}{B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= F -3.8e-307)
   (/ (sqrt (* -2.0 (* B_m F))) (- B_m))
   (/ (sqrt (* (- C (hypot C B_m)) (* F -2.0))) B_m)))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= -3.8e-307) {
		tmp = sqrt((-2.0 * (B_m * F))) / -B_m;
	} else {
		tmp = sqrt(((C - hypot(C, B_m)) * (F * -2.0))) / B_m;
	}
	return tmp;
}

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= -3.8e-307) {
		tmp = Math.sqrt((-2.0 * (B_m * F))) / -B_m;
	} else {
		tmp = Math.sqrt(((C - Math.hypot(C, B_m)) * (F * -2.0))) / B_m;
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if F <= -3.8e-307:
		tmp = math.sqrt((-2.0 * (B_m * F))) / -B_m
	else:
		tmp = math.sqrt(((C - math.hypot(C, B_m)) * (F * -2.0))) / B_m
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (F <= -3.8e-307)
		tmp = Float64(sqrt(Float64(-2.0 * Float64(B_m * F))) / Float64(-B_m));
	else
		tmp = Float64(sqrt(Float64(Float64(C - hypot(C, B_m)) * Float64(F * -2.0))) / B_m);
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (F <= -3.8e-307)
		tmp = sqrt((-2.0 * (B_m * F))) / -B_m;
	else
		tmp = sqrt(((C - hypot(C, B_m)) * (F * -2.0))) / B_m;
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[F, -3.8e-307], N[(N[Sqrt[N[(-2.0 * N[(B$95$m * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision], N[(N[Sqrt[N[(N[(C - N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision] * N[(F * -2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / B$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -3.79999999999999985e-307

   1. Initial program 13.7%

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

   3. Taylor expanded in A around 0 7.0%

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

      1. mul-1-neg 7.0%

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

      2. +-commutative 7.0%

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

      3. unpow2 7.0%

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

      4. unpow2 7.0%

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

      5. hypot-define 20.4%

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

   5. Simplified 20.4%

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

      1. neg-sub0 20.4%

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

      2. associate-*l/ 20.4%

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

      3. pow1/2 20.4%

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

      4. pow1/2 20.4%

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

      5. pow-prod-down 20.5%

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

   7. Applied egg-rr 20.5%

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

      1. neg-sub0 20.5%

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

      2. distribute-neg-frac2 20.5%

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

      3. unpow1/2 20.5%

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

      4. associate-*r* 20.5%

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

   9. Simplified 20.5%

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

   10. Taylor expanded in C around 0 17.0%

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

   if -3.79999999999999985e-307 < F

   1. Initial program 30.6%

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

   3. Taylor expanded in A around 0 11.7%

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

      1. mul-1-neg 11.7%

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

      2. +-commutative 11.7%

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

      3. unpow2 11.7%

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

      4. unpow2 11.7%

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

      5. hypot-define 23.1%

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

   5. Simplified 23.1%

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

      1. neg-sub0 23.1%

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

      2. associate-*l/ 23.1%

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

      3. pow1/2 23.1%

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

      4. pow1/2 23.3%

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

      5. pow-prod-down 23.3%

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

   7. Applied egg-rr 23.3%

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

      1. neg-sub0 23.3%

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

      2. distribute-neg-frac2 23.3%

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

      3. unpow1/2 23.1%

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

      4. associate-*r* 23.1%

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

   9. Simplified 23.1%

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

   11. Applied egg-rr 25.2%

       \[\leadsto \color{blue}{0 + \frac{\sqrt{-2 \cdot \left(F \cdot \left(C - \mathsf{hypot}\left(C, B\right)\right)\right)}}{B}} \]
   12. Step-by-step derivation

       1. +-lft-identity 25.2%

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

       2. associate-*r* 25.2%

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

       3. *-commutative 25.2%

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

   13. Simplified 25.2%

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

4. Final simplification 19.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -3.8 \cdot 10^{-307}:\\ \;\;\;\;\frac{\sqrt{-2 \cdot \left(B \cdot F\right)}}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(C - \mathsf{hypot}\left(C, B\right)\right) \cdot \left(F \cdot -2\right)}}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 33.6% accurate, 3.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;F \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sqrt{-2 \cdot \left(B\_m \cdot F\right)}}{-B\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\frac{{B\_m}^{2} \cdot F}{-C}}}{B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= F -1e-310)
   (/ (sqrt (* -2.0 (* B_m F))) (- B_m))
   (/ (sqrt (/ (* (pow B_m 2.0) F) (- C))) 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 (F <= -1e-310) {
		tmp = sqrt((-2.0 * (B_m * F))) / -B_m;
	} else {
		tmp = sqrt(((pow(B_m, 2.0) * F) / -C)) / B_m;
	}
	return tmp;
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (f <= (-1d-310)) then
        tmp = sqrt(((-2.0d0) * (b_m * f))) / -b_m
    else
        tmp = sqrt((((b_m ** 2.0d0) * f) / -c)) / b_m
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= -1e-310) {
		tmp = Math.sqrt((-2.0 * (B_m * F))) / -B_m;
	} else {
		tmp = Math.sqrt(((Math.pow(B_m, 2.0) * F) / -C)) / B_m;
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if F <= -1e-310:
		tmp = math.sqrt((-2.0 * (B_m * F))) / -B_m
	else:
		tmp = math.sqrt(((math.pow(B_m, 2.0) * F) / -C)) / 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 (F <= -1e-310)
		tmp = Float64(sqrt(Float64(-2.0 * Float64(B_m * F))) / Float64(-B_m));
	else
		tmp = Float64(sqrt(Float64(Float64((B_m ^ 2.0) * F) / Float64(-C))) / B_m);
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (F <= -1e-310)
		tmp = sqrt((-2.0 * (B_m * F))) / -B_m;
	else
		tmp = sqrt((((B_m ^ 2.0) * F) / -C)) / B_m;
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[F, -1e-310], N[(N[Sqrt[N[(-2.0 * N[(B$95$m * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision], N[(N[Sqrt[N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] * F), $MachinePrecision] / (-C)), $MachinePrecision]], $MachinePrecision] / B$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -9.999999999999969e-311

   1. Initial program 13.6%

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

   3. Taylor expanded in A around 0 7.0%

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

      1. mul-1-neg 7.0%

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

      2. +-commutative 7.0%

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

      3. unpow2 7.0%

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

      4. unpow2 7.0%

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

      5. hypot-define 20.3%

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

   5. Simplified 20.3%

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

      1. neg-sub0 20.3%

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

      2. associate-*l/ 20.3%

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

      3. pow1/2 20.3%

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

      4. pow1/2 20.3%

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

      5. pow-prod-down 20.4%

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

   7. Applied egg-rr 20.4%

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

      1. neg-sub0 20.4%

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

      2. distribute-neg-frac2 20.4%

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

      3. unpow1/2 20.4%

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

      4. associate-*r* 20.4%

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

   9. Simplified 20.4%

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

   10. Taylor expanded in C around 0 16.9%

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

   if -9.999999999999969e-311 < F

   1. Initial program 31.0%

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

   3. Taylor expanded in A around 0 11.8%

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

      1. mul-1-neg 11.8%

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

      2. +-commutative 11.8%

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

      3. unpow2 11.8%

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

      4. unpow2 11.8%

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

      5. hypot-define 23.4%

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

   5. Simplified 23.4%

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

      1. neg-sub0 23.4%

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

      2. associate-*l/ 23.4%

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

      3. pow1/2 23.4%

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

      4. pow1/2 23.6%

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

      5. pow-prod-down 23.6%

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

   7. Applied egg-rr 23.6%

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

      1. neg-sub0 23.6%

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

      2. distribute-neg-frac2 23.6%

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

      3. unpow1/2 23.4%

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

      4. associate-*r* 23.4%

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

   9. Simplified 23.4%

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

   10. Taylor expanded in C around inf 39.6%

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

   12. Applied egg-rr 30.2%

       \[\leadsto \frac{\sqrt{-1 \cdot \frac{{B}^{2} \cdot F}{C}}}{\color{blue}{0 + B}} \]
   13. Step-by-step derivation

       1. +-lft-identity 30.2%

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

   14. Simplified 30.2%

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

4. Final simplification 20.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\frac{\sqrt{-2 \cdot \left(B \cdot F\right)}}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\frac{{B}^{2} \cdot F}{-C}}}{B}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 28.2% accurate, 3.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;F \leq -3.8 \cdot 10^{-307}:\\ \;\;\;\;\frac{\sqrt{-2 \cdot \left(B\_m \cdot F\right)}}{-B\_m}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F \cdot \left(C \cdot \frac{-4}{{B\_m}^{-2}}\right)}\\ \end{array} \end{array} \]

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 (<= F -3.8e-307)
   (/ (sqrt (* -2.0 (* B_m F))) (- B_m))
   (sqrt (* F (* C (/ -4.0 (pow B_m -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) {
	double tmp;
	if (F <= -3.8e-307) {
		tmp = sqrt((-2.0 * (B_m * F))) / -B_m;
	} else {
		tmp = sqrt((F * (C * (-4.0 / pow(B_m, -2.0)))));
	}
	return tmp;
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (f <= (-3.8d-307)) then
        tmp = sqrt(((-2.0d0) * (b_m * f))) / -b_m
    else
        tmp = sqrt((f * (c * ((-4.0d0) / (b_m ** (-2.0d0))))))
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= -3.8e-307) {
		tmp = Math.sqrt((-2.0 * (B_m * F))) / -B_m;
	} else {
		tmp = Math.sqrt((F * (C * (-4.0 / Math.pow(B_m, -2.0)))));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if F <= -3.8e-307:
		tmp = math.sqrt((-2.0 * (B_m * F))) / -B_m
	else:
		tmp = math.sqrt((F * (C * (-4.0 / math.pow(B_m, -2.0)))))
	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 (F <= -3.8e-307)
		tmp = Float64(sqrt(Float64(-2.0 * Float64(B_m * F))) / Float64(-B_m));
	else
		tmp = sqrt(Float64(F * Float64(C * Float64(-4.0 / (B_m ^ -2.0)))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (F <= -3.8e-307)
		tmp = sqrt((-2.0 * (B_m * F))) / -B_m;
	else
		tmp = sqrt((F * (C * (-4.0 / (B_m ^ -2.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[F, -3.8e-307], N[(N[Sqrt[N[(-2.0 * N[(B$95$m * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision], N[Sqrt[N[(F * N[(C * N[(-4.0 / N[Power[B$95$m, -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < -3.79999999999999985e-307

   1. Initial program 13.7%

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

   3. Taylor expanded in A around 0 7.0%

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

      1. mul-1-neg 7.0%

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

      2. +-commutative 7.0%

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

      3. unpow2 7.0%

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

      4. unpow2 7.0%

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

      5. hypot-define 20.4%

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

   5. Simplified 20.4%

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

      1. neg-sub0 20.4%

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

      2. associate-*l/ 20.4%

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

      3. pow1/2 20.4%

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

      4. pow1/2 20.4%

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

      5. pow-prod-down 20.5%

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

   7. Applied egg-rr 20.5%

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

      1. neg-sub0 20.5%

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

      2. distribute-neg-frac2 20.5%

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

      3. unpow1/2 20.5%

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

      4. associate-*r* 20.5%

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

   9. Simplified 20.5%

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

   10. Taylor expanded in C around 0 17.0%

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

   if -3.79999999999999985e-307 < F

   1. Initial program 30.6%

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

   3. Taylor expanded in A around 0 11.7%

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

      1. mul-1-neg 11.7%

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

      2. +-commutative 11.7%

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

      3. unpow2 11.7%

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

      4. unpow2 11.7%

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

      5. hypot-define 23.1%

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

   5. Simplified 23.1%

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

   6. Taylor expanded in C around -inf 0.0%

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

      1. unpow2 0.0%

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

      2. rem-square-sqrt 4.0%

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

      3. unpow2 4.0%

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

      4. rem-square-sqrt 4.0%

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

      5. metadata-eval 4.0%

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

      6. *-commutative 4.0%

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

   8. Simplified 4.0%

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

   10. Applied egg-rr 29.9%

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

       1. associate-*l* 31.7%

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

   12. Simplified 31.7%

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

Alternative 11: 28.1% accurate, 5.6× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;F \leq 1.2 \cdot 10^{-102}:\\ \;\;\;\;\frac{\sqrt{-2 \cdot \left(B\_m \cdot F\right)}}{-B\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{B\_m} \cdot {\left(C \cdot F\right)}^{-0.5}\\ \end{array} \end{array} \]

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 (<= F 1.2e-102)
   (/ (sqrt (* -2.0 (* B_m F))) (- B_m))
   (* (/ -2.0 B_m) (pow (* C F) -0.5))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= 1.2e-102) {
		tmp = sqrt((-2.0 * (B_m * F))) / -B_m;
	} else {
		tmp = (-2.0 / B_m) * pow((C * F), -0.5);
	}
	return tmp;
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (f <= 1.2d-102) then
        tmp = sqrt(((-2.0d0) * (b_m * f))) / -b_m
    else
        tmp = ((-2.0d0) / b_m) * ((c * f) ** (-0.5d0))
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= 1.2e-102) {
		tmp = Math.sqrt((-2.0 * (B_m * F))) / -B_m;
	} else {
		tmp = (-2.0 / B_m) * Math.pow((C * F), -0.5);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if F <= 1.2e-102:
		tmp = math.sqrt((-2.0 * (B_m * F))) / -B_m
	else:
		tmp = (-2.0 / B_m) * math.pow((C * F), -0.5)
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (F <= 1.2e-102)
		tmp = Float64(sqrt(Float64(-2.0 * Float64(B_m * F))) / Float64(-B_m));
	else
		tmp = Float64(Float64(-2.0 / B_m) * (Float64(C * F) ^ -0.5));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (F <= 1.2e-102)
		tmp = sqrt((-2.0 * (B_m * F))) / -B_m;
	else
		tmp = (-2.0 / B_m) * ((C * F) ^ -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[F, 1.2e-102], N[(N[Sqrt[N[(-2.0 * N[(B$95$m * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision], N[(N[(-2.0 / B$95$m), $MachinePrecision] * N[Power[N[(C * F), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < 1.2e-102

   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 A around 0 8.9%

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

      1. mul-1-neg 8.9%

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

      2. +-commutative 8.9%

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

      3. unpow2 8.9%

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

      4. unpow2 8.9%

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

      5. hypot-define 21.7%

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

   5. Simplified 21.7%

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

      1. neg-sub0 21.7%

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

      2. associate-*l/ 21.7%

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

      3. pow1/2 21.7%

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

      4. pow1/2 21.7%

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

      5. pow-prod-down 21.8%

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

   7. Applied egg-rr 21.8%

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

      1. neg-sub0 21.8%

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

      2. distribute-neg-frac2 21.8%

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

      3. unpow1/2 21.8%

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

      4. associate-*r* 21.8%

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

   9. Simplified 21.8%

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

   10. Taylor expanded in C around 0 18.3%

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

   if 1.2e-102 < F

   1. Initial program 26.8%

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

   3. Taylor expanded in A around 0 4.5%

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

      1. mul-1-neg 4.5%

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

      2. +-commutative 4.5%

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

      3. unpow2 4.5%

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

      4. unpow2 4.5%

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

      5. hypot-define 17.8%

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

   5. Simplified 17.8%

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

   6. Taylor expanded in C around -inf 0.0%

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

      1. unpow2 0.0%

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

      2. rem-square-sqrt 0.4%

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

      3. unpow2 0.4%

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

      4. rem-square-sqrt 0.4%

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

      5. metadata-eval 0.4%

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

      6. *-commutative 0.4%

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

   8. Simplified 0.4%

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

   10. Applied egg-rr 9.7%

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

4. Final simplification 16.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 1.2 \cdot 10^{-102}:\\ \;\;\;\;\frac{\sqrt{-2 \cdot \left(B \cdot F\right)}}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{-2}{B} \cdot {\left(C \cdot F\right)}^{-0.5}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 3.8% accurate, 5.9× speedup

Math

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

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 (* (/ -2.0 B_m) (pow (* C F) -0.5)))

C

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

Fortran

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

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	return (-2.0 / B_m) * Math.pow((C * F), -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 (-2.0 / B_m) * math.pow((C * F), -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(Float64(-2.0 / B_m) * (Float64(C * F) ^ -0.5))
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp = code(A, B_m, C, F)
	tmp = (-2.0 / B_m) * ((C * F) ^ -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[(N[(-2.0 / B$95$m), $MachinePrecision] * N[Power[N[(C * F), $MachinePrecision], -0.5], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 17.7%

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

3. Taylor expanded in A around 0 8.1%

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

   1. mul-1-neg 8.1%

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

   2. +-commutative 8.1%

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

   3. unpow2 8.1%

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

   4. unpow2 8.1%

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

   5. hypot-define 21.0%

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

5. Simplified 21.0%

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

6. Taylor expanded in C around -inf 0.0%

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

   1. unpow2 0.0%

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

   2. rem-square-sqrt 3.8%

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

   3. unpow2 3.8%

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

   4. rem-square-sqrt 3.9%

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

   5. metadata-eval 3.9%

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

   6. *-commutative 3.9%

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

8. Simplified 3.9%

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

10. Applied egg-rr 3.0%

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

11. Final simplification 3.0%

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

Alternative 13: 1.2% accurate, 5.9× speedup

Math

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

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 (* (/ -2.0 B_m) (sqrt (* C 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 (-2.0 / B_m) * sqrt((C * 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 = ((-2.0d0) / b_m) * sqrt((c * 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 (-2.0 / B_m) * Math.sqrt((C * 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 (-2.0 / B_m) * math.sqrt((C * 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(Float64(-2.0 / B_m) * sqrt(Float64(C * 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 = (-2.0 / B_m) * sqrt((C * 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[(N[(-2.0 / B$95$m), $MachinePrecision] * N[Sqrt[N[(C * F), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 17.7%

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

3. Taylor expanded in A around 0 8.1%

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

   1. mul-1-neg 8.1%

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

   2. +-commutative 8.1%

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

   3. unpow2 8.1%

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

   4. unpow2 8.1%

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

   5. hypot-define 21.0%

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

5. Simplified 21.0%

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

6. Taylor expanded in C around -inf 0.0%

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

   1. unpow2 0.0%

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

   2. rem-square-sqrt 3.8%

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

   3. unpow2 3.8%

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

   4. rem-square-sqrt 3.9%

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

   5. metadata-eval 3.9%

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

   6. *-commutative 3.9%

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

8. Simplified 3.9%

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

9. Final simplification 3.9%

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

Reproduce

herbie shell --seed 2024170 
(FPCore (A B C F)
  :name "ABCF->ab-angle b"
  :precision binary64
  (/ (- (sqrt (* (* 2.0 (* (- (pow B 2.0) (* (* 4.0 A) C)) F)) (- (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0))))))) (- (pow B 2.0) (* (* 4.0 A) C))))