ABCF->ab-angle a

Percentage Accurate: 19.4% → 63.8%

Time: 23.3s

Alternatives: 12

Speedup: 6.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 19.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 63.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := \mathsf{hypot}\left(B\_m, A - C\right)\\ t_2 := -t\_0\\ t_3 := \left(4 \cdot A\right) \cdot C\\ t_4 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_3\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_3 - {B\_m}^{2}}\\ t_5 := F \cdot t\_0\\ \mathbf{if}\;t\_4 \leq -\infty:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{F \cdot \frac{\left(A + C\right) + t\_1}{\mathsf{fma}\left(-4, A \cdot C, {B\_m}^{2}\right)}}\right)\\ \mathbf{elif}\;t\_4 \leq -2 \cdot 10^{-194}:\\ \;\;\;\;\frac{\sqrt{t\_5 \cdot \left(2 \cdot \left(A + \left(C + t\_1\right)\right)\right)}}{t\_2}\\ \mathbf{elif}\;t\_4 \leq 0:\\ \;\;\;\;\frac{\sqrt{t\_5 \cdot \left(4 \cdot C - \frac{{B\_m}^{2}}{A}\right)}}{t\_2}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;\frac{\sqrt{2 \cdot t\_5} \cdot \sqrt{A + \left(C + \mathsf{hypot}\left(A - C, B\_m\right)\right)}}{t\_2}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(C, B\_m\right)} \cdot \sqrt{F}\right) \cdot \left(-\frac{\sqrt{2}}{B\_m}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* A (* C -4.0))))
        (t_1 (hypot B_m (- A C)))
        (t_2 (- t_0))
        (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 (* F t_0)))
   (if (<= t_4 (- INFINITY))
     (*
      (sqrt 2.0)
      (- (sqrt (* F (/ (+ (+ A C) t_1) (fma -4.0 (* A C) (pow B_m 2.0)))))))
     (if (<= t_4 -2e-194)
       (/ (sqrt (* t_5 (* 2.0 (+ A (+ C t_1))))) t_2)
       (if (<= t_4 0.0)
         (/ (sqrt (* t_5 (- (* 4.0 C) (/ (pow B_m 2.0) A)))) t_2)
         (if (<= t_4 INFINITY)
           (/
            (* (sqrt (* 2.0 t_5)) (sqrt (+ A (+ C (hypot (- A C) B_m)))))
            t_2)
           (*
            (* (sqrt (+ C (hypot C B_m))) (sqrt F))
            (- (/ (sqrt 2.0) B_m)))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (A * (C * -4.0)));
	double t_1 = hypot(B_m, (A - C));
	double t_2 = -t_0;
	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 = F * t_0;
	double tmp;
	if (t_4 <= -((double) INFINITY)) {
		tmp = sqrt(2.0) * -sqrt((F * (((A + C) + t_1) / fma(-4.0, (A * C), pow(B_m, 2.0)))));
	} else if (t_4 <= -2e-194) {
		tmp = sqrt((t_5 * (2.0 * (A + (C + t_1))))) / t_2;
	} else if (t_4 <= 0.0) {
		tmp = sqrt((t_5 * ((4.0 * C) - (pow(B_m, 2.0) / A)))) / t_2;
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = (sqrt((2.0 * t_5)) * sqrt((A + (C + hypot((A - C), B_m))))) / t_2;
	} else {
		tmp = (sqrt((C + hypot(C, B_m))) * sqrt(F)) * -(sqrt(2.0) / B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	t_1 = hypot(B_m, Float64(A - C))
	t_2 = Float64(-t_0)
	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(F * t_0)
	tmp = 0.0
	if (t_4 <= Float64(-Inf))
		tmp = Float64(sqrt(2.0) * Float64(-sqrt(Float64(F * Float64(Float64(Float64(A + C) + t_1) / fma(-4.0, Float64(A * C), (B_m ^ 2.0)))))));
	elseif (t_4 <= -2e-194)
		tmp = Float64(sqrt(Float64(t_5 * Float64(2.0 * Float64(A + Float64(C + t_1))))) / t_2);
	elseif (t_4 <= 0.0)
		tmp = Float64(sqrt(Float64(t_5 * Float64(Float64(4.0 * C) - Float64((B_m ^ 2.0) / A)))) / t_2);
	elseif (t_4 <= Inf)
		tmp = Float64(Float64(sqrt(Float64(2.0 * t_5)) * sqrt(Float64(A + Float64(C + hypot(Float64(A - C), B_m))))) / t_2);
	else
		tmp = Float64(Float64(sqrt(Float64(C + hypot(C, B_m))) * sqrt(F)) * Float64(-Float64(sqrt(2.0) / B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]}, Block[{t$95$2 = (-t$95$0)}, 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[(F * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$4, (-Infinity)], N[(N[Sqrt[2.0], $MachinePrecision] * (-N[Sqrt[N[(F * N[(N[(N[(A + C), $MachinePrecision] + t$95$1), $MachinePrecision] / N[(-4.0 * N[(A * C), $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], If[LessEqual[t$95$4, -2e-194], N[(N[Sqrt[N[(t$95$5 * N[(2.0 * N[(A + N[(C + t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[t$95$4, 0.0], N[(N[Sqrt[N[(t$95$5 * N[(N[(4.0 * C), $MachinePrecision] - N[(N[Power[B$95$m, 2.0], $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[(N[(N[Sqrt[N[(2.0 * t$95$5), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(A + N[(C + N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], N[(N[(N[Sqrt[N[(C + N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] * (-N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision])), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 5 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 3.2%

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

   3. Taylor expanded in F around 0 21.3%

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

   5. Simplified 66.8%

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

   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))) < -2.00000000000000004e-194

   1. Initial program 98.3%

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

   3. Simplified 98.3%

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

   if -2.00000000000000004e-194 < (/.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))) < -0.0

   1. Initial program 3.7%

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

   3. Simplified 5.6%

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

   5. Taylor expanded in A around -inf 24.8%

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

   if -0.0 < (/.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))) < +inf.0

   1. Initial program 51.8%

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

   3. Simplified 59.8%

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

      1. associate-*r* 59.8%

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

      2. associate-+r+ 59.8%

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

      3. hypot-undefine 51.8%

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

      4. unpow2 51.8%

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

      5. unpow2 51.8%

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

      6. +-commutative 51.8%

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

      7. sqrt-prod 55.7%

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

      8. *-commutative 55.7%

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

      9. associate-+l+ 55.7%

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

   6. Applied egg-rr 83.4%

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

   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 1.9%

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

      1. mul-1-neg 1.9%

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

      2. *-commutative 1.9%

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

      3. *-commutative 1.9%

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

      4. +-commutative 1.9%

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

      5. unpow2 1.9%

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

      6. unpow2 1.9%

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

      7. hypot-define 14.8%

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

   5. Simplified 14.8%

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

      1. sqrt-prod 26.2%

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

   7. Applied egg-rr 26.2%

      \[\leadsto -\color{blue}{\left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \sqrt{F}\right)} \cdot \frac{\sqrt{2}}{B} \]
3. Recombined 5 regimes into one program.

4. Final simplification 51.6%

   \[\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:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{F \cdot \frac{\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)}{\mathsf{fma}\left(-4, A \cdot C, {B}^{2}\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 -2 \cdot 10^{-194}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot \left(2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;\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 0:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot \left(4 \cdot C - \frac{{B}^{2}}{A}\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 \infty:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)} \cdot \sqrt{A + \left(C + \mathsf{hypot}\left(A - C, B\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \sqrt{F}\right) \cdot \left(-\frac{\sqrt{2}}{B}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 59.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;B\_m \leq 2 \cdot 10^{-55}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot t\_0\right) \cdot \left(4 \cdot C\right)}}{-t\_0}\\ \mathbf{elif}\;B\_m \leq 1.45 \cdot 10^{+82}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{F \cdot \frac{\left(A + C\right) + \mathsf{hypot}\left(B\_m, A - C\right)}{\mathsf{fma}\left(-4, A \cdot C, {B\_m}^{2}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(C, B\_m\right)} \cdot \sqrt{F}\right) \cdot \left(-\frac{\sqrt{2}}{B\_m}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* A (* C -4.0)))))
   (if (<= B_m 2e-55)
     (/ (sqrt (* (* F t_0) (* 4.0 C))) (- t_0))
     (if (<= B_m 1.45e+82)
       (*
        (sqrt 2.0)
        (-
         (sqrt
          (*
           F
           (/
            (+ (+ A C) (hypot B_m (- A C)))
            (fma -4.0 (* A C) (pow B_m 2.0)))))))
       (* (* (sqrt (+ C (hypot C B_m))) (sqrt F)) (- (/ (sqrt 2.0) B_m)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (A * (C * -4.0)));
	double tmp;
	if (B_m <= 2e-55) {
		tmp = sqrt(((F * t_0) * (4.0 * C))) / -t_0;
	} else if (B_m <= 1.45e+82) {
		tmp = sqrt(2.0) * -sqrt((F * (((A + C) + hypot(B_m, (A - C))) / fma(-4.0, (A * C), pow(B_m, 2.0)))));
	} else {
		tmp = (sqrt((C + hypot(C, B_m))) * sqrt(F)) * -(sqrt(2.0) / B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	tmp = 0.0
	if (B_m <= 2e-55)
		tmp = Float64(sqrt(Float64(Float64(F * t_0) * Float64(4.0 * C))) / Float64(-t_0));
	elseif (B_m <= 1.45e+82)
		tmp = Float64(sqrt(2.0) * Float64(-sqrt(Float64(F * Float64(Float64(Float64(A + C) + hypot(B_m, Float64(A - C))) / fma(-4.0, Float64(A * C), (B_m ^ 2.0)))))));
	else
		tmp = Float64(Float64(sqrt(Float64(C + hypot(C, B_m))) * sqrt(F)) * Float64(-Float64(sqrt(2.0) / B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 2e-55], N[(N[Sqrt[N[(N[(F * t$95$0), $MachinePrecision] * N[(4.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], If[LessEqual[B$95$m, 1.45e+82], N[(N[Sqrt[2.0], $MachinePrecision] * (-N[Sqrt[N[(F * N[(N[(N[(A + C), $MachinePrecision] + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] / N[(-4.0 * N[(A * C), $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(N[(N[Sqrt[N[(C + N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] * (-N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision])), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if B < 1.99999999999999999e-55

   1. Initial program 22.9%

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

   3. Simplified 28.9%

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

   5. Taylor expanded in A around -inf 17.6%

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

      1. *-commutative 17.6%

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

   7. Simplified 17.6%

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

   if 1.99999999999999999e-55 < B < 1.4500000000000001e82

   1. Initial program 51.0%

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

   3. Taylor expanded in F around 0 54.8%

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

   5. Simplified 67.4%

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

   if 1.4500000000000001e82 < B

   1. Initial program 3.2%

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

   3. Taylor expanded in A around 0 13.9%

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

      1. mul-1-neg 13.9%

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

      2. *-commutative 13.9%

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

      3. *-commutative 13.9%

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

      4. +-commutative 13.9%

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

      5. unpow2 13.9%

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

      6. unpow2 13.9%

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

      7. hypot-define 43.0%

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

   5. Simplified 43.0%

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

      1. sqrt-prod 73.4%

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

   7. Applied egg-rr 73.4%

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

4. Final simplification 31.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 2 \cdot 10^{-55}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot \left(4 \cdot C\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 1.45 \cdot 10^{+82}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{F \cdot \frac{\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)}{\mathsf{fma}\left(-4, A \cdot C, {B}^{2}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \sqrt{F}\right) \cdot \left(-\frac{\sqrt{2}}{B}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 58.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;B\_m \leq 2 \cdot 10^{-40}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot t\_0\right) \cdot \left(4 \cdot C\right)}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(C, B\_m\right)} \cdot \sqrt{F}\right) \cdot \left(-\frac{\sqrt{2}}{B\_m}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* A (* C -4.0)))))
   (if (<= B_m 2e-40)
     (/ (sqrt (* (* F t_0) (* 4.0 C))) (- t_0))
     (* (* (sqrt (+ C (hypot C B_m))) (sqrt F)) (- (/ (sqrt 2.0) B_m))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (A * (C * -4.0)));
	double tmp;
	if (B_m <= 2e-40) {
		tmp = sqrt(((F * t_0) * (4.0 * C))) / -t_0;
	} else {
		tmp = (sqrt((C + hypot(C, B_m))) * sqrt(F)) * -(sqrt(2.0) / B_m);
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	tmp = 0.0
	if (B_m <= 2e-40)
		tmp = Float64(sqrt(Float64(Float64(F * t_0) * Float64(4.0 * C))) / Float64(-t_0));
	else
		tmp = Float64(Float64(sqrt(Float64(C + hypot(C, B_m))) * sqrt(F)) * Float64(-Float64(sqrt(2.0) / B_m)));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 2e-40], N[(N[Sqrt[N[(N[(F * t$95$0), $MachinePrecision] * N[(4.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], N[(N[(N[Sqrt[N[(C + N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] * (-N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if B < 1.9999999999999999e-40

   1. Initial program 23.7%

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

   3. Simplified 29.4%

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

   5. Taylor expanded in A around -inf 18.0%

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

      1. *-commutative 18.0%

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

   7. Simplified 18.0%

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

   if 1.9999999999999999e-40 < B

   1. Initial program 17.3%

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

   3. Taylor expanded in A around 0 25.1%

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

      1. mul-1-neg 25.1%

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

      2. *-commutative 25.1%

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

      3. *-commutative 25.1%

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

      4. +-commutative 25.1%

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

      5. unpow2 25.1%

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

      6. unpow2 25.1%

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

      7. hypot-define 46.1%

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

   5. Simplified 46.1%

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

      1. sqrt-prod 68.1%

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

   7. Applied egg-rr 68.1%

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

4. Final simplification 29.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 2 \cdot 10^{-40}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot \left(4 \cdot C\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{C + \mathsf{hypot}\left(C, B\right)} \cdot \sqrt{F}\right) \cdot \left(-\frac{\sqrt{2}}{B}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 53.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;B\_m \leq 6.2 \cdot 10^{-49}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot t\_0\right) \cdot \left(4 \cdot C\right)}}{-t\_0}\\ \mathbf{elif}\;B\_m \leq 6.4 \cdot 10^{+86}:\\ \;\;\;\;\frac{B\_m \cdot {\left(2 \cdot \left(F \cdot \left(C + \mathsf{hypot}\left(C, B\_m\right)\right)\right)\right)}^{0.5}}{\left(4 \cdot A\right) \cdot C - {B\_m}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B\_m} \cdot \left(\sqrt{F} \cdot \left(-\sqrt{B\_m + C}\right)\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* A (* C -4.0)))))
   (if (<= B_m 6.2e-49)
     (/ (sqrt (* (* F t_0) (* 4.0 C))) (- t_0))
     (if (<= B_m 6.4e+86)
       (/
        (* B_m (pow (* 2.0 (* F (+ C (hypot C B_m)))) 0.5))
        (- (* (* 4.0 A) C) (pow B_m 2.0)))
       (* (/ (sqrt 2.0) B_m) (* (sqrt F) (- (sqrt (+ B_m C)))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (A * (C * -4.0)));
	double tmp;
	if (B_m <= 6.2e-49) {
		tmp = sqrt(((F * t_0) * (4.0 * C))) / -t_0;
	} else if (B_m <= 6.4e+86) {
		tmp = (B_m * pow((2.0 * (F * (C + hypot(C, B_m)))), 0.5)) / (((4.0 * A) * C) - pow(B_m, 2.0));
	} else {
		tmp = (sqrt(2.0) / B_m) * (sqrt(F) * -sqrt((B_m + C)));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	tmp = 0.0
	if (B_m <= 6.2e-49)
		tmp = Float64(sqrt(Float64(Float64(F * t_0) * Float64(4.0 * C))) / Float64(-t_0));
	elseif (B_m <= 6.4e+86)
		tmp = Float64(Float64(B_m * (Float64(2.0 * Float64(F * Float64(C + hypot(C, B_m)))) ^ 0.5)) / Float64(Float64(Float64(4.0 * A) * C) - (B_m ^ 2.0)));
	else
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(sqrt(F) * Float64(-sqrt(Float64(B_m + C)))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 6.2e-49], N[(N[Sqrt[N[(N[(F * t$95$0), $MachinePrecision] * N[(4.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], If[LessEqual[B$95$m, 6.4e+86], N[(N[(B$95$m * N[Power[N[(2.0 * N[(F * N[(C + N[Sqrt[C ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision]), $MachinePrecision] / N[(N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision] - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[(N[Sqrt[F], $MachinePrecision] * (-N[Sqrt[N[(B$95$m + C), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if B < 6.2e-49

   1. Initial program 23.0%

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

   3. Simplified 28.8%

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

   5. Taylor expanded in A around -inf 17.2%

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

      1. *-commutative 17.2%

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

   7. Simplified 17.2%

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

   if 6.2e-49 < B < 6.4000000000000001e86

   1. Initial program 56.0%

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

   3. Taylor expanded in A around 0 56.0%

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

      1. mul-1-neg 56.0%

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

      2. associate-*l* 56.2%

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

      3. *-commutative 56.2%

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

   5. Simplified 56.2%

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

      1. *-commutative 56.2%

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

      2. pow1/2 56.2%

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

      3. pow1/2 56.2%

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

      4. pow-prod-down 56.3%

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

      5. *-commutative 56.3%

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

      6. +-commutative 56.3%

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

      7. unpow2 56.3%

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

      8. unpow2 56.3%

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

      9. hypot-undefine 56.3%

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

   7. Applied egg-rr 56.3%

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

   if 6.4000000000000001e86 < B

   1. Initial program 3.2%

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

   3. Taylor expanded in A around 0 13.9%

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

      1. mul-1-neg 13.9%

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

      2. *-commutative 13.9%

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

      3. *-commutative 13.9%

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

      4. +-commutative 13.9%

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

      5. unpow2 13.9%

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

      6. unpow2 13.9%

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

      7. hypot-define 43.0%

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

   5. Simplified 43.0%

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

      1. sqrt-prod 73.4%

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

   7. Applied egg-rr 73.4%

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

   8. Taylor expanded in C around 0 65.2%

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

4. Final simplification 28.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 6.2 \cdot 10^{-49}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot \left(4 \cdot C\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;B \leq 6.4 \cdot 10^{+86}:\\ \;\;\;\;\frac{B \cdot {\left(2 \cdot \left(F \cdot \left(C + \mathsf{hypot}\left(C, B\right)\right)\right)\right)}^{0.5}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{F} \cdot \left(-\sqrt{B + C}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 39.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;F \leq 2.5 \cdot 10^{+106}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(2 \cdot \left(C + \mathsf{hypot}\left(B\_m, C\right)\right)\right)}}{-B\_m}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B\_m} \cdot \left(\sqrt{F} \cdot \left(-\sqrt{B\_m + C}\right)\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= F 2.5e+106)
   (/ (sqrt (* F (* 2.0 (+ C (hypot B_m C))))) (- B_m))
   (* (/ (sqrt 2.0) B_m) (* (sqrt F) (- (sqrt (+ B_m C)))))))

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 <= 2.5e+106) {
		tmp = sqrt((F * (2.0 * (C + hypot(B_m, C))))) / -B_m;
	} else {
		tmp = (sqrt(2.0) / B_m) * (sqrt(F) * -sqrt((B_m + C)));
	}
	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 <= 2.5e+106) {
		tmp = Math.sqrt((F * (2.0 * (C + Math.hypot(B_m, C))))) / -B_m;
	} else {
		tmp = (Math.sqrt(2.0) / B_m) * (Math.sqrt(F) * -Math.sqrt((B_m + C)));
	}
	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 <= 2.5e+106:
		tmp = math.sqrt((F * (2.0 * (C + math.hypot(B_m, C))))) / -B_m
	else:
		tmp = (math.sqrt(2.0) / B_m) * (math.sqrt(F) * -math.sqrt((B_m + C)))
	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 <= 2.5e+106)
		tmp = Float64(sqrt(Float64(F * Float64(2.0 * Float64(C + hypot(B_m, C))))) / Float64(-B_m));
	else
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(sqrt(F) * Float64(-sqrt(Float64(B_m + C)))));
	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 <= 2.5e+106)
		tmp = sqrt((F * (2.0 * (C + hypot(B_m, C))))) / -B_m;
	else
		tmp = (sqrt(2.0) / B_m) * (sqrt(F) * -sqrt((B_m + C)));
	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, 2.5e+106], N[(N[Sqrt[N[(F * N[(2.0 * N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[(N[Sqrt[F], $MachinePrecision] * (-N[Sqrt[N[(B$95$m + C), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < 2.4999999999999999e106

   1. Initial program 24.9%

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

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

      3. *-commutative 9.0%

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

      4. +-commutative 9.0%

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

      5. unpow2 9.0%

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

      6. unpow2 9.0%

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

      7. hypot-define 16.8%

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

   5. Simplified 16.8%

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

      1. neg-sub0 16.8%

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

      2. associate-*r/ 16.8%

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

   7. Applied egg-rr 16.9%

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

      1. neg-sub0 16.9%

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

      2. distribute-neg-frac2 16.9%

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

      3. associate-*l* 16.9%

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

      4. hypot-undefine 9.1%

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

      5. unpow2 9.1%

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

      6. unpow2 9.1%

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

      7. +-commutative 9.1%

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

      8. unpow2 9.1%

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

      9. unpow2 9.1%

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

      10. hypot-undefine 16.9%

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

   9. Simplified 16.9%

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

   if 2.4999999999999999e106 < F

   1. Initial program 15.5%

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

   3. Taylor expanded in A around 0 12.6%

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

      1. mul-1-neg 12.6%

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

      2. *-commutative 12.6%

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

      3. *-commutative 12.6%

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

      4. +-commutative 12.6%

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

      5. unpow2 12.6%

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

      6. unpow2 12.6%

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

      7. hypot-define 14.1%

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

   5. Simplified 14.1%

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

      1. sqrt-prod 28.4%

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

   7. Applied egg-rr 28.4%

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

   8. Taylor expanded in C around 0 24.9%

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

4. Final simplification 19.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 2.5 \cdot 10^{+106}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(2 \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)\right)}}{-B}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{F} \cdot \left(-\sqrt{B + C}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 38.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;F \leq 7 \cdot 10^{+135}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(2 \cdot \left(C + \mathsf{hypot}\left(B\_m, C\right)\right)\right)}}{-B\_m}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \frac{\sqrt{F}}{-\sqrt{B\_m}}\\ \end{array} \end{array} \]

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 7e+135)
   (/ (sqrt (* F (* 2.0 (+ C (hypot B_m C))))) (- B_m))
   (* (sqrt 2.0) (/ (sqrt F) (- (sqrt 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 <= 7e+135) {
		tmp = sqrt((F * (2.0 * (C + hypot(B_m, C))))) / -B_m;
	} else {
		tmp = sqrt(2.0) * (sqrt(F) / -sqrt(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 <= 7e+135) {
		tmp = Math.sqrt((F * (2.0 * (C + Math.hypot(B_m, C))))) / -B_m;
	} else {
		tmp = Math.sqrt(2.0) * (Math.sqrt(F) / -Math.sqrt(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 <= 7e+135:
		tmp = math.sqrt((F * (2.0 * (C + math.hypot(B_m, C))))) / -B_m
	else:
		tmp = math.sqrt(2.0) * (math.sqrt(F) / -math.sqrt(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 <= 7e+135)
		tmp = Float64(sqrt(Float64(F * Float64(2.0 * Float64(C + hypot(B_m, C))))) / Float64(-B_m));
	else
		tmp = Float64(sqrt(2.0) * Float64(sqrt(F) / Float64(-sqrt(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 <= 7e+135)
		tmp = sqrt((F * (2.0 * (C + hypot(B_m, C))))) / -B_m;
	else
		tmp = sqrt(2.0) * (sqrt(F) / -sqrt(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, 7e+135], N[(N[Sqrt[N[(F * N[(2.0 * N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sqrt[F], $MachinePrecision] / (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if F < 7.0000000000000005e135

   1. Initial program 23.5%

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

   3. Taylor expanded in A around 0 9.7%

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

      1. mul-1-neg 9.7%

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

      2. *-commutative 9.7%

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

      3. *-commutative 9.7%

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

      4. +-commutative 9.7%

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

      5. unpow2 9.7%

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

      6. unpow2 9.7%

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

      7. hypot-define 17.0%

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

   5. Simplified 17.0%

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

      1. neg-sub0 17.0%

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

      2. associate-*r/ 17.0%

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

   7. Applied egg-rr 17.1%

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

      1. neg-sub0 17.1%

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

      2. distribute-neg-frac2 17.1%

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

      3. associate-*l* 17.1%

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

      4. hypot-undefine 9.7%

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

      5. unpow2 9.7%

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

      6. unpow2 9.7%

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

      7. +-commutative 9.7%

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

      8. unpow2 9.7%

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

      9. unpow2 9.7%

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

      10. hypot-undefine 17.1%

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

   9. Simplified 17.1%

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

   if 7.0000000000000005e135 < F

   1. Initial program 17.9%

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

   3. Taylor expanded in B around inf 23.1%

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

      1. mul-1-neg 23.1%

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

      2. *-commutative 23.1%

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

   5. Simplified 23.1%

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

      1. sqrt-div 25.6%

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

   7. Applied egg-rr 25.6%

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

4. Final simplification 19.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 7 \cdot 10^{+135}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(2 \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)\right)}}{-B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \frac{\sqrt{F}}{-\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 38.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;F \leq 1.25 \cdot 10^{+139}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(2 \cdot \left(C + \mathsf{hypot}\left(B\_m, C\right)\right)\right)}}{-B\_m}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{F \cdot \frac{2}{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= F 1.25e+139)
   (/ (sqrt (* F (* 2.0 (+ C (hypot B_m C))))) (- B_m))
   (- (sqrt (* F (/ 2.0 B_m))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= 1.25e+139) {
		tmp = sqrt((F * (2.0 * (C + hypot(B_m, C))))) / -B_m;
	} else {
		tmp = -sqrt((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 <= 1.25e+139) {
		tmp = Math.sqrt((F * (2.0 * (C + Math.hypot(B_m, C))))) / -B_m;
	} else {
		tmp = -Math.sqrt((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 <= 1.25e+139:
		tmp = math.sqrt((F * (2.0 * (C + math.hypot(B_m, C))))) / -B_m
	else:
		tmp = -math.sqrt((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 <= 1.25e+139)
		tmp = Float64(sqrt(Float64(F * Float64(2.0 * Float64(C + hypot(B_m, C))))) / Float64(-B_m));
	else
		tmp = Float64(-sqrt(Float64(F * Float64(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 <= 1.25e+139)
		tmp = sqrt((F * (2.0 * (C + hypot(B_m, C))))) / -B_m;
	else
		tmp = -sqrt((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, 1.25e+139], N[(N[Sqrt[N[(F * N[(2.0 * N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision], (-N[Sqrt[N[(F * N[(2.0 / B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if F < 1.25000000000000007e139

   1. Initial program 23.4%

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

   3. Taylor expanded in A around 0 9.7%

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

      1. mul-1-neg 9.7%

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

      2. *-commutative 9.7%

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

      3. *-commutative 9.7%

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

      4. +-commutative 9.7%

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

      5. unpow2 9.7%

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

      6. unpow2 9.7%

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

      7. hypot-define 17.0%

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

   5. Simplified 17.0%

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

      1. neg-sub0 17.0%

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

      2. associate-*r/ 17.0%

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

   7. Applied egg-rr 17.0%

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

      1. neg-sub0 17.0%

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

      2. distribute-neg-frac2 17.0%

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

      3. associate-*l* 17.0%

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

      4. hypot-undefine 9.7%

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

      5. unpow2 9.7%

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

      6. unpow2 9.7%

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

      7. +-commutative 9.7%

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

      8. unpow2 9.7%

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

      9. unpow2 9.7%

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

      10. hypot-undefine 17.0%

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

   9. Simplified 17.0%

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

   if 1.25000000000000007e139 < F

   1. Initial program 18.2%

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

   3. Taylor expanded in B around inf 23.4%

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

      1. mul-1-neg 23.4%

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

      2. *-commutative 23.4%

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

   5. Simplified 23.4%

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

      1. add-cbrt-cube 23.2%

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

      2. pow1/3 23.4%

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

      3. rem-square-sqrt 23.4%

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

   7. Applied egg-rr 23.4%

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

      1. unpow1/3 23.2%

         \[\leadsto -\color{blue}{\sqrt[3]{2 \cdot \sqrt{2}}} \cdot \sqrt{\frac{F}{B}} \]

      2. *-commutative 23.2%

         \[\leadsto -\sqrt[3]{\color{blue}{\sqrt{2} \cdot 2}} \cdot \sqrt{\frac{F}{B}} \]

      3. unpow1/2 23.2%

         \[\leadsto -\sqrt[3]{\color{blue}{{2}^{0.5}} \cdot 2} \cdot \sqrt{\frac{F}{B}} \]

      4. pow-plus 23.2%

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

      5. metadata-eval 23.2%

         \[\leadsto -\sqrt[3]{{2}^{\color{blue}{1.5}}} \cdot \sqrt{\frac{F}{B}} \]

   9. Simplified 23.2%

      \[\leadsto -\color{blue}{\sqrt[3]{{2}^{1.5}}} \cdot \sqrt{\frac{F}{B}} \]
   10. Step-by-step derivation

       1. neg-sub0 23.2%

          \[\leadsto \color{blue}{0 - \sqrt[3]{{2}^{1.5}} \cdot \sqrt{\frac{F}{B}}} \]

       2. *-commutative 23.2%

          \[\leadsto 0 - \color{blue}{\sqrt{\frac{F}{B}} \cdot \sqrt[3]{{2}^{1.5}}} \]

       3. pow1/3 23.4%

          \[\leadsto 0 - \sqrt{\frac{F}{B}} \cdot \color{blue}{{\left({2}^{1.5}\right)}^{0.3333333333333333}} \]

       4. pow-pow 23.4%

          \[\leadsto 0 - \sqrt{\frac{F}{B}} \cdot \color{blue}{{2}^{\left(1.5 \cdot 0.3333333333333333\right)}} \]

       5. metadata-eval 23.4%

          \[\leadsto 0 - \sqrt{\frac{F}{B}} \cdot {2}^{\color{blue}{0.5}} \]

       6. pow1/2 23.4%

          \[\leadsto 0 - \sqrt{\frac{F}{B}} \cdot \color{blue}{\sqrt{2}} \]

       7. sqrt-prod 23.5%

          \[\leadsto 0 - \color{blue}{\sqrt{\frac{F}{B} \cdot 2}} \]

       8. *-commutative 23.5%

          \[\leadsto 0 - \sqrt{\color{blue}{2 \cdot \frac{F}{B}}} \]

   11. Applied egg-rr 23.5%

       \[\leadsto \color{blue}{0 - \sqrt{2 \cdot \frac{F}{B}}} \]
   12. Step-by-step derivation

       1. neg-sub0 23.5%

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

       2. associate-*r/ 23.5%

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

       3. *-commutative 23.5%

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

       4. associate-/l* 23.5%

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

   13. Simplified 23.5%

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

4. Final simplification 18.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 1.25 \cdot 10^{+139}:\\ \;\;\;\;\frac{\sqrt{F \cdot \left(2 \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)\right)}}{-B}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{F \cdot \frac{2}{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 36.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;F \leq 1.56 \cdot 10^{+35}:\\ \;\;\;\;\sqrt{F \cdot \left(B\_m + C\right)} \cdot \left(-\frac{\sqrt{2}}{B\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{F \cdot \frac{2}{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= F 1.56e+35)
   (* (sqrt (* F (+ B_m C))) (- (/ (sqrt 2.0) B_m)))
   (- (sqrt (* F (/ 2.0 B_m))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= 1.56e+35) {
		tmp = sqrt((F * (B_m + C))) * -(sqrt(2.0) / B_m);
	} else {
		tmp = -sqrt((F * (2.0 / 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 <= 1.56d+35) then
        tmp = sqrt((f * (b_m + c))) * -(sqrt(2.0d0) / b_m)
    else
        tmp = -sqrt((f * (2.0d0 / 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 <= 1.56e+35) {
		tmp = Math.sqrt((F * (B_m + C))) * -(Math.sqrt(2.0) / B_m);
	} else {
		tmp = -Math.sqrt((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 <= 1.56e+35:
		tmp = math.sqrt((F * (B_m + C))) * -(math.sqrt(2.0) / B_m)
	else:
		tmp = -math.sqrt((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 <= 1.56e+35)
		tmp = Float64(sqrt(Float64(F * Float64(B_m + C))) * Float64(-Float64(sqrt(2.0) / B_m)));
	else
		tmp = Float64(-sqrt(Float64(F * Float64(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 <= 1.56e+35)
		tmp = sqrt((F * (B_m + C))) * -(sqrt(2.0) / B_m);
	else
		tmp = -sqrt((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, 1.56e+35], N[(N[Sqrt[N[(F * N[(B$95$m + C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision])), $MachinePrecision], (-N[Sqrt[N[(F * N[(2.0 / B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if F < 1.56000000000000008e35

   1. Initial program 25.5%

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

   3. Taylor expanded in A around 0 8.7%

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

      1. mul-1-neg 8.7%

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

      2. *-commutative 8.7%

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

      3. *-commutative 8.7%

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

      4. +-commutative 8.7%

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

      5. unpow2 8.7%

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

      6. unpow2 8.7%

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

      7. hypot-define 17.4%

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

   5. Simplified 17.4%

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

   6. Taylor expanded in C around 0 13.4%

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

   if 1.56000000000000008e35 < F

   1. Initial program 16.5%

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

   3. Taylor expanded in B around inf 21.2%

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

      1. mul-1-neg 21.2%

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

      2. *-commutative 21.2%

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

   5. Simplified 21.2%

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

      1. add-cbrt-cube 20.9%

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

      2. pow1/3 21.2%

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

      3. rem-square-sqrt 21.2%

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

   7. Applied egg-rr 21.2%

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

      1. unpow1/3 20.9%

         \[\leadsto -\color{blue}{\sqrt[3]{2 \cdot \sqrt{2}}} \cdot \sqrt{\frac{F}{B}} \]

      2. *-commutative 20.9%

         \[\leadsto -\sqrt[3]{\color{blue}{\sqrt{2} \cdot 2}} \cdot \sqrt{\frac{F}{B}} \]

      3. unpow1/2 20.9%

         \[\leadsto -\sqrt[3]{\color{blue}{{2}^{0.5}} \cdot 2} \cdot \sqrt{\frac{F}{B}} \]

      4. pow-plus 20.9%

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

      5. metadata-eval 20.9%

         \[\leadsto -\sqrt[3]{{2}^{\color{blue}{1.5}}} \cdot \sqrt{\frac{F}{B}} \]

   9. Simplified 20.9%

      \[\leadsto -\color{blue}{\sqrt[3]{{2}^{1.5}}} \cdot \sqrt{\frac{F}{B}} \]
   10. Step-by-step derivation

       1. neg-sub0 20.9%

          \[\leadsto \color{blue}{0 - \sqrt[3]{{2}^{1.5}} \cdot \sqrt{\frac{F}{B}}} \]

       2. *-commutative 20.9%

          \[\leadsto 0 - \color{blue}{\sqrt{\frac{F}{B}} \cdot \sqrt[3]{{2}^{1.5}}} \]

       3. pow1/3 21.2%

          \[\leadsto 0 - \sqrt{\frac{F}{B}} \cdot \color{blue}{{\left({2}^{1.5}\right)}^{0.3333333333333333}} \]

       4. pow-pow 21.2%

          \[\leadsto 0 - \sqrt{\frac{F}{B}} \cdot \color{blue}{{2}^{\left(1.5 \cdot 0.3333333333333333\right)}} \]

       5. metadata-eval 21.2%

          \[\leadsto 0 - \sqrt{\frac{F}{B}} \cdot {2}^{\color{blue}{0.5}} \]

       6. pow1/2 21.2%

          \[\leadsto 0 - \sqrt{\frac{F}{B}} \cdot \color{blue}{\sqrt{2}} \]

       7. sqrt-prod 21.2%

          \[\leadsto 0 - \color{blue}{\sqrt{\frac{F}{B} \cdot 2}} \]

       8. *-commutative 21.2%

          \[\leadsto 0 - \sqrt{\color{blue}{2 \cdot \frac{F}{B}}} \]

   11. Applied egg-rr 21.2%

       \[\leadsto \color{blue}{0 - \sqrt{2 \cdot \frac{F}{B}}} \]
   12. Step-by-step derivation

       1. neg-sub0 21.2%

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

       2. associate-*r/ 21.2%

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

       3. *-commutative 21.2%

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

       4. associate-/l* 21.2%

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

   13. Simplified 21.2%

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

4. Final simplification 16.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 1.56 \cdot 10^{+35}:\\ \;\;\;\;\sqrt{F \cdot \left(B + C\right)} \cdot \left(-\frac{\sqrt{2}}{B}\right)\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{F \cdot \frac{2}{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 35.9% accurate, 3.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;F \leq 6.2 \cdot 10^{-34}:\\ \;\;\;\;\sqrt{B\_m \cdot F} \cdot \left(-\frac{\sqrt{2}}{B\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{F \cdot \frac{2}{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= F 6.2e-34)
   (* (sqrt (* B_m F)) (- (/ (sqrt 2.0) B_m)))
   (- (sqrt (* F (/ 2.0 B_m))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (F <= 6.2e-34) {
		tmp = sqrt((B_m * F)) * -(sqrt(2.0) / B_m);
	} else {
		tmp = -sqrt((F * (2.0 / 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 <= 6.2d-34) then
        tmp = sqrt((b_m * f)) * -(sqrt(2.0d0) / b_m)
    else
        tmp = -sqrt((f * (2.0d0 / 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 <= 6.2e-34) {
		tmp = Math.sqrt((B_m * F)) * -(Math.sqrt(2.0) / B_m);
	} else {
		tmp = -Math.sqrt((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 <= 6.2e-34:
		tmp = math.sqrt((B_m * F)) * -(math.sqrt(2.0) / B_m)
	else:
		tmp = -math.sqrt((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 <= 6.2e-34)
		tmp = Float64(sqrt(Float64(B_m * F)) * Float64(-Float64(sqrt(2.0) / B_m)));
	else
		tmp = Float64(-sqrt(Float64(F * Float64(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 <= 6.2e-34)
		tmp = sqrt((B_m * F)) * -(sqrt(2.0) / B_m);
	else
		tmp = -sqrt((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, 6.2e-34], N[(N[Sqrt[N[(B$95$m * F), $MachinePrecision]], $MachinePrecision] * (-N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision])), $MachinePrecision], (-N[Sqrt[N[(F * N[(2.0 / B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if F < 6.1999999999999996e-34

   1. Initial program 25.1%

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

   3. Taylor expanded in A around 0 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%

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

      3. *-commutative 8.1%

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

      4. +-commutative 8.1%

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

      5. unpow2 8.1%

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

      6. unpow2 8.1%

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

      7. hypot-define 17.1%

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

   5. Simplified 17.1%

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

   6. Taylor expanded in C around 0 14.3%

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

      1. *-commutative 14.3%

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

   8. Simplified 14.3%

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

   if 6.1999999999999996e-34 < F

   1. Initial program 18.5%

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

   3. Taylor expanded in B around inf 19.3%

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

      1. mul-1-neg 19.3%

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

      2. *-commutative 19.3%

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

   5. Simplified 19.3%

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

      1. add-cbrt-cube 19.1%

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

      2. pow1/3 19.3%

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

      3. rem-square-sqrt 19.3%

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

   7. Applied egg-rr 19.3%

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

      1. unpow1/3 19.1%

         \[\leadsto -\color{blue}{\sqrt[3]{2 \cdot \sqrt{2}}} \cdot \sqrt{\frac{F}{B}} \]

      2. *-commutative 19.1%

         \[\leadsto -\sqrt[3]{\color{blue}{\sqrt{2} \cdot 2}} \cdot \sqrt{\frac{F}{B}} \]

      3. unpow1/2 19.1%

         \[\leadsto -\sqrt[3]{\color{blue}{{2}^{0.5}} \cdot 2} \cdot \sqrt{\frac{F}{B}} \]

      4. pow-plus 19.1%

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

      5. metadata-eval 19.1%

         \[\leadsto -\sqrt[3]{{2}^{\color{blue}{1.5}}} \cdot \sqrt{\frac{F}{B}} \]

   9. Simplified 19.1%

      \[\leadsto -\color{blue}{\sqrt[3]{{2}^{1.5}}} \cdot \sqrt{\frac{F}{B}} \]
   10. Step-by-step derivation

       1. neg-sub0 19.1%

          \[\leadsto \color{blue}{0 - \sqrt[3]{{2}^{1.5}} \cdot \sqrt{\frac{F}{B}}} \]

       2. *-commutative 19.1%

          \[\leadsto 0 - \color{blue}{\sqrt{\frac{F}{B}} \cdot \sqrt[3]{{2}^{1.5}}} \]

       3. pow1/3 19.3%

          \[\leadsto 0 - \sqrt{\frac{F}{B}} \cdot \color{blue}{{\left({2}^{1.5}\right)}^{0.3333333333333333}} \]

       4. pow-pow 19.3%

          \[\leadsto 0 - \sqrt{\frac{F}{B}} \cdot \color{blue}{{2}^{\left(1.5 \cdot 0.3333333333333333\right)}} \]

       5. metadata-eval 19.3%

          \[\leadsto 0 - \sqrt{\frac{F}{B}} \cdot {2}^{\color{blue}{0.5}} \]

       6. pow1/2 19.3%

          \[\leadsto 0 - \sqrt{\frac{F}{B}} \cdot \color{blue}{\sqrt{2}} \]

       7. sqrt-prod 19.3%

          \[\leadsto 0 - \color{blue}{\sqrt{\frac{F}{B} \cdot 2}} \]

       8. *-commutative 19.3%

          \[\leadsto 0 - \sqrt{\color{blue}{2 \cdot \frac{F}{B}}} \]

   11. Applied egg-rr 19.3%

       \[\leadsto \color{blue}{0 - \sqrt{2 \cdot \frac{F}{B}}} \]
   12. Step-by-step derivation

       1. neg-sub0 19.3%

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

       2. associate-*r/ 19.3%

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

       3. *-commutative 19.3%

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

       4. associate-/l* 19.3%

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

   13. Simplified 19.3%

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

4. Final simplification 16.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 6.2 \cdot 10^{-34}:\\ \;\;\;\;\sqrt{B \cdot F} \cdot \left(-\frac{\sqrt{2}}{B}\right)\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{F \cdot \frac{2}{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 28.3% accurate, 6.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 22.2%

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

3. Taylor expanded in B around inf 14.5%

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

   1. mul-1-neg 14.5%

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

   2. *-commutative 14.5%

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

5. Simplified 14.5%

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

   1. neg-sub0 14.5%

      \[\leadsto \color{blue}{0 - \sqrt{2} \cdot \sqrt{\frac{F}{B}}} \]

   2. sqrt-unprod 14.6%

      \[\leadsto 0 - \color{blue}{\sqrt{2 \cdot \frac{F}{B}}} \]

7. Applied egg-rr 14.6%

   \[\leadsto \color{blue}{0 - \sqrt{2 \cdot \frac{F}{B}}} \]
8. Step-by-step derivation

   1. neg-sub0 14.6%

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

   2. *-commutative 14.6%

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

9. Simplified 14.6%

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

10. Final simplification 14.6%

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

Alternative 11: 28.3% accurate, 6.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 22.2%

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

3. Taylor expanded in B around inf 14.5%

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

   1. mul-1-neg 14.5%

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

   2. *-commutative 14.5%

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

5. Simplified 14.5%

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

   1. add-cbrt-cube 14.4%

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

   2. pow1/3 14.5%

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

   3. rem-square-sqrt 14.5%

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

7. Applied egg-rr 14.5%

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

   1. unpow1/3 14.4%

      \[\leadsto -\color{blue}{\sqrt[3]{2 \cdot \sqrt{2}}} \cdot \sqrt{\frac{F}{B}} \]

   2. *-commutative 14.4%

      \[\leadsto -\sqrt[3]{\color{blue}{\sqrt{2} \cdot 2}} \cdot \sqrt{\frac{F}{B}} \]

   3. unpow1/2 14.4%

      \[\leadsto -\sqrt[3]{\color{blue}{{2}^{0.5}} \cdot 2} \cdot \sqrt{\frac{F}{B}} \]

   4. pow-plus 14.4%

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

   5. metadata-eval 14.4%

      \[\leadsto -\sqrt[3]{{2}^{\color{blue}{1.5}}} \cdot \sqrt{\frac{F}{B}} \]

9. Simplified 14.4%

   \[\leadsto -\color{blue}{\sqrt[3]{{2}^{1.5}}} \cdot \sqrt{\frac{F}{B}} \]
10. Step-by-step derivation

    1. neg-sub0 14.4%

       \[\leadsto \color{blue}{0 - \sqrt[3]{{2}^{1.5}} \cdot \sqrt{\frac{F}{B}}} \]

    2. *-commutative 14.4%

       \[\leadsto 0 - \color{blue}{\sqrt{\frac{F}{B}} \cdot \sqrt[3]{{2}^{1.5}}} \]

    3. pow1/3 14.5%

       \[\leadsto 0 - \sqrt{\frac{F}{B}} \cdot \color{blue}{{\left({2}^{1.5}\right)}^{0.3333333333333333}} \]

    4. pow-pow 14.5%

       \[\leadsto 0 - \sqrt{\frac{F}{B}} \cdot \color{blue}{{2}^{\left(1.5 \cdot 0.3333333333333333\right)}} \]

    5. metadata-eval 14.5%

       \[\leadsto 0 - \sqrt{\frac{F}{B}} \cdot {2}^{\color{blue}{0.5}} \]

    6. pow1/2 14.5%

       \[\leadsto 0 - \sqrt{\frac{F}{B}} \cdot \color{blue}{\sqrt{2}} \]

    7. sqrt-prod 14.6%

       \[\leadsto 0 - \color{blue}{\sqrt{\frac{F}{B} \cdot 2}} \]

    8. *-commutative 14.6%

       \[\leadsto 0 - \sqrt{\color{blue}{2 \cdot \frac{F}{B}}} \]

11. Applied egg-rr 14.6%

    \[\leadsto \color{blue}{0 - \sqrt{2 \cdot \frac{F}{B}}} \]
12. Step-by-step derivation

    1. neg-sub0 14.6%

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

    2. associate-*r/ 14.6%

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

    3. *-commutative 14.6%

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

    4. associate-/l* 14.6%

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

13. Simplified 14.6%

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

Alternative 12: 2.4% accurate, 6.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 22.2%

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

3. Taylor expanded in B around inf 14.5%

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

   1. mul-1-neg 14.5%

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

   2. *-commutative 14.5%

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

5. Simplified 14.5%

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

   1. *-un-lft-identity 14.5%

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

   2. add-sqr-sqrt 0.8%

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

   3. sqrt-unprod 2.0%

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

   4. sqr-neg 2.0%

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

   5. sqrt-unprod 2.0%

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

   6. sqrt-unprod 2.0%

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

   7. add-sqr-sqrt 2.0%

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

7. Applied egg-rr 2.0%

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

   1. *-lft-identity 2.0%

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

   2. *-commutative 2.0%

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

9. Simplified 2.0%

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

10. Taylor expanded in F around 0 2.0%

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

    1. associate-*r/ 2.0%

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

    2. *-commutative 2.0%

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

    3. associate-/l* 2.0%

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

12. Simplified 2.0%

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

Reproduce

herbie shell --seed 2024152 
(FPCore (A B C F)
  :name "ABCF->ab-angle a"
  :precision binary64
  (/ (- (sqrt (* (* 2.0 (* (- (pow B 2.0) (* (* 4.0 A) C)) F)) (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0))))))) (- (pow B 2.0) (* (* 4.0 A) C))))