ABCF->ab-angle b

Percentage Accurate: 18.8% → 53.2%

Time: 24.4s

Alternatives: 12

Speedup: 5.6×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 18.8% 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: 53.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := A + \left(C - \mathsf{hypot}\left(B\_m, A - C\right)\right)\\ t_2 := -t\_0\\ t_3 := \left(4 \cdot A\right) \cdot C\\ t_4 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_3\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_3 - {B\_m}^{2}}\\ t_5 := F \cdot t\_0\\ \mathbf{if}\;t\_4 \leq -\infty:\\ \;\;\;\;-\sqrt{2 \cdot \left(F \cdot \frac{t\_1}{\mathsf{fma}\left(-4, A \cdot C, {B\_m}^{2}\right)}\right)}\\ \mathbf{elif}\;t\_4 \leq -2 \cdot 10^{-134}:\\ \;\;\;\;\frac{\sqrt{t\_5 \cdot \left(2 \cdot t\_1\right)}}{t\_2}\\ \mathbf{elif}\;t\_4 \leq 10^{+180}:\\ \;\;\;\;\frac{\sqrt{t\_5 \cdot \left(2 \cdot \left(A + A\right) - \frac{{B\_m}^{2}}{C}\right)}}{t\_2}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;\frac{-1}{\frac{t\_0}{\sqrt{A \cdot -8} \cdot \sqrt{\left(C \cdot F\right) \cdot \left(A + A\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot F\right) \cdot \left(A - \mathsf{hypot}\left(B\_m, A\right)\right)}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* A (* C -4.0))))
        (t_1 (+ A (- C (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 (* F (/ t_1 (fma -4.0 (* A C) (pow B_m 2.0)))))))
     (if (<= t_4 -2e-134)
       (/ (sqrt (* t_5 (* 2.0 t_1))) t_2)
       (if (<= t_4 1e+180)
         (/ (sqrt (* t_5 (- (* 2.0 (+ A A)) (/ (pow B_m 2.0) C)))) t_2)
         (if (<= t_4 INFINITY)
           (/ -1.0 (/ t_0 (* (sqrt (* A -8.0)) (sqrt (* (* C F) (+ A A))))))
           (/ (sqrt (* (* 2.0 F) (- A (hypot B_m A)))) (- B_m))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (A * (C * -4.0)));
	double t_1 = A + (C - 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 * (F * (t_1 / fma(-4.0, (A * C), pow(B_m, 2.0))))));
	} else if (t_4 <= -2e-134) {
		tmp = sqrt((t_5 * (2.0 * t_1))) / t_2;
	} else if (t_4 <= 1e+180) {
		tmp = sqrt((t_5 * ((2.0 * (A + A)) - (pow(B_m, 2.0) / C)))) / t_2;
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = -1.0 / (t_0 / (sqrt((A * -8.0)) * sqrt(((C * F) * (A + A)))));
	} else {
		tmp = sqrt(((2.0 * F) * (A - hypot(B_m, A)))) / -B_m;
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	t_1 = Float64(A + Float64(C - 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(Float64(2.0 * Float64(F * Float64(t_1 / fma(-4.0, Float64(A * C), (B_m ^ 2.0)))))));
	elseif (t_4 <= -2e-134)
		tmp = Float64(sqrt(Float64(t_5 * Float64(2.0 * t_1))) / t_2);
	elseif (t_4 <= 1e+180)
		tmp = Float64(sqrt(Float64(t_5 * Float64(Float64(2.0 * Float64(A + A)) - Float64((B_m ^ 2.0) / C)))) / t_2);
	elseif (t_4 <= Inf)
		tmp = Float64(-1.0 / Float64(t_0 / Float64(sqrt(Float64(A * -8.0)) * sqrt(Float64(Float64(C * F) * Float64(A + A))))));
	else
		tmp = Float64(sqrt(Float64(Float64(2.0 * F) * Float64(A - hypot(B_m, A)))) / Float64(-B_m));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(A + N[(C - N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $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[Sqrt[N[(2.0 * N[(F * N[(t$95$1 / 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-134], N[(N[Sqrt[N[(t$95$5 * N[(2.0 * t$95$1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[t$95$4, 1e+180], N[(N[Sqrt[N[(t$95$5 * N[(N[(2.0 * N[(A + A), $MachinePrecision]), $MachinePrecision] - N[(N[Power[B$95$m, 2.0], $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[(-1.0 / N[(t$95$0 / N[(N[Sqrt[N[(A * -8.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(N[(C * F), $MachinePrecision] * N[(A + A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] * N[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $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 3.3%

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

   3. Taylor expanded in F around 0 21.7%

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

   5. Simplified 74.7%

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

      1. sqrt-unprod 74.9%

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

      2. associate--r- 73.7%

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

      3. *-commutative 73.7%

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

   7. Applied egg-rr 73.7%

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

   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))) < -2.00000000000000008e-134

   1. Initial program 98.7%

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

   3. Simplified 98.7%

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

   if -2.00000000000000008e-134 < (/.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))) < 1e180

   1. Initial program 19.8%

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

   3. Simplified 23.6%

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

   5. Taylor expanded in C around inf 38.5%

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

   if 1e180 < (/.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 12.7%

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

   3. Simplified 43.1%

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

      1. clear-num 43.3%

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

      2. inv-pow 43.3%

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

   6. Applied egg-rr 43.3%

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

      1. unpow-1 43.3%

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

      2. associate-*l* 25.8%

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

      3. associate-+r- 25.8%

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

      4. +-commutative 25.8%

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

      5. associate-+l- 25.8%

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

   8. Simplified 25.8%

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

   9. Taylor expanded in C around inf 17.6%

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

       1. mul-1-neg 17.6%

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

   11. Simplified 17.6%

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

       1. pow1/2 17.6%

          \[\leadsto \frac{1}{\frac{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}{\color{blue}{{\left(-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A - \left(-A\right)\right)\right)\right)\right)\right)}^{0.5}}}} \]

       2. associate-*r* 17.6%

          \[\leadsto \frac{1}{\frac{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}{{\color{blue}{\left(\left(-8 \cdot A\right) \cdot \left(C \cdot \left(F \cdot \left(A - \left(-A\right)\right)\right)\right)\right)}}^{0.5}}} \]

       3. unpow-prod-down 17.9%

          \[\leadsto \frac{1}{\frac{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}{\color{blue}{{\left(-8 \cdot A\right)}^{0.5} \cdot {\left(C \cdot \left(F \cdot \left(A - \left(-A\right)\right)\right)\right)}^{0.5}}}} \]

       4. pow1/2 17.9%

          \[\leadsto \frac{1}{\frac{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}{{\left(-8 \cdot A\right)}^{0.5} \cdot \color{blue}{\sqrt{C \cdot \left(F \cdot \left(A - \left(-A\right)\right)\right)}}}} \]

   13. Applied egg-rr 17.9%

       \[\leadsto \frac{1}{\frac{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}{\color{blue}{{\left(-8 \cdot A\right)}^{0.5} \cdot \sqrt{C \cdot \left(F \cdot \left(A - \left(-A\right)\right)\right)}}}} \]
   14. Step-by-step derivation

       1. unpow1/2 17.9%

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

       2. *-commutative 17.9%

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

       3. associate-*r* 35.2%

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

       4. *-commutative 35.2%

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

   15. Simplified 35.2%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in C around 0 1.9%

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

      1. mul-1-neg 1.9%

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

      2. +-commutative 1.9%

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

      3. unpow2 1.9%

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

      4. unpow2 1.9%

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

      5. hypot-define 17.8%

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

   5. Simplified 17.8%

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

      1. neg-sub0 17.8%

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

      2. associate-*l/ 17.8%

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

      3. pow1/2 17.8%

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

      4. pow1/2 17.8%

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

      5. pow-prod-down 17.9%

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

   7. Applied egg-rr 17.9%

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

      1. neg-sub0 17.9%

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

      2. distribute-neg-frac2 17.9%

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

      3. unpow1/2 17.9%

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

      4. associate-*r* 17.9%

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

      5. *-commutative 17.9%

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

   9. Simplified 17.9%

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

4. Final simplification 42.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -\infty:\\ \;\;\;\;-\sqrt{2 \cdot \left(F \cdot \frac{A + \left(C - \mathsf{hypot}\left(B, A - C\right)\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^{-134}:\\ \;\;\;\;\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 10^{+180}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot \left(2 \cdot \left(A + A\right) - \frac{{B}^{2}}{C}\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{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{-1}{\frac{\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}{\sqrt{A \cdot -8} \cdot \sqrt{\left(C \cdot F\right) \cdot \left(A + A\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot F\right) \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 49.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 3.9 \cdot 10^{-259}:\\ \;\;\;\;\frac{\sqrt{\left(4 \cdot A\right) \cdot \left(F \cdot \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\right)}}{4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B\_m \leq 3.7 \cdot 10^{-72}:\\ \;\;\;\;\frac{\sqrt{-F}}{-\sqrt{C}}\\ \mathbf{elif}\;B\_m \leq 6.4 \cdot 10^{+153}:\\ \;\;\;\;-\sqrt{2 \cdot \left(F \cdot \frac{A + \left(C - \mathsf{hypot}\left(B\_m, A - C\right)\right)}{\mathsf{fma}\left(-4, A \cdot C, {B\_m}^{2}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot F\right) \cdot \left(A - \mathsf{hypot}\left(B\_m, A\right)\right)}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 3.9e-259)
   (/
    (sqrt (* (* 4.0 A) (* F (fma B_m B_m (* A (* C -4.0))))))
    (* 4.0 (* A C)))
   (if (<= B_m 3.7e-72)
     (/ (sqrt (- F)) (- (sqrt C)))
     (if (<= B_m 6.4e+153)
       (-
        (sqrt
         (*
          2.0
          (*
           F
           (/
            (+ A (- C (hypot B_m (- A C))))
            (fma -4.0 (* A C) (pow B_m 2.0)))))))
       (/ (sqrt (* (* 2.0 F) (- A (hypot B_m A)))) (- B_m))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 3.9e-259) {
		tmp = sqrt(((4.0 * A) * (F * fma(B_m, B_m, (A * (C * -4.0)))))) / (4.0 * (A * C));
	} else if (B_m <= 3.7e-72) {
		tmp = sqrt(-F) / -sqrt(C);
	} else if (B_m <= 6.4e+153) {
		tmp = -sqrt((2.0 * (F * ((A + (C - hypot(B_m, (A - C)))) / fma(-4.0, (A * C), pow(B_m, 2.0))))));
	} else {
		tmp = sqrt(((2.0 * F) * (A - hypot(B_m, A)))) / -B_m;
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 3.9e-259)
		tmp = Float64(sqrt(Float64(Float64(4.0 * A) * Float64(F * fma(B_m, B_m, Float64(A * Float64(C * -4.0)))))) / Float64(4.0 * Float64(A * C)));
	elseif (B_m <= 3.7e-72)
		tmp = Float64(sqrt(Float64(-F)) / Float64(-sqrt(C)));
	elseif (B_m <= 6.4e+153)
		tmp = Float64(-sqrt(Float64(2.0 * Float64(F * Float64(Float64(A + Float64(C - hypot(B_m, Float64(A - C)))) / fma(-4.0, Float64(A * C), (B_m ^ 2.0)))))));
	else
		tmp = Float64(sqrt(Float64(Float64(2.0 * F) * Float64(A - hypot(B_m, A)))) / Float64(-B_m));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 3.9e-259], N[(N[Sqrt[N[(N[(4.0 * A), $MachinePrecision] * N[(F * N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 3.7e-72], N[(N[Sqrt[(-F)], $MachinePrecision] / (-N[Sqrt[C], $MachinePrecision])), $MachinePrecision], If[LessEqual[B$95$m, 6.4e+153], (-N[Sqrt[N[(2.0 * N[(F * N[(N[(A + N[(C - N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-4.0 * N[(A * C), $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), N[(N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] * N[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if B < 3.90000000000000016e-259

   1. Initial program 17.9%

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

   3. Simplified 25.5%

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

   5. Taylor expanded in A around -inf 15.2%

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

   6. Taylor expanded in B around 0 13.4%

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

   if 3.90000000000000016e-259 < B < 3.6999999999999998e-72

   1. Initial program 19.4%

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

   3. Taylor expanded in F around 0 11.2%

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

   5. Simplified 23.9%

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

   6. Taylor expanded in A around -inf 23.2%

      \[\leadsto -\sqrt{\color{blue}{-0.5 \cdot \frac{F}{C}}} \cdot \sqrt{2} \]
   7. Step-by-step derivation

      1. associate-*r/ 23.2%

         \[\leadsto -\sqrt{\color{blue}{\frac{-0.5 \cdot F}{C}}} \cdot \sqrt{2} \]

      2. sqrt-div 29.4%

         \[\leadsto -\color{blue}{\frac{\sqrt{-0.5 \cdot F}}{\sqrt{C}}} \cdot \sqrt{2} \]

   8. Applied egg-rr 29.4%

      \[\leadsto -\color{blue}{\frac{\sqrt{-0.5 \cdot F}}{\sqrt{C}}} \cdot \sqrt{2} \]
   9. Step-by-step derivation

      1. *-commutative 29.4%

         \[\leadsto -\frac{\sqrt{\color{blue}{F \cdot -0.5}}}{\sqrt{C}} \cdot \sqrt{2} \]

   10. Simplified 29.4%

       \[\leadsto -\color{blue}{\frac{\sqrt{F \cdot -0.5}}{\sqrt{C}}} \cdot \sqrt{2} \]
   11. Step-by-step derivation

       1. associate-*l/ 29.4%

          \[\leadsto -\color{blue}{\frac{\sqrt{F \cdot -0.5} \cdot \sqrt{2}}{\sqrt{C}}} \]

       2. pow1/2 29.4%

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

       3. pow1/2 29.4%

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

       4. pow-prod-down 29.5%

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

   12. Applied egg-rr 29.5%

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

       1. unpow1/2 29.5%

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

       2. associate-*l* 29.5%

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

       3. metadata-eval 29.5%

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

       4. *-commutative 29.5%

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

       5. mul-1-neg 29.5%

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

   14. Simplified 29.5%

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

   if 3.6999999999999998e-72 < B < 6.4000000000000003e153

   1. Initial program 25.4%

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

   3. Taylor expanded in F around 0 30.8%

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

   5. Simplified 51.1%

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

      1. sqrt-unprod 51.3%

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

      2. associate--r- 50.3%

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

      3. *-commutative 50.3%

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

   7. Applied egg-rr 50.3%

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

   if 6.4000000000000003e153 < B

   1. Initial program 0.0%

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

   3. Taylor expanded in C around 0 2.5%

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

      1. mul-1-neg 2.5%

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

      2. +-commutative 2.5%

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

      3. unpow2 2.5%

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

      4. unpow2 2.5%

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

      5. hypot-define 47.6%

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

   5. Simplified 47.6%

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

      1. neg-sub0 47.6%

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

      2. associate-*l/ 47.5%

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

      3. pow1/2 47.5%

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

      4. pow1/2 47.5%

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

      5. pow-prod-down 47.7%

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

   7. Applied egg-rr 47.7%

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

      1. neg-sub0 47.7%

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

      2. distribute-neg-frac2 47.7%

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

      3. unpow1/2 47.7%

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

      4. associate-*r* 47.7%

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

      5. *-commutative 47.7%

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

   9. Simplified 47.7%

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

4. Final simplification 27.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 3.9 \cdot 10^{-259}:\\ \;\;\;\;\frac{\sqrt{\left(4 \cdot A\right) \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)}}{4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 3.7 \cdot 10^{-72}:\\ \;\;\;\;\frac{\sqrt{-F}}{-\sqrt{C}}\\ \mathbf{elif}\;B \leq 6.4 \cdot 10^{+153}:\\ \;\;\;\;-\sqrt{2 \cdot \left(F \cdot \frac{A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)}{\mathsf{fma}\left(-4, A \cdot C, {B}^{2}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot F\right) \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 47.6% accurate, 2.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 5.5 \cdot 10^{-259}:\\ \;\;\;\;\frac{\sqrt{\left(4 \cdot A\right) \cdot \left(F \cdot \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\right)}}{4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B\_m \leq 1.9 \cdot 10^{+23}:\\ \;\;\;\;\frac{\sqrt{-F}}{-\sqrt{C}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot F\right) \cdot \left(A - \mathsf{hypot}\left(B\_m, A\right)\right)}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 5.5e-259)
   (/
    (sqrt (* (* 4.0 A) (* F (fma B_m B_m (* A (* C -4.0))))))
    (* 4.0 (* A C)))
   (if (<= B_m 1.9e+23)
     (/ (sqrt (- F)) (- (sqrt C)))
     (/ (sqrt (* (* 2.0 F) (- A (hypot B_m A)))) (- B_m)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 5.5e-259) {
		tmp = sqrt(((4.0 * A) * (F * fma(B_m, B_m, (A * (C * -4.0)))))) / (4.0 * (A * C));
	} else if (B_m <= 1.9e+23) {
		tmp = sqrt(-F) / -sqrt(C);
	} else {
		tmp = sqrt(((2.0 * F) * (A - hypot(B_m, A)))) / -B_m;
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 5.5e-259)
		tmp = Float64(sqrt(Float64(Float64(4.0 * A) * Float64(F * fma(B_m, B_m, Float64(A * Float64(C * -4.0)))))) / Float64(4.0 * Float64(A * C)));
	elseif (B_m <= 1.9e+23)
		tmp = Float64(sqrt(Float64(-F)) / Float64(-sqrt(C)));
	else
		tmp = Float64(sqrt(Float64(Float64(2.0 * F) * Float64(A - hypot(B_m, A)))) / Float64(-B_m));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 5.5e-259], N[(N[Sqrt[N[(N[(4.0 * A), $MachinePrecision] * N[(F * N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 1.9e+23], N[(N[Sqrt[(-F)], $MachinePrecision] / (-N[Sqrt[C], $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] * N[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if B < 5.50000000000000038e-259

   1. Initial program 17.9%

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

   3. Simplified 25.5%

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

   5. Taylor expanded in A around -inf 15.2%

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

   6. Taylor expanded in B around 0 13.4%

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

   if 5.50000000000000038e-259 < B < 1.89999999999999987e23

   1. Initial program 23.9%

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

   3. Taylor expanded in F around 0 18.5%

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

   5. Simplified 31.3%

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

   6. Taylor expanded in A around -inf 18.7%

      \[\leadsto -\sqrt{\color{blue}{-0.5 \cdot \frac{F}{C}}} \cdot \sqrt{2} \]
   7. Step-by-step derivation

      1. associate-*r/ 18.7%

         \[\leadsto -\sqrt{\color{blue}{\frac{-0.5 \cdot F}{C}}} \cdot \sqrt{2} \]

      2. sqrt-div 23.7%

         \[\leadsto -\color{blue}{\frac{\sqrt{-0.5 \cdot F}}{\sqrt{C}}} \cdot \sqrt{2} \]

   8. Applied egg-rr 23.7%

      \[\leadsto -\color{blue}{\frac{\sqrt{-0.5 \cdot F}}{\sqrt{C}}} \cdot \sqrt{2} \]
   9. Step-by-step derivation

      1. *-commutative 23.7%

         \[\leadsto -\frac{\sqrt{\color{blue}{F \cdot -0.5}}}{\sqrt{C}} \cdot \sqrt{2} \]

   10. Simplified 23.7%

       \[\leadsto -\color{blue}{\frac{\sqrt{F \cdot -0.5}}{\sqrt{C}}} \cdot \sqrt{2} \]
   11. Step-by-step derivation

       1. associate-*l/ 23.7%

          \[\leadsto -\color{blue}{\frac{\sqrt{F \cdot -0.5} \cdot \sqrt{2}}{\sqrt{C}}} \]

       2. pow1/2 23.7%

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

       3. pow1/2 23.7%

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

       4. pow-prod-down 23.8%

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

   12. Applied egg-rr 23.8%

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

       1. unpow1/2 23.8%

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

       2. associate-*l* 23.8%

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

       3. metadata-eval 23.8%

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

       4. *-commutative 23.8%

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

       5. mul-1-neg 23.8%

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

   14. Simplified 23.8%

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

   if 1.89999999999999987e23 < B

   1. Initial program 6.3%

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

   3. Taylor expanded in C around 0 11.9%

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

      1. mul-1-neg 11.9%

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

      2. +-commutative 11.9%

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

      3. unpow2 11.9%

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

      4. unpow2 11.9%

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

      5. hypot-define 43.2%

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

   5. Simplified 43.2%

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

      1. neg-sub0 43.2%

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

      2. associate-*l/ 43.1%

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

      3. pow1/2 43.1%

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

      4. pow1/2 43.1%

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

      5. pow-prod-down 43.2%

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

   7. Applied egg-rr 43.2%

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

      1. neg-sub0 43.2%

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

      2. distribute-neg-frac2 43.2%

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

      3. unpow1/2 43.2%

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

      4. associate-*r* 43.2%

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

      5. *-commutative 43.2%

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

   9. Simplified 43.2%

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

4. Final simplification 22.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 5.5 \cdot 10^{-259}:\\ \;\;\;\;\frac{\sqrt{\left(4 \cdot A\right) \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)}}{4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 1.9 \cdot 10^{+23}:\\ \;\;\;\;\frac{\sqrt{-F}}{-\sqrt{C}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot F\right) \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 47.4% accurate, 2.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 3.4 \cdot 10^{-259}:\\ \;\;\;\;\frac{1}{\frac{4 \cdot \left(A \cdot C\right)}{\sqrt{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A + A\right)\right)\right)\right)}}}\\ \mathbf{elif}\;B\_m \leq 1.65 \cdot 10^{+22}:\\ \;\;\;\;\frac{\sqrt{-F}}{-\sqrt{C}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot F\right) \cdot \left(A - \mathsf{hypot}\left(B\_m, A\right)\right)}}{-B\_m}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 3.4e-259)
   (/ 1.0 (/ (* 4.0 (* A C)) (sqrt (* -8.0 (* A (* C (* F (+ A A))))))))
   (if (<= B_m 1.65e+22)
     (/ (sqrt (- F)) (- (sqrt C)))
     (/ (sqrt (* (* 2.0 F) (- A (hypot B_m A)))) (- B_m)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 3.4e-259) {
		tmp = 1.0 / ((4.0 * (A * C)) / sqrt((-8.0 * (A * (C * (F * (A + A)))))));
	} else if (B_m <= 1.65e+22) {
		tmp = sqrt(-F) / -sqrt(C);
	} else {
		tmp = sqrt(((2.0 * F) * (A - hypot(B_m, A)))) / -B_m;
	}
	return tmp;
}

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 3.4e-259) {
		tmp = 1.0 / ((4.0 * (A * C)) / Math.sqrt((-8.0 * (A * (C * (F * (A + A)))))));
	} else if (B_m <= 1.65e+22) {
		tmp = Math.sqrt(-F) / -Math.sqrt(C);
	} else {
		tmp = Math.sqrt(((2.0 * F) * (A - Math.hypot(B_m, A)))) / -B_m;
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 3.4e-259:
		tmp = 1.0 / ((4.0 * (A * C)) / math.sqrt((-8.0 * (A * (C * (F * (A + A)))))))
	elif B_m <= 1.65e+22:
		tmp = math.sqrt(-F) / -math.sqrt(C)
	else:
		tmp = math.sqrt(((2.0 * F) * (A - math.hypot(B_m, A)))) / -B_m
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 3.4e-259)
		tmp = Float64(1.0 / Float64(Float64(4.0 * Float64(A * C)) / sqrt(Float64(-8.0 * Float64(A * Float64(C * Float64(F * Float64(A + A))))))));
	elseif (B_m <= 1.65e+22)
		tmp = Float64(sqrt(Float64(-F)) / Float64(-sqrt(C)));
	else
		tmp = Float64(sqrt(Float64(Float64(2.0 * F) * Float64(A - hypot(B_m, A)))) / Float64(-B_m));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (B_m <= 3.4e-259)
		tmp = 1.0 / ((4.0 * (A * C)) / sqrt((-8.0 * (A * (C * (F * (A + A)))))));
	elseif (B_m <= 1.65e+22)
		tmp = sqrt(-F) / -sqrt(C);
	else
		tmp = sqrt(((2.0 * F) * (A - hypot(B_m, A)))) / -B_m;
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 3.4e-259], N[(1.0 / N[(N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(-8.0 * N[(A * N[(C * N[(F * N[(A + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 1.65e+22], N[(N[Sqrt[(-F)], $MachinePrecision] / (-N[Sqrt[C], $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] * N[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if B < 3.40000000000000012e-259

   1. Initial program 17.9%

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

   3. Simplified 25.5%

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

      1. clear-num 25.5%

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

      2. inv-pow 25.5%

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

   6. Applied egg-rr 24.5%

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

      1. unpow-1 24.5%

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

      2. associate-*l* 21.7%

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

      3. associate-+r- 22.5%

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

      4. +-commutative 22.5%

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

      5. associate-+l- 22.2%

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

   8. Simplified 22.2%

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

   9. Taylor expanded in C around inf 13.6%

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

       1. mul-1-neg 13.6%

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

   11. Simplified 13.6%

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

   12. Taylor expanded in B around 0 13.9%

       \[\leadsto \frac{1}{\frac{\color{blue}{4 \cdot \left(A \cdot C\right)}}{\sqrt{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A - \left(-A\right)\right)\right)\right)\right)}}} \]

   if 3.40000000000000012e-259 < B < 1.6499999999999999e22

   1. Initial program 23.9%

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

   3. Taylor expanded in F around 0 18.5%

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

   5. Simplified 31.3%

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

   6. Taylor expanded in A around -inf 18.7%

      \[\leadsto -\sqrt{\color{blue}{-0.5 \cdot \frac{F}{C}}} \cdot \sqrt{2} \]
   7. Step-by-step derivation

      1. associate-*r/ 18.7%

         \[\leadsto -\sqrt{\color{blue}{\frac{-0.5 \cdot F}{C}}} \cdot \sqrt{2} \]

      2. sqrt-div 23.7%

         \[\leadsto -\color{blue}{\frac{\sqrt{-0.5 \cdot F}}{\sqrt{C}}} \cdot \sqrt{2} \]

   8. Applied egg-rr 23.7%

      \[\leadsto -\color{blue}{\frac{\sqrt{-0.5 \cdot F}}{\sqrt{C}}} \cdot \sqrt{2} \]
   9. Step-by-step derivation

      1. *-commutative 23.7%

         \[\leadsto -\frac{\sqrt{\color{blue}{F \cdot -0.5}}}{\sqrt{C}} \cdot \sqrt{2} \]

   10. Simplified 23.7%

       \[\leadsto -\color{blue}{\frac{\sqrt{F \cdot -0.5}}{\sqrt{C}}} \cdot \sqrt{2} \]
   11. Step-by-step derivation

       1. associate-*l/ 23.7%

          \[\leadsto -\color{blue}{\frac{\sqrt{F \cdot -0.5} \cdot \sqrt{2}}{\sqrt{C}}} \]

       2. pow1/2 23.7%

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

       3. pow1/2 23.7%

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

       4. pow-prod-down 23.8%

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

   12. Applied egg-rr 23.8%

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

       1. unpow1/2 23.8%

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

       2. associate-*l* 23.8%

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

       3. metadata-eval 23.8%

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

       4. *-commutative 23.8%

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

       5. mul-1-neg 23.8%

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

   14. Simplified 23.8%

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

   if 1.6499999999999999e22 < B

   1. Initial program 6.3%

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

   3. Taylor expanded in C around 0 11.9%

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

      1. mul-1-neg 11.9%

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

      2. +-commutative 11.9%

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

      3. unpow2 11.9%

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

      4. unpow2 11.9%

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

      5. hypot-define 43.2%

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

   5. Simplified 43.2%

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

      1. neg-sub0 43.2%

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

      2. associate-*l/ 43.1%

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

      3. pow1/2 43.1%

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

      4. pow1/2 43.1%

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

      5. pow-prod-down 43.2%

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

   7. Applied egg-rr 43.2%

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

      1. neg-sub0 43.2%

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

      2. distribute-neg-frac2 43.2%

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

      3. unpow1/2 43.2%

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

      4. associate-*r* 43.2%

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

      5. *-commutative 43.2%

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

   9. Simplified 43.2%

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

4. Final simplification 22.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 3.4 \cdot 10^{-259}:\\ \;\;\;\;\frac{1}{\frac{4 \cdot \left(A \cdot C\right)}{\sqrt{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A + A\right)\right)\right)\right)}}}\\ \mathbf{elif}\;B \leq 1.65 \cdot 10^{+22}:\\ \;\;\;\;\frac{\sqrt{-F}}{-\sqrt{C}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot F\right) \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 44.7% accurate, 2.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 2.5 \cdot 10^{-258}:\\ \;\;\;\;\frac{1}{\frac{4 \cdot \left(A \cdot C\right)}{\sqrt{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A + A\right)\right)\right)\right)}}}\\ \mathbf{elif}\;B\_m \leq 2.5 \cdot 10^{+53}:\\ \;\;\;\;\frac{\sqrt{-F}}{-\sqrt{C}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\frac{F}{B\_m}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 2.5e-258)
   (/ 1.0 (/ (* 4.0 (* A C)) (sqrt (* -8.0 (* A (* C (* F (+ A A))))))))
   (if (<= B_m 2.5e+53)
     (/ (sqrt (- F)) (- (sqrt C)))
     (* (sqrt (- (/ F B_m))) (- (sqrt 2.0))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 2.5e-258) {
		tmp = 1.0 / ((4.0 * (A * C)) / sqrt((-8.0 * (A * (C * (F * (A + A)))))));
	} else if (B_m <= 2.5e+53) {
		tmp = sqrt(-F) / -sqrt(C);
	} else {
		tmp = sqrt(-(F / B_m)) * -sqrt(2.0);
	}
	return tmp;
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (b_m <= 2.5d-258) then
        tmp = 1.0d0 / ((4.0d0 * (a * c)) / sqrt(((-8.0d0) * (a * (c * (f * (a + a)))))))
    else if (b_m <= 2.5d+53) then
        tmp = sqrt(-f) / -sqrt(c)
    else
        tmp = sqrt(-(f / b_m)) * -sqrt(2.0d0)
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 2.5e-258) {
		tmp = 1.0 / ((4.0 * (A * C)) / Math.sqrt((-8.0 * (A * (C * (F * (A + A)))))));
	} else if (B_m <= 2.5e+53) {
		tmp = Math.sqrt(-F) / -Math.sqrt(C);
	} else {
		tmp = Math.sqrt(-(F / B_m)) * -Math.sqrt(2.0);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 2.5e-258:
		tmp = 1.0 / ((4.0 * (A * C)) / math.sqrt((-8.0 * (A * (C * (F * (A + A)))))))
	elif B_m <= 2.5e+53:
		tmp = math.sqrt(-F) / -math.sqrt(C)
	else:
		tmp = math.sqrt(-(F / B_m)) * -math.sqrt(2.0)
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 2.5e-258)
		tmp = Float64(1.0 / Float64(Float64(4.0 * Float64(A * C)) / sqrt(Float64(-8.0 * Float64(A * Float64(C * Float64(F * Float64(A + A))))))));
	elseif (B_m <= 2.5e+53)
		tmp = Float64(sqrt(Float64(-F)) / Float64(-sqrt(C)));
	else
		tmp = Float64(sqrt(Float64(-Float64(F / B_m))) * Float64(-sqrt(2.0)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (B_m <= 2.5e-258)
		tmp = 1.0 / ((4.0 * (A * C)) / sqrt((-8.0 * (A * (C * (F * (A + A)))))));
	elseif (B_m <= 2.5e+53)
		tmp = sqrt(-F) / -sqrt(C);
	else
		tmp = sqrt(-(F / B_m)) * -sqrt(2.0);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 2.5e-258], N[(1.0 / N[(N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision] / N[Sqrt[N[(-8.0 * N[(A * N[(C * N[(F * N[(A + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 2.5e+53], N[(N[Sqrt[(-F)], $MachinePrecision] / (-N[Sqrt[C], $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[(-N[(F / B$95$m), $MachinePrecision])], $MachinePrecision] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if B < 2.4999999999999999e-258

   1. Initial program 17.9%

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

   3. Simplified 25.5%

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

      1. clear-num 25.5%

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

      2. inv-pow 25.5%

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

   6. Applied egg-rr 24.5%

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

      1. unpow-1 24.5%

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

      2. associate-*l* 21.7%

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

      3. associate-+r- 22.5%

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

      4. +-commutative 22.5%

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

      5. associate-+l- 22.2%

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

   8. Simplified 22.2%

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

   9. Taylor expanded in C around inf 13.6%

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

       1. mul-1-neg 13.6%

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

   11. Simplified 13.6%

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

   12. Taylor expanded in B around 0 13.9%

       \[\leadsto \frac{1}{\frac{\color{blue}{4 \cdot \left(A \cdot C\right)}}{\sqrt{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A - \left(-A\right)\right)\right)\right)\right)}}} \]

   if 2.4999999999999999e-258 < B < 2.5000000000000002e53

   1. Initial program 24.5%

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

   3. Taylor expanded in F around 0 19.6%

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

   5. Simplified 32.5%

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

   6. Taylor expanded in A around -inf 22.3%

      \[\leadsto -\sqrt{\color{blue}{-0.5 \cdot \frac{F}{C}}} \cdot \sqrt{2} \]
   7. Step-by-step derivation

      1. associate-*r/ 22.3%

         \[\leadsto -\sqrt{\color{blue}{\frac{-0.5 \cdot F}{C}}} \cdot \sqrt{2} \]

      2. sqrt-div 26.8%

         \[\leadsto -\color{blue}{\frac{\sqrt{-0.5 \cdot F}}{\sqrt{C}}} \cdot \sqrt{2} \]

   8. Applied egg-rr 26.8%

      \[\leadsto -\color{blue}{\frac{\sqrt{-0.5 \cdot F}}{\sqrt{C}}} \cdot \sqrt{2} \]
   9. Step-by-step derivation

      1. *-commutative 26.8%

         \[\leadsto -\frac{\sqrt{\color{blue}{F \cdot -0.5}}}{\sqrt{C}} \cdot \sqrt{2} \]

   10. Simplified 26.8%

       \[\leadsto -\color{blue}{\frac{\sqrt{F \cdot -0.5}}{\sqrt{C}}} \cdot \sqrt{2} \]
   11. Step-by-step derivation

       1. associate-*l/ 26.8%

          \[\leadsto -\color{blue}{\frac{\sqrt{F \cdot -0.5} \cdot \sqrt{2}}{\sqrt{C}}} \]

       2. pow1/2 26.8%

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

       3. pow1/2 26.8%

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

       4. pow-prod-down 26.8%

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

   12. Applied egg-rr 26.8%

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

       1. unpow1/2 26.8%

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

       2. associate-*l* 26.8%

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

       3. metadata-eval 26.8%

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

       4. *-commutative 26.8%

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

       5. mul-1-neg 26.8%

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

   14. Simplified 26.8%

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

   if 2.5000000000000002e53 < B

   1. Initial program 2.7%

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

   3. Taylor expanded in F around 0 8.8%

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

   5. Simplified 22.4%

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

   6. Taylor expanded in B around inf 51.6%

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

      1. associate-*r/ 51.6%

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

      2. mul-1-neg 51.6%

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

   8. Simplified 51.6%

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

4. Final simplification 24.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 2.5 \cdot 10^{-258}:\\ \;\;\;\;\frac{1}{\frac{4 \cdot \left(A \cdot C\right)}{\sqrt{-8 \cdot \left(A \cdot \left(C \cdot \left(F \cdot \left(A + A\right)\right)\right)\right)}}}\\ \mathbf{elif}\;B \leq 2.5 \cdot 10^{+53}:\\ \;\;\;\;\frac{\sqrt{-F}}{-\sqrt{C}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{-\frac{F}{B}} \cdot \left(-\sqrt{2}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 44.6% accurate, 3.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;C \leq -2 \cdot 10^{-285}:\\ \;\;\;\;\sqrt{2 \cdot \frac{F \cdot -0.5}{C}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{-F}}{-\sqrt{C}}\\ \end{array} \end{array} \]

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 (<= C -2e-285)
   (sqrt (* 2.0 (/ (* F -0.5) C)))
   (/ (sqrt (- F)) (- (sqrt 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 (C <= -2e-285) {
		tmp = sqrt((2.0 * ((F * -0.5) / C)));
	} else {
		tmp = sqrt(-F) / -sqrt(C);
	}
	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 (c <= (-2d-285)) then
        tmp = sqrt((2.0d0 * ((f * (-0.5d0)) / c)))
    else
        tmp = sqrt(-f) / -sqrt(c)
    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 (C <= -2e-285) {
		tmp = Math.sqrt((2.0 * ((F * -0.5) / C)));
	} else {
		tmp = Math.sqrt(-F) / -Math.sqrt(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 C <= -2e-285:
		tmp = math.sqrt((2.0 * ((F * -0.5) / C)))
	else:
		tmp = math.sqrt(-F) / -math.sqrt(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 (C <= -2e-285)
		tmp = sqrt(Float64(2.0 * Float64(Float64(F * -0.5) / C)));
	else
		tmp = Float64(sqrt(Float64(-F)) / Float64(-sqrt(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 (C <= -2e-285)
		tmp = sqrt((2.0 * ((F * -0.5) / C)));
	else
		tmp = sqrt(-F) / -sqrt(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[C, -2e-285], N[Sqrt[N[(2.0 * N[(N[(F * -0.5), $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Sqrt[(-F)], $MachinePrecision] / (-N[Sqrt[C], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if C < -2.00000000000000015e-285

   1. Initial program 14.9%

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

   3. Taylor expanded in F around 0 14.1%

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

   5. Simplified 29.2%

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

   6. Taylor expanded in A around -inf 0.8%

      \[\leadsto -\sqrt{\color{blue}{-0.5 \cdot \frac{F}{C}}} \cdot \sqrt{2} \]
   7. Step-by-step derivation

      1. add-sqr-sqrt 0.5%

         \[\leadsto \color{blue}{\sqrt{-\sqrt{-0.5 \cdot \frac{F}{C}} \cdot \sqrt{2}} \cdot \sqrt{-\sqrt{-0.5 \cdot \frac{F}{C}} \cdot \sqrt{2}}} \]

      2. sqrt-unprod 8.0%

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

      3. sqr-neg 8.0%

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

      4. sqrt-unprod 8.0%

         \[\leadsto \color{blue}{\sqrt{\sqrt{-0.5 \cdot \frac{F}{C}} \cdot \sqrt{2}} \cdot \sqrt{\sqrt{-0.5 \cdot \frac{F}{C}} \cdot \sqrt{2}}} \]

      5. add-sqr-sqrt 8.0%

         \[\leadsto \color{blue}{\sqrt{-0.5 \cdot \frac{F}{C}} \cdot \sqrt{2}} \]

      6. *-un-lft-identity 8.0%

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

      7. sqrt-unprod 8.0%

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

      8. associate-*r/ 8.0%

         \[\leadsto 1 \cdot \sqrt{\color{blue}{\frac{-0.5 \cdot F}{C}} \cdot 2} \]

   8. Applied egg-rr 8.0%

      \[\leadsto \color{blue}{1 \cdot \sqrt{\frac{-0.5 \cdot F}{C} \cdot 2}} \]

   if -2.00000000000000015e-285 < C

   1. Initial program 19.3%

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

   3. Taylor expanded in F around 0 19.6%

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

   5. Simplified 30.2%

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

   6. Taylor expanded in A around -inf 31.8%

      \[\leadsto -\sqrt{\color{blue}{-0.5 \cdot \frac{F}{C}}} \cdot \sqrt{2} \]
   7. Step-by-step derivation

      1. associate-*r/ 31.8%

         \[\leadsto -\sqrt{\color{blue}{\frac{-0.5 \cdot F}{C}}} \cdot \sqrt{2} \]

      2. sqrt-div 38.7%

         \[\leadsto -\color{blue}{\frac{\sqrt{-0.5 \cdot F}}{\sqrt{C}}} \cdot \sqrt{2} \]

   8. Applied egg-rr 38.7%

      \[\leadsto -\color{blue}{\frac{\sqrt{-0.5 \cdot F}}{\sqrt{C}}} \cdot \sqrt{2} \]
   9. Step-by-step derivation

      1. *-commutative 38.7%

         \[\leadsto -\frac{\sqrt{\color{blue}{F \cdot -0.5}}}{\sqrt{C}} \cdot \sqrt{2} \]

   10. Simplified 38.7%

       \[\leadsto -\color{blue}{\frac{\sqrt{F \cdot -0.5}}{\sqrt{C}}} \cdot \sqrt{2} \]
   11. Step-by-step derivation

       1. associate-*l/ 38.7%

          \[\leadsto -\color{blue}{\frac{\sqrt{F \cdot -0.5} \cdot \sqrt{2}}{\sqrt{C}}} \]

       2. pow1/2 38.7%

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

       3. pow1/2 38.7%

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

       4. pow-prod-down 38.8%

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

   12. Applied egg-rr 38.8%

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

       1. unpow1/2 38.8%

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

       2. associate-*l* 38.8%

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

       3. metadata-eval 38.8%

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

       4. *-commutative 38.8%

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

       5. mul-1-neg 38.8%

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

   14. Simplified 38.8%

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

4. Final simplification 22.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq -2 \cdot 10^{-285}:\\ \;\;\;\;\sqrt{2 \cdot \frac{F \cdot -0.5}{C}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{-F}}{-\sqrt{C}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 36.7% accurate, 5.6× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := 2 \cdot \frac{F \cdot -0.5}{C}\\ \mathbf{if}\;F \leq 1.7 \cdot 10^{-255}:\\ \;\;\;\;-{t\_0}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{t\_0}\\ \end{array} \end{array} \]

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 (* 2.0 (/ (* F -0.5) C))))
   (if (<= F 1.7e-255) (- (pow t_0 0.5)) (sqrt t_0))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = 2.0 * ((F * -0.5) / C);
	double tmp;
	if (F <= 1.7e-255) {
		tmp = -pow(t_0, 0.5);
	} else {
		tmp = sqrt(t_0);
	}
	return tmp;
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 2.0d0 * ((f * (-0.5d0)) / c)
    if (f <= 1.7d-255) then
        tmp = -(t_0 ** 0.5d0)
    else
        tmp = sqrt(t_0)
    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 t_0 = 2.0 * ((F * -0.5) / C);
	double tmp;
	if (F <= 1.7e-255) {
		tmp = -Math.pow(t_0, 0.5);
	} else {
		tmp = Math.sqrt(t_0);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	t_0 = 2.0 * ((F * -0.5) / C)
	tmp = 0
	if F <= 1.7e-255:
		tmp = -math.pow(t_0, 0.5)
	else:
		tmp = math.sqrt(t_0)
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(2.0 * Float64(Float64(F * -0.5) / C))
	tmp = 0.0
	if (F <= 1.7e-255)
		tmp = Float64(-(t_0 ^ 0.5));
	else
		tmp = sqrt(t_0);
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	t_0 = 2.0 * ((F * -0.5) / C);
	tmp = 0.0;
	if (F <= 1.7e-255)
		tmp = -(t_0 ^ 0.5);
	else
		tmp = sqrt(t_0);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(2.0 * N[(N[(F * -0.5), $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[F, 1.7e-255], (-N[Power[t$95$0, 0.5], $MachinePrecision]), N[Sqrt[t$95$0], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if F < 1.69999999999999992e-255

   1. Initial program 16.2%

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

   3. Taylor expanded in F around 0 19.4%

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

   5. Simplified 34.1%

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

   6. Taylor expanded in A around -inf 18.1%

      \[\leadsto -\sqrt{\color{blue}{-0.5 \cdot \frac{F}{C}}} \cdot \sqrt{2} \]
   7. Step-by-step derivation

      1. sqrt-unprod 18.2%

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

      2. pow1/2 18.4%

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

      3. associate-*r/ 18.4%

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

   8. Applied egg-rr 18.4%

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

   if 1.69999999999999992e-255 < F

   1. Initial program 22.3%

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

   3. Taylor expanded in F around 0 0.4%

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

   5. Simplified 1.5%

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

   6. Taylor expanded in A around -inf 1.7%

      \[\leadsto -\sqrt{\color{blue}{-0.5 \cdot \frac{F}{C}}} \cdot \sqrt{2} \]
   7. Step-by-step derivation

      1. add-sqr-sqrt 0.6%

         \[\leadsto \color{blue}{\sqrt{-\sqrt{-0.5 \cdot \frac{F}{C}} \cdot \sqrt{2}} \cdot \sqrt{-\sqrt{-0.5 \cdot \frac{F}{C}} \cdot \sqrt{2}}} \]

      2. sqrt-unprod 28.7%

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

      3. sqr-neg 28.7%

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

      4. sqrt-unprod 28.6%

         \[\leadsto \color{blue}{\sqrt{\sqrt{-0.5 \cdot \frac{F}{C}} \cdot \sqrt{2}} \cdot \sqrt{\sqrt{-0.5 \cdot \frac{F}{C}} \cdot \sqrt{2}}} \]

      5. add-sqr-sqrt 28.7%

         \[\leadsto \color{blue}{\sqrt{-0.5 \cdot \frac{F}{C}} \cdot \sqrt{2}} \]

      6. *-un-lft-identity 28.7%

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

      7. sqrt-unprod 28.8%

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

      8. associate-*r/ 28.8%

         \[\leadsto 1 \cdot \sqrt{\color{blue}{\frac{-0.5 \cdot F}{C}} \cdot 2} \]

   8. Applied egg-rr 28.8%

      \[\leadsto \color{blue}{1 \cdot \sqrt{\frac{-0.5 \cdot F}{C} \cdot 2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 19.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;F \leq 1.7 \cdot 10^{-255}:\\ \;\;\;\;-{\left(2 \cdot \frac{F \cdot -0.5}{C}\right)}^{0.5}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \frac{F \cdot -0.5}{C}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 30.6% accurate, 5.6× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;C \leq 1.38 \cdot 10^{-306}:\\ \;\;\;\;\sqrt{A \cdot F} \cdot \frac{-2}{B\_m}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{2 \cdot \left(F \cdot -0.5\right)}{C}}\\ \end{array} \end{array} \]

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 (<= C 1.38e-306)
   (* (sqrt (* A F)) (/ -2.0 B_m))
   (- (sqrt (/ (* 2.0 (* F -0.5)) 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 (C <= 1.38e-306) {
		tmp = sqrt((A * F)) * (-2.0 / B_m);
	} else {
		tmp = -sqrt(((2.0 * (F * -0.5)) / C));
	}
	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 (c <= 1.38d-306) then
        tmp = sqrt((a * f)) * ((-2.0d0) / b_m)
    else
        tmp = -sqrt(((2.0d0 * (f * (-0.5d0))) / c))
    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 (C <= 1.38e-306) {
		tmp = Math.sqrt((A * F)) * (-2.0 / B_m);
	} else {
		tmp = -Math.sqrt(((2.0 * (F * -0.5)) / 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 C <= 1.38e-306:
		tmp = math.sqrt((A * F)) * (-2.0 / B_m)
	else:
		tmp = -math.sqrt(((2.0 * (F * -0.5)) / 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 (C <= 1.38e-306)
		tmp = Float64(sqrt(Float64(A * F)) * Float64(-2.0 / B_m));
	else
		tmp = Float64(-sqrt(Float64(Float64(2.0 * Float64(F * -0.5)) / 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 (C <= 1.38e-306)
		tmp = sqrt((A * F)) * (-2.0 / B_m);
	else
		tmp = -sqrt(((2.0 * (F * -0.5)) / 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[C, 1.38e-306], N[(N[Sqrt[N[(A * F), $MachinePrecision]], $MachinePrecision] * N[(-2.0 / B$95$m), $MachinePrecision]), $MachinePrecision], (-N[Sqrt[N[(N[(2.0 * N[(F * -0.5), $MachinePrecision]), $MachinePrecision] / C), $MachinePrecision]], $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if C < 1.3799999999999999e-306

   1. Initial program 15.1%

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

   3. Taylor expanded in C around 0 6.1%

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

      1. mul-1-neg 6.1%

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

      2. +-commutative 6.1%

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

      3. unpow2 6.1%

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

      4. unpow2 6.1%

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

      5. hypot-define 14.5%

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

   5. Simplified 14.5%

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

   6. Taylor expanded in A around -inf 0.0%

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

      1. *-commutative 0.0%

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

      2. unpow2 0.0%

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

      3. unpow2 0.0%

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

      4. rem-square-sqrt 2.8%

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

      5. rem-square-sqrt 2.8%

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

      6. metadata-eval 2.8%

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

   8. Simplified 2.8%

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

   if 1.3799999999999999e-306 < C

   1. Initial program 19.2%

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

   3. Taylor expanded in F around 0 18.7%

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

   5. Simplified 29.6%

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

   6. Taylor expanded in A around -inf 33.1%

      \[\leadsto -\sqrt{\color{blue}{-0.5 \cdot \frac{F}{C}}} \cdot \sqrt{2} \]
   7. Step-by-step derivation

      1. pow1 33.1%

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

      2. sqrt-unprod 33.3%

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

      3. associate-*r/ 33.3%

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

   8. Applied egg-rr 33.3%

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

      1. unpow1 33.3%

         \[\leadsto -\color{blue}{\sqrt{\frac{-0.5 \cdot F}{C} \cdot 2}} \]

      2. associate-*l/ 33.3%

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

      3. *-commutative 33.3%

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

   10. Simplified 33.3%

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

4. Final simplification 17.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;C \leq 1.38 \cdot 10^{-306}:\\ \;\;\;\;\sqrt{A \cdot F} \cdot \frac{-2}{B}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{2 \cdot \left(F \cdot -0.5\right)}{C}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 9.0% accurate, 5.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 17.0%

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

3. Taylor expanded in C around 0 6.1%

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

   1. mul-1-neg 6.1%

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

   2. +-commutative 6.1%

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

   3. unpow2 6.1%

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

   4. unpow2 6.1%

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

   5. hypot-define 13.6%

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

5. Simplified 13.6%

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

6. Taylor expanded in A around -inf 0.0%

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

   1. *-commutative 0.0%

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

   2. unpow2 0.0%

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

   3. unpow2 0.0%

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

   4. rem-square-sqrt 2.9%

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

   5. rem-square-sqrt 2.9%

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

   6. metadata-eval 2.9%

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

8. Simplified 2.9%

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

9. Final simplification 2.9%

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

Alternative 10: 9.0% accurate, 5.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 17.0%

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

3. Simplified 23.0%

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

5. Taylor expanded in A around -inf 15.5%

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

6. Taylor expanded in B around inf 2.9%

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

   1. associate-*r/ 2.9%

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

   2. *-rgt-identity 2.9%

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

   3. *-commutative 2.9%

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

8. Simplified 2.9%

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

9. Final simplification 2.9%

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

Alternative 11: 1.7% accurate, 6.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 17.0%

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

3. Taylor expanded in B around -inf 0.0%

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

   1. mul-1-neg 0.0%

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

   2. unpow2 0.0%

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

   3. rem-square-sqrt 1.9%

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

5. Simplified 1.9%

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

6. Taylor expanded in F around 0 1.9%

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

   1. sqrt-unprod 1.9%

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

   2. pow1/2 2.1%

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

8. Applied egg-rr 2.1%

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

9. Final simplification 2.1%

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

Alternative 12: 1.6% accurate, 6.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 17.0%

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

3. Taylor expanded in B around -inf 0.0%

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

   1. mul-1-neg 0.0%

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

   2. unpow2 0.0%

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

   3. rem-square-sqrt 1.9%

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

5. Simplified 1.9%

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

6. Taylor expanded in F around 0 1.9%

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

   1. sqrt-unprod 1.9%

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

8. Applied egg-rr 1.9%

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

9. Taylor expanded in F around 0 1.9%

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

    1. associate-*r/ 1.9%

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

    2. *-commutative 1.9%

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

    3. associate-/l* 1.9%

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

11. Simplified 1.9%

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

Reproduce

herbie shell --seed 2024139 
(FPCore (A B C F)
  :name "ABCF->ab-angle b"
  :precision binary64
  (/ (- (sqrt (* (* 2.0 (* (- (pow B 2.0) (* (* 4.0 A) C)) F)) (- (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0))))))) (- (pow B 2.0) (* (* 4.0 A) C))))