ABCF->ab-angle b

Percentage Accurate: 19.5% → 62.3%

Time: 27.0s

Alternatives: 14

Speedup: 6.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 19.5% 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: 62.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := \mathsf{hypot}\left(B\_m, A - C\right)\\ t_2 := -t\_0\\ t_3 := \left(4 \cdot A\right) \cdot C\\ t_4 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_3\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_3 - {B\_m}^{2}}\\ t_5 := F \cdot t\_0\\ \mathbf{if}\;t\_4 \leq -\infty:\\ \;\;\;\;-\sqrt{2 \cdot \left(F \cdot \frac{\left(A + C\right) - t\_1}{\mathsf{fma}\left(B\_m, B\_m, -4 \cdot \left(A \cdot C\right)\right)}\right)}\\ \mathbf{elif}\;t\_4 \leq -1 \cdot 10^{-189}:\\ \;\;\;\;\frac{\sqrt{t\_5 \cdot \left(2 \cdot \left(A + \left(C - t\_1\right)\right)\right)}}{t\_2}\\ \mathbf{elif}\;t\_4 \leq 5 \cdot 10^{+131}:\\ \;\;\;\;\frac{\sqrt{t\_5 \cdot \left(2 \cdot \left(A + \left(A + -0.5 \cdot \frac{{B\_m}^{2}}{C}\right)\right)\right)}}{t\_2}\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;\frac{\sqrt{F}}{\sqrt{-C}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B\_m} \cdot \left(\sqrt{-F} \cdot \left(-\sqrt{B\_m}\right)\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* A (* C -4.0))))
        (t_1 (hypot B_m (- A C)))
        (t_2 (- t_0))
        (t_3 (* (* 4.0 A) C))
        (t_4
         (/
          (sqrt
           (*
            (* 2.0 (* (- (pow B_m 2.0) t_3) F))
            (- (+ A C) (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))))))
          (- t_3 (pow B_m 2.0))))
        (t_5 (* F t_0)))
   (if (<= t_4 (- INFINITY))
     (-
      (sqrt (* 2.0 (* F (/ (- (+ A C) t_1) (fma B_m B_m (* -4.0 (* A C))))))))
     (if (<= t_4 -1e-189)
       (/ (sqrt (* t_5 (* 2.0 (+ A (- C t_1))))) t_2)
       (if (<= t_4 5e+131)
         (/
          (sqrt (* t_5 (* 2.0 (+ A (+ A (* -0.5 (/ (pow B_m 2.0) C)))))))
          t_2)
         (if (<= t_4 INFINITY)
           (/ (sqrt F) (sqrt (- C)))
           (* (/ (sqrt 2.0) B_m) (* (sqrt (- F)) (- (sqrt B_m))))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (A * (C * -4.0)));
	double t_1 = hypot(B_m, (A - C));
	double t_2 = -t_0;
	double t_3 = (4.0 * A) * C;
	double t_4 = sqrt(((2.0 * ((pow(B_m, 2.0) - t_3) * F)) * ((A + C) - sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / (t_3 - pow(B_m, 2.0));
	double t_5 = F * t_0;
	double tmp;
	if (t_4 <= -((double) INFINITY)) {
		tmp = -sqrt((2.0 * (F * (((A + C) - t_1) / fma(B_m, B_m, (-4.0 * (A * C)))))));
	} else if (t_4 <= -1e-189) {
		tmp = sqrt((t_5 * (2.0 * (A + (C - t_1))))) / t_2;
	} else if (t_4 <= 5e+131) {
		tmp = sqrt((t_5 * (2.0 * (A + (A + (-0.5 * (pow(B_m, 2.0) / C))))))) / t_2;
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = sqrt(F) / sqrt(-C);
	} else {
		tmp = (sqrt(2.0) / B_m) * (sqrt(-F) * -sqrt(B_m));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	t_1 = hypot(B_m, Float64(A - C))
	t_2 = Float64(-t_0)
	t_3 = Float64(Float64(4.0 * A) * C)
	t_4 = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_3) * F)) * Float64(Float64(A + C) - sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / Float64(t_3 - (B_m ^ 2.0)))
	t_5 = Float64(F * t_0)
	tmp = 0.0
	if (t_4 <= Float64(-Inf))
		tmp = Float64(-sqrt(Float64(2.0 * Float64(F * Float64(Float64(Float64(A + C) - t_1) / fma(B_m, B_m, Float64(-4.0 * Float64(A * C))))))));
	elseif (t_4 <= -1e-189)
		tmp = Float64(sqrt(Float64(t_5 * Float64(2.0 * Float64(A + Float64(C - t_1))))) / t_2);
	elseif (t_4 <= 5e+131)
		tmp = Float64(sqrt(Float64(t_5 * Float64(2.0 * Float64(A + Float64(A + Float64(-0.5 * Float64((B_m ^ 2.0) / C))))))) / t_2);
	elseif (t_4 <= Inf)
		tmp = Float64(sqrt(F) / sqrt(Float64(-C)));
	else
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(sqrt(Float64(-F)) * Float64(-sqrt(B_m))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]}, Block[{t$95$2 = (-t$95$0)}, Block[{t$95$3 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$4 = N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$3), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(N[(A + C), $MachinePrecision] - N[Sqrt[N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$3 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(F * t$95$0), $MachinePrecision]}, If[LessEqual[t$95$4, (-Infinity)], (-N[Sqrt[N[(2.0 * N[(F * N[(N[(N[(A + C), $MachinePrecision] - t$95$1), $MachinePrecision] / N[(B$95$m * B$95$m + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), If[LessEqual[t$95$4, -1e-189], N[(N[Sqrt[N[(t$95$5 * N[(2.0 * N[(A + N[(C - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[t$95$4, 5e+131], N[(N[Sqrt[N[(t$95$5 * N[(2.0 * N[(A + N[(A + N[(-0.5 * N[(N[Power[B$95$m, 2.0], $MachinePrecision] / C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[t$95$4, Infinity], N[(N[Sqrt[F], $MachinePrecision] / N[Sqrt[(-C)], $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[(N[Sqrt[(-F)], $MachinePrecision] * (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]]]]]]]]]

Derivation

1. Split input into 5 regimes

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

   1. Initial program 3.3%

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

   3. Taylor expanded in F around 0 18.9%

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

      1. pow1 18.9%

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

   5. Applied egg-rr 67.6%

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

      1. unpow1 67.6%

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

   7. Simplified 67.6%

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

   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))) < -1.00000000000000007e-189

   1. Initial program 98.1%

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

   3. Simplified 98.1%

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

   if -1.00000000000000007e-189 < (/.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))) < 4.99999999999999995e131

   1. Initial program 22.3%

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

   3. Simplified 24.2%

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

   5. Taylor expanded in C around inf 35.4%

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

      1. mul-1-neg 35.4%

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

   7. Simplified 35.4%

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

   if 4.99999999999999995e131 < (/.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 29.2%

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

   3. Taylor expanded in F around 0 0.5%

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

      1. pow1 0.5%

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

   5. Applied egg-rr 1.1%

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

      1. unpow1 1.1%

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

   7. Simplified 1.1%

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

   8. Taylor expanded in A around -inf 1.6%

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

      1. mul-1-neg 1.6%

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

   10. Simplified 1.6%

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

       1. add-sqr-sqrt 0.9%

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

       2. sqrt-unprod 30.0%

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

       3. sqr-neg 30.0%

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

       4. add-sqr-sqrt 30.0%

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

       5. distribute-neg-frac2 30.0%

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

       6. sqrt-div 61.6%

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

   12. Applied egg-rr 61.6%

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

   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 1.8%

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

      1. mul-1-neg 1.8%

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

      2. +-commutative 1.8%

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

      3. unpow2 1.8%

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

      4. unpow2 1.8%

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

      5. hypot-define 15.4%

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

   5. Simplified 15.4%

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

   6. Taylor expanded in A around 0 14.3%

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

      1. mul-1-neg 14.3%

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

   8. Simplified 14.3%

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

      1. pow1/2 14.4%

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

      2. distribute-rgt-neg-in 14.4%

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

      3. unpow-prod-down 24.1%

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

      4. pow1/2 24.1%

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

   10. Applied egg-rr 24.1%

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

       1. unpow1/2 24.1%

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

       2. *-commutative 24.1%

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

   12. Simplified 24.1%

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

4. Final simplification 48.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -\infty:\\ \;\;\;\;-\sqrt{2 \cdot \left(F \cdot \frac{\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -1 \cdot 10^{-189}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot \left(2 \cdot \left(A + \left(C - \mathsf{hypot}\left(B, A - C\right)\right)\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq 5 \cdot 10^{+131}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot \left(2 \cdot \left(A + \left(A + -0.5 \cdot \frac{{B}^{2}}{C}\right)\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) - \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{F}}{\sqrt{-C}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{-F} \cdot \left(-\sqrt{B}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 55.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 5 \cdot 10^{-86}:\\ \;\;\;\;\frac{\sqrt{-8 \cdot \left(\left(A \cdot C\right) \cdot \left(F \cdot \left(A + A\right)\right)\right)}}{4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;{B\_m}^{2} \leq 5 \cdot 10^{+300}:\\ \;\;\;\;-\sqrt{2 \cdot \left(F \cdot \frac{\left(A + C\right) - \mathsf{hypot}\left(B\_m, A - C\right)}{\mathsf{fma}\left(B\_m, B\_m, -4 \cdot \left(A \cdot C\right)\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B\_m} \cdot \left(\sqrt{-F} \cdot \left(-\sqrt{B\_m}\right)\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 5e-86)
   (/ (sqrt (* -8.0 (* (* A C) (* F (+ A A))))) (* 4.0 (* A C)))
   (if (<= (pow B_m 2.0) 5e+300)
     (-
      (sqrt
       (*
        2.0
        (*
         F
         (/ (- (+ A C) (hypot B_m (- A C))) (fma B_m B_m (* -4.0 (* A C))))))))
     (* (/ (sqrt 2.0) B_m) (* (sqrt (- F)) (- (sqrt B_m)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (pow(B_m, 2.0) <= 5e-86) {
		tmp = sqrt((-8.0 * ((A * C) * (F * (A + A))))) / (4.0 * (A * C));
	} else if (pow(B_m, 2.0) <= 5e+300) {
		tmp = -sqrt((2.0 * (F * (((A + C) - hypot(B_m, (A - C))) / fma(B_m, B_m, (-4.0 * (A * C)))))));
	} else {
		tmp = (sqrt(2.0) / B_m) * (sqrt(-F) * -sqrt(B_m));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 5e-86)
		tmp = Float64(sqrt(Float64(-8.0 * Float64(Float64(A * C) * Float64(F * Float64(A + A))))) / Float64(4.0 * Float64(A * C)));
	elseif ((B_m ^ 2.0) <= 5e+300)
		tmp = Float64(-sqrt(Float64(2.0 * Float64(F * Float64(Float64(Float64(A + C) - hypot(B_m, Float64(A - C))) / fma(B_m, B_m, Float64(-4.0 * Float64(A * C))))))));
	else
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(sqrt(Float64(-F)) * Float64(-sqrt(B_m))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e-86], N[(N[Sqrt[N[(-8.0 * N[(N[(A * C), $MachinePrecision] * N[(F * N[(A + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e+300], (-N[Sqrt[N[(2.0 * N[(F * N[(N[(N[(A + C), $MachinePrecision] - N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] / N[(B$95$m * B$95$m + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[(N[Sqrt[(-F)], $MachinePrecision] * (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 4.9999999999999999e-86

   1. Initial program 20.4%

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

   3. Simplified 23.1%

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

   5. Taylor expanded in C around inf 22.2%

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

   6. Taylor expanded in C around inf 21.9%

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

      1. associate-*r* 21.8%

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

      2. mul-1-neg 21.8%

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

   8. Simplified 21.8%

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

   if 4.9999999999999999e-86 < (pow.f64 B #s(literal 2 binary64)) < 5.00000000000000026e300

   1. Initial program 40.2%

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

   3. Taylor expanded in F around 0 43.9%

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

      1. pow1 43.9%

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

   5. Applied egg-rr 65.2%

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

      1. unpow1 65.2%

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

   7. Simplified 65.2%

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

   if 5.00000000000000026e300 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 1.4%

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

   3. Taylor expanded in C around 0 3.0%

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

      1. mul-1-neg 3.0%

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

      2. +-commutative 3.0%

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

      3. unpow2 3.0%

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

      4. unpow2 3.0%

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

      5. hypot-define 23.3%

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

   5. Simplified 23.3%

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

   6. Taylor expanded in A around 0 21.9%

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

      1. mul-1-neg 21.9%

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

   8. Simplified 21.9%

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

      1. pow1/2 22.1%

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

      2. distribute-rgt-neg-in 22.1%

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

      3. unpow-prod-down 37.3%

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

      4. pow1/2 37.3%

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

   10. Applied egg-rr 37.3%

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

       1. unpow1/2 37.3%

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

       2. *-commutative 37.3%

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

   12. Simplified 37.3%

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

4. Final simplification 39.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 5 \cdot 10^{-86}:\\ \;\;\;\;\frac{\sqrt{-8 \cdot \left(\left(A \cdot C\right) \cdot \left(F \cdot \left(A + A\right)\right)\right)}}{4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;{B}^{2} \leq 5 \cdot 10^{+300}:\\ \;\;\;\;-\sqrt{2 \cdot \left(F \cdot \frac{\left(A + C\right) - \mathsf{hypot}\left(B, A - C\right)}{\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{-F} \cdot \left(-\sqrt{B}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 53.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 5 \cdot 10^{-86}:\\ \;\;\;\;\frac{\sqrt{-8 \cdot \left(\left(A \cdot C\right) \cdot \left(F \cdot \left(A + A\right)\right)\right)}}{4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;{B\_m}^{2} \leq 5 \cdot 10^{+300}:\\ \;\;\;\;-\sqrt{2 \cdot \left(F \cdot \frac{A - \mathsf{hypot}\left(B\_m, A\right)}{{B\_m}^{2}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B\_m} \cdot \left(\sqrt{-F} \cdot \left(-\sqrt{B\_m}\right)\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 5e-86)
   (/ (sqrt (* -8.0 (* (* A C) (* F (+ A A))))) (* 4.0 (* A C)))
   (if (<= (pow B_m 2.0) 5e+300)
     (- (sqrt (* 2.0 (* F (/ (- A (hypot B_m A)) (pow B_m 2.0))))))
     (* (/ (sqrt 2.0) B_m) (* (sqrt (- F)) (- (sqrt B_m)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (pow(B_m, 2.0) <= 5e-86) {
		tmp = sqrt((-8.0 * ((A * C) * (F * (A + A))))) / (4.0 * (A * C));
	} else if (pow(B_m, 2.0) <= 5e+300) {
		tmp = -sqrt((2.0 * (F * ((A - hypot(B_m, A)) / pow(B_m, 2.0)))));
	} else {
		tmp = (sqrt(2.0) / B_m) * (sqrt(-F) * -sqrt(B_m));
	}
	return tmp;
}

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (Math.pow(B_m, 2.0) <= 5e-86) {
		tmp = Math.sqrt((-8.0 * ((A * C) * (F * (A + A))))) / (4.0 * (A * C));
	} else if (Math.pow(B_m, 2.0) <= 5e+300) {
		tmp = -Math.sqrt((2.0 * (F * ((A - Math.hypot(B_m, A)) / Math.pow(B_m, 2.0)))));
	} else {
		tmp = (Math.sqrt(2.0) / B_m) * (Math.sqrt(-F) * -Math.sqrt(B_m));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if math.pow(B_m, 2.0) <= 5e-86:
		tmp = math.sqrt((-8.0 * ((A * C) * (F * (A + A))))) / (4.0 * (A * C))
	elif math.pow(B_m, 2.0) <= 5e+300:
		tmp = -math.sqrt((2.0 * (F * ((A - math.hypot(B_m, A)) / math.pow(B_m, 2.0)))))
	else:
		tmp = (math.sqrt(2.0) / B_m) * (math.sqrt(-F) * -math.sqrt(B_m))
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 5e-86)
		tmp = Float64(sqrt(Float64(-8.0 * Float64(Float64(A * C) * Float64(F * Float64(A + A))))) / Float64(4.0 * Float64(A * C)));
	elseif ((B_m ^ 2.0) <= 5e+300)
		tmp = Float64(-sqrt(Float64(2.0 * Float64(F * Float64(Float64(A - hypot(B_m, A)) / (B_m ^ 2.0))))));
	else
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(sqrt(Float64(-F)) * Float64(-sqrt(B_m))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if ((B_m ^ 2.0) <= 5e-86)
		tmp = sqrt((-8.0 * ((A * C) * (F * (A + A))))) / (4.0 * (A * C));
	elseif ((B_m ^ 2.0) <= 5e+300)
		tmp = -sqrt((2.0 * (F * ((A - hypot(B_m, A)) / (B_m ^ 2.0)))));
	else
		tmp = (sqrt(2.0) / B_m) * (sqrt(-F) * -sqrt(B_m));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e-86], N[(N[Sqrt[N[(-8.0 * N[(N[(A * C), $MachinePrecision] * N[(F * N[(A + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e+300], (-N[Sqrt[N[(2.0 * N[(F * N[(N[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision] / N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[(N[Sqrt[(-F)], $MachinePrecision] * (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 4.9999999999999999e-86

   1. Initial program 20.4%

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

   3. Simplified 23.1%

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

   5. Taylor expanded in C around inf 22.2%

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

   6. Taylor expanded in C around inf 21.9%

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

      1. associate-*r* 21.8%

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

      2. mul-1-neg 21.8%

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

   8. Simplified 21.8%

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

   if 4.9999999999999999e-86 < (pow.f64 B #s(literal 2 binary64)) < 5.00000000000000026e300

   1. Initial program 40.2%

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

   3. Taylor expanded in F around 0 43.9%

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

      1. pow1 43.9%

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

   5. Applied egg-rr 65.2%

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

      1. unpow1 65.2%

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

   7. Simplified 65.2%

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

   8. Taylor expanded in C around 0 38.1%

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

      1. associate-/l* 43.2%

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

      2. +-commutative 43.2%

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

      3. unpow2 43.2%

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

      4. unpow2 43.2%

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

      5. hypot-undefine 49.9%

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

   10. Simplified 49.9%

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

   if 5.00000000000000026e300 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 1.4%

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

   3. Taylor expanded in C around 0 3.0%

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

      1. mul-1-neg 3.0%

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

      2. +-commutative 3.0%

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

      3. unpow2 3.0%

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

      4. unpow2 3.0%

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

      5. hypot-define 23.3%

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

   5. Simplified 23.3%

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

   6. Taylor expanded in A around 0 21.9%

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

      1. mul-1-neg 21.9%

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

   8. Simplified 21.9%

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

      1. pow1/2 22.1%

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

      2. distribute-rgt-neg-in 22.1%

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

      3. unpow-prod-down 37.3%

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

      4. pow1/2 37.3%

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

   10. Applied egg-rr 37.3%

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

       1. unpow1/2 37.3%

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

       2. *-commutative 37.3%

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

   12. Simplified 37.3%

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

4. Final simplification 34.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 5 \cdot 10^{-86}:\\ \;\;\;\;\frac{\sqrt{-8 \cdot \left(\left(A \cdot C\right) \cdot \left(F \cdot \left(A + A\right)\right)\right)}}{4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;{B}^{2} \leq 5 \cdot 10^{+300}:\\ \;\;\;\;-\sqrt{2 \cdot \left(F \cdot \frac{A - \mathsf{hypot}\left(B, A\right)}{{B}^{2}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{-F} \cdot \left(-\sqrt{B}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 52.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 1.8 \cdot 10^{-42}:\\ \;\;\;\;\frac{\sqrt{-8 \cdot \left(\left(A \cdot C\right) \cdot \left(F \cdot \left(A + A\right)\right)\right)}}{4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B\_m \leq 6.5 \cdot 10^{+174}:\\ \;\;\;\;\frac{-1}{\frac{B\_m}{\sqrt{\left(A - \mathsf{hypot}\left(B\_m, A\right)\right) \cdot \left(2 \cdot F\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B\_m} \cdot \left(\sqrt{-F} \cdot \left(-\sqrt{B\_m}\right)\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 1.8e-42)
   (/ (sqrt (* -8.0 (* (* A C) (* F (+ A A))))) (* 4.0 (* A C)))
   (if (<= B_m 6.5e+174)
     (/ -1.0 (/ B_m (sqrt (* (- A (hypot B_m A)) (* 2.0 F)))))
     (* (/ (sqrt 2.0) B_m) (* (sqrt (- F)) (- (sqrt B_m)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 1.8e-42) {
		tmp = sqrt((-8.0 * ((A * C) * (F * (A + A))))) / (4.0 * (A * C));
	} else if (B_m <= 6.5e+174) {
		tmp = -1.0 / (B_m / sqrt(((A - hypot(B_m, A)) * (2.0 * F))));
	} else {
		tmp = (sqrt(2.0) / B_m) * (sqrt(-F) * -sqrt(B_m));
	}
	return tmp;
}

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 1.8e-42) {
		tmp = Math.sqrt((-8.0 * ((A * C) * (F * (A + A))))) / (4.0 * (A * C));
	} else if (B_m <= 6.5e+174) {
		tmp = -1.0 / (B_m / Math.sqrt(((A - Math.hypot(B_m, A)) * (2.0 * F))));
	} else {
		tmp = (Math.sqrt(2.0) / B_m) * (Math.sqrt(-F) * -Math.sqrt(B_m));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 1.8e-42:
		tmp = math.sqrt((-8.0 * ((A * C) * (F * (A + A))))) / (4.0 * (A * C))
	elif B_m <= 6.5e+174:
		tmp = -1.0 / (B_m / math.sqrt(((A - math.hypot(B_m, A)) * (2.0 * F))))
	else:
		tmp = (math.sqrt(2.0) / B_m) * (math.sqrt(-F) * -math.sqrt(B_m))
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 1.8e-42)
		tmp = Float64(sqrt(Float64(-8.0 * Float64(Float64(A * C) * Float64(F * Float64(A + A))))) / Float64(4.0 * Float64(A * C)));
	elseif (B_m <= 6.5e+174)
		tmp = Float64(-1.0 / Float64(B_m / sqrt(Float64(Float64(A - hypot(B_m, A)) * Float64(2.0 * F)))));
	else
		tmp = Float64(Float64(sqrt(2.0) / B_m) * Float64(sqrt(Float64(-F)) * Float64(-sqrt(B_m))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (B_m <= 1.8e-42)
		tmp = sqrt((-8.0 * ((A * C) * (F * (A + A))))) / (4.0 * (A * C));
	elseif (B_m <= 6.5e+174)
		tmp = -1.0 / (B_m / sqrt(((A - hypot(B_m, A)) * (2.0 * F))));
	else
		tmp = (sqrt(2.0) / B_m) * (sqrt(-F) * -sqrt(B_m));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 1.8e-42], N[(N[Sqrt[N[(-8.0 * N[(N[(A * C), $MachinePrecision] * N[(F * N[(A + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 6.5e+174], N[(-1.0 / N[(B$95$m / N[Sqrt[N[(N[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision] * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Sqrt[2.0], $MachinePrecision] / B$95$m), $MachinePrecision] * N[(N[Sqrt[(-F)], $MachinePrecision] * (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if B < 1.8000000000000001e-42

   1. Initial program 21.1%

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

   3. Simplified 20.0%

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

   5. Taylor expanded in C around inf 14.5%

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

   6. Taylor expanded in C around inf 14.9%

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

      1. associate-*r* 14.8%

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

      2. mul-1-neg 14.8%

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

   8. Simplified 14.8%

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

   if 1.8000000000000001e-42 < B < 6.5000000000000001e174

   1. Initial program 34.9%

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

   3. Taylor expanded in C around 0 35.1%

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

      1. mul-1-neg 35.1%

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

      2. +-commutative 35.1%

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

      3. unpow2 35.1%

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

      4. unpow2 35.1%

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

      5. hypot-define 39.5%

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

   5. Simplified 39.5%

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

      1. add-exp-log 36.8%

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

      2. associate-*l/ 36.8%

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

      3. pow1/2 36.8%

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

      4. pow1/2 36.8%

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

      5. pow-prod-down 36.8%

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

   7. Applied egg-rr 36.8%

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

      1. rem-exp-log 39.7%

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

      2. clear-num 39.7%

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

      3. unpow1/2 39.7%

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

      4. associate-*r* 39.7%

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

   9. Applied egg-rr 39.7%

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

   if 6.5000000000000001e174 < 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 42.8%

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

   5. Simplified 42.8%

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

   6. Taylor expanded in A around 0 42.8%

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

      1. mul-1-neg 42.8%

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

   8. Simplified 42.8%

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

      1. pow1/2 42.8%

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

      2. distribute-rgt-neg-in 42.8%

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

      3. unpow-prod-down 74.9%

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

      4. pow1/2 74.9%

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

   10. Applied egg-rr 74.9%

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

       1. unpow1/2 74.9%

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

       2. *-commutative 74.9%

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

   12. Simplified 74.9%

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

4. Final simplification 27.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.8 \cdot 10^{-42}:\\ \;\;\;\;\frac{\sqrt{-8 \cdot \left(\left(A \cdot C\right) \cdot \left(F \cdot \left(A + A\right)\right)\right)}}{4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 6.5 \cdot 10^{+174}:\\ \;\;\;\;\frac{-1}{\frac{B}{\sqrt{\left(A - \mathsf{hypot}\left(B, A\right)\right) \cdot \left(2 \cdot F\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2}}{B} \cdot \left(\sqrt{-F} \cdot \left(-\sqrt{B}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 46.2% accurate, 2.8× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 1.05 \cdot 10^{-42}:\\ \;\;\;\;\frac{\sqrt{-8 \cdot \left(\left(A \cdot C\right) \cdot \left(F \cdot \left(A + A\right)\right)\right)}}{4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B\_m \leq 5.5 \cdot 10^{+188}:\\ \;\;\;\;\frac{-1}{\frac{B\_m}{\sqrt{\left(A - \mathsf{hypot}\left(B\_m, A\right)\right) \cdot \left(2 \cdot F\right)}}}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{2 \cdot \frac{\mathsf{fma}\left(-1, F, F \cdot \frac{A + C}{B\_m}\right)}{B\_m}}\\ \end{array} \end{array} \]

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 1.05e-42)
   (/ (sqrt (* -8.0 (* (* A C) (* F (+ A A))))) (* 4.0 (* A C)))
   (if (<= B_m 5.5e+188)
     (/ -1.0 (/ B_m (sqrt (* (- A (hypot B_m A)) (* 2.0 F)))))
     (- (sqrt (* 2.0 (/ (fma -1.0 F (* F (/ (+ A C) B_m))) B_m)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 1.05e-42) {
		tmp = sqrt((-8.0 * ((A * C) * (F * (A + A))))) / (4.0 * (A * C));
	} else if (B_m <= 5.5e+188) {
		tmp = -1.0 / (B_m / sqrt(((A - hypot(B_m, A)) * (2.0 * F))));
	} else {
		tmp = -sqrt((2.0 * (fma(-1.0, F, (F * ((A + C) / B_m))) / B_m)));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 1.05e-42)
		tmp = Float64(sqrt(Float64(-8.0 * Float64(Float64(A * C) * Float64(F * Float64(A + A))))) / Float64(4.0 * Float64(A * C)));
	elseif (B_m <= 5.5e+188)
		tmp = Float64(-1.0 / Float64(B_m / sqrt(Float64(Float64(A - hypot(B_m, A)) * Float64(2.0 * F)))));
	else
		tmp = Float64(-sqrt(Float64(2.0 * Float64(fma(-1.0, F, Float64(F * Float64(Float64(A + C) / B_m))) / 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, 1.05e-42], N[(N[Sqrt[N[(-8.0 * N[(N[(A * C), $MachinePrecision] * N[(F * N[(A + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 5.5e+188], N[(-1.0 / N[(B$95$m / N[Sqrt[N[(N[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision] * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[Sqrt[N[(2.0 * N[(N[(-1.0 * F + N[(F * N[(N[(A + C), $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])]]

Derivation

1. Split input into 3 regimes

2. if B < 1.05000000000000003e-42

   1. Initial program 21.1%

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

   3. Simplified 20.0%

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

   5. Taylor expanded in C around inf 14.5%

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

   6. Taylor expanded in C around inf 14.9%

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

      1. associate-*r* 14.8%

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

      2. mul-1-neg 14.8%

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

   8. Simplified 14.8%

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

   if 1.05000000000000003e-42 < B < 5.50000000000000013e188

   1. Initial program 33.5%

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

   3. Taylor expanded in C around 0 33.8%

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

      1. mul-1-neg 33.8%

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

      2. +-commutative 33.8%

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

      3. unpow2 33.8%

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

      4. unpow2 33.8%

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

      5. hypot-define 38.3%

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

   5. Simplified 38.3%

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

      1. add-exp-log 35.7%

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

      2. associate-*l/ 35.7%

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

      3. pow1/2 35.7%

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

      4. pow1/2 35.7%

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

      5. pow-prod-down 35.7%

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

   7. Applied egg-rr 35.7%

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

      1. rem-exp-log 38.5%

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

      2. clear-num 38.5%

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

      3. unpow1/2 38.4%

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

      4. associate-*r* 38.4%

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

   9. Applied egg-rr 38.4%

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

   if 5.50000000000000013e188 < 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 F around 0 0.0%

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

      1. pow1 0.0%

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

   5. Applied egg-rr 4.5%

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

      1. unpow1 4.5%

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

   7. Simplified 4.5%

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

   8. Taylor expanded in B around inf 39.2%

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

      1. fma-define 39.2%

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

      2. associate-/l* 57.8%

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

   10. Simplified 57.8%

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

4. Final simplification 25.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.05 \cdot 10^{-42}:\\ \;\;\;\;\frac{\sqrt{-8 \cdot \left(\left(A \cdot C\right) \cdot \left(F \cdot \left(A + A\right)\right)\right)}}{4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 5.5 \cdot 10^{+188}:\\ \;\;\;\;\frac{-1}{\frac{B}{\sqrt{\left(A - \mathsf{hypot}\left(B, A\right)\right) \cdot \left(2 \cdot F\right)}}}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{2 \cdot \frac{\mathsf{fma}\left(-1, F, F \cdot \frac{A + C}{B}\right)}{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 46.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 2.1 \cdot 10^{-42}:\\ \;\;\;\;\frac{\sqrt{-8 \cdot \left(\left(A \cdot C\right) \cdot \left(F \cdot \left(A + A\right)\right)\right)}}{4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B\_m \leq 2.6 \cdot 10^{+174}:\\ \;\;\;\;\frac{-1}{\frac{B\_m}{\sqrt{\left(A - \mathsf{hypot}\left(B\_m, A\right)\right) \cdot \left(2 \cdot F\right)}}}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{-2 \cdot \frac{F}{B\_m}}\\ \end{array} \end{array} \]

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 2.1e-42)
   (/ (sqrt (* -8.0 (* (* A C) (* F (+ A A))))) (* 4.0 (* A C)))
   (if (<= B_m 2.6e+174)
     (/ -1.0 (/ B_m (sqrt (* (- A (hypot B_m A)) (* 2.0 F)))))
     (- (sqrt (* -2.0 (/ F B_m)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 2.1e-42) {
		tmp = sqrt((-8.0 * ((A * C) * (F * (A + A))))) / (4.0 * (A * C));
	} else if (B_m <= 2.6e+174) {
		tmp = -1.0 / (B_m / sqrt(((A - hypot(B_m, A)) * (2.0 * F))));
	} else {
		tmp = -sqrt((-2.0 * (F / 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 <= 2.1e-42) {
		tmp = Math.sqrt((-8.0 * ((A * C) * (F * (A + A))))) / (4.0 * (A * C));
	} else if (B_m <= 2.6e+174) {
		tmp = -1.0 / (B_m / Math.sqrt(((A - Math.hypot(B_m, A)) * (2.0 * F))));
	} else {
		tmp = -Math.sqrt((-2.0 * (F / 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 <= 2.1e-42:
		tmp = math.sqrt((-8.0 * ((A * C) * (F * (A + A))))) / (4.0 * (A * C))
	elif B_m <= 2.6e+174:
		tmp = -1.0 / (B_m / math.sqrt(((A - math.hypot(B_m, A)) * (2.0 * F))))
	else:
		tmp = -math.sqrt((-2.0 * (F / B_m)))
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 2.1e-42)
		tmp = Float64(sqrt(Float64(-8.0 * Float64(Float64(A * C) * Float64(F * Float64(A + A))))) / Float64(4.0 * Float64(A * C)));
	elseif (B_m <= 2.6e+174)
		tmp = Float64(-1.0 / Float64(B_m / sqrt(Float64(Float64(A - hypot(B_m, A)) * Float64(2.0 * F)))));
	else
		tmp = Float64(-sqrt(Float64(-2.0 * Float64(F / B_m))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (B_m <= 2.1e-42)
		tmp = sqrt((-8.0 * ((A * C) * (F * (A + A))))) / (4.0 * (A * C));
	elseif (B_m <= 2.6e+174)
		tmp = -1.0 / (B_m / sqrt(((A - hypot(B_m, A)) * (2.0 * F))));
	else
		tmp = -sqrt((-2.0 * (F / 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, 2.1e-42], N[(N[Sqrt[N[(-8.0 * N[(N[(A * C), $MachinePrecision] * N[(F * N[(A + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 2.6e+174], N[(-1.0 / N[(B$95$m / N[Sqrt[N[(N[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision] * N[(2.0 * F), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[Sqrt[N[(-2.0 * N[(F / B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])]]

Derivation

1. Split input into 3 regimes

2. if B < 2.10000000000000006e-42

   1. Initial program 21.1%

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

   3. Simplified 20.0%

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

   5. Taylor expanded in C around inf 14.5%

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

   6. Taylor expanded in C around inf 14.9%

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

      1. associate-*r* 14.8%

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

      2. mul-1-neg 14.8%

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

   8. Simplified 14.8%

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

   if 2.10000000000000006e-42 < B < 2.5999999999999999e174

   1. Initial program 34.9%

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

   3. Taylor expanded in C around 0 35.1%

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

      1. mul-1-neg 35.1%

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

      2. +-commutative 35.1%

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

      3. unpow2 35.1%

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

      4. unpow2 35.1%

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

      5. hypot-define 39.5%

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

   5. Simplified 39.5%

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

      1. add-exp-log 36.8%

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

      2. associate-*l/ 36.8%

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

      3. pow1/2 36.8%

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

      4. pow1/2 36.8%

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

      5. pow-prod-down 36.8%

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

   7. Applied egg-rr 36.8%

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

      1. rem-exp-log 39.7%

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

      2. clear-num 39.7%

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

      3. unpow1/2 39.7%

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

      4. associate-*r* 39.7%

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

   9. Applied egg-rr 39.7%

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

   if 2.5999999999999999e174 < 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 F around 0 0.0%

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

      1. pow1 0.0%

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

   5. Applied egg-rr 4.6%

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

      1. unpow1 4.6%

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

   7. Simplified 4.6%

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

   8. Taylor expanded in B around inf 54.0%

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

4. Final simplification 24.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 2.1 \cdot 10^{-42}:\\ \;\;\;\;\frac{\sqrt{-8 \cdot \left(\left(A \cdot C\right) \cdot \left(F \cdot \left(A + A\right)\right)\right)}}{4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 2.6 \cdot 10^{+174}:\\ \;\;\;\;\frac{-1}{\frac{B}{\sqrt{\left(A - \mathsf{hypot}\left(B, A\right)\right) \cdot \left(2 \cdot F\right)}}}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{-2 \cdot \frac{F}{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 46.1% accurate, 2.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 2.6 \cdot 10^{-41}:\\ \;\;\;\;\frac{\sqrt{-8 \cdot \left(\left(A \cdot C\right) \cdot \left(F \cdot \left(A + A\right)\right)\right)}}{4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B\_m \leq 3.9 \cdot 10^{+176}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(A - \mathsf{hypot}\left(B\_m, A\right)\right)\right)}}{-B\_m}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{-2 \cdot \frac{F}{B\_m}}\\ \end{array} \end{array} \]

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 2.6e-41)
   (/ (sqrt (* -8.0 (* (* A C) (* F (+ A A))))) (* 4.0 (* A C)))
   (if (<= B_m 3.9e+176)
     (/ (sqrt (* 2.0 (* F (- A (hypot B_m A))))) (- B_m))
     (- (sqrt (* -2.0 (/ F B_m)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 2.6e-41) {
		tmp = sqrt((-8.0 * ((A * C) * (F * (A + A))))) / (4.0 * (A * C));
	} else if (B_m <= 3.9e+176) {
		tmp = sqrt((2.0 * (F * (A - hypot(B_m, A))))) / -B_m;
	} else {
		tmp = -sqrt((-2.0 * (F / 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 <= 2.6e-41) {
		tmp = Math.sqrt((-8.0 * ((A * C) * (F * (A + A))))) / (4.0 * (A * C));
	} else if (B_m <= 3.9e+176) {
		tmp = Math.sqrt((2.0 * (F * (A - Math.hypot(B_m, A))))) / -B_m;
	} else {
		tmp = -Math.sqrt((-2.0 * (F / 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 <= 2.6e-41:
		tmp = math.sqrt((-8.0 * ((A * C) * (F * (A + A))))) / (4.0 * (A * C))
	elif B_m <= 3.9e+176:
		tmp = math.sqrt((2.0 * (F * (A - math.hypot(B_m, A))))) / -B_m
	else:
		tmp = -math.sqrt((-2.0 * (F / B_m)))
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 2.6e-41)
		tmp = Float64(sqrt(Float64(-8.0 * Float64(Float64(A * C) * Float64(F * Float64(A + A))))) / Float64(4.0 * Float64(A * C)));
	elseif (B_m <= 3.9e+176)
		tmp = Float64(sqrt(Float64(2.0 * Float64(F * Float64(A - hypot(B_m, A))))) / Float64(-B_m));
	else
		tmp = Float64(-sqrt(Float64(-2.0 * Float64(F / B_m))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (B_m <= 2.6e-41)
		tmp = sqrt((-8.0 * ((A * C) * (F * (A + A))))) / (4.0 * (A * C));
	elseif (B_m <= 3.9e+176)
		tmp = sqrt((2.0 * (F * (A - hypot(B_m, A))))) / -B_m;
	else
		tmp = -sqrt((-2.0 * (F / 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, 2.6e-41], N[(N[Sqrt[N[(-8.0 * N[(N[(A * C), $MachinePrecision] * N[(F * N[(A + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 3.9e+176], N[(N[Sqrt[N[(2.0 * N[(F * N[(A - N[Sqrt[B$95$m ^ 2 + A ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-B$95$m)), $MachinePrecision], (-N[Sqrt[N[(-2.0 * N[(F / B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])]]

Derivation

1. Split input into 3 regimes

2. if B < 2.5999999999999999e-41

   1. Initial program 21.1%

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

   3. Simplified 20.0%

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

   5. Taylor expanded in C around inf 14.5%

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

   6. Taylor expanded in C around inf 14.9%

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

      1. associate-*r* 14.8%

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

      2. mul-1-neg 14.8%

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

   8. Simplified 14.8%

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

   if 2.5999999999999999e-41 < B < 3.9000000000000001e176

   1. Initial program 34.9%

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

   3. Taylor expanded in C around 0 35.1%

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

      1. mul-1-neg 35.1%

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

      2. +-commutative 35.1%

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

      3. unpow2 35.1%

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

      4. unpow2 35.1%

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

      5. hypot-define 39.5%

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

   5. Simplified 39.5%

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

      1. neg-sub0 39.5%

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

      2. associate-*l/ 39.6%

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

      3. pow1/2 39.6%

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

      4. pow1/2 39.6%

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

      5. pow-prod-down 39.7%

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

   7. Applied egg-rr 39.7%

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

      1. neg-sub0 39.7%

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

      2. distribute-neg-frac2 39.7%

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

      3. unpow1/2 39.7%

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

   9. Simplified 39.7%

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

   if 3.9000000000000001e176 < 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 F around 0 0.0%

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

      1. pow1 0.0%

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

   5. Applied egg-rr 4.6%

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

      1. unpow1 4.6%

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

   7. Simplified 4.6%

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

   8. Taylor expanded in B around inf 54.0%

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

4. Final simplification 24.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 2.6 \cdot 10^{-41}:\\ \;\;\;\;\frac{\sqrt{-8 \cdot \left(\left(A \cdot C\right) \cdot \left(F \cdot \left(A + A\right)\right)\right)}}{4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 3.9 \cdot 10^{+176}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \left(A - \mathsf{hypot}\left(B, A\right)\right)\right)}}{-B}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{-2 \cdot \frac{F}{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 42.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 1.25 \cdot 10^{-23}:\\ \;\;\;\;\frac{\sqrt{-8 \cdot \left(\left(A \cdot C\right) \cdot \left(F \cdot \left(A + A\right)\right)\right)}}{4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B\_m \leq 7.4 \cdot 10^{+173}:\\ \;\;\;\;\left(\sqrt{2} \cdot \sqrt{B\_m \cdot \left(-F\right)}\right) \cdot \frac{-1}{B\_m}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{-2 \cdot \frac{F}{B\_m}}\\ \end{array} \end{array} \]

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 1.25e-23)
   (/ (sqrt (* -8.0 (* (* A C) (* F (+ A A))))) (* 4.0 (* A C)))
   (if (<= B_m 7.4e+173)
     (* (* (sqrt 2.0) (sqrt (* B_m (- F)))) (/ -1.0 B_m))
     (- (sqrt (* -2.0 (/ F B_m)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 1.25e-23) {
		tmp = sqrt((-8.0 * ((A * C) * (F * (A + A))))) / (4.0 * (A * C));
	} else if (B_m <= 7.4e+173) {
		tmp = (sqrt(2.0) * sqrt((B_m * -F))) * (-1.0 / B_m);
	} else {
		tmp = -sqrt((-2.0 * (F / B_m)));
	}
	return tmp;
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (b_m <= 1.25d-23) then
        tmp = sqrt(((-8.0d0) * ((a * c) * (f * (a + a))))) / (4.0d0 * (a * c))
    else if (b_m <= 7.4d+173) then
        tmp = (sqrt(2.0d0) * sqrt((b_m * -f))) * ((-1.0d0) / b_m)
    else
        tmp = -sqrt(((-2.0d0) * (f / b_m)))
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 1.25e-23) {
		tmp = Math.sqrt((-8.0 * ((A * C) * (F * (A + A))))) / (4.0 * (A * C));
	} else if (B_m <= 7.4e+173) {
		tmp = (Math.sqrt(2.0) * Math.sqrt((B_m * -F))) * (-1.0 / B_m);
	} else {
		tmp = -Math.sqrt((-2.0 * (F / 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 <= 1.25e-23:
		tmp = math.sqrt((-8.0 * ((A * C) * (F * (A + A))))) / (4.0 * (A * C))
	elif B_m <= 7.4e+173:
		tmp = (math.sqrt(2.0) * math.sqrt((B_m * -F))) * (-1.0 / B_m)
	else:
		tmp = -math.sqrt((-2.0 * (F / 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 <= 1.25e-23)
		tmp = Float64(sqrt(Float64(-8.0 * Float64(Float64(A * C) * Float64(F * Float64(A + A))))) / Float64(4.0 * Float64(A * C)));
	elseif (B_m <= 7.4e+173)
		tmp = Float64(Float64(sqrt(2.0) * sqrt(Float64(B_m * Float64(-F)))) * Float64(-1.0 / B_m));
	else
		tmp = Float64(-sqrt(Float64(-2.0 * Float64(F / 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 <= 1.25e-23)
		tmp = sqrt((-8.0 * ((A * C) * (F * (A + A))))) / (4.0 * (A * C));
	elseif (B_m <= 7.4e+173)
		tmp = (sqrt(2.0) * sqrt((B_m * -F))) * (-1.0 / B_m);
	else
		tmp = -sqrt((-2.0 * (F / 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, 1.25e-23], N[(N[Sqrt[N[(-8.0 * N[(N[(A * C), $MachinePrecision] * N[(F * N[(A + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 7.4e+173], N[(N[(N[Sqrt[2.0], $MachinePrecision] * N[Sqrt[N[(B$95$m * (-F)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-1.0 / B$95$m), $MachinePrecision]), $MachinePrecision], (-N[Sqrt[N[(-2.0 * N[(F / B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])]]

Derivation

1. Split input into 3 regimes

2. if B < 1.2500000000000001e-23

   1. Initial program 20.6%

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

   3. Simplified 20.1%

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

   5. Taylor expanded in C around inf 14.3%

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

   6. Taylor expanded in C around inf 15.7%

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

      1. associate-*r* 15.6%

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

      2. mul-1-neg 15.6%

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

   8. Simplified 15.6%

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

   if 1.2500000000000001e-23 < B < 7.39999999999999972e173

   1. Initial program 38.5%

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

   3. Taylor expanded in C around 0 38.7%

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

      1. mul-1-neg 38.7%

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

      2. +-commutative 38.7%

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

      3. unpow2 38.7%

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

      4. unpow2 38.7%

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

      5. hypot-define 43.5%

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

   5. Simplified 43.5%

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

      1. clear-num 43.5%

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

      2. inv-pow 43.5%

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

   7. Applied egg-rr 43.5%

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

      1. unpow-1 43.5%

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

   9. Simplified 43.5%

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

       1. distribute-rgt-neg-in 43.5%

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

       2. associate-/r/ 43.4%

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

   11. Applied egg-rr 43.4%

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

       1. associate-*l* 43.6%

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

   13. Simplified 43.6%

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

   14. Taylor expanded in A around 0 40.0%

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

       1. mul-1-neg 40.0%

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

   16. Simplified 40.0%

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

   if 7.39999999999999972e173 < 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 F around 0 0.0%

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

      1. pow1 0.0%

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

   5. Applied egg-rr 4.6%

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

      1. unpow1 4.6%

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

   7. Simplified 4.6%

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

   8. Taylor expanded in B around inf 54.0%

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

4. Final simplification 24.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.25 \cdot 10^{-23}:\\ \;\;\;\;\frac{\sqrt{-8 \cdot \left(\left(A \cdot C\right) \cdot \left(F \cdot \left(A + A\right)\right)\right)}}{4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 7.4 \cdot 10^{+173}:\\ \;\;\;\;\left(\sqrt{2} \cdot \sqrt{B \cdot \left(-F\right)}\right) \cdot \frac{-1}{B}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{-2 \cdot \frac{F}{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 42.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 4.5 \cdot 10^{-24}:\\ \;\;\;\;\frac{\sqrt{-8 \cdot \left(\left(A \cdot C\right) \cdot \left(F \cdot \left(A + A\right)\right)\right)}}{4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B\_m \leq 6 \cdot 10^{+174}:\\ \;\;\;\;\sqrt{B\_m \cdot \left(-F\right)} \cdot \frac{\sqrt{2}}{-B\_m}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{-2 \cdot \frac{F}{B\_m}}\\ \end{array} \end{array} \]

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 4.5e-24)
   (/ (sqrt (* -8.0 (* (* A C) (* F (+ A A))))) (* 4.0 (* A C)))
   (if (<= B_m 6e+174)
     (* (sqrt (* B_m (- F))) (/ (sqrt 2.0) (- B_m)))
     (- (sqrt (* -2.0 (/ F B_m)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 4.5e-24) {
		tmp = sqrt((-8.0 * ((A * C) * (F * (A + A))))) / (4.0 * (A * C));
	} else if (B_m <= 6e+174) {
		tmp = sqrt((B_m * -F)) * (sqrt(2.0) / -B_m);
	} else {
		tmp = -sqrt((-2.0 * (F / B_m)));
	}
	return tmp;
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (b_m <= 4.5d-24) then
        tmp = sqrt(((-8.0d0) * ((a * c) * (f * (a + a))))) / (4.0d0 * (a * c))
    else if (b_m <= 6d+174) then
        tmp = sqrt((b_m * -f)) * (sqrt(2.0d0) / -b_m)
    else
        tmp = -sqrt(((-2.0d0) * (f / b_m)))
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 4.5e-24) {
		tmp = Math.sqrt((-8.0 * ((A * C) * (F * (A + A))))) / (4.0 * (A * C));
	} else if (B_m <= 6e+174) {
		tmp = Math.sqrt((B_m * -F)) * (Math.sqrt(2.0) / -B_m);
	} else {
		tmp = -Math.sqrt((-2.0 * (F / 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 <= 4.5e-24:
		tmp = math.sqrt((-8.0 * ((A * C) * (F * (A + A))))) / (4.0 * (A * C))
	elif B_m <= 6e+174:
		tmp = math.sqrt((B_m * -F)) * (math.sqrt(2.0) / -B_m)
	else:
		tmp = -math.sqrt((-2.0 * (F / 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 <= 4.5e-24)
		tmp = Float64(sqrt(Float64(-8.0 * Float64(Float64(A * C) * Float64(F * Float64(A + A))))) / Float64(4.0 * Float64(A * C)));
	elseif (B_m <= 6e+174)
		tmp = Float64(sqrt(Float64(B_m * Float64(-F))) * Float64(sqrt(2.0) / Float64(-B_m)));
	else
		tmp = Float64(-sqrt(Float64(-2.0 * Float64(F / 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 <= 4.5e-24)
		tmp = sqrt((-8.0 * ((A * C) * (F * (A + A))))) / (4.0 * (A * C));
	elseif (B_m <= 6e+174)
		tmp = sqrt((B_m * -F)) * (sqrt(2.0) / -B_m);
	else
		tmp = -sqrt((-2.0 * (F / 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, 4.5e-24], N[(N[Sqrt[N[(-8.0 * N[(N[(A * C), $MachinePrecision] * N[(F * N[(A + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[B$95$m, 6e+174], N[(N[Sqrt[N[(B$95$m * (-F)), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision]), $MachinePrecision], (-N[Sqrt[N[(-2.0 * N[(F / B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])]]

Derivation

1. Split input into 3 regimes

2. if B < 4.4999999999999997e-24

   1. Initial program 20.6%

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

   3. Simplified 20.1%

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

   5. Taylor expanded in C around inf 14.3%

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

   6. Taylor expanded in C around inf 15.7%

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

      1. associate-*r* 15.6%

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

      2. mul-1-neg 15.6%

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

   8. Simplified 15.6%

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

   if 4.4999999999999997e-24 < B < 6e174

   1. Initial program 38.5%

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

   3. Taylor expanded in C around 0 38.7%

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

      1. mul-1-neg 38.7%

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

      2. +-commutative 38.7%

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

      3. unpow2 38.7%

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

      4. unpow2 38.7%

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

      5. hypot-define 43.5%

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

   5. Simplified 43.5%

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

   6. Taylor expanded in A around 0 40.0%

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

      1. mul-1-neg 40.0%

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

   8. Simplified 40.0%

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

   if 6e174 < 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 F around 0 0.0%

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

      1. pow1 0.0%

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

   5. Applied egg-rr 4.6%

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

      1. unpow1 4.6%

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

   7. Simplified 4.6%

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

   8. Taylor expanded in B around inf 54.0%

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

4. Final simplification 24.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 4.5 \cdot 10^{-24}:\\ \;\;\;\;\frac{\sqrt{-8 \cdot \left(\left(A \cdot C\right) \cdot \left(F \cdot \left(A + A\right)\right)\right)}}{4 \cdot \left(A \cdot C\right)}\\ \mathbf{elif}\;B \leq 6 \cdot 10^{+174}:\\ \;\;\;\;\sqrt{B \cdot \left(-F\right)} \cdot \frac{\sqrt{2}}{-B}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{-2 \cdot \frac{F}{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 43.0% accurate, 5.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 4.6 \cdot 10^{-23}:\\ \;\;\;\;\frac{\sqrt{-8 \cdot \left(\left(A \cdot C\right) \cdot \left(F \cdot \left(A + A\right)\right)\right)}}{4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{-2 \cdot \frac{F}{B\_m}}\\ \end{array} \end{array} \]

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 4.6e-23)
   (/ (sqrt (* -8.0 (* (* A C) (* F (+ A A))))) (* 4.0 (* A C)))
   (- (sqrt (* -2.0 (/ F B_m))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 4.6e-23) {
		tmp = sqrt((-8.0 * ((A * C) * (F * (A + A))))) / (4.0 * (A * C));
	} else {
		tmp = -sqrt((-2.0 * (F / B_m)));
	}
	return tmp;
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (b_m <= 4.6d-23) then
        tmp = sqrt(((-8.0d0) * ((a * c) * (f * (a + a))))) / (4.0d0 * (a * c))
    else
        tmp = -sqrt(((-2.0d0) * (f / b_m)))
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 4.6e-23) {
		tmp = Math.sqrt((-8.0 * ((A * C) * (F * (A + A))))) / (4.0 * (A * C));
	} else {
		tmp = -Math.sqrt((-2.0 * (F / 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 <= 4.6e-23:
		tmp = math.sqrt((-8.0 * ((A * C) * (F * (A + A))))) / (4.0 * (A * C))
	else:
		tmp = -math.sqrt((-2.0 * (F / 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 <= 4.6e-23)
		tmp = Float64(sqrt(Float64(-8.0 * Float64(Float64(A * C) * Float64(F * Float64(A + A))))) / Float64(4.0 * Float64(A * C)));
	else
		tmp = Float64(-sqrt(Float64(-2.0 * Float64(F / 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 <= 4.6e-23)
		tmp = sqrt((-8.0 * ((A * C) * (F * (A + A))))) / (4.0 * (A * C));
	else
		tmp = -sqrt((-2.0 * (F / 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, 4.6e-23], N[(N[Sqrt[N[(-8.0 * N[(N[(A * C), $MachinePrecision] * N[(F * N[(A + A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], (-N[Sqrt[N[(-2.0 * N[(F / B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if B < 4.6000000000000002e-23

   1. Initial program 20.6%

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

   3. Simplified 20.1%

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

   5. Taylor expanded in C around inf 14.3%

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

   6. Taylor expanded in C around inf 15.7%

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

      1. associate-*r* 15.6%

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

      2. mul-1-neg 15.6%

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

   8. Simplified 15.6%

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

   if 4.6000000000000002e-23 < B

   1. Initial program 21.2%

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

   3. Taylor expanded in F around 0 21.2%

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

      1. pow1 21.2%

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

   5. Applied egg-rr 33.7%

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

      1. unpow1 33.7%

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

   7. Simplified 33.7%

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

   8. Taylor expanded in B around inf 45.2%

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

4. Final simplification 24.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 4.6 \cdot 10^{-23}:\\ \;\;\;\;\frac{\sqrt{-8 \cdot \left(\left(A \cdot C\right) \cdot \left(F \cdot \left(A + A\right)\right)\right)}}{4 \cdot \left(A \cdot C\right)}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{-2 \cdot \frac{F}{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 40.6% accurate, 5.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 1.4 \cdot 10^{-7}:\\ \;\;\;\;-\sqrt{\frac{F}{-C}}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{-2 \cdot \frac{F}{B\_m}}\\ \end{array} \end{array} \]

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 1.4e-7) (- (sqrt (/ F (- C)))) (- (sqrt (* -2.0 (/ F B_m))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 1.4e-7) {
		tmp = -sqrt((F / -C));
	} else {
		tmp = -sqrt((-2.0 * (F / B_m)));
	}
	return tmp;
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (b_m <= 1.4d-7) then
        tmp = -sqrt((f / -c))
    else
        tmp = -sqrt(((-2.0d0) * (f / b_m)))
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 1.4e-7) {
		tmp = -Math.sqrt((F / -C));
	} else {
		tmp = -Math.sqrt((-2.0 * (F / 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 <= 1.4e-7:
		tmp = -math.sqrt((F / -C))
	else:
		tmp = -math.sqrt((-2.0 * (F / 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 <= 1.4e-7)
		tmp = Float64(-sqrt(Float64(F / Float64(-C))));
	else
		tmp = Float64(-sqrt(Float64(-2.0 * Float64(F / 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 <= 1.4e-7)
		tmp = -sqrt((F / -C));
	else
		tmp = -sqrt((-2.0 * (F / 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, 1.4e-7], (-N[Sqrt[N[(F / (-C)), $MachinePrecision]], $MachinePrecision]), (-N[Sqrt[N[(-2.0 * N[(F / B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if B < 1.4000000000000001e-7

   1. Initial program 21.3%

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

   3. Taylor expanded in F around 0 18.3%

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

      1. pow1 18.3%

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

   5. Applied egg-rr 30.3%

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

      1. unpow1 30.3%

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

   7. Simplified 30.3%

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

   8. Taylor expanded in A around -inf 15.6%

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

      1. mul-1-neg 15.6%

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

   10. Simplified 15.6%

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

   if 1.4000000000000001e-7 < B

   1. Initial program 19.6%

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

   3. Taylor expanded in F around 0 19.6%

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

      1. pow1 19.6%

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

   5. Applied egg-rr 32.8%

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

      1. unpow1 32.8%

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

   7. Simplified 32.8%

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

   8. Taylor expanded in B around inf 46.0%

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

4. Final simplification 24.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1.4 \cdot 10^{-7}:\\ \;\;\;\;-\sqrt{\frac{F}{-C}}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{-2 \cdot \frac{F}{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 27.5% accurate, 6.0× speedup

Math

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

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 (- 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) {
	return -sqrt((F / -C));
}

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 / -c))
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 / -C));
}

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 / -C))

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	return Float64(-sqrt(Float64(F / Float64(-C))))
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 / -C));
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 / (-C)), $MachinePrecision]], $MachinePrecision])

Derivation

1. Initial program 20.8%

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

3. Taylor expanded in F around 0 18.7%

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

   1. pow1 18.7%

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

5. Applied egg-rr 31.0%

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

   1. unpow1 31.0%

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

7. Simplified 31.0%

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

8. Taylor expanded in A around -inf 13.0%

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

   1. mul-1-neg 13.0%

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

10. Simplified 13.0%

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

11. Final simplification 13.0%

    \[\leadsto -\sqrt{\frac{F}{-C}} \]
12. Add Preprocessing

Alternative 13: 1.3% accurate, 6.1× speedup

Math

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

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 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) {
	return -sqrt((F / C));
}

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 / c))
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 / C));
}

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 / C))

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	return Float64(-sqrt(Float64(F / C)))
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 / C));
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 / C), $MachinePrecision]], $MachinePrecision])

Derivation

1. Initial program 20.8%

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

3. Taylor expanded in F around 0 18.7%

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

   1. pow1 18.7%

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

5. Applied egg-rr 31.0%

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

   1. unpow1 31.0%

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

7. Simplified 31.0%

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

8. Taylor expanded in A around -inf 13.0%

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

   1. mul-1-neg 13.0%

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

10. Simplified 13.0%

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

    1. neg-sub0 13.0%

       \[\leadsto \color{blue}{0 - \sqrt{-\frac{F}{C}}} \]

    2. add-sqr-sqrt 13.0%

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

    3. sqrt-unprod 9.9%

       \[\leadsto 0 - \sqrt{\color{blue}{\sqrt{\left(-\frac{F}{C}\right) \cdot \left(-\frac{F}{C}\right)}}} \]

    4. sqr-neg 9.9%

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

    5. sqrt-unprod 2.9%

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

    6. add-sqr-sqrt 2.9%

       \[\leadsto 0 - \sqrt{\color{blue}{\frac{F}{C}}} \]

12. Applied egg-rr 2.9%

    \[\leadsto \color{blue}{0 - \sqrt{\frac{F}{C}}} \]
13. Step-by-step derivation

    1. neg-sub0 2.9%

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

14. Simplified 2.9%

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

Alternative 14: 0.9% accurate, 6.2× speedup

Math

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

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 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) {
	return sqrt((F / C));
}

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 / c))
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 / C));
}

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 / C))

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 / C))
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 / C));
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 / C), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 20.8%

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

3. Taylor expanded in F around 0 18.7%

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

   1. pow1 18.7%

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

5. Applied egg-rr 31.0%

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

   1. unpow1 31.0%

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

7. Simplified 31.0%

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

8. Taylor expanded in A around -inf 13.0%

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

   1. mul-1-neg 13.0%

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

10. Simplified 13.0%

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

    1. *-un-lft-identity 13.0%

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

    2. add-sqr-sqrt 0.7%

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

    3. sqrt-unprod 9.0%

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

    4. sqr-neg 9.0%

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

    5. sqrt-unprod 6.9%

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

    6. sqr-neg 6.9%

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

    7. sqrt-unprod 1.2%

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

    8. add-sqr-sqrt 1.2%

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

12. Applied egg-rr 1.2%

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

    1. *-lft-identity 1.2%

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

14. Simplified 1.2%

    \[\leadsto \color{blue}{\sqrt{\frac{F}{C}}} \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024118 
(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))))