ABCF->ab-angle a

Percentage Accurate: 20.6% → 56.9%

Time: 21.5s

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: 20.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 56.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := -t\_0\\ t_2 := \left(4 \cdot A\right) \cdot C\\ t_3 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_2\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_2 - {B\_m}^{2}}\\ t_4 := \frac{\sqrt{2 \cdot \left(F \cdot t\_0\right)} \cdot \sqrt{A + \left(C + \mathsf{hypot}\left(A - C, B\_m\right)\right)}}{t\_1}\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;\frac{\sqrt{F}}{-\sqrt{-A}}\\ \mathbf{elif}\;t\_3 \leq -1 \cdot 10^{-201}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;\frac{\sqrt{A \cdot -4} \cdot \sqrt{\left(C \cdot F\right) \cdot \left(4 \cdot C\right)}}{t\_1}\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;t\_4\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-\sqrt{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* A (* C -4.0))))
        (t_1 (- t_0))
        (t_2 (* (* 4.0 A) C))
        (t_3
         (/
          (sqrt
           (*
            (* 2.0 (* (- (pow B_m 2.0) t_2) F))
            (+ (+ A C) (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))))))
          (- t_2 (pow B_m 2.0))))
        (t_4
         (/
          (* (sqrt (* 2.0 (* F t_0))) (sqrt (+ A (+ C (hypot (- A C) B_m)))))
          t_1)))
   (if (<= t_3 (- INFINITY))
     (/ (sqrt F) (- (sqrt (- A))))
     (if (<= t_3 -1e-201)
       t_4
       (if (<= t_3 0.0)
         (/ (* (sqrt (* A -4.0)) (sqrt (* (* C F) (* 4.0 C)))) t_1)
         (if (<= t_3 INFINITY) t_4 (/ (sqrt (* 2.0 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 = -t_0;
	double t_2 = (4.0 * A) * C;
	double t_3 = sqrt(((2.0 * ((pow(B_m, 2.0) - t_2) * F)) * ((A + C) + sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / (t_2 - pow(B_m, 2.0));
	double t_4 = (sqrt((2.0 * (F * t_0))) * sqrt((A + (C + hypot((A - C), B_m))))) / t_1;
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = sqrt(F) / -sqrt(-A);
	} else if (t_3 <= -1e-201) {
		tmp = t_4;
	} else if (t_3 <= 0.0) {
		tmp = (sqrt((A * -4.0)) * sqrt(((C * F) * (4.0 * C)))) / t_1;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = t_4;
	} else {
		tmp = sqrt((2.0 * 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 = Float64(-t_0)
	t_2 = Float64(Float64(4.0 * A) * C)
	t_3 = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_2) * F)) * Float64(Float64(A + C) + sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / Float64(t_2 - (B_m ^ 2.0)))
	t_4 = Float64(Float64(sqrt(Float64(2.0 * Float64(F * t_0))) * sqrt(Float64(A + Float64(C + hypot(Float64(A - C), B_m))))) / t_1)
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(sqrt(F) / Float64(-sqrt(Float64(-A))));
	elseif (t_3 <= -1e-201)
		tmp = t_4;
	elseif (t_3 <= 0.0)
		tmp = Float64(Float64(sqrt(Float64(A * -4.0)) * sqrt(Float64(Float64(C * F) * Float64(4.0 * C)))) / t_1);
	elseif (t_3 <= Inf)
		tmp = t_4;
	else
		tmp = Float64(sqrt(Float64(2.0 * 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 = (-t$95$0)}, Block[{t$95$2 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$3 = N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$2), $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$2 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[Sqrt[N[(2.0 * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(A + N[(C + N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], N[(N[Sqrt[F], $MachinePrecision] / (-N[Sqrt[(-A)], $MachinePrecision])), $MachinePrecision], If[LessEqual[t$95$3, -1e-201], t$95$4, If[LessEqual[t$95$3, 0.0], N[(N[(N[Sqrt[N[(A * -4.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(N[(C * F), $MachinePrecision] * N[(4.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[t$95$3, Infinity], t$95$4, N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] / (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision]]]]]]]]]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 3.0%

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

   3. Taylor expanded in F around 0 15.7%

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

   5. Simplified 48.7%

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

      1. *-commutative 48.7%

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

      2. pow1/2 48.7%

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

      3. pow1/2 48.7%

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

      4. pow-prod-down 49.0%

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

      5. +-commutative 49.0%

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

      6. +-commutative 49.0%

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

      7. *-commutative 49.0%

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

   7. Applied egg-rr 49.0%

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

      1. unpow1/2 49.0%

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

   9. Simplified 49.0%

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

   10. Taylor expanded in A around -inf 32.3%

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

       1. mul-1-neg 32.3%

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

   12. Simplified 32.3%

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

       1. distribute-neg-frac2 32.3%

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

       2. sqrt-div 42.7%

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

   14. Applied egg-rr 42.7%

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

   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))) < -9.99999999999999946e-202 or -0.0 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < +inf.0

   1. Initial program 75.9%

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

   3. Simplified 83.8%

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

      1. associate-*r* 83.8%

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

      2. associate-+r+ 83.8%

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

      3. hypot-undefine 75.9%

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

      4. unpow2 75.9%

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

      5. unpow2 75.9%

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

      6. +-commutative 75.9%

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

      7. sqrt-prod 76.0%

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

      8. *-commutative 76.0%

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

      9. associate-+l+ 76.0%

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

   6. Applied egg-rr 93.5%

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

   if -9.99999999999999946e-202 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -0.0

   1. Initial program 21.9%

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

   3. Simplified 21.8%

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

   5. Taylor expanded in B around 0 22.0%

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

      1. associate-*r* 22.0%

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

   7. Simplified 22.0%

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

   8. Taylor expanded in A around -inf 35.8%

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

      1. pow1/2 35.8%

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

      2. associate-*l* 38.9%

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

      3. unpow-prod-down 40.4%

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

      4. pow1/2 40.4%

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

      5. *-commutative 40.4%

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

      6. associate-*r* 40.4%

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

      7. metadata-eval 40.4%

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

   10. Applied egg-rr 40.4%

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

       1. unpow1/2 40.4%

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

   12. Simplified 40.4%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in B around inf 13.5%

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

      1. mul-1-neg 13.5%

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

      2. *-commutative 13.5%

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

   5. Simplified 13.5%

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

      1. sqrt-div 20.0%

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

   7. Applied egg-rr 20.0%

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

      1. associate-*r/ 19.9%

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

      2. pow1/2 19.9%

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

      3. pow1/2 19.9%

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

      4. pow-prod-down 20.0%

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

      5. *-commutative 20.0%

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

      6. pow1/2 20.0%

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

      7. *-commutative 20.0%

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

   9. Applied egg-rr 20.0%

      \[\leadsto -\color{blue}{\frac{\sqrt{2 \cdot F}}{\sqrt{B}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 41.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -\infty:\\ \;\;\;\;\frac{\sqrt{F}}{-\sqrt{-A}}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -1 \cdot 10^{-201}:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)} \cdot \sqrt{A + \left(C + \mathsf{hypot}\left(A - C, B\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq 0:\\ \;\;\;\;\frac{\sqrt{A \cdot -4} \cdot \sqrt{\left(C \cdot F\right) \cdot \left(4 \cdot C\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)} \cdot \sqrt{A + \left(C + \mathsf{hypot}\left(A - C, B\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 52.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 9 \cdot 10^{+55}:\\ \;\;\;\;\frac{{\left(\mathsf{fma}\left(B\_m, B\_m, -4 \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot \left(4 \cdot C\right)\right)\right)}^{0.5}}{-\mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;{B\_m}^{2} \leq 10^{+293}:\\ \;\;\;\;\sqrt{\frac{2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B\_m, A - C\right)\right)\right)}{\mathsf{fma}\left(-4, A \cdot C, {B\_m}^{2}\right)}} \cdot \left(-\sqrt{F}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-\sqrt{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 9e+55)
   (/
    (pow (* (fma B_m B_m (* -4.0 (* A C))) (* F (* 4.0 C))) 0.5)
    (- (fma B_m B_m (* A (* C -4.0)))))
   (if (<= (pow B_m 2.0) 1e+293)
     (*
      (sqrt
       (/
        (* 2.0 (+ A (+ C (hypot B_m (- A C)))))
        (fma -4.0 (* A C) (pow B_m 2.0))))
      (- (sqrt F)))
     (/ (sqrt (* 2.0 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) <= 9e+55) {
		tmp = pow((fma(B_m, B_m, (-4.0 * (A * C))) * (F * (4.0 * C))), 0.5) / -fma(B_m, B_m, (A * (C * -4.0)));
	} else if (pow(B_m, 2.0) <= 1e+293) {
		tmp = sqrt(((2.0 * (A + (C + hypot(B_m, (A - C))))) / fma(-4.0, (A * C), pow(B_m, 2.0)))) * -sqrt(F);
	} else {
		tmp = sqrt((2.0 * 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) <= 9e+55)
		tmp = Float64((Float64(fma(B_m, B_m, Float64(-4.0 * Float64(A * C))) * Float64(F * Float64(4.0 * C))) ^ 0.5) / Float64(-fma(B_m, B_m, Float64(A * Float64(C * -4.0)))));
	elseif ((B_m ^ 2.0) <= 1e+293)
		tmp = Float64(sqrt(Float64(Float64(2.0 * Float64(A + Float64(C + hypot(B_m, Float64(A - C))))) / fma(-4.0, Float64(A * C), (B_m ^ 2.0)))) * Float64(-sqrt(F)));
	else
		tmp = Float64(sqrt(Float64(2.0 * 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], 9e+55], N[(N[Power[N[(N[(B$95$m * B$95$m + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(F * N[(4.0 * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision] / (-N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e+293], N[(N[Sqrt[N[(N[(2.0 * N[(A + N[(C + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-4.0 * N[(A * C), $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * (-N[Sqrt[F], $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] / (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 8.99999999999999996e55

   1. Initial program 23.2%

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

   3. Simplified 33.8%

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

   5. Taylor expanded in A around -inf 30.6%

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

      1. *-commutative 30.6%

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

   7. Simplified 30.6%

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

      1. pow1/2 30.6%

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

      2. associate-*l* 30.0%

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

      3. associate-*r* 30.0%

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

   9. Applied egg-rr 30.0%

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

   if 8.99999999999999996e55 < (pow.f64 B #s(literal 2 binary64)) < 9.9999999999999992e292

   1. Initial program 33.3%

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

   3. Taylor expanded in F around 0 32.0%

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

   5. Simplified 57.6%

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

      1. *-commutative 57.6%

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

      2. pow1/2 57.6%

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

      3. pow1/2 57.6%

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

      4. pow-prod-down 57.8%

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

      5. +-commutative 57.8%

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

      6. +-commutative 57.8%

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

      7. *-commutative 57.8%

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

   7. Applied egg-rr 57.8%

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

      1. unpow1/2 57.8%

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

   9. Simplified 57.8%

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

       1. pow1/2 57.8%

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

       2. associate-*l* 57.8%

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

       3. unpow-prod-down 60.2%

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

       4. pow1/2 60.2%

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

   11. Applied egg-rr 60.2%

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

       1. unpow1/2 60.2%

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

       2. associate-*l/ 60.2%

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

       3. +-commutative 60.2%

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

       4. hypot-undefine 49.5%

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

       5. unpow2 49.5%

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

       6. unpow2 49.5%

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

       7. associate-+r+ 49.4%

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

       8. unpow2 49.4%

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

       9. unpow2 49.4%

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

       10. hypot-undefine 60.1%

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

   13. Simplified 60.1%

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

   if 9.9999999999999992e292 < (pow.f64 B #s(literal 2 binary64))

   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 B around inf 22.5%

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

      1. mul-1-neg 22.5%

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

      2. *-commutative 22.5%

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

   5. Simplified 22.5%

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

      1. sqrt-div 34.0%

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

   7. Applied egg-rr 34.0%

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

      1. associate-*r/ 33.8%

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

      2. pow1/2 33.8%

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

      3. pow1/2 33.8%

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

      4. pow-prod-down 34.0%

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

      5. *-commutative 34.0%

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

      6. pow1/2 34.0%

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

      7. *-commutative 34.0%

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

   9. Applied egg-rr 34.0%

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

4. Final simplification 36.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 9 \cdot 10^{+55}:\\ \;\;\;\;\frac{{\left(\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot \left(4 \cdot C\right)\right)\right)}^{0.5}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;{B}^{2} \leq 10^{+293}:\\ \;\;\;\;\sqrt{\frac{2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)}{\mathsf{fma}\left(-4, A \cdot C, {B}^{2}\right)}} \cdot \left(-\sqrt{F}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 52.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 2 \cdot 10^{-77}:\\ \;\;\;\;\frac{\sqrt{\left(4 \cdot C\right) \cdot \left(F \cdot t\_0\right)}}{-t\_0}\\ \mathbf{elif}\;{B\_m}^{2} \leq 5 \cdot 10^{+166}:\\ \;\;\;\;-\sqrt{2 \cdot \left(F \cdot \frac{\left(A + C\right) + \mathsf{hypot}\left(B\_m, A - C\right)}{\mathsf{fma}\left(-4, A \cdot C, {B\_m}^{2}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-\sqrt{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* A (* C -4.0)))))
   (if (<= (pow B_m 2.0) 2e-77)
     (/ (sqrt (* (* 4.0 C) (* F t_0))) (- t_0))
     (if (<= (pow B_m 2.0) 5e+166)
       (-
        (sqrt
         (*
          2.0
          (*
           F
           (/
            (+ (+ A C) (hypot B_m (- A C)))
            (fma -4.0 (* A C) (pow B_m 2.0)))))))
       (/ (sqrt (* 2.0 F)) (- (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 tmp;
	if (pow(B_m, 2.0) <= 2e-77) {
		tmp = sqrt(((4.0 * C) * (F * t_0))) / -t_0;
	} else if (pow(B_m, 2.0) <= 5e+166) {
		tmp = -sqrt((2.0 * (F * (((A + C) + hypot(B_m, (A - C))) / fma(-4.0, (A * C), pow(B_m, 2.0))))));
	} else {
		tmp = sqrt((2.0 * F)) / -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)))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 2e-77)
		tmp = Float64(sqrt(Float64(Float64(4.0 * C) * Float64(F * t_0))) / Float64(-t_0));
	elseif ((B_m ^ 2.0) <= 5e+166)
		tmp = Float64(-sqrt(Float64(2.0 * Float64(F * Float64(Float64(Float64(A + C) + hypot(B_m, Float64(A - C))) / fma(-4.0, Float64(A * C), (B_m ^ 2.0)))))));
	else
		tmp = Float64(sqrt(Float64(2.0 * 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]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e-77], N[(N[Sqrt[N[(N[(4.0 * C), $MachinePrecision] * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 5e+166], (-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[(-4.0 * N[(A * C), $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] / (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 1.9999999999999999e-77

   1. Initial program 21.9%

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

   3. Simplified 31.4%

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

   5. Taylor expanded in A around -inf 31.0%

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

      1. *-commutative 31.0%

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

   7. Simplified 31.0%

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

   if 1.9999999999999999e-77 < (pow.f64 B #s(literal 2 binary64)) < 5.0000000000000002e166

   1. Initial program 33.7%

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

   3. Taylor expanded in F around 0 31.9%

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

   5. Simplified 53.4%

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

      1. *-commutative 53.4%

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

      2. pow1/2 53.4%

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

      3. pow1/2 53.4%

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

      4. pow-prod-down 53.6%

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

      5. +-commutative 53.6%

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

      6. +-commutative 53.6%

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

      7. *-commutative 53.6%

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

   7. Applied egg-rr 53.6%

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

      1. unpow1/2 53.6%

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

   9. Simplified 53.6%

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

   if 5.0000000000000002e166 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 8.9%

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

   3. Taylor expanded in B around inf 22.7%

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

      1. mul-1-neg 22.7%

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

      2. *-commutative 22.7%

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

   5. Simplified 22.7%

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

      1. sqrt-div 36.6%

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

   7. Applied egg-rr 36.6%

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

      1. associate-*r/ 36.5%

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

      2. pow1/2 36.5%

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

      3. pow1/2 36.5%

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

      4. pow-prod-down 36.7%

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

      5. *-commutative 36.7%

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

      6. pow1/2 36.7%

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

      7. *-commutative 36.7%

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

   9. Applied egg-rr 36.7%

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

4. Final simplification 36.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 2 \cdot 10^{-77}:\\ \;\;\;\;\frac{\sqrt{\left(4 \cdot C\right) \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;{B}^{2} \leq 5 \cdot 10^{+166}:\\ \;\;\;\;-\sqrt{2 \cdot \left(F \cdot \frac{\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)}{\mathsf{fma}\left(-4, A \cdot C, {B}^{2}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 49.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 9 \cdot 10^{+55}:\\ \;\;\;\;\frac{{\left(\mathsf{fma}\left(B\_m, B\_m, -4 \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot \left(4 \cdot C\right)\right)\right)}^{0.5}}{-\mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-\sqrt{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 9e+55)
   (/
    (pow (* (fma B_m B_m (* -4.0 (* A C))) (* F (* 4.0 C))) 0.5)
    (- (fma B_m B_m (* A (* C -4.0)))))
   (/ (sqrt (* 2.0 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) <= 9e+55) {
		tmp = pow((fma(B_m, B_m, (-4.0 * (A * C))) * (F * (4.0 * C))), 0.5) / -fma(B_m, B_m, (A * (C * -4.0)));
	} else {
		tmp = sqrt((2.0 * 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) <= 9e+55)
		tmp = Float64((Float64(fma(B_m, B_m, Float64(-4.0 * Float64(A * C))) * Float64(F * Float64(4.0 * C))) ^ 0.5) / Float64(-fma(B_m, B_m, Float64(A * Float64(C * -4.0)))));
	else
		tmp = Float64(sqrt(Float64(2.0 * 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], 9e+55], N[(N[Power[N[(N[(B$95$m * B$95$m + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(F * N[(4.0 * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.5], $MachinePrecision] / (-N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] / (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 8.99999999999999996e55

   1. Initial program 23.2%

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

   3. Simplified 33.8%

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

   5. Taylor expanded in A around -inf 30.6%

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

      1. *-commutative 30.6%

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

   7. Simplified 30.6%

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

      1. pow1/2 30.6%

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

      2. associate-*l* 30.0%

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

      3. associate-*r* 30.0%

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

   9. Applied egg-rr 30.0%

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

   if 8.99999999999999996e55 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 13.9%

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

   3. Taylor expanded in B around inf 22.8%

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

      1. mul-1-neg 22.8%

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

      2. *-commutative 22.8%

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

   5. Simplified 22.8%

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

      1. sqrt-div 34.5%

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

   7. Applied egg-rr 34.5%

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

      1. associate-*r/ 34.4%

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

      2. pow1/2 34.4%

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

      3. pow1/2 34.4%

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

      4. pow-prod-down 34.6%

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

      5. *-commutative 34.6%

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

      6. pow1/2 34.6%

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

      7. *-commutative 34.6%

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

   9. Applied egg-rr 34.6%

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

4. Final simplification 32.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 9 \cdot 10^{+55}:\\ \;\;\;\;\frac{{\left(\mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right) \cdot \left(F \cdot \left(4 \cdot C\right)\right)\right)}^{0.5}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 48.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 9 \cdot 10^{+55}:\\ \;\;\;\;\frac{\sqrt{\left(4 \cdot C\right) \cdot \left(F \cdot t\_0\right)}}{-t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-\sqrt{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* A (* C -4.0)))))
   (if (<= (pow B_m 2.0) 9e+55)
     (/ (sqrt (* (* 4.0 C) (* F t_0))) (- t_0))
     (/ (sqrt (* 2.0 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 tmp;
	if (pow(B_m, 2.0) <= 9e+55) {
		tmp = sqrt(((4.0 * C) * (F * t_0))) / -t_0;
	} else {
		tmp = sqrt((2.0 * 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)))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 9e+55)
		tmp = Float64(sqrt(Float64(Float64(4.0 * C) * Float64(F * t_0))) / Float64(-t_0));
	else
		tmp = Float64(sqrt(Float64(2.0 * 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]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 9e+55], N[(N[Sqrt[N[(N[(4.0 * C), $MachinePrecision] * N[(F * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] / (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 8.99999999999999996e55

   1. Initial program 23.2%

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

   3. Simplified 33.8%

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

   5. Taylor expanded in A around -inf 30.6%

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

      1. *-commutative 30.6%

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

   7. Simplified 30.6%

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

   if 8.99999999999999996e55 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 13.9%

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

   3. Taylor expanded in B around inf 22.8%

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

      1. mul-1-neg 22.8%

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

      2. *-commutative 22.8%

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

   5. Simplified 22.8%

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

      1. sqrt-div 34.5%

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

   7. Applied egg-rr 34.5%

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

      1. associate-*r/ 34.4%

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

      2. pow1/2 34.4%

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

      3. pow1/2 34.4%

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

      4. pow-prod-down 34.6%

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

      5. *-commutative 34.6%

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

      6. pow1/2 34.6%

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

      7. *-commutative 34.6%

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

   9. Applied egg-rr 34.6%

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

4. Final simplification 32.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 9 \cdot 10^{+55}:\\ \;\;\;\;\frac{\sqrt{\left(4 \cdot C\right) \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 48.6% accurate, 1.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 10^{-129}:\\ \;\;\;\;\frac{\sqrt{\left(4 \cdot C\right) \cdot \left(F \cdot \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\right)}}{A \cdot \left(4 \cdot C\right)}\\ \mathbf{elif}\;{B\_m}^{2} \leq 2 \cdot 10^{+97}:\\ \;\;\;\;\frac{\sqrt{F}}{-\sqrt{-A}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-\sqrt{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 1e-129)
   (/
    (sqrt (* (* 4.0 C) (* F (fma B_m B_m (* A (* C -4.0))))))
    (* A (* 4.0 C)))
   (if (<= (pow B_m 2.0) 2e+97)
     (/ (sqrt F) (- (sqrt (- A))))
     (/ (sqrt (* 2.0 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) <= 1e-129) {
		tmp = sqrt(((4.0 * C) * (F * fma(B_m, B_m, (A * (C * -4.0)))))) / (A * (4.0 * C));
	} else if (pow(B_m, 2.0) <= 2e+97) {
		tmp = sqrt(F) / -sqrt(-A);
	} else {
		tmp = sqrt((2.0 * 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) <= 1e-129)
		tmp = Float64(sqrt(Float64(Float64(4.0 * C) * Float64(F * fma(B_m, B_m, Float64(A * Float64(C * -4.0)))))) / Float64(A * Float64(4.0 * C)));
	elseif ((B_m ^ 2.0) <= 2e+97)
		tmp = Float64(sqrt(F) / Float64(-sqrt(Float64(-A))));
	else
		tmp = Float64(sqrt(Float64(2.0 * 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], 1e-129], N[(N[Sqrt[N[(N[(4.0 * C), $MachinePrecision] * N[(F * N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(A * N[(4.0 * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+97], N[(N[Sqrt[F], $MachinePrecision] / (-N[Sqrt[(-A)], $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] / (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 9.9999999999999993e-130

   1. Initial program 21.8%

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

   3. Simplified 30.4%

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

   5. Taylor expanded in A around -inf 29.2%

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

      1. *-commutative 29.2%

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

   7. Simplified 29.2%

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

   8. Taylor expanded in B around 0 29.1%

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

      1. metadata-eval 29.1%

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

      2. distribute-lft-neg-in 29.1%

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

      3. *-commutative 29.1%

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

      4. associate-*l* 29.1%

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

      5. *-commutative 29.1%

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

      6. distribute-rgt-neg-in 29.1%

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

      7. distribute-lft-neg-in 29.1%

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

      8. metadata-eval 29.1%

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

      9. *-commutative 29.1%

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

   10. Simplified 29.1%

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

   if 9.9999999999999993e-130 < (pow.f64 B #s(literal 2 binary64)) < 2.0000000000000001e97

   1. Initial program 29.0%

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

   3. Taylor expanded in F around 0 27.1%

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

   5. Simplified 44.6%

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

      1. *-commutative 44.6%

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

      2. pow1/2 44.6%

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

      3. pow1/2 44.6%

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

      4. pow-prod-down 44.8%

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

      5. +-commutative 44.8%

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

      6. +-commutative 44.8%

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

      7. *-commutative 44.8%

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

   7. Applied egg-rr 44.8%

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

      1. unpow1/2 44.8%

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

   9. Simplified 44.8%

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

   10. Taylor expanded in A around -inf 23.9%

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

       1. mul-1-neg 23.9%

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

   12. Simplified 23.9%

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

       1. distribute-neg-frac2 23.9%

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

       2. sqrt-div 29.8%

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

   14. Applied egg-rr 29.8%

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

   if 2.0000000000000001e97 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 12.0%

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

   3. Taylor expanded in B around inf 22.4%

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

      1. mul-1-neg 22.4%

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

      2. *-commutative 22.4%

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

   5. Simplified 22.4%

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

      1. sqrt-div 35.0%

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

   7. Applied egg-rr 35.0%

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

      1. associate-*r/ 34.9%

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

      2. pow1/2 34.9%

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

      3. pow1/2 34.9%

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

      4. pow-prod-down 35.1%

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

      5. *-commutative 35.1%

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

      6. pow1/2 35.1%

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

      7. *-commutative 35.1%

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

   9. Applied egg-rr 35.1%

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

4. Final simplification 31.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 10^{-129}:\\ \;\;\;\;\frac{\sqrt{\left(4 \cdot C\right) \cdot \left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)}}{A \cdot \left(4 \cdot C\right)}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{+97}:\\ \;\;\;\;\frac{\sqrt{F}}{-\sqrt{-A}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 48.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 2 \cdot 10^{+97}:\\ \;\;\;\;\frac{\sqrt{F}}{-\sqrt{-A}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-\sqrt{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 2e+97)
   (/ (sqrt F) (- (sqrt (- A))))
   (/ (sqrt (* 2.0 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) <= 2e+97) {
		tmp = sqrt(F) / -sqrt(-A);
	} else {
		tmp = sqrt((2.0 * F)) / -sqrt(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 ** 2.0d0) <= 2d+97) then
        tmp = sqrt(f) / -sqrt(-a)
    else
        tmp = sqrt((2.0d0 * f)) / -sqrt(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 (Math.pow(B_m, 2.0) <= 2e+97) {
		tmp = Math.sqrt(F) / -Math.sqrt(-A);
	} else {
		tmp = Math.sqrt((2.0 * 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) <= 2e+97:
		tmp = math.sqrt(F) / -math.sqrt(-A)
	else:
		tmp = math.sqrt((2.0 * 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) <= 2e+97)
		tmp = Float64(sqrt(F) / Float64(-sqrt(Float64(-A))));
	else
		tmp = Float64(sqrt(Float64(2.0 * 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) <= 2e+97)
		tmp = sqrt(F) / -sqrt(-A);
	else
		tmp = sqrt((2.0 * 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], 2e+97], N[(N[Sqrt[F], $MachinePrecision] / (-N[Sqrt[(-A)], $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] / (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 2.0000000000000001e97

   1. Initial program 24.0%

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

   3. Taylor expanded in F around 0 18.6%

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

   5. Simplified 29.4%

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

      1. *-commutative 29.4%

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

      2. pow1/2 29.4%

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

      3. pow1/2 29.4%

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

      4. pow-prod-down 29.5%

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

      5. +-commutative 29.5%

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

      6. +-commutative 29.5%

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

      7. *-commutative 29.5%

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

   7. Applied egg-rr 29.5%

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

      1. unpow1/2 29.5%

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

   9. Simplified 29.5%

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

   10. Taylor expanded in A around -inf 21.6%

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

       1. mul-1-neg 21.6%

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

   12. Simplified 21.6%

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

       1. distribute-neg-frac2 21.6%

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

       2. sqrt-div 22.9%

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

   14. Applied egg-rr 22.9%

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

   if 2.0000000000000001e97 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 12.0%

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

   3. Taylor expanded in B around inf 22.4%

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

      1. mul-1-neg 22.4%

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

      2. *-commutative 22.4%

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

   5. Simplified 22.4%

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

      1. sqrt-div 35.0%

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

   7. Applied egg-rr 35.0%

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

      1. associate-*r/ 34.9%

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

      2. pow1/2 34.9%

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

      3. pow1/2 34.9%

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

      4. pow-prod-down 35.1%

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

      5. *-commutative 35.1%

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

      6. pow1/2 35.1%

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

      7. *-commutative 35.1%

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

   9. Applied egg-rr 35.1%

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

4. Final simplification 27.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 2 \cdot 10^{+97}:\\ \;\;\;\;\frac{\sqrt{F}}{-\sqrt{-A}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 48.7% accurate, 3.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 5.5 \cdot 10^{+48}:\\ \;\;\;\;\frac{\sqrt{F}}{-\sqrt{-A}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F} \cdot \left(-\sqrt{\frac{2}{B\_m}}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 5.5e+48)
   (/ (sqrt F) (- (sqrt (- A))))
   (* (sqrt F) (- (sqrt (/ 2.0 B_m))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 5.5e+48) {
		tmp = sqrt(F) / -sqrt(-A);
	} else {
		tmp = sqrt(F) * -sqrt((2.0 / B_m));
	}
	return tmp;
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (b_m <= 5.5d+48) then
        tmp = sqrt(f) / -sqrt(-a)
    else
        tmp = sqrt(f) * -sqrt((2.0d0 / b_m))
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 5.5e+48) {
		tmp = Math.sqrt(F) / -Math.sqrt(-A);
	} else {
		tmp = Math.sqrt(F) * -Math.sqrt((2.0 / B_m));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 5.5e+48:
		tmp = math.sqrt(F) / -math.sqrt(-A)
	else:
		tmp = math.sqrt(F) * -math.sqrt((2.0 / B_m))
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 5.5e+48)
		tmp = Float64(sqrt(F) / Float64(-sqrt(Float64(-A))));
	else
		tmp = Float64(sqrt(F) * Float64(-sqrt(Float64(2.0 / B_m))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (B_m <= 5.5e+48)
		tmp = sqrt(F) / -sqrt(-A);
	else
		tmp = sqrt(F) * -sqrt((2.0 / B_m));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 5.5e+48], N[(N[Sqrt[F], $MachinePrecision] / (-N[Sqrt[(-A)], $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[F], $MachinePrecision] * (-N[Sqrt[N[(2.0 / B$95$m), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 5.5000000000000002e48

   1. Initial program 18.9%

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

   3. Taylor expanded in F around 0 15.4%

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

   5. Simplified 26.8%

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

      1. *-commutative 26.8%

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

      2. pow1/2 26.8%

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

      3. pow1/2 26.8%

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

      4. pow-prod-down 26.9%

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

      5. +-commutative 26.9%

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

      6. +-commutative 26.9%

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

      7. *-commutative 26.9%

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

   7. Applied egg-rr 26.9%

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

      1. unpow1/2 26.9%

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

   9. Simplified 26.9%

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

   10. Taylor expanded in A around -inf 19.4%

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

       1. mul-1-neg 19.4%

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

   12. Simplified 19.4%

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

       1. distribute-neg-frac2 19.4%

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

       2. sqrt-div 19.8%

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

   14. Applied egg-rr 19.8%

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

   if 5.5000000000000002e48 < B

   1. Initial program 19.8%

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

   3. Simplified 23.2%

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

      1. add-cube-cbrt 23.1%

         \[\leadsto \frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \color{blue}{\left(\left(\sqrt[3]{2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt[3]{2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)}\right) \cdot \sqrt[3]{2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)}\right)}}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      2. pow3 23.1%

         \[\leadsto \frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \color{blue}{{\left(\sqrt[3]{2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)}\right)}^{3}}}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      3. hypot-undefine 19.7%

         \[\leadsto \frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot {\left(\sqrt[3]{2 \cdot \left(A + \left(C + \color{blue}{\sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}}\right)\right)}\right)}^{3}}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      4. unpow2 19.7%

         \[\leadsto \frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot {\left(\sqrt[3]{2 \cdot \left(A + \left(C + \sqrt{\color{blue}{{B}^{2}} + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)}\right)}^{3}}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      5. unpow2 19.7%

         \[\leadsto \frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot {\left(\sqrt[3]{2 \cdot \left(A + \left(C + \sqrt{{B}^{2} + \color{blue}{{\left(A - C\right)}^{2}}}\right)\right)}\right)}^{3}}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      6. +-commutative 19.7%

         \[\leadsto \frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot {\left(\sqrt[3]{2 \cdot \left(A + \left(C + \sqrt{\color{blue}{{\left(A - C\right)}^{2} + {B}^{2}}}\right)\right)}\right)}^{3}}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      7. unpow2 19.7%

         \[\leadsto \frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot {\left(\sqrt[3]{2 \cdot \left(A + \left(C + \sqrt{\color{blue}{\left(A - C\right) \cdot \left(A - C\right)} + {B}^{2}}\right)\right)}\right)}^{3}}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      8. unpow2 19.7%

         \[\leadsto \frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot {\left(\sqrt[3]{2 \cdot \left(A + \left(C + \sqrt{\left(A - C\right) \cdot \left(A - C\right) + \color{blue}{B \cdot B}}\right)\right)}\right)}^{3}}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      9. hypot-define 23.1%

         \[\leadsto \frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot {\left(\sqrt[3]{2 \cdot \left(A + \left(C + \color{blue}{\mathsf{hypot}\left(A - C, B\right)}\right)\right)}\right)}^{3}}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   6. Applied egg-rr 23.1%

      \[\leadsto \frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \color{blue}{{\left(\sqrt[3]{2 \cdot \left(A + \left(C + \mathsf{hypot}\left(A - C, B\right)\right)\right)}\right)}^{3}}}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   7. Taylor expanded in B around inf 37.9%

      \[\leadsto \color{blue}{-1 \cdot \sqrt{\frac{F \cdot {\left(\sqrt[3]{2}\right)}^{3}}{B}}} \]
   8. Step-by-step derivation

      1. mul-1-neg 37.9%

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

      2. rem-cube-cbrt 38.7%

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

   9. Simplified 38.7%

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

       1. pow1/2 38.7%

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

       2. associate-/l* 38.7%

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

       3. unpow-prod-down 64.6%

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

       4. pow1/2 64.6%

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

   11. Applied egg-rr 64.6%

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

       1. unpow1/2 64.6%

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

   13. Simplified 64.6%

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

4. Final simplification 29.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 5.5 \cdot 10^{+48}:\\ \;\;\;\;\frac{\sqrt{F}}{-\sqrt{-A}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{F} \cdot \left(-\sqrt{\frac{2}{B}}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 40.3% accurate, 3.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;A \leq -6 \cdot 10^{-34}:\\ \;\;\;\;\frac{\sqrt{F}}{-\sqrt{-A}}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{2 \cdot \left(F + F \cdot \frac{A + C}{B\_m}\right)}{B\_m}}\\ \end{array} \end{array} \]

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 (<= A -6e-34)
   (/ (sqrt F) (- (sqrt (- A))))
   (- (sqrt (/ (* 2.0 (+ 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 (A <= -6e-34) {
		tmp = sqrt(F) / -sqrt(-A);
	} else {
		tmp = -sqrt(((2.0 * (F + (F * ((A + C) / B_m)))) / 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 (a <= (-6d-34)) then
        tmp = sqrt(f) / -sqrt(-a)
    else
        tmp = -sqrt(((2.0d0 * (f + (f * ((a + c) / b_m)))) / 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 (A <= -6e-34) {
		tmp = Math.sqrt(F) / -Math.sqrt(-A);
	} else {
		tmp = -Math.sqrt(((2.0 * (F + (F * ((A + C) / B_m)))) / B_m));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if A <= -6e-34:
		tmp = math.sqrt(F) / -math.sqrt(-A)
	else:
		tmp = -math.sqrt(((2.0 * (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 (A <= -6e-34)
		tmp = Float64(sqrt(F) / Float64(-sqrt(Float64(-A))));
	else
		tmp = Float64(-sqrt(Float64(Float64(2.0 * Float64(F + Float64(F * Float64(Float64(A + C) / B_m)))) / 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 (A <= -6e-34)
		tmp = sqrt(F) / -sqrt(-A);
	else
		tmp = -sqrt(((2.0 * (F + (F * ((A + C) / B_m)))) / B_m));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[A, -6e-34], N[(N[Sqrt[F], $MachinePrecision] / (-N[Sqrt[(-A)], $MachinePrecision])), $MachinePrecision], (-N[Sqrt[N[(N[(2.0 * N[(F + N[(F * N[(N[(A + C), $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / B$95$m), $MachinePrecision]], $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if A < -6e-34

   1. Initial program 12.3%

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

   3. Taylor expanded in F around 0 12.3%

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

   5. Simplified 29.1%

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

      1. *-commutative 29.1%

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

      2. pow1/2 29.1%

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

      3. pow1/2 29.1%

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

      4. pow-prod-down 29.0%

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

      5. +-commutative 29.0%

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

      6. +-commutative 29.0%

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

      7. *-commutative 29.0%

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

   7. Applied egg-rr 29.0%

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

      1. unpow1/2 29.0%

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

   9. Simplified 29.0%

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

   10. Taylor expanded in A around -inf 40.7%

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

       1. mul-1-neg 40.7%

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

   12. Simplified 40.7%

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

       1. distribute-neg-frac2 40.7%

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

       2. sqrt-div 43.4%

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

   14. Applied egg-rr 43.4%

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

   if -6e-34 < A

   1. Initial program 22.7%

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

   3. Taylor expanded in F around 0 17.4%

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

   5. Simplified 27.6%

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

      1. *-commutative 27.6%

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

      2. pow1/2 27.6%

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

      3. pow1/2 27.6%

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

      4. pow-prod-down 27.8%

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

      5. +-commutative 27.8%

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

      6. +-commutative 27.8%

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

      7. *-commutative 27.8%

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

   7. Applied egg-rr 27.8%

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

      1. unpow1/2 27.8%

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

   9. Simplified 27.8%

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

   10. Taylor expanded in B around inf 14.0%

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

       1. distribute-lft-out 14.0%

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

       2. associate-/l* 14.7%

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

   12. Simplified 14.7%

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

4. Final simplification 24.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -6 \cdot 10^{-34}:\\ \;\;\;\;\frac{\sqrt{F}}{-\sqrt{-A}}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{2 \cdot \left(F + F \cdot \frac{A + C}{B}\right)}{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 37.8% accurate, 3.0× speedup

Math

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

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 (<= A -6.2e-34)
   (- (sqrt (fabs (/ F A))))
   (- (sqrt (/ (* 2.0 (+ 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 (A <= -6.2e-34) {
		tmp = -sqrt(fabs((F / A)));
	} else {
		tmp = -sqrt(((2.0 * (F + (F * ((A + C) / B_m)))) / 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 (a <= (-6.2d-34)) then
        tmp = -sqrt(abs((f / a)))
    else
        tmp = -sqrt(((2.0d0 * (f + (f * ((a + c) / b_m)))) / 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 (A <= -6.2e-34) {
		tmp = -Math.sqrt(Math.abs((F / A)));
	} else {
		tmp = -Math.sqrt(((2.0 * (F + (F * ((A + C) / B_m)))) / B_m));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if A <= -6.2e-34:
		tmp = -math.sqrt(math.fabs((F / A)))
	else:
		tmp = -math.sqrt(((2.0 * (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 (A <= -6.2e-34)
		tmp = Float64(-sqrt(abs(Float64(F / A))));
	else
		tmp = Float64(-sqrt(Float64(Float64(2.0 * Float64(F + Float64(F * Float64(Float64(A + C) / B_m)))) / 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 (A <= -6.2e-34)
		tmp = -sqrt(abs((F / A)));
	else
		tmp = -sqrt(((2.0 * (F + (F * ((A + C) / B_m)))) / B_m));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[A, -6.2e-34], (-N[Sqrt[N[Abs[N[(F / A), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), (-N[Sqrt[N[(N[(2.0 * N[(F + N[(F * N[(N[(A + C), $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / B$95$m), $MachinePrecision]], $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if A < -6.1999999999999996e-34

   1. Initial program 12.3%

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

   3. Taylor expanded in F around 0 12.3%

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

   5. Simplified 29.1%

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

      1. *-commutative 29.1%

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

      2. pow1/2 29.1%

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

      3. pow1/2 29.1%

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

      4. pow-prod-down 29.0%

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

      5. +-commutative 29.0%

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

      6. +-commutative 29.0%

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

      7. *-commutative 29.0%

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

   7. Applied egg-rr 29.0%

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

      1. unpow1/2 29.0%

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

   9. Simplified 29.0%

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

   10. Taylor expanded in A around -inf 40.7%

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

       1. mul-1-neg 40.7%

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

   12. Simplified 40.7%

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

       1. add-sqr-sqrt 40.7%

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

       2. pow1/2 40.7%

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

       3. pow1/2 40.7%

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

       4. pow-prod-down 31.4%

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

       5. pow2 31.4%

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

       6. distribute-neg-frac 31.4%

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

   14. Applied egg-rr 31.4%

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

       1. unpow1/2 31.4%

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

       2. unpow2 31.4%

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

       3. rem-sqrt-square 41.1%

          \[\leadsto -\sqrt{\color{blue}{\left|\frac{-F}{A}\right|}} \]

   16. Simplified 41.1%

       \[\leadsto -\sqrt{\color{blue}{\left|\frac{-F}{A}\right|}} \]

   if -6.1999999999999996e-34 < A

   1. Initial program 22.7%

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

   3. Taylor expanded in F around 0 17.4%

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

   5. Simplified 27.6%

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

      1. *-commutative 27.6%

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

      2. pow1/2 27.6%

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

      3. pow1/2 27.6%

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

      4. pow-prod-down 27.8%

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

      5. +-commutative 27.8%

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

      6. +-commutative 27.8%

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

      7. *-commutative 27.8%

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

   7. Applied egg-rr 27.8%

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

      1. unpow1/2 27.8%

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

   9. Simplified 27.8%

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

   10. Taylor expanded in B around inf 14.0%

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

       1. distribute-lft-out 14.0%

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

       2. associate-/l* 14.7%

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

   12. Simplified 14.7%

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

4. Final simplification 23.9%

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

Alternative 11: 37.6% accurate, 5.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;A \leq -6 \cdot 10^{-34}:\\ \;\;\;\;-\sqrt{\frac{F}{-A}}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{2 \cdot \left(F + F \cdot \frac{A + C}{B\_m}\right)}{B\_m}}\\ \end{array} \end{array} \]

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 (<= A -6e-34)
   (- (sqrt (/ F (- A))))
   (- (sqrt (/ (* 2.0 (+ 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 (A <= -6e-34) {
		tmp = -sqrt((F / -A));
	} else {
		tmp = -sqrt(((2.0 * (F + (F * ((A + C) / B_m)))) / 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 (a <= (-6d-34)) then
        tmp = -sqrt((f / -a))
    else
        tmp = -sqrt(((2.0d0 * (f + (f * ((a + c) / b_m)))) / 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 (A <= -6e-34) {
		tmp = -Math.sqrt((F / -A));
	} else {
		tmp = -Math.sqrt(((2.0 * (F + (F * ((A + C) / B_m)))) / B_m));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if A <= -6e-34:
		tmp = -math.sqrt((F / -A))
	else:
		tmp = -math.sqrt(((2.0 * (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 (A <= -6e-34)
		tmp = Float64(-sqrt(Float64(F / Float64(-A))));
	else
		tmp = Float64(-sqrt(Float64(Float64(2.0 * Float64(F + Float64(F * Float64(Float64(A + C) / B_m)))) / 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 (A <= -6e-34)
		tmp = -sqrt((F / -A));
	else
		tmp = -sqrt(((2.0 * (F + (F * ((A + C) / B_m)))) / B_m));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[A, -6e-34], (-N[Sqrt[N[(F / (-A)), $MachinePrecision]], $MachinePrecision]), (-N[Sqrt[N[(N[(2.0 * N[(F + N[(F * N[(N[(A + C), $MachinePrecision] / B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / B$95$m), $MachinePrecision]], $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if A < -6e-34

   1. Initial program 12.3%

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

   3. Taylor expanded in F around 0 12.3%

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

   5. Simplified 29.1%

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

      1. *-commutative 29.1%

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

      2. pow1/2 29.1%

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

      3. pow1/2 29.1%

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

      4. pow-prod-down 29.0%

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

      5. +-commutative 29.0%

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

      6. +-commutative 29.0%

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

      7. *-commutative 29.0%

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

   7. Applied egg-rr 29.0%

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

      1. unpow1/2 29.0%

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

   9. Simplified 29.0%

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

   10. Taylor expanded in A around -inf 40.7%

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

       1. mul-1-neg 40.7%

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

   12. Simplified 40.7%

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

   if -6e-34 < A

   1. Initial program 22.7%

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

   3. Taylor expanded in F around 0 17.4%

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

   5. Simplified 27.6%

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

      1. *-commutative 27.6%

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

      2. pow1/2 27.6%

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

      3. pow1/2 27.6%

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

      4. pow-prod-down 27.8%

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

      5. +-commutative 27.8%

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

      6. +-commutative 27.8%

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

      7. *-commutative 27.8%

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

   7. Applied egg-rr 27.8%

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

      1. unpow1/2 27.8%

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

   9. Simplified 27.8%

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

   10. Taylor expanded in B around inf 14.0%

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

       1. distribute-lft-out 14.0%

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

       2. associate-/l* 14.7%

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

   12. Simplified 14.7%

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

4. Final simplification 23.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;A \leq -6 \cdot 10^{-34}:\\ \;\;\;\;-\sqrt{\frac{F}{-A}}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{2 \cdot \left(F + F \cdot \frac{A + C}{B}\right)}{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 39.2% accurate, 5.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 9 \cdot 10^{+29}:\\ \;\;\;\;-\sqrt{\frac{F}{-A}}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{2 \cdot F}{B\_m}}\\ \end{array} \end{array} \]

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 9e+29) (- (sqrt (/ F (- A)))) (- (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 <= 9e+29) {
		tmp = -sqrt((F / -A));
	} 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 <= 9d+29) then
        tmp = -sqrt((f / -a))
    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 <= 9e+29) {
		tmp = -Math.sqrt((F / -A));
	} 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 <= 9e+29:
		tmp = -math.sqrt((F / -A))
	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 <= 9e+29)
		tmp = Float64(-sqrt(Float64(F / Float64(-A))));
	else
		tmp = Float64(-sqrt(Float64(Float64(2.0 * 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 <= 9e+29)
		tmp = -sqrt((F / -A));
	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, 9e+29], (-N[Sqrt[N[(F / (-A)), $MachinePrecision]], $MachinePrecision]), (-N[Sqrt[N[(N[(2.0 * F), $MachinePrecision] / B$95$m), $MachinePrecision]], $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if B < 9.0000000000000005e29

   1. Initial program 18.2%

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

   3. Taylor expanded in F around 0 14.7%

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

   5. Simplified 26.2%

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

      1. *-commutative 26.2%

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

      2. pow1/2 26.2%

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

      3. pow1/2 26.2%

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

      4. pow-prod-down 26.3%

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

      5. +-commutative 26.3%

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

      6. +-commutative 26.3%

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

      7. *-commutative 26.3%

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

   7. Applied egg-rr 26.3%

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

      1. unpow1/2 26.3%

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

   9. Simplified 26.3%

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

   10. Taylor expanded in A around -inf 19.7%

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

       1. mul-1-neg 19.7%

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

   12. Simplified 19.7%

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

   if 9.0000000000000005e29 < B

   1. Initial program 21.9%

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

   3. Simplified 25.0%

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

      1. add-cube-cbrt 24.9%

         \[\leadsto \frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \color{blue}{\left(\left(\sqrt[3]{2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)} \cdot \sqrt[3]{2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)}\right) \cdot \sqrt[3]{2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)}\right)}}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      2. pow3 24.9%

         \[\leadsto \frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \color{blue}{{\left(\sqrt[3]{2 \cdot \left(A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)\right)}\right)}^{3}}}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      3. hypot-undefine 21.7%

         \[\leadsto \frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot {\left(\sqrt[3]{2 \cdot \left(A + \left(C + \color{blue}{\sqrt{B \cdot B + \left(A - C\right) \cdot \left(A - C\right)}}\right)\right)}\right)}^{3}}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      4. unpow2 21.7%

         \[\leadsto \frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot {\left(\sqrt[3]{2 \cdot \left(A + \left(C + \sqrt{\color{blue}{{B}^{2}} + \left(A - C\right) \cdot \left(A - C\right)}\right)\right)}\right)}^{3}}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      5. unpow2 21.7%

         \[\leadsto \frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot {\left(\sqrt[3]{2 \cdot \left(A + \left(C + \sqrt{{B}^{2} + \color{blue}{{\left(A - C\right)}^{2}}}\right)\right)}\right)}^{3}}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      6. +-commutative 21.7%

         \[\leadsto \frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot {\left(\sqrt[3]{2 \cdot \left(A + \left(C + \sqrt{\color{blue}{{\left(A - C\right)}^{2} + {B}^{2}}}\right)\right)}\right)}^{3}}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      7. unpow2 21.7%

         \[\leadsto \frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot {\left(\sqrt[3]{2 \cdot \left(A + \left(C + \sqrt{\color{blue}{\left(A - C\right) \cdot \left(A - C\right)} + {B}^{2}}\right)\right)}\right)}^{3}}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      8. unpow2 21.7%

         \[\leadsto \frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot {\left(\sqrt[3]{2 \cdot \left(A + \left(C + \sqrt{\left(A - C\right) \cdot \left(A - C\right) + \color{blue}{B \cdot B}}\right)\right)}\right)}^{3}}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

      9. hypot-define 24.9%

         \[\leadsto \frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot {\left(\sqrt[3]{2 \cdot \left(A + \left(C + \color{blue}{\mathsf{hypot}\left(A - C, B\right)}\right)\right)}\right)}^{3}}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   6. Applied egg-rr 24.9%

      \[\leadsto \frac{\sqrt{\left(\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right) \cdot F\right) \cdot \color{blue}{{\left(\sqrt[3]{2 \cdot \left(A + \left(C + \mathsf{hypot}\left(A - C, B\right)\right)\right)}\right)}^{3}}}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)} \]

   7. Taylor expanded in B around inf 38.8%

      \[\leadsto \color{blue}{-1 \cdot \sqrt{\frac{F \cdot {\left(\sqrt[3]{2}\right)}^{3}}{B}}} \]
   8. Step-by-step derivation

      1. mul-1-neg 38.8%

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

      2. rem-cube-cbrt 39.7%

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

   9. Simplified 39.7%

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

4. Final simplification 24.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 9 \cdot 10^{+29}:\\ \;\;\;\;-\sqrt{\frac{F}{-A}}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{\frac{2 \cdot F}{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 39.3% accurate, 5.7× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 8.8 \cdot 10^{+29}:\\ \;\;\;\;-\sqrt{\frac{F}{-A}}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{F \cdot \frac{2}{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 8.8e+29) (- (sqrt (/ F (- A)))) (- (sqrt (* F (/ 2.0 B_m))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 8.8e+29) {
		tmp = -sqrt((F / -A));
	} else {
		tmp = -sqrt((F * (2.0 / B_m)));
	}
	return tmp;
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (b_m <= 8.8d+29) then
        tmp = -sqrt((f / -a))
    else
        tmp = -sqrt((f * (2.0d0 / b_m)))
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 8.8e+29) {
		tmp = -Math.sqrt((F / -A));
	} else {
		tmp = -Math.sqrt((F * (2.0 / B_m)));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 8.8e+29:
		tmp = -math.sqrt((F / -A))
	else:
		tmp = -math.sqrt((F * (2.0 / B_m)))
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 8.8e+29)
		tmp = Float64(-sqrt(Float64(F / Float64(-A))));
	else
		tmp = Float64(-sqrt(Float64(F * Float64(2.0 / B_m))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (B_m <= 8.8e+29)
		tmp = -sqrt((F / -A));
	else
		tmp = -sqrt((F * (2.0 / B_m)));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 8.8e+29], (-N[Sqrt[N[(F / (-A)), $MachinePrecision]], $MachinePrecision]), (-N[Sqrt[N[(F * N[(2.0 / B$95$m), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if B < 8.8000000000000005e29

   1. Initial program 18.2%

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

   3. Taylor expanded in F around 0 14.7%

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

   5. Simplified 26.2%

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

      1. *-commutative 26.2%

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

      2. pow1/2 26.2%

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

      3. pow1/2 26.2%

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

      4. pow-prod-down 26.3%

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

      5. +-commutative 26.3%

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

      6. +-commutative 26.3%

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

      7. *-commutative 26.3%

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

   7. Applied egg-rr 26.3%

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

      1. unpow1/2 26.3%

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

   9. Simplified 26.3%

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

   10. Taylor expanded in A around -inf 19.7%

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

       1. mul-1-neg 19.7%

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

   12. Simplified 19.7%

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

   if 8.8000000000000005e29 < B

   1. Initial program 21.9%

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

   3. Taylor expanded in B around inf 39.5%

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

      1. mul-1-neg 39.5%

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

      2. *-commutative 39.5%

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

   5. Simplified 39.5%

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

      1. sqrt-div 63.8%

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

   7. Applied egg-rr 63.8%

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

      1. neg-sub0 63.8%

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

      2. associate-*r/ 63.6%

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

      3. pow1/2 63.6%

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

      4. pow1/2 63.6%

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

      5. pow-prod-down 64.0%

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

      6. *-commutative 64.0%

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

      7. pow1/2 64.0%

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

      8. sqrt-div 39.7%

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

      9. associate-/l* 39.6%

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

   9. Applied egg-rr 39.6%

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

       1. neg-sub0 39.6%

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

   11. Simplified 39.6%

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

4. Final simplification 24.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 8.8 \cdot 10^{+29}:\\ \;\;\;\;-\sqrt{\frac{F}{-A}}\\ \mathbf{else}:\\ \;\;\;\;-\sqrt{F \cdot \frac{2}{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 28.1% accurate, 6.0× speedup

Math

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

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 (- A)))))

C

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

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 / -a))
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 / -A));
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 19.1%

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

3. Taylor expanded in F around 0 15.6%

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

5. Simplified 28.1%

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

   1. *-commutative 28.1%

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

   2. pow1/2 28.1%

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

   3. pow1/2 28.1%

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

   4. pow-prod-down 28.2%

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

   5. +-commutative 28.2%

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

   6. +-commutative 28.2%

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

   7. *-commutative 28.2%

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

7. Applied egg-rr 28.2%

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

   1. unpow1/2 28.2%

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

9. Simplified 28.2%

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

10. Taylor expanded in A around -inf 16.4%

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

    1. mul-1-neg 16.4%

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

12. Simplified 16.4%

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

13. Final simplification 16.4%

    \[\leadsto -\sqrt{\frac{F}{-A}} \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024173 
(FPCore (A B C F)
  :name "ABCF->ab-angle a"
  :precision binary64
  (/ (- (sqrt (* (* 2.0 (* (- (pow B 2.0) (* (* 4.0 A) C)) F)) (+ (+ A C) (sqrt (+ (pow (- A C) 2.0) (pow B 2.0))))))) (- (pow B 2.0) (* (* 4.0 A) C))))