ABCF->ab-angle a

Percentage Accurate: 19.7% → 63.3%

Time: 52.9s

Alternatives: 14

Speedup: 6.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 19.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 63.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ t_1 := t\_0 - {B\_m}^{2}\\ t_2 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_0\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_1}\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{-173}:\\ \;\;\;\;\left(\sqrt{F} \cdot \sqrt{\frac{\left(A + C\right) + \mathsf{hypot}\left(B\_m, A - C\right)}{\mathsf{fma}\left(-4, A \cdot C, {B\_m}^{2}\right)}}\right) \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(2, C, -0.5 \cdot \frac{{B\_m}^{2}}{A}\right)} \cdot \sqrt{2 \cdot \left(F \cdot \left({B\_m}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right)}}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{F} \cdot \sqrt{C + \mathsf{hypot}\left(B\_m, C\right)}\right) \cdot \frac{-1}{B\_m \cdot {2}^{-0.5}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (* (* 4.0 A) C))
        (t_1 (- t_0 (pow B_m 2.0)))
        (t_2
         (/
          (sqrt
           (*
            (* 2.0 (* (- (pow B_m 2.0) t_0) F))
            (+ (+ A C) (sqrt (+ (pow B_m 2.0) (pow (- A C) 2.0))))))
          t_1)))
   (if (<= t_2 -1e-173)
     (*
      (*
       (sqrt F)
       (sqrt
        (/ (+ (+ A C) (hypot B_m (- A C))) (fma -4.0 (* A C) (pow B_m 2.0)))))
      (- (sqrt 2.0)))
     (if (<= t_2 INFINITY)
       (/
        (*
         (sqrt (fma 2.0 C (* -0.5 (/ (pow B_m 2.0) A))))
         (sqrt (* 2.0 (* F (- (pow B_m 2.0) (* 4.0 (* A C)))))))
        t_1)
       (*
        (* (sqrt F) (sqrt (+ C (hypot B_m C))))
        (/ -1.0 (* B_m (pow 2.0 -0.5))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = (4.0 * A) * C;
	double t_1 = t_0 - pow(B_m, 2.0);
	double t_2 = sqrt(((2.0 * ((pow(B_m, 2.0) - t_0) * F)) * ((A + C) + sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / t_1;
	double tmp;
	if (t_2 <= -1e-173) {
		tmp = (sqrt(F) * sqrt((((A + C) + hypot(B_m, (A - C))) / fma(-4.0, (A * C), pow(B_m, 2.0))))) * -sqrt(2.0);
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = (sqrt(fma(2.0, C, (-0.5 * (pow(B_m, 2.0) / A)))) * sqrt((2.0 * (F * (pow(B_m, 2.0) - (4.0 * (A * C))))))) / t_1;
	} else {
		tmp = (sqrt(F) * sqrt((C + hypot(B_m, C)))) * (-1.0 / (B_m * pow(2.0, -0.5)));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(Float64(4.0 * A) * C)
	t_1 = Float64(t_0 - (B_m ^ 2.0))
	t_2 = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_0) * F)) * Float64(Float64(A + C) + sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / t_1)
	tmp = 0.0
	if (t_2 <= -1e-173)
		tmp = Float64(Float64(sqrt(F) * sqrt(Float64(Float64(Float64(A + C) + hypot(B_m, Float64(A - C))) / fma(-4.0, Float64(A * C), (B_m ^ 2.0))))) * Float64(-sqrt(2.0)));
	elseif (t_2 <= Inf)
		tmp = Float64(Float64(sqrt(fma(2.0, C, Float64(-0.5 * Float64((B_m ^ 2.0) / A)))) * sqrt(Float64(2.0 * Float64(F * Float64((B_m ^ 2.0) - Float64(4.0 * Float64(A * C))))))) / t_1);
	else
		tmp = Float64(Float64(sqrt(F) * sqrt(Float64(C + hypot(B_m, C)))) * Float64(-1.0 / Float64(B_m * (2.0 ^ -0.5))));
	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[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $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] / t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, -1e-173], N[(N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[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] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], If[LessEqual[t$95$2, Infinity], N[(N[(N[Sqrt[N[(2.0 * C + N[(-0.5 * N[(N[Power[B$95$m, 2.0], $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(2.0 * N[(F * N[(N[Power[B$95$m, 2.0], $MachinePrecision] - N[(4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-1.0 / N[(B$95$m * N[Power[2.0, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))) < -1e-173

   1. Initial program 43.4%

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

   3. Taylor expanded in F around 0 48.5%

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

      1. mul-1-neg 48.5%

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

      2. unpow2 48.5%

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

      3. unpow2 48.5%

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

      4. hypot-undefine 61.6%

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

      5. cancel-sign-sub-inv 61.6%

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

      6. metadata-eval 61.6%

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

   5. Simplified 61.6%

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

      1. pow1/2 61.6%

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

      2. associate-/l* 67.3%

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

      3. unpow-prod-down 80.1%

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

      4. pow1/2 80.1%

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

      5. associate-+r+ 79.2%

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

      6. +-commutative 79.2%

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

      7. fma-define 79.2%

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

   7. Applied egg-rr 79.2%

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

      1. unpow1/2 79.2%

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

   9. Simplified 79.2%

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

   if -1e-173 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 2 (*.f64 (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) 2) (pow.f64 B 2))))))) (-.f64 (pow.f64 B 2) (*.f64 (*.f64 4 A) C))) < +inf.0

   1. Initial program 12.7%

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

   3. Taylor expanded in A around -inf 33.9%

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

      1. pow1/2 33.9%

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

      2. *-commutative 33.9%

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

      3. unpow-prod-down 28.7%

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

      4. pow1/2 28.6%

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

      5. +-commutative 28.6%

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

      6. fma-define 28.6%

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

      7. pow1/2 28.6%

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

      8. associate-*l* 28.6%

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

   5. Applied egg-rr 28.6%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in A around 0 2.1%

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

      1. mul-1-neg 2.1%

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

   5. Simplified 2.1%

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

      1. pow1/2 2.2%

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

      2. *-commutative 2.2%

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

      3. unpow-prod-down 2.1%

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

      4. pow1/2 2.1%

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

      5. unpow2 2.1%

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

      6. unpow2 2.1%

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

      7. hypot-define 26.3%

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

      8. pow1/2 26.3%

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

   7. Applied egg-rr 26.3%

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

      1. clear-num 26.3%

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

      2. inv-pow 26.3%

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

   9. Applied egg-rr 26.3%

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

       1. unpow-1 26.3%

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

   11. Simplified 26.3%

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

       1. div-inv 26.2%

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

       2. pow1/2 26.2%

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

       3. metadata-eval 26.2%

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

       4. metadata-eval 26.2%

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

       5. sqrt-pow2 26.2%

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

       6. pow-flip 26.2%

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

       7. sqrt-pow2 26.3%

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

       8. metadata-eval 26.3%

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

       9. metadata-eval 26.3%

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

       10. metadata-eval 26.3%

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

   13. Applied egg-rr 26.3%

       \[\leadsto -\frac{1}{\color{blue}{B \cdot {2}^{-0.5}}} \cdot \left(\sqrt{C + \mathsf{hypot}\left(B, C\right)} \cdot \sqrt{F}\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 43.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -1 \cdot 10^{-173}:\\ \;\;\;\;\left(\sqrt{F} \cdot \sqrt{\frac{\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)}{\mathsf{fma}\left(-4, A \cdot C, {B}^{2}\right)}}\right) \cdot \left(-\sqrt{2}\right)\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(2, C, -0.5 \cdot \frac{{B}^{2}}{A}\right)} \cdot \sqrt{2 \cdot \left(F \cdot \left({B}^{2} - 4 \cdot \left(A \cdot C\right)\right)\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{F} \cdot \sqrt{C + \mathsf{hypot}\left(B, C\right)}\right) \cdot \frac{-1}{B \cdot {2}^{-0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 59.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ t_1 := t\_0 - {B\_m}^{2}\\ t_2 := 2 \cdot \left(\left({B\_m}^{2} - t\_0\right) \cdot F\right)\\ \mathbf{if}\;B\_m \leq 3.6 \cdot 10^{-227}:\\ \;\;\;\;\frac{\sqrt{t\_2 \cdot \left(2 \cdot C\right)}}{t\_1}\\ \mathbf{elif}\;B\_m \leq 4.2 \cdot 10^{-142}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{-0.5 \cdot \frac{F}{A}}\right)\\ \mathbf{elif}\;B\_m \leq 2.7 \cdot 10^{-39}:\\ \;\;\;\;\frac{\sqrt{t\_2 \cdot \left(-0.5 \cdot \frac{{B\_m}^{2}}{A} + 2 \cdot C\right)}}{t\_1}\\ \mathbf{elif}\;B\_m \leq 4.2 \cdot 10^{+151}:\\ \;\;\;\;\left(\sqrt{F} \cdot \sqrt{\frac{\left(A + C\right) + \mathsf{hypot}\left(B\_m, A - C\right)}{\mathsf{fma}\left(-4, A \cdot C, {B\_m}^{2}\right)}}\right) \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{F} \cdot \sqrt{C + \mathsf{hypot}\left(B\_m, C\right)}\right) \cdot \frac{-1}{B\_m \cdot {2}^{-0.5}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (* (* 4.0 A) C))
        (t_1 (- t_0 (pow B_m 2.0)))
        (t_2 (* 2.0 (* (- (pow B_m 2.0) t_0) F))))
   (if (<= B_m 3.6e-227)
     (/ (sqrt (* t_2 (* 2.0 C))) t_1)
     (if (<= B_m 4.2e-142)
       (* (sqrt 2.0) (- (sqrt (* -0.5 (/ F A)))))
       (if (<= B_m 2.7e-39)
         (/ (sqrt (* t_2 (+ (* -0.5 (/ (pow B_m 2.0) A)) (* 2.0 C)))) t_1)
         (if (<= B_m 4.2e+151)
           (*
            (*
             (sqrt F)
             (sqrt
              (/
               (+ (+ A C) (hypot B_m (- A C)))
               (fma -4.0 (* A C) (pow B_m 2.0)))))
            (- (sqrt 2.0)))
           (*
            (* (sqrt F) (sqrt (+ C (hypot B_m C))))
            (/ -1.0 (* B_m (pow 2.0 -0.5))))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = (4.0 * A) * C;
	double t_1 = t_0 - pow(B_m, 2.0);
	double t_2 = 2.0 * ((pow(B_m, 2.0) - t_0) * F);
	double tmp;
	if (B_m <= 3.6e-227) {
		tmp = sqrt((t_2 * (2.0 * C))) / t_1;
	} else if (B_m <= 4.2e-142) {
		tmp = sqrt(2.0) * -sqrt((-0.5 * (F / A)));
	} else if (B_m <= 2.7e-39) {
		tmp = sqrt((t_2 * ((-0.5 * (pow(B_m, 2.0) / A)) + (2.0 * C)))) / t_1;
	} else if (B_m <= 4.2e+151) {
		tmp = (sqrt(F) * sqrt((((A + C) + hypot(B_m, (A - C))) / fma(-4.0, (A * C), pow(B_m, 2.0))))) * -sqrt(2.0);
	} else {
		tmp = (sqrt(F) * sqrt((C + hypot(B_m, C)))) * (-1.0 / (B_m * pow(2.0, -0.5)));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(Float64(4.0 * A) * C)
	t_1 = Float64(t_0 - (B_m ^ 2.0))
	t_2 = Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_0) * F))
	tmp = 0.0
	if (B_m <= 3.6e-227)
		tmp = Float64(sqrt(Float64(t_2 * Float64(2.0 * C))) / t_1);
	elseif (B_m <= 4.2e-142)
		tmp = Float64(sqrt(2.0) * Float64(-sqrt(Float64(-0.5 * Float64(F / A)))));
	elseif (B_m <= 2.7e-39)
		tmp = Float64(sqrt(Float64(t_2 * Float64(Float64(-0.5 * Float64((B_m ^ 2.0) / A)) + Float64(2.0 * C)))) / t_1);
	elseif (B_m <= 4.2e+151)
		tmp = Float64(Float64(sqrt(F) * sqrt(Float64(Float64(Float64(A + C) + hypot(B_m, Float64(A - C))) / fma(-4.0, Float64(A * C), (B_m ^ 2.0))))) * Float64(-sqrt(2.0)));
	else
		tmp = Float64(Float64(sqrt(F) * sqrt(Float64(C + hypot(B_m, C)))) * Float64(-1.0 / Float64(B_m * (2.0 ^ -0.5))));
	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[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 3.6e-227], N[(N[Sqrt[N[(t$95$2 * N[(2.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[B$95$m, 4.2e-142], N[(N[Sqrt[2.0], $MachinePrecision] * (-N[Sqrt[N[(-0.5 * N[(F / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], If[LessEqual[B$95$m, 2.7e-39], N[(N[Sqrt[N[(t$95$2 * N[(N[(-0.5 * N[(N[Power[B$95$m, 2.0], $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision] + N[(2.0 * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[B$95$m, 4.2e+151], N[(N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[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] * (-N[Sqrt[2.0], $MachinePrecision])), $MachinePrecision], N[(N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-1.0 / N[(B$95$m * N[Power[2.0, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if B < 3.5999999999999999e-227

   1. Initial program 17.7%

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

   3. Taylor expanded in A around -inf 16.9%

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

   if 3.5999999999999999e-227 < B < 4.1999999999999999e-142

   1. Initial program 12.1%

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

   3. Taylor expanded in F around 0 18.5%

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

      1. mul-1-neg 18.5%

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

      2. unpow2 18.5%

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

      3. unpow2 18.5%

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

      4. hypot-undefine 19.7%

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

      5. cancel-sign-sub-inv 19.7%

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

      6. metadata-eval 19.7%

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

   5. Simplified 19.7%

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

   6. Taylor expanded in A around -inf 21.0%

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

   if 4.1999999999999999e-142 < B < 2.7000000000000001e-39

   1. Initial program 26.8%

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

   3. Taylor expanded in A around -inf 32.3%

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

   if 2.7000000000000001e-39 < B < 4.2000000000000001e151

   1. Initial program 35.7%

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

   3. Taylor expanded in F around 0 36.2%

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

      1. mul-1-neg 36.2%

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

      2. unpow2 36.2%

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

      3. unpow2 36.2%

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

      4. hypot-undefine 48.7%

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

      5. cancel-sign-sub-inv 48.7%

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

      6. metadata-eval 48.7%

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

   5. Simplified 48.7%

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

      1. pow1/2 48.7%

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

      2. associate-/l* 52.5%

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

      3. unpow-prod-down 60.1%

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

      4. pow1/2 60.1%

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

      5. associate-+r+ 59.5%

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

      6. +-commutative 59.5%

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

      7. fma-define 59.5%

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

   7. Applied egg-rr 59.5%

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

      1. unpow1/2 59.5%

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

   9. Simplified 59.5%

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

   if 4.2000000000000001e151 < B

   1. Initial program 0.0%

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

   3. Taylor expanded in A around 0 2.6%

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

      1. mul-1-neg 2.6%

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

   5. Simplified 2.6%

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

      1. pow1/2 2.7%

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

      2. *-commutative 2.7%

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

      3. unpow-prod-down 2.6%

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

      4. pow1/2 2.6%

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

      5. unpow2 2.6%

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

      6. unpow2 2.6%

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

      7. hypot-define 65.3%

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

      8. pow1/2 65.3%

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

   7. Applied egg-rr 65.3%

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

      1. clear-num 65.1%

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

      2. inv-pow 65.1%

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

   9. Applied egg-rr 65.1%

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

       1. unpow-1 65.1%

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

   11. Simplified 65.1%

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

       1. div-inv 64.9%

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

       2. pow1/2 64.9%

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

       3. metadata-eval 64.9%

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

       4. metadata-eval 64.9%

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

       5. sqrt-pow2 64.9%

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

       6. pow-flip 64.9%

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

       7. sqrt-pow2 65.3%

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

       8. metadata-eval 65.3%

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

       9. metadata-eval 65.3%

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

       10. metadata-eval 65.3%

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

   13. Applied egg-rr 65.3%

       \[\leadsto -\frac{1}{\color{blue}{B \cdot {2}^{-0.5}}} \cdot \left(\sqrt{C + \mathsf{hypot}\left(B, C\right)} \cdot \sqrt{F}\right) \]
3. Recombined 5 regimes into one program.

4. Final simplification 29.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 3.6 \cdot 10^{-227}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;B \leq 4.2 \cdot 10^{-142}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{-0.5 \cdot \frac{F}{A}}\right)\\ \mathbf{elif}\;B \leq 2.7 \cdot 10^{-39}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(-0.5 \cdot \frac{{B}^{2}}{A} + 2 \cdot C\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;B \leq 4.2 \cdot 10^{+151}:\\ \;\;\;\;\left(\sqrt{F} \cdot \sqrt{\frac{\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)}{\mathsf{fma}\left(-4, A \cdot C, {B}^{2}\right)}}\right) \cdot \left(-\sqrt{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{F} \cdot \sqrt{C + \mathsf{hypot}\left(B, C\right)}\right) \cdot \frac{-1}{B \cdot {2}^{-0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 56.6% accurate, 1.4× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ t_1 := t\_0 - {B\_m}^{2}\\ t_2 := 2 \cdot \left(\left({B\_m}^{2} - t\_0\right) \cdot F\right)\\ \mathbf{if}\;B\_m \leq 10^{-226}:\\ \;\;\;\;\frac{\sqrt{t\_2 \cdot \left(2 \cdot C\right)}}{t\_1}\\ \mathbf{elif}\;B\_m \leq 1.08 \cdot 10^{-144}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{-0.5 \cdot \frac{F}{A}}\right)\\ \mathbf{elif}\;B\_m \leq 9.2 \cdot 10^{+23}:\\ \;\;\;\;\frac{\sqrt{t\_2 \cdot \left(-0.5 \cdot \frac{{B\_m}^{2}}{A} + 2 \cdot C\right)}}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{F} \cdot \sqrt{C + \mathsf{hypot}\left(B\_m, C\right)}\right) \cdot \frac{\sqrt{2}}{-B\_m}\\ \end{array} \end{array} \]

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
 (let* ((t_0 (* (* 4.0 A) C))
        (t_1 (- t_0 (pow B_m 2.0)))
        (t_2 (* 2.0 (* (- (pow B_m 2.0) t_0) F))))
   (if (<= B_m 1e-226)
     (/ (sqrt (* t_2 (* 2.0 C))) t_1)
     (if (<= B_m 1.08e-144)
       (* (sqrt 2.0) (- (sqrt (* -0.5 (/ F A)))))
       (if (<= B_m 9.2e+23)
         (/ (sqrt (* t_2 (+ (* -0.5 (/ (pow B_m 2.0) A)) (* 2.0 C)))) t_1)
         (*
          (* (sqrt F) (sqrt (+ C (hypot B_m C))))
          (/ (sqrt 2.0) (- B_m))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = (4.0 * A) * C;
	double t_1 = t_0 - pow(B_m, 2.0);
	double t_2 = 2.0 * ((pow(B_m, 2.0) - t_0) * F);
	double tmp;
	if (B_m <= 1e-226) {
		tmp = sqrt((t_2 * (2.0 * C))) / t_1;
	} else if (B_m <= 1.08e-144) {
		tmp = sqrt(2.0) * -sqrt((-0.5 * (F / A)));
	} else if (B_m <= 9.2e+23) {
		tmp = sqrt((t_2 * ((-0.5 * (pow(B_m, 2.0) / A)) + (2.0 * C)))) / t_1;
	} else {
		tmp = (sqrt(F) * sqrt((C + hypot(B_m, C)))) * (sqrt(2.0) / -B_m);
	}
	return tmp;
}

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double t_0 = (4.0 * A) * C;
	double t_1 = t_0 - Math.pow(B_m, 2.0);
	double t_2 = 2.0 * ((Math.pow(B_m, 2.0) - t_0) * F);
	double tmp;
	if (B_m <= 1e-226) {
		tmp = Math.sqrt((t_2 * (2.0 * C))) / t_1;
	} else if (B_m <= 1.08e-144) {
		tmp = Math.sqrt(2.0) * -Math.sqrt((-0.5 * (F / A)));
	} else if (B_m <= 9.2e+23) {
		tmp = Math.sqrt((t_2 * ((-0.5 * (Math.pow(B_m, 2.0) / A)) + (2.0 * C)))) / t_1;
	} else {
		tmp = (Math.sqrt(F) * Math.sqrt((C + Math.hypot(B_m, C)))) * (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):
	t_0 = (4.0 * A) * C
	t_1 = t_0 - math.pow(B_m, 2.0)
	t_2 = 2.0 * ((math.pow(B_m, 2.0) - t_0) * F)
	tmp = 0
	if B_m <= 1e-226:
		tmp = math.sqrt((t_2 * (2.0 * C))) / t_1
	elif B_m <= 1.08e-144:
		tmp = math.sqrt(2.0) * -math.sqrt((-0.5 * (F / A)))
	elif B_m <= 9.2e+23:
		tmp = math.sqrt((t_2 * ((-0.5 * (math.pow(B_m, 2.0) / A)) + (2.0 * C)))) / t_1
	else:
		tmp = (math.sqrt(F) * math.sqrt((C + math.hypot(B_m, C)))) * (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)
	t_0 = Float64(Float64(4.0 * A) * C)
	t_1 = Float64(t_0 - (B_m ^ 2.0))
	t_2 = Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_0) * F))
	tmp = 0.0
	if (B_m <= 1e-226)
		tmp = Float64(sqrt(Float64(t_2 * Float64(2.0 * C))) / t_1);
	elseif (B_m <= 1.08e-144)
		tmp = Float64(sqrt(2.0) * Float64(-sqrt(Float64(-0.5 * Float64(F / A)))));
	elseif (B_m <= 9.2e+23)
		tmp = Float64(sqrt(Float64(t_2 * Float64(Float64(-0.5 * Float64((B_m ^ 2.0) / A)) + Float64(2.0 * C)))) / t_1);
	else
		tmp = Float64(Float64(sqrt(F) * sqrt(Float64(C + hypot(B_m, C)))) * Float64(sqrt(2.0) / Float64(-B_m)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	t_0 = (4.0 * A) * C;
	t_1 = t_0 - (B_m ^ 2.0);
	t_2 = 2.0 * (((B_m ^ 2.0) - t_0) * F);
	tmp = 0.0;
	if (B_m <= 1e-226)
		tmp = sqrt((t_2 * (2.0 * C))) / t_1;
	elseif (B_m <= 1.08e-144)
		tmp = sqrt(2.0) * -sqrt((-0.5 * (F / A)));
	elseif (B_m <= 9.2e+23)
		tmp = sqrt((t_2 * ((-0.5 * ((B_m ^ 2.0) / A)) + (2.0 * C)))) / t_1;
	else
		tmp = (sqrt(F) * sqrt((C + hypot(B_m, C)))) * (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_] := Block[{t$95$0 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 1e-226], N[(N[Sqrt[N[(t$95$2 * N[(2.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[B$95$m, 1.08e-144], N[(N[Sqrt[2.0], $MachinePrecision] * (-N[Sqrt[N[(-0.5 * N[(F / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], If[LessEqual[B$95$m, 9.2e+23], N[(N[Sqrt[N[(t$95$2 * N[(N[(-0.5 * N[(N[Power[B$95$m, 2.0], $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision] + N[(2.0 * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if B < 9.99999999999999921e-227

   1. Initial program 17.7%

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

   3. Taylor expanded in A around -inf 16.9%

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

   if 9.99999999999999921e-227 < B < 1.08e-144

   1. Initial program 12.1%

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

   3. Taylor expanded in F around 0 18.5%

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

      1. mul-1-neg 18.5%

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

      2. unpow2 18.5%

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

      3. unpow2 18.5%

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

      4. hypot-undefine 19.7%

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

      5. cancel-sign-sub-inv 19.7%

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

      6. metadata-eval 19.7%

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

   5. Simplified 19.7%

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

   6. Taylor expanded in A around -inf 21.0%

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

   if 1.08e-144 < B < 9.2000000000000002e23

   1. Initial program 32.1%

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

   3. Taylor expanded in A around -inf 31.1%

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

   if 9.2000000000000002e23 < B

   1. Initial program 10.0%

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

   3. Taylor expanded in A around 0 15.7%

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

      1. mul-1-neg 15.7%

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

   5. Simplified 15.7%

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

      1. pow1/2 15.7%

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

      2. *-commutative 15.7%

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

      3. unpow-prod-down 15.6%

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

      4. pow1/2 15.6%

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

      5. unpow2 15.6%

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

      6. unpow2 15.6%

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

      7. hypot-define 58.1%

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

      8. pow1/2 58.1%

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

   7. Applied egg-rr 58.1%

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

4. Final simplification 27.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 10^{-226}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;B \leq 1.08 \cdot 10^{-144}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{-0.5 \cdot \frac{F}{A}}\right)\\ \mathbf{elif}\;B \leq 9.2 \cdot 10^{+23}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(-0.5 \cdot \frac{{B}^{2}}{A} + 2 \cdot C\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{F} \cdot \sqrt{C + \mathsf{hypot}\left(B, C\right)}\right) \cdot \frac{\sqrt{2}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 57.8% accurate, 1.4× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ t_1 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_0\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{t\_0 - {B\_m}^{2}}\\ \mathbf{if}\;B\_m \leq 4.05 \cdot 10^{-227}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;B\_m \leq 2.2 \cdot 10^{-142}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{-0.5 \cdot \frac{F}{A}}\right)\\ \mathbf{elif}\;B\_m \leq 1.4 \cdot 10^{-26}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;B\_m \leq 5.5 \cdot 10^{+142}:\\ \;\;\;\;-\sqrt{2 \cdot \left(F \cdot \frac{\left(A + C\right) + \mathsf{hypot}\left(B\_m, A - C\right)}{\mathsf{fma}\left(-4, A \cdot C, {B\_m}^{2}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{F} \cdot \sqrt{C + \mathsf{hypot}\left(B\_m, C\right)}\right) \cdot \frac{-1}{B\_m \cdot {2}^{-0.5}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (* (* 4.0 A) C))
        (t_1
         (/
          (sqrt (* (* 2.0 (* (- (pow B_m 2.0) t_0) F)) (* 2.0 C)))
          (- t_0 (pow B_m 2.0)))))
   (if (<= B_m 4.05e-227)
     t_1
     (if (<= B_m 2.2e-142)
       (* (sqrt 2.0) (- (sqrt (* -0.5 (/ F A)))))
       (if (<= B_m 1.4e-26)
         t_1
         (if (<= B_m 5.5e+142)
           (-
            (sqrt
             (*
              2.0
              (*
               F
               (/
                (+ (+ A C) (hypot B_m (- A C)))
                (fma -4.0 (* A C) (pow B_m 2.0)))))))
           (*
            (* (sqrt F) (sqrt (+ C (hypot B_m C))))
            (/ -1.0 (* B_m (pow 2.0 -0.5))))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = (4.0 * A) * C;
	double t_1 = sqrt(((2.0 * ((pow(B_m, 2.0) - t_0) * F)) * (2.0 * C))) / (t_0 - pow(B_m, 2.0));
	double tmp;
	if (B_m <= 4.05e-227) {
		tmp = t_1;
	} else if (B_m <= 2.2e-142) {
		tmp = sqrt(2.0) * -sqrt((-0.5 * (F / A)));
	} else if (B_m <= 1.4e-26) {
		tmp = t_1;
	} else if (B_m <= 5.5e+142) {
		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(F) * sqrt((C + hypot(B_m, C)))) * (-1.0 / (B_m * pow(2.0, -0.5)));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(Float64(4.0 * A) * C)
	t_1 = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_0) * F)) * Float64(2.0 * C))) / Float64(t_0 - (B_m ^ 2.0)))
	tmp = 0.0
	if (B_m <= 4.05e-227)
		tmp = t_1;
	elseif (B_m <= 2.2e-142)
		tmp = Float64(sqrt(2.0) * Float64(-sqrt(Float64(-0.5 * Float64(F / A)))));
	elseif (B_m <= 1.4e-26)
		tmp = t_1;
	elseif (B_m <= 5.5e+142)
		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(Float64(sqrt(F) * sqrt(Float64(C + hypot(B_m, C)))) * Float64(-1.0 / Float64(B_m * (2.0 ^ -0.5))));
	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[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(2.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 4.05e-227], t$95$1, If[LessEqual[B$95$m, 2.2e-142], N[(N[Sqrt[2.0], $MachinePrecision] * (-N[Sqrt[N[(-0.5 * N[(F / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], If[LessEqual[B$95$m, 1.4e-26], t$95$1, If[LessEqual[B$95$m, 5.5e+142], (-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[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(-1.0 / N[(B$95$m * N[Power[2.0, -0.5], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if B < 4.04999999999999993e-227 or 2.20000000000000016e-142 < B < 1.4000000000000001e-26

   1. Initial program 19.1%

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

   3. Taylor expanded in A around -inf 19.0%

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

   if 4.04999999999999993e-227 < B < 2.20000000000000016e-142

   1. Initial program 12.1%

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

   3. Taylor expanded in F around 0 18.5%

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

      1. mul-1-neg 18.5%

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

      2. unpow2 18.5%

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

      3. unpow2 18.5%

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

      4. hypot-undefine 19.7%

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

      5. cancel-sign-sub-inv 19.7%

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

      6. metadata-eval 19.7%

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

   5. Simplified 19.7%

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

   6. Taylor expanded in A around -inf 21.0%

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

   if 1.4000000000000001e-26 < B < 5.50000000000000035e142

   1. Initial program 35.7%

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

   3. Taylor expanded in F around 0 40.4%

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

      1. mul-1-neg 40.4%

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

      2. unpow2 40.4%

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

      3. unpow2 40.4%

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

      4. hypot-undefine 56.8%

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

      5. cancel-sign-sub-inv 56.8%

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

      6. metadata-eval 56.8%

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

   5. Simplified 56.8%

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

      1. pow1 56.8%

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

      2. sqrt-unprod 56.5%

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

      3. associate-/l* 61.8%

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

      4. associate-+r+ 61.8%

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

      5. +-commutative 61.8%

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

      6. fma-define 61.8%

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

   7. Applied egg-rr 61.8%

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

      1. unpow1 61.8%

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

   9. Simplified 61.8%

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

   if 5.50000000000000035e142 < B

   1. Initial program 2.8%

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

   3. Taylor expanded in A around 0 7.8%

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

      1. mul-1-neg 7.8%

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

   5. Simplified 7.8%

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

      1. pow1/2 7.8%

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

      2. *-commutative 7.8%

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

      3. unpow-prod-down 7.8%

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

      4. pow1/2 7.8%

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

      5. unpow2 7.8%

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

      6. unpow2 7.8%

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

      7. hypot-define 65.6%

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

      8. pow1/2 65.6%

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

   7. Applied egg-rr 65.6%

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

      1. clear-num 65.4%

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

      2. inv-pow 65.4%

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

   9. Applied egg-rr 65.4%

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

       1. unpow-1 65.4%

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

   11. Simplified 65.4%

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

       1. div-inv 65.2%

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

       2. pow1/2 65.2%

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

       3. metadata-eval 65.2%

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

       4. metadata-eval 65.2%

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

       5. sqrt-pow2 65.2%

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

       6. pow-flip 65.2%

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

       7. sqrt-pow2 65.5%

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

       8. metadata-eval 65.5%

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

       9. metadata-eval 65.5%

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

       10. metadata-eval 65.5%

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

   13. Applied egg-rr 65.5%

       \[\leadsto -\frac{1}{\color{blue}{B \cdot {2}^{-0.5}}} \cdot \left(\sqrt{C + \mathsf{hypot}\left(B, C\right)} \cdot \sqrt{F}\right) \]
3. Recombined 4 regimes into one program.

4. Final simplification 29.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 4.05 \cdot 10^{-227}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;B \leq 2.2 \cdot 10^{-142}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{-0.5 \cdot \frac{F}{A}}\right)\\ \mathbf{elif}\;B \leq 1.4 \cdot 10^{-26}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;B \leq 5.5 \cdot 10^{+142}:\\ \;\;\;\;-\sqrt{2 \cdot \left(F \cdot \frac{\left(A + C\right) + \mathsf{hypot}\left(B, A - C\right)}{\mathsf{fma}\left(-4, A \cdot C, {B}^{2}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{F} \cdot \sqrt{C + \mathsf{hypot}\left(B, C\right)}\right) \cdot \frac{-1}{B \cdot {2}^{-0.5}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 57.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ t_1 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_0\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{t\_0 - {B\_m}^{2}}\\ \mathbf{if}\;B\_m \leq 4.6 \cdot 10^{-227}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;B\_m \leq 2.2 \cdot 10^{-140}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{-0.5 \cdot \frac{F}{A}}\right)\\ \mathbf{elif}\;B\_m \leq 3.8 \cdot 10^{-51}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{F} \cdot \sqrt{C + \mathsf{hypot}\left(B\_m, C\right)}\right) \cdot \frac{\sqrt{2}}{-B\_m}\\ \end{array} \end{array} \]

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
 (let* ((t_0 (* (* 4.0 A) C))
        (t_1
         (/
          (sqrt (* (* 2.0 (* (- (pow B_m 2.0) t_0) F)) (* 2.0 C)))
          (- t_0 (pow B_m 2.0)))))
   (if (<= B_m 4.6e-227)
     t_1
     (if (<= B_m 2.2e-140)
       (* (sqrt 2.0) (- (sqrt (* -0.5 (/ F A)))))
       (if (<= B_m 3.8e-51)
         t_1
         (*
          (* (sqrt F) (sqrt (+ C (hypot B_m C))))
          (/ (sqrt 2.0) (- B_m))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = (4.0 * A) * C;
	double t_1 = sqrt(((2.0 * ((pow(B_m, 2.0) - t_0) * F)) * (2.0 * C))) / (t_0 - pow(B_m, 2.0));
	double tmp;
	if (B_m <= 4.6e-227) {
		tmp = t_1;
	} else if (B_m <= 2.2e-140) {
		tmp = sqrt(2.0) * -sqrt((-0.5 * (F / A)));
	} else if (B_m <= 3.8e-51) {
		tmp = t_1;
	} else {
		tmp = (sqrt(F) * sqrt((C + hypot(B_m, C)))) * (sqrt(2.0) / -B_m);
	}
	return tmp;
}

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double t_0 = (4.0 * A) * C;
	double t_1 = Math.sqrt(((2.0 * ((Math.pow(B_m, 2.0) - t_0) * F)) * (2.0 * C))) / (t_0 - Math.pow(B_m, 2.0));
	double tmp;
	if (B_m <= 4.6e-227) {
		tmp = t_1;
	} else if (B_m <= 2.2e-140) {
		tmp = Math.sqrt(2.0) * -Math.sqrt((-0.5 * (F / A)));
	} else if (B_m <= 3.8e-51) {
		tmp = t_1;
	} else {
		tmp = (Math.sqrt(F) * Math.sqrt((C + Math.hypot(B_m, C)))) * (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):
	t_0 = (4.0 * A) * C
	t_1 = math.sqrt(((2.0 * ((math.pow(B_m, 2.0) - t_0) * F)) * (2.0 * C))) / (t_0 - math.pow(B_m, 2.0))
	tmp = 0
	if B_m <= 4.6e-227:
		tmp = t_1
	elif B_m <= 2.2e-140:
		tmp = math.sqrt(2.0) * -math.sqrt((-0.5 * (F / A)))
	elif B_m <= 3.8e-51:
		tmp = t_1
	else:
		tmp = (math.sqrt(F) * math.sqrt((C + math.hypot(B_m, C)))) * (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)
	t_0 = Float64(Float64(4.0 * A) * C)
	t_1 = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_0) * F)) * Float64(2.0 * C))) / Float64(t_0 - (B_m ^ 2.0)))
	tmp = 0.0
	if (B_m <= 4.6e-227)
		tmp = t_1;
	elseif (B_m <= 2.2e-140)
		tmp = Float64(sqrt(2.0) * Float64(-sqrt(Float64(-0.5 * Float64(F / A)))));
	elseif (B_m <= 3.8e-51)
		tmp = t_1;
	else
		tmp = Float64(Float64(sqrt(F) * sqrt(Float64(C + hypot(B_m, C)))) * Float64(sqrt(2.0) / Float64(-B_m)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	t_0 = (4.0 * A) * C;
	t_1 = sqrt(((2.0 * (((B_m ^ 2.0) - t_0) * F)) * (2.0 * C))) / (t_0 - (B_m ^ 2.0));
	tmp = 0.0;
	if (B_m <= 4.6e-227)
		tmp = t_1;
	elseif (B_m <= 2.2e-140)
		tmp = sqrt(2.0) * -sqrt((-0.5 * (F / A)));
	elseif (B_m <= 3.8e-51)
		tmp = t_1;
	else
		tmp = (sqrt(F) * sqrt((C + hypot(B_m, C)))) * (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_] := Block[{t$95$0 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(2.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 4.6e-227], t$95$1, If[LessEqual[B$95$m, 2.2e-140], N[(N[Sqrt[2.0], $MachinePrecision] * (-N[Sqrt[N[(-0.5 * N[(F / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], If[LessEqual[B$95$m, 3.8e-51], t$95$1, N[(N[(N[Sqrt[F], $MachinePrecision] * N[Sqrt[N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[2.0], $MachinePrecision] / (-B$95$m)), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if B < 4.60000000000000024e-227 or 2.1999999999999999e-140 < B < 3.80000000000000003e-51

   1. Initial program 18.7%

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

   3. Taylor expanded in A around -inf 18.6%

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

   if 4.60000000000000024e-227 < B < 2.1999999999999999e-140

   1. Initial program 12.1%

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

   3. Taylor expanded in F around 0 18.5%

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

      1. mul-1-neg 18.5%

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

      2. unpow2 18.5%

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

      3. unpow2 18.5%

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

      4. hypot-undefine 19.7%

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

      5. cancel-sign-sub-inv 19.7%

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

      6. metadata-eval 19.7%

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

   5. Simplified 19.7%

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

   6. Taylor expanded in A around -inf 21.0%

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

   if 3.80000000000000003e-51 < B

   1. Initial program 15.6%

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

   3. Taylor expanded in A around 0 19.6%

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

      1. mul-1-neg 19.6%

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

   5. Simplified 19.6%

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

      1. pow1/2 19.6%

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

      2. *-commutative 19.6%

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

      3. unpow-prod-down 19.5%

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

      4. pow1/2 19.5%

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

      5. unpow2 19.5%

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

      6. unpow2 19.5%

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

      7. hypot-define 54.0%

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

      8. pow1/2 54.0%

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

   7. Applied egg-rr 54.0%

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

4. Final simplification 27.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 4.6 \cdot 10^{-227}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;B \leq 2.2 \cdot 10^{-140}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{-0.5 \cdot \frac{F}{A}}\right)\\ \mathbf{elif}\;B \leq 3.8 \cdot 10^{-51}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\left(\sqrt{F} \cdot \sqrt{C + \mathsf{hypot}\left(B, C\right)}\right) \cdot \frac{\sqrt{2}}{-B}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 51.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ t_1 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_0\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{t\_0 - {B\_m}^{2}}\\ \mathbf{if}\;B\_m \leq 9 \cdot 10^{-227}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;B\_m \leq 6 \cdot 10^{-142}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{-0.5 \cdot \frac{F}{A}}\right)\\ \mathbf{elif}\;B\_m \leq 2.15 \cdot 10^{-26}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot F} \cdot \left(-{B\_m}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (* (* 4.0 A) C))
        (t_1
         (/
          (sqrt (* (* 2.0 (* (- (pow B_m 2.0) t_0) F)) (* 2.0 C)))
          (- t_0 (pow B_m 2.0)))))
   (if (<= B_m 9e-227)
     t_1
     (if (<= B_m 6e-142)
       (* (sqrt 2.0) (- (sqrt (* -0.5 (/ F A)))))
       (if (<= B_m 2.15e-26) t_1 (* (sqrt (* 2.0 F)) (- (pow B_m -0.5))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = (4.0 * A) * C;
	double t_1 = sqrt(((2.0 * ((pow(B_m, 2.0) - t_0) * F)) * (2.0 * C))) / (t_0 - pow(B_m, 2.0));
	double tmp;
	if (B_m <= 9e-227) {
		tmp = t_1;
	} else if (B_m <= 6e-142) {
		tmp = sqrt(2.0) * -sqrt((-0.5 * (F / A)));
	} else if (B_m <= 2.15e-26) {
		tmp = t_1;
	} else {
		tmp = sqrt((2.0 * F)) * -pow(B_m, -0.5);
	}
	return tmp;
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (4.0d0 * a) * c
    t_1 = sqrt(((2.0d0 * (((b_m ** 2.0d0) - t_0) * f)) * (2.0d0 * c))) / (t_0 - (b_m ** 2.0d0))
    if (b_m <= 9d-227) then
        tmp = t_1
    else if (b_m <= 6d-142) then
        tmp = sqrt(2.0d0) * -sqrt(((-0.5d0) * (f / a)))
    else if (b_m <= 2.15d-26) then
        tmp = t_1
    else
        tmp = sqrt((2.0d0 * f)) * -(b_m ** (-0.5d0))
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double t_0 = (4.0 * A) * C;
	double t_1 = Math.sqrt(((2.0 * ((Math.pow(B_m, 2.0) - t_0) * F)) * (2.0 * C))) / (t_0 - Math.pow(B_m, 2.0));
	double tmp;
	if (B_m <= 9e-227) {
		tmp = t_1;
	} else if (B_m <= 6e-142) {
		tmp = Math.sqrt(2.0) * -Math.sqrt((-0.5 * (F / A)));
	} else if (B_m <= 2.15e-26) {
		tmp = t_1;
	} else {
		tmp = Math.sqrt((2.0 * F)) * -Math.pow(B_m, -0.5);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	t_0 = (4.0 * A) * C
	t_1 = math.sqrt(((2.0 * ((math.pow(B_m, 2.0) - t_0) * F)) * (2.0 * C))) / (t_0 - math.pow(B_m, 2.0))
	tmp = 0
	if B_m <= 9e-227:
		tmp = t_1
	elif B_m <= 6e-142:
		tmp = math.sqrt(2.0) * -math.sqrt((-0.5 * (F / A)))
	elif B_m <= 2.15e-26:
		tmp = t_1
	else:
		tmp = math.sqrt((2.0 * F)) * -math.pow(B_m, -0.5)
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = Float64(Float64(4.0 * A) * C)
	t_1 = Float64(sqrt(Float64(Float64(2.0 * Float64(Float64((B_m ^ 2.0) - t_0) * F)) * Float64(2.0 * C))) / Float64(t_0 - (B_m ^ 2.0)))
	tmp = 0.0
	if (B_m <= 9e-227)
		tmp = t_1;
	elseif (B_m <= 6e-142)
		tmp = Float64(sqrt(2.0) * Float64(-sqrt(Float64(-0.5 * Float64(F / A)))));
	elseif (B_m <= 2.15e-26)
		tmp = t_1;
	else
		tmp = Float64(sqrt(Float64(2.0 * F)) * Float64(-(B_m ^ -0.5)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	t_0 = (4.0 * A) * C;
	t_1 = sqrt(((2.0 * (((B_m ^ 2.0) - t_0) * F)) * (2.0 * C))) / (t_0 - (B_m ^ 2.0));
	tmp = 0.0;
	if (B_m <= 9e-227)
		tmp = t_1;
	elseif (B_m <= 6e-142)
		tmp = sqrt(2.0) * -sqrt((-0.5 * (F / A)));
	elseif (B_m <= 2.15e-26)
		tmp = t_1;
	else
		tmp = sqrt((2.0 * F)) * -(B_m ^ -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(N[(4.0 * A), $MachinePrecision] * C), $MachinePrecision]}, Block[{t$95$1 = N[(N[Sqrt[N[(N[(2.0 * N[(N[(N[Power[B$95$m, 2.0], $MachinePrecision] - t$95$0), $MachinePrecision] * F), $MachinePrecision]), $MachinePrecision] * N[(2.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[B$95$m, 9e-227], t$95$1, If[LessEqual[B$95$m, 6e-142], N[(N[Sqrt[2.0], $MachinePrecision] * (-N[Sqrt[N[(-0.5 * N[(F / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], If[LessEqual[B$95$m, 2.15e-26], t$95$1, N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] * (-N[Power[B$95$m, -0.5], $MachinePrecision])), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if B < 8.99999999999999986e-227 or 6.0000000000000002e-142 < B < 2.14999999999999994e-26

   1. Initial program 19.1%

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

   3. Taylor expanded in A around -inf 19.0%

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

   if 8.99999999999999986e-227 < B < 6.0000000000000002e-142

   1. Initial program 12.1%

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

   3. Taylor expanded in F around 0 18.5%

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

      1. mul-1-neg 18.5%

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

      2. unpow2 18.5%

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

      3. unpow2 18.5%

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

      4. hypot-undefine 19.7%

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

      5. cancel-sign-sub-inv 19.7%

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

      6. metadata-eval 19.7%

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

   5. Simplified 19.7%

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

   6. Taylor expanded in A around -inf 21.0%

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

   if 2.14999999999999994e-26 < B

   1. Initial program 13.7%

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

   3. Taylor expanded in B around inf 46.9%

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

      1. mul-1-neg 46.9%

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

   5. Simplified 46.9%

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

      1. pow1 46.9%

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

      2. sqrt-unprod 47.1%

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

   7. Applied egg-rr 47.1%

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

      1. unpow1 47.1%

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

   9. Simplified 47.1%

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

       1. *-un-lft-identity 47.1%

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

       2. associate-*l/ 47.2%

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

   11. Applied egg-rr 47.2%

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

       1. *-lft-identity 47.2%

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

   13. Simplified 47.2%

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

       1. div-inv 47.1%

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

       2. sqrt-prod 56.8%

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

       3. inv-pow 56.8%

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

       4. sqrt-pow1 56.8%

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

       5. metadata-eval 56.8%

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

   15. Applied egg-rr 56.8%

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

4. Final simplification 27.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 9 \cdot 10^{-227}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{elif}\;B \leq 6 \cdot 10^{-142}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{-0.5 \cdot \frac{F}{A}}\right)\\ \mathbf{elif}\;B \leq 2.15 \cdot 10^{-26}:\\ \;\;\;\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(2 \cdot C\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot F} \cdot \left(-{B}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 48.9% accurate, 3.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 2.5 \cdot 10^{+25}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{-0.5 \cdot \frac{F}{A}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot F} \cdot \left(-{B\_m}^{-0.5}\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 2.5e+25)
   (* (sqrt 2.0) (- (sqrt (* -0.5 (/ F A)))))
   (* (sqrt (* 2.0 F)) (- (pow B_m -0.5)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 2.5e+25) {
		tmp = sqrt(2.0) * -sqrt((-0.5 * (F / A)));
	} else {
		tmp = sqrt((2.0 * F)) * -pow(B_m, -0.5);
	}
	return tmp;
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (b_m <= 2.5d+25) then
        tmp = sqrt(2.0d0) * -sqrt(((-0.5d0) * (f / a)))
    else
        tmp = sqrt((2.0d0 * f)) * -(b_m ** (-0.5d0))
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 2.5e+25) {
		tmp = Math.sqrt(2.0) * -Math.sqrt((-0.5 * (F / A)));
	} else {
		tmp = Math.sqrt((2.0 * F)) * -Math.pow(B_m, -0.5);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 2.5e+25:
		tmp = math.sqrt(2.0) * -math.sqrt((-0.5 * (F / A)))
	else:
		tmp = math.sqrt((2.0 * F)) * -math.pow(B_m, -0.5)
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 2.5e+25)
		tmp = Float64(sqrt(2.0) * Float64(-sqrt(Float64(-0.5 * Float64(F / A)))));
	else
		tmp = Float64(sqrt(Float64(2.0 * F)) * Float64(-(B_m ^ -0.5)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (B_m <= 2.5e+25)
		tmp = sqrt(2.0) * -sqrt((-0.5 * (F / A)));
	else
		tmp = sqrt((2.0 * F)) * -(B_m ^ -0.5);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 2.5e+25], N[(N[Sqrt[2.0], $MachinePrecision] * (-N[Sqrt[N[(-0.5 * N[(F / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] * (-N[Power[B$95$m, -0.5], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 2.50000000000000012e25

   1. Initial program 19.3%

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

   3. Taylor expanded in F around 0 21.3%

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

      1. mul-1-neg 21.3%

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

      2. unpow2 21.3%

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

      3. unpow2 21.3%

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

      4. hypot-undefine 25.9%

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

      5. cancel-sign-sub-inv 25.9%

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

      6. metadata-eval 25.9%

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

   5. Simplified 25.9%

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

   6. Taylor expanded in A around -inf 17.4%

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

   if 2.50000000000000012e25 < B

   1. Initial program 10.0%

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

   3. Taylor expanded in B around inf 47.1%

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

      1. mul-1-neg 47.1%

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

   5. Simplified 47.1%

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

      1. pow1 47.1%

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

      2. sqrt-unprod 47.3%

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

   7. Applied egg-rr 47.3%

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

      1. unpow1 47.3%

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

   9. Simplified 47.3%

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

       1. *-un-lft-identity 47.3%

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

       2. associate-*l/ 47.4%

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

   11. Applied egg-rr 47.4%

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

       1. *-lft-identity 47.4%

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

   13. Simplified 47.4%

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

       1. div-inv 47.3%

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

       2. sqrt-prod 57.9%

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

       3. inv-pow 57.9%

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

       4. sqrt-pow1 58.0%

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

       5. metadata-eval 58.0%

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

   15. Applied egg-rr 58.0%

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

4. Final simplification 25.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 2.5 \cdot 10^{+25}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{-0.5 \cdot \frac{F}{A}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot F} \cdot \left(-{B}^{-0.5}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 35.8% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 17.4%

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

3. Taylor expanded in B around inf 12.5%

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

   1. mul-1-neg 12.5%

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

5. Simplified 12.5%

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

   1. pow1 12.5%

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

   2. sqrt-unprod 12.5%

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

7. Applied egg-rr 12.5%

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

   1. unpow1 12.5%

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

9. Simplified 12.5%

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

    1. *-un-lft-identity 12.5%

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

    2. associate-*l/ 12.6%

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

11. Applied egg-rr 12.6%

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

    1. *-lft-identity 12.6%

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

13. Simplified 12.6%

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

    1. div-inv 12.5%

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

    2. sqrt-prod 15.1%

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

    3. inv-pow 15.1%

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

    4. sqrt-pow1 15.1%

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

    5. metadata-eval 15.1%

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

15. Applied egg-rr 15.1%

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

16. Final simplification 15.1%

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

Alternative 9: 35.8% accurate, 3.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 17.4%

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

3. Taylor expanded in B around inf 12.5%

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

   1. mul-1-neg 12.5%

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

5. Simplified 12.5%

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

   1. pow1 12.5%

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

   2. sqrt-unprod 12.5%

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

7. Applied egg-rr 12.5%

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

   1. unpow1 12.5%

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

9. Simplified 12.5%

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

    1. *-un-lft-identity 12.5%

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

    2. associate-*l/ 12.6%

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

11. Applied egg-rr 12.6%

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

    1. *-lft-identity 12.6%

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

13. Simplified 12.6%

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

    1. metadata-eval 12.6%

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

    2. metadata-eval 12.6%

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

    3. sqrt-pow2 12.5%

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

    4. associate-/l* 12.4%

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

    5. sqrt-prod 15.0%

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

    6. sqrt-pow2 15.1%

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

    7. metadata-eval 15.1%

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

    8. metadata-eval 15.1%

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

15. Applied egg-rr 15.1%

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

16. Final simplification 15.1%

    \[\leadsto \sqrt{\frac{2}{B}} \cdot \left(-\sqrt{F}\right) \]
17. Add Preprocessing

Alternative 10: 35.8% accurate, 3.1× speedup

Math

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

FPCore

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 17.4%

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

3. Taylor expanded in B around inf 12.5%

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

   1. mul-1-neg 12.5%

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

5. Simplified 12.5%

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

   1. pow1 12.5%

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

   2. sqrt-unprod 12.5%

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

7. Applied egg-rr 12.5%

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

   1. unpow1 12.5%

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

9. Simplified 12.5%

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

    1. *-un-lft-identity 12.5%

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

    2. associate-*l/ 12.6%

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

11. Applied egg-rr 12.6%

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

    1. *-lft-identity 12.6%

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

13. Simplified 12.6%

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

    1. sqrt-div 15.1%

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

15. Applied egg-rr 15.1%

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

16. Final simplification 15.1%

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

Alternative 11: 27.5% accurate, 5.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 17.4%

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

3. Taylor expanded in B around inf 12.5%

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

   1. mul-1-neg 12.5%

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

5. Simplified 12.5%

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

   1. pow1 12.5%

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

   2. sqrt-unprod 12.5%

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

7. Applied egg-rr 12.5%

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

   1. unpow1 12.5%

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

9. Simplified 12.5%

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

    1. pow1/2 12.7%

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

    2. associate-*l/ 12.8%

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

11. Applied egg-rr 12.8%

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

12. Final simplification 12.8%

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

Alternative 12: 27.5% accurate, 6.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 17.4%

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

3. Taylor expanded in B around inf 12.5%

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

   1. mul-1-neg 12.5%

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

5. Simplified 12.5%

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

   1. pow1 12.5%

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

   2. sqrt-unprod 12.5%

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

7. Applied egg-rr 12.5%

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

   1. unpow1 12.5%

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

9. Simplified 12.5%

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

    1. *-un-lft-identity 12.5%

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

    2. associate-*l/ 12.6%

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

11. Applied egg-rr 12.6%

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

    1. *-lft-identity 12.6%

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

13. Simplified 12.6%

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

    1. *-un-lft-identity 12.6%

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

    2. metadata-eval 12.6%

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

    3. metadata-eval 12.6%

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

    4. sqrt-pow2 12.5%

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

    5. associate-/l* 12.4%

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

    6. sqrt-pow2 12.5%

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

    7. metadata-eval 12.5%

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

    8. metadata-eval 12.5%

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

15. Applied egg-rr 12.5%

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

    1. *-lft-identity 12.5%

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

17. Simplified 12.5%

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

18. Final simplification 12.5%

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

Alternative 13: 27.5% accurate, 6.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 17.4%

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

3. Taylor expanded in B around inf 12.5%

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

   1. mul-1-neg 12.5%

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

5. Simplified 12.5%

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

   1. pow1 12.5%

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

   2. sqrt-unprod 12.5%

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

7. Applied egg-rr 12.5%

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

   1. unpow1 12.5%

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

9. Simplified 12.5%

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

10. Final simplification 12.5%

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

Alternative 14: 27.5% accurate, 6.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 17.4%

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

3. Taylor expanded in B around inf 12.5%

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

   1. mul-1-neg 12.5%

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

5. Simplified 12.5%

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

   1. pow1 12.5%

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

   2. sqrt-unprod 12.5%

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

7. Applied egg-rr 12.5%

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

   1. unpow1 12.5%

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

9. Simplified 12.5%

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

    1. *-un-lft-identity 12.5%

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

    2. associate-*l/ 12.6%

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

11. Applied egg-rr 12.6%

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

    1. *-lft-identity 12.6%

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

13. Simplified 12.6%

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

14. Final simplification 12.6%

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

Reproduce

herbie shell --seed 2024055 
(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))))