ABCF->ab-angle a

Percentage Accurate: 19.0% → 60.9%

Time: 26.1s

Alternatives: 16

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: 19.0% 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: 60.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \left(4 \cdot A\right) \cdot C\\ t_1 := \frac{\sqrt{\left(2 \cdot \left(\left({B\_m}^{2} - t\_0\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B\_m}^{2} + {\left(A - C\right)}^{2}}\right)}}{t\_0 - {B\_m}^{2}}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-210}:\\ \;\;\;\;\left(\left(-\sqrt{2}\right) \cdot \sqrt{F}\right) \cdot \sqrt{\frac{C + \left(A + \mathsf{hypot}\left(B\_m, A - C\right)\right)}{\mathsf{fma}\left(-4, A \cdot C, {B\_m}^{2}\right)}}\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B\_m, B\_m, -4 \cdot \left(A \cdot C\right)\right)\right)} \cdot \sqrt{-0.5 \cdot \frac{{B\_m}^{2}}{A} + 2 \cdot C}}{-\mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left({F}^{0.25} \cdot \left({F}^{0.25} \cdot \left(-{B\_m}^{-0.5}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (* (* 4.0 A) C))
        (t_1
         (/
          (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_0 (pow B_m 2.0)))))
   (if (<= t_1 -1e-210)
     (*
      (* (- (sqrt 2.0)) (sqrt F))
      (sqrt
       (/ (+ C (+ A (hypot B_m (- A C)))) (fma -4.0 (* A C) (pow B_m 2.0)))))
     (if (<= t_1 INFINITY)
       (/
        (*
         (sqrt (* 2.0 (* F (fma B_m B_m (* -4.0 (* A C))))))
         (sqrt (+ (* -0.5 (/ (pow B_m 2.0) A)) (* 2.0 C))))
        (- (fma B_m B_m (* A (* C -4.0)))))
       (* (sqrt 2.0) (* (pow F 0.25) (* (pow F 0.25) (- (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)) * ((A + C) + sqrt((pow(B_m, 2.0) + pow((A - C), 2.0)))))) / (t_0 - pow(B_m, 2.0));
	double tmp;
	if (t_1 <= -1e-210) {
		tmp = (-sqrt(2.0) * sqrt(F)) * sqrt(((C + (A + hypot(B_m, (A - C)))) / fma(-4.0, (A * C), pow(B_m, 2.0))));
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (sqrt((2.0 * (F * fma(B_m, B_m, (-4.0 * (A * C)))))) * sqrt(((-0.5 * (pow(B_m, 2.0) / A)) + (2.0 * C)))) / -fma(B_m, B_m, (A * (C * -4.0)));
	} else {
		tmp = sqrt(2.0) * (pow(F, 0.25) * (pow(F, 0.25) * -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(Float64(A + C) + sqrt(Float64((B_m ^ 2.0) + (Float64(A - C) ^ 2.0)))))) / Float64(t_0 - (B_m ^ 2.0)))
	tmp = 0.0
	if (t_1 <= -1e-210)
		tmp = Float64(Float64(Float64(-sqrt(2.0)) * sqrt(F)) * sqrt(Float64(Float64(C + Float64(A + hypot(B_m, Float64(A - C)))) / fma(-4.0, Float64(A * C), (B_m ^ 2.0)))));
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(sqrt(Float64(2.0 * Float64(F * fma(B_m, B_m, Float64(-4.0 * Float64(A * C)))))) * sqrt(Float64(Float64(-0.5 * Float64((B_m ^ 2.0) / A)) + Float64(2.0 * C)))) / Float64(-fma(B_m, B_m, Float64(A * Float64(C * -4.0)))));
	else
		tmp = Float64(sqrt(2.0) * Float64((F ^ 0.25) * Float64((F ^ 0.25) * Float64(-(B_m ^ -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[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(N[Power[B$95$m, 2.0], $MachinePrecision] + N[Power[N[(A - C), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(t$95$0 - N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-210], N[(N[((-N[Sqrt[2.0], $MachinePrecision]) * N[Sqrt[F], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(N[(C + N[(A + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-4.0 * N[(A * C), $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, Infinity], N[(N[(N[Sqrt[N[(2.0 * N[(F * N[(B$95$m * B$95$m + N[(-4.0 * N[(A * C), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(N[(-0.5 * N[(N[Power[B$95$m, 2.0], $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision] + N[(2.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / (-N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Power[F, 0.25], $MachinePrecision] * N[(N[Power[F, 0.25], $MachinePrecision] * (-N[Power[B$95$m, -0.5], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < -1e-210

   1. Initial program 46.6%

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

   3. Taylor expanded in F around 0 44.4%

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

   5. Simplified 67.0%

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

      1. distribute-lft-neg-in 67.0%

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

      2. sqrt-prod 81.8%

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

      3. associate-*r* 81.7%

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

      4. +-commutative 81.7%

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

      5. associate-+l+ 81.6%

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

      6. *-commutative 81.6%

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

   7. Applied egg-rr 81.6%

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

   if -1e-210 < (/.f64 (neg.f64 (sqrt.f64 (*.f64 (*.f64 #s(literal 2 binary64) (*.f64 (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C)) F)) (+.f64 (+.f64 A C) (sqrt.f64 (+.f64 (pow.f64 (-.f64 A C) #s(literal 2 binary64)) (pow.f64 B #s(literal 2 binary64)))))))) (-.f64 (pow.f64 B #s(literal 2 binary64)) (*.f64 (*.f64 #s(literal 4 binary64) A) C))) < +inf.0

   1. Initial program 18.8%

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

   3. Simplified 29.9%

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

      1. associate-*r* 29.9%

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

      2. associate-+r+ 27.5%

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

      3. hypot-undefine 18.8%

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

      4. unpow2 18.8%

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

      5. unpow2 18.8%

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

      6. +-commutative 18.8%

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

      7. sqrt-prod 20.8%

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

      8. *-commutative 20.8%

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

      9. associate-*r* 20.8%

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

      10. associate-+l+ 23.3%

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

   6. Applied egg-rr 37.7%

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

   7. Taylor expanded in A around -inf 28.4%

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

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

   1. Initial program 0.0%

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

   3. Taylor expanded in B around inf 17.1%

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

      1. mul-1-neg 17.1%

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

      2. *-commutative 17.1%

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

   5. Simplified 17.1%

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

      1. add-cbrt-cube 14.2%

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

      2. pow1/3 13.4%

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

      3. add-sqr-sqrt 13.4%

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

      4. pow1 13.4%

         \[\leadsto -\sqrt{2} \cdot {\left(\color{blue}{{\left(\frac{F}{B}\right)}^{1}} \cdot \sqrt{\frac{F}{B}}\right)}^{0.3333333333333333} \]

      5. pow1/2 13.5%

         \[\leadsto -\sqrt{2} \cdot {\left({\left(\frac{F}{B}\right)}^{1} \cdot \color{blue}{{\left(\frac{F}{B}\right)}^{0.5}}\right)}^{0.3333333333333333} \]

      6. pow-prod-up 13.5%

         \[\leadsto -\sqrt{2} \cdot {\color{blue}{\left({\left(\frac{F}{B}\right)}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333} \]

      7. metadata-eval 13.5%

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

   7. Applied egg-rr 13.5%

      \[\leadsto -\sqrt{2} \cdot \color{blue}{{\left({\left(\frac{F}{B}\right)}^{1.5}\right)}^{0.3333333333333333}} \]
   8. Step-by-step derivation

      1. pow-pow 17.2%

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

      2. metadata-eval 17.2%

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

      3. pow1/2 17.1%

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

      4. div-inv 17.1%

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

      5. sqrt-unprod 20.8%

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

      6. *-commutative 20.8%

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

      7. add-sqr-sqrt 20.8%

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

      8. associate-*r* 20.8%

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

      9. inv-pow 20.8%

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

      10. sqrt-pow1 20.8%

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

      11. metadata-eval 20.8%

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

      12. pow1/2 20.8%

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

      13. metadata-eval 20.8%

          \[\leadsto -\sqrt{2} \cdot \left(\left({B}^{-0.5} \cdot \sqrt{{F}^{\color{blue}{\left(1.5 \cdot 0.3333333333333333\right)}}}\right) \cdot \sqrt{\sqrt{F}}\right) \]

      14. sqrt-pow1 20.8%

          \[\leadsto -\sqrt{2} \cdot \left(\left({B}^{-0.5} \cdot \color{blue}{{F}^{\left(\frac{1.5 \cdot 0.3333333333333333}{2}\right)}}\right) \cdot \sqrt{\sqrt{F}}\right) \]

      15. metadata-eval 20.8%

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

      16. metadata-eval 20.8%

          \[\leadsto -\sqrt{2} \cdot \left(\left({B}^{-0.5} \cdot {F}^{\color{blue}{0.25}}\right) \cdot \sqrt{\sqrt{F}}\right) \]

      17. pow1/2 20.8%

          \[\leadsto -\sqrt{2} \cdot \left(\left({B}^{-0.5} \cdot {F}^{0.25}\right) \cdot \sqrt{\color{blue}{{F}^{0.5}}}\right) \]

      18. metadata-eval 20.8%

          \[\leadsto -\sqrt{2} \cdot \left(\left({B}^{-0.5} \cdot {F}^{0.25}\right) \cdot \sqrt{{F}^{\color{blue}{\left(1.5 \cdot 0.3333333333333333\right)}}}\right) \]

      19. sqrt-pow1 20.8%

          \[\leadsto -\sqrt{2} \cdot \left(\left({B}^{-0.5} \cdot {F}^{0.25}\right) \cdot \color{blue}{{F}^{\left(\frac{1.5 \cdot 0.3333333333333333}{2}\right)}}\right) \]

      20. metadata-eval 20.8%

          \[\leadsto -\sqrt{2} \cdot \left(\left({B}^{-0.5} \cdot {F}^{0.25}\right) \cdot {F}^{\left(\frac{\color{blue}{0.5}}{2}\right)}\right) \]

      21. metadata-eval 20.8%

          \[\leadsto -\sqrt{2} \cdot \left(\left({B}^{-0.5} \cdot {F}^{0.25}\right) \cdot {F}^{\color{blue}{0.25}}\right) \]

   9. Applied egg-rr 20.8%

      \[\leadsto -\sqrt{2} \cdot \color{blue}{\left(\left({B}^{-0.5} \cdot {F}^{0.25}\right) \cdot {F}^{0.25}\right)} \]
   10. Step-by-step derivation

       1. *-commutative 20.8%

          \[\leadsto -\sqrt{2} \cdot \color{blue}{\left({F}^{0.25} \cdot \left({B}^{-0.5} \cdot {F}^{0.25}\right)\right)} \]

   11. Simplified 20.8%

       \[\leadsto -\sqrt{2} \cdot \color{blue}{\left({F}^{0.25} \cdot \left({B}^{-0.5} \cdot {F}^{0.25}\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 44.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq -1 \cdot 10^{-210}:\\ \;\;\;\;\left(\left(-\sqrt{2}\right) \cdot \sqrt{F}\right) \cdot \sqrt{\frac{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)}{\mathsf{fma}\left(-4, A \cdot C, {B}^{2}\right)}}\\ \mathbf{elif}\;\frac{\sqrt{\left(2 \cdot \left(\left({B}^{2} - \left(4 \cdot A\right) \cdot C\right) \cdot F\right)\right) \cdot \left(\left(A + C\right) + \sqrt{{B}^{2} + {\left(A - C\right)}^{2}}\right)}}{\left(4 \cdot A\right) \cdot C - {B}^{2}} \leq \infty:\\ \;\;\;\;\frac{\sqrt{2 \cdot \left(F \cdot \mathsf{fma}\left(B, B, -4 \cdot \left(A \cdot C\right)\right)\right)} \cdot \sqrt{-0.5 \cdot \frac{{B}^{2}}{A} + 2 \cdot C}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left({F}^{0.25} \cdot \left({F}^{0.25} \cdot \left(-{B}^{-0.5}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 54.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ t_1 := -t\_0\\ \mathbf{if}\;{B\_m}^{2} \leq 2 \cdot 10^{-86}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot t\_0\right) \cdot \left(4 \cdot C - \frac{{B\_m}^{2}}{A}\right)}}{t\_1}\\ \mathbf{elif}\;{B\_m}^{2} \leq 2 \cdot 10^{+194}:\\ \;\;\;\;\sqrt{F \cdot \left(2 \cdot t\_0\right)} \cdot \frac{\sqrt{\left(A + C\right) + \mathsf{hypot}\left(A - C, B\_m\right)}}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left({F}^{0.25} \cdot \left({F}^{0.25} \cdot \left(-{B\_m}^{-0.5}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* A (* C -4.0)))) (t_1 (- t_0)))
   (if (<= (pow B_m 2.0) 2e-86)
     (/ (sqrt (* (* F t_0) (- (* 4.0 C) (/ (pow B_m 2.0) A)))) t_1)
     (if (<= (pow B_m 2.0) 2e+194)
       (*
        (sqrt (* F (* 2.0 t_0)))
        (/ (sqrt (+ (+ A C) (hypot (- A C) B_m))) t_1))
       (* (sqrt 2.0) (* (pow F 0.25) (* (pow F 0.25) (- (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 = fma(B_m, B_m, (A * (C * -4.0)));
	double t_1 = -t_0;
	double tmp;
	if (pow(B_m, 2.0) <= 2e-86) {
		tmp = sqrt(((F * t_0) * ((4.0 * C) - (pow(B_m, 2.0) / A)))) / t_1;
	} else if (pow(B_m, 2.0) <= 2e+194) {
		tmp = sqrt((F * (2.0 * t_0))) * (sqrt(((A + C) + hypot((A - C), B_m))) / t_1);
	} else {
		tmp = sqrt(2.0) * (pow(F, 0.25) * (pow(F, 0.25) * -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 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	t_1 = Float64(-t_0)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 2e-86)
		tmp = Float64(sqrt(Float64(Float64(F * t_0) * Float64(Float64(4.0 * C) - Float64((B_m ^ 2.0) / A)))) / t_1);
	elseif ((B_m ^ 2.0) <= 2e+194)
		tmp = Float64(sqrt(Float64(F * Float64(2.0 * t_0))) * Float64(sqrt(Float64(Float64(A + C) + hypot(Float64(A - C), B_m))) / t_1));
	else
		tmp = Float64(sqrt(2.0) * Float64((F ^ 0.25) * Float64((F ^ 0.25) * Float64(-(B_m ^ -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[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = (-t$95$0)}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e-86], N[(N[Sqrt[N[(N[(F * t$95$0), $MachinePrecision] * N[(N[(4.0 * C), $MachinePrecision] - N[(N[Power[B$95$m, 2.0], $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+194], N[(N[Sqrt[N[(F * N[(2.0 * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[Sqrt[N[(N[(A + C), $MachinePrecision] + N[Sqrt[N[(A - C), $MachinePrecision] ^ 2 + B$95$m ^ 2], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Power[F, 0.25], $MachinePrecision] * N[(N[Power[F, 0.25], $MachinePrecision] * (-N[Power[B$95$m, -0.5], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 2.00000000000000017e-86

   1. Initial program 21.6%

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

   3. Simplified 29.3%

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

   5. Taylor expanded in A around -inf 22.8%

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

   if 2.00000000000000017e-86 < (pow.f64 B #s(literal 2 binary64)) < 1.99999999999999989e194

   1. Initial program 39.7%

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

   3. Simplified 51.7%

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

      1. associate-*r* 51.7%

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

      2. associate-+r+ 50.8%

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

      3. hypot-undefine 39.7%

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

      4. unpow2 39.7%

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

      5. unpow2 39.7%

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

      6. +-commutative 39.7%

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

      7. sqrt-prod 41.9%

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

      8. *-commutative 41.9%

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

      9. associate-*r* 41.9%

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

      10. associate-+l+ 42.6%

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

   6. Applied egg-rr 61.7%

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

      1. associate-/l* 61.9%

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

      2. associate-*l* 61.9%

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

      3. associate-*r* 61.9%

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

      4. associate-+r+ 61.0%

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

   8. Applied egg-rr 61.0%

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

   if 1.99999999999999989e194 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 6.7%

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

   3. Taylor expanded in B around inf 24.4%

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

      1. mul-1-neg 24.4%

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

      2. *-commutative 24.4%

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

   5. Simplified 24.4%

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

      1. add-cbrt-cube 20.8%

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

      2. pow1/3 19.6%

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

      3. add-sqr-sqrt 19.6%

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

      4. pow1 19.6%

         \[\leadsto -\sqrt{2} \cdot {\left(\color{blue}{{\left(\frac{F}{B}\right)}^{1}} \cdot \sqrt{\frac{F}{B}}\right)}^{0.3333333333333333} \]

      5. pow1/2 19.6%

         \[\leadsto -\sqrt{2} \cdot {\left({\left(\frac{F}{B}\right)}^{1} \cdot \color{blue}{{\left(\frac{F}{B}\right)}^{0.5}}\right)}^{0.3333333333333333} \]

      6. pow-prod-up 19.6%

         \[\leadsto -\sqrt{2} \cdot {\color{blue}{\left({\left(\frac{F}{B}\right)}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333} \]

      7. metadata-eval 19.6%

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

   7. Applied egg-rr 19.6%

      \[\leadsto -\sqrt{2} \cdot \color{blue}{{\left({\left(\frac{F}{B}\right)}^{1.5}\right)}^{0.3333333333333333}} \]
   8. Step-by-step derivation

      1. pow-pow 24.4%

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

      2. metadata-eval 24.4%

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

      3. pow1/2 24.4%

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

      4. div-inv 24.4%

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

      5. sqrt-unprod 28.9%

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

      6. *-commutative 28.9%

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

      7. add-sqr-sqrt 28.8%

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

      8. associate-*r* 28.8%

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

      9. inv-pow 28.8%

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

      10. sqrt-pow1 28.9%

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

      11. metadata-eval 28.9%

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

      12. pow1/2 28.9%

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

      13. metadata-eval 28.9%

          \[\leadsto -\sqrt{2} \cdot \left(\left({B}^{-0.5} \cdot \sqrt{{F}^{\color{blue}{\left(1.5 \cdot 0.3333333333333333\right)}}}\right) \cdot \sqrt{\sqrt{F}}\right) \]

      14. sqrt-pow1 28.9%

          \[\leadsto -\sqrt{2} \cdot \left(\left({B}^{-0.5} \cdot \color{blue}{{F}^{\left(\frac{1.5 \cdot 0.3333333333333333}{2}\right)}}\right) \cdot \sqrt{\sqrt{F}}\right) \]

      15. metadata-eval 28.9%

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

      16. metadata-eval 28.9%

          \[\leadsto -\sqrt{2} \cdot \left(\left({B}^{-0.5} \cdot {F}^{\color{blue}{0.25}}\right) \cdot \sqrt{\sqrt{F}}\right) \]

      17. pow1/2 28.9%

          \[\leadsto -\sqrt{2} \cdot \left(\left({B}^{-0.5} \cdot {F}^{0.25}\right) \cdot \sqrt{\color{blue}{{F}^{0.5}}}\right) \]

      18. metadata-eval 28.9%

          \[\leadsto -\sqrt{2} \cdot \left(\left({B}^{-0.5} \cdot {F}^{0.25}\right) \cdot \sqrt{{F}^{\color{blue}{\left(1.5 \cdot 0.3333333333333333\right)}}}\right) \]

      19. sqrt-pow1 28.9%

          \[\leadsto -\sqrt{2} \cdot \left(\left({B}^{-0.5} \cdot {F}^{0.25}\right) \cdot \color{blue}{{F}^{\left(\frac{1.5 \cdot 0.3333333333333333}{2}\right)}}\right) \]

      20. metadata-eval 28.9%

          \[\leadsto -\sqrt{2} \cdot \left(\left({B}^{-0.5} \cdot {F}^{0.25}\right) \cdot {F}^{\left(\frac{\color{blue}{0.5}}{2}\right)}\right) \]

      21. metadata-eval 28.9%

          \[\leadsto -\sqrt{2} \cdot \left(\left({B}^{-0.5} \cdot {F}^{0.25}\right) \cdot {F}^{\color{blue}{0.25}}\right) \]

   9. Applied egg-rr 28.9%

      \[\leadsto -\sqrt{2} \cdot \color{blue}{\left(\left({B}^{-0.5} \cdot {F}^{0.25}\right) \cdot {F}^{0.25}\right)} \]
   10. Step-by-step derivation

       1. *-commutative 28.9%

          \[\leadsto -\sqrt{2} \cdot \color{blue}{\left({F}^{0.25} \cdot \left({B}^{-0.5} \cdot {F}^{0.25}\right)\right)} \]

   11. Simplified 28.9%

       \[\leadsto -\sqrt{2} \cdot \color{blue}{\left({F}^{0.25} \cdot \left({B}^{-0.5} \cdot {F}^{0.25}\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 33.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 2 \cdot 10^{-86}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot \left(4 \cdot C - \frac{{B}^{2}}{A}\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{+194}:\\ \;\;\;\;\sqrt{F \cdot \left(2 \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right)} \cdot \frac{\sqrt{\left(A + C\right) + \mathsf{hypot}\left(A - C, B\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left({F}^{0.25} \cdot \left({F}^{0.25} \cdot \left(-{B}^{-0.5}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 54.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 2 \cdot 10^{-86}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot t\_0\right) \cdot \left(4 \cdot C - \frac{{B\_m}^{2}}{A}\right)}}{-t\_0}\\ \mathbf{elif}\;{B\_m}^{2} \leq 2 \cdot 10^{+202}:\\ \;\;\;\;-\sqrt{2 \cdot \left(F \cdot \frac{C + \left(A + \mathsf{hypot}\left(B\_m, A - C\right)\right)}{\mathsf{fma}\left(-4, A \cdot C, {B\_m}^{2}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left({F}^{0.25} \cdot \left({F}^{0.25} \cdot \left(-{B\_m}^{-0.5}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* A (* C -4.0)))))
   (if (<= (pow B_m 2.0) 2e-86)
     (/ (sqrt (* (* F t_0) (- (* 4.0 C) (/ (pow B_m 2.0) A)))) (- t_0))
     (if (<= (pow B_m 2.0) 2e+202)
       (-
        (sqrt
         (*
          2.0
          (*
           F
           (/
            (+ C (+ A (hypot B_m (- A C))))
            (fma -4.0 (* A C) (pow B_m 2.0)))))))
       (* (sqrt 2.0) (* (pow F 0.25) (* (pow F 0.25) (- (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 = fma(B_m, B_m, (A * (C * -4.0)));
	double tmp;
	if (pow(B_m, 2.0) <= 2e-86) {
		tmp = sqrt(((F * t_0) * ((4.0 * C) - (pow(B_m, 2.0) / A)))) / -t_0;
	} else if (pow(B_m, 2.0) <= 2e+202) {
		tmp = -sqrt((2.0 * (F * ((C + (A + hypot(B_m, (A - C)))) / fma(-4.0, (A * C), pow(B_m, 2.0))))));
	} else {
		tmp = sqrt(2.0) * (pow(F, 0.25) * (pow(F, 0.25) * -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 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 2e-86)
		tmp = Float64(sqrt(Float64(Float64(F * t_0) * Float64(Float64(4.0 * C) - Float64((B_m ^ 2.0) / A)))) / Float64(-t_0));
	elseif ((B_m ^ 2.0) <= 2e+202)
		tmp = Float64(-sqrt(Float64(2.0 * Float64(F * Float64(Float64(C + Float64(A + hypot(B_m, Float64(A - C)))) / fma(-4.0, Float64(A * C), (B_m ^ 2.0)))))));
	else
		tmp = Float64(sqrt(2.0) * Float64((F ^ 0.25) * Float64((F ^ 0.25) * Float64(-(B_m ^ -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[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e-86], N[(N[Sqrt[N[(N[(F * t$95$0), $MachinePrecision] * N[(N[(4.0 * C), $MachinePrecision] - N[(N[Power[B$95$m, 2.0], $MachinePrecision] / A), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+202], (-N[Sqrt[N[(2.0 * N[(F * N[(N[(C + N[(A + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-4.0 * N[(A * C), $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Power[F, 0.25], $MachinePrecision] * N[(N[Power[F, 0.25], $MachinePrecision] * (-N[Power[B$95$m, -0.5], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 2.00000000000000017e-86

   1. Initial program 21.6%

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

   3. Simplified 29.3%

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

   5. Taylor expanded in A around -inf 22.8%

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

   if 2.00000000000000017e-86 < (pow.f64 B #s(literal 2 binary64)) < 1.9999999999999998e202

   1. Initial program 40.1%

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

   3. Taylor expanded in F around 0 40.5%

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

   5. Simplified 56.9%

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

      1. neg-sub0 56.9%

         \[\leadsto \color{blue}{0 - \sqrt{2} \cdot \sqrt{F \cdot \frac{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)}{\mathsf{fma}\left(-4, C \cdot A, {B}^{2}\right)}}} \]

      2. sqrt-unprod 57.2%

         \[\leadsto 0 - \color{blue}{\sqrt{2 \cdot \left(F \cdot \frac{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)}{\mathsf{fma}\left(-4, C \cdot A, {B}^{2}\right)}\right)}} \]

      3. associate-*r* 57.2%

         \[\leadsto 0 - \sqrt{\color{blue}{\left(2 \cdot F\right) \cdot \frac{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)}{\mathsf{fma}\left(-4, C \cdot A, {B}^{2}\right)}}} \]

      4. +-commutative 57.2%

         \[\leadsto 0 - \sqrt{\left(2 \cdot F\right) \cdot \frac{\color{blue}{\left(C + \mathsf{hypot}\left(B, A - C\right)\right) + A}}{\mathsf{fma}\left(-4, C \cdot A, {B}^{2}\right)}} \]

      5. associate-+l+ 56.9%

         \[\leadsto 0 - \sqrt{\left(2 \cdot F\right) \cdot \frac{\color{blue}{C + \left(\mathsf{hypot}\left(B, A - C\right) + A\right)}}{\mathsf{fma}\left(-4, C \cdot A, {B}^{2}\right)}} \]

      6. *-commutative 56.9%

         \[\leadsto 0 - \sqrt{\left(2 \cdot F\right) \cdot \frac{C + \left(\mathsf{hypot}\left(B, A - C\right) + A\right)}{\mathsf{fma}\left(-4, \color{blue}{A \cdot C}, {B}^{2}\right)}} \]

   7. Applied egg-rr 56.9%

      \[\leadsto \color{blue}{0 - \sqrt{\left(2 \cdot F\right) \cdot \frac{C + \left(\mathsf{hypot}\left(B, A - C\right) + A\right)}{\mathsf{fma}\left(-4, A \cdot C, {B}^{2}\right)}}} \]

   9. Simplified 56.9%

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

   if 1.9999999999999998e202 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 5.5%

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

   3. Taylor expanded in B around inf 23.6%

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

      1. mul-1-neg 23.6%

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

      2. *-commutative 23.6%

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

   5. Simplified 23.6%

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

      1. add-cbrt-cube 19.9%

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

      2. pow1/3 18.8%

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

      3. add-sqr-sqrt 18.8%

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

      4. pow1 18.8%

         \[\leadsto -\sqrt{2} \cdot {\left(\color{blue}{{\left(\frac{F}{B}\right)}^{1}} \cdot \sqrt{\frac{F}{B}}\right)}^{0.3333333333333333} \]

      5. pow1/2 18.8%

         \[\leadsto -\sqrt{2} \cdot {\left({\left(\frac{F}{B}\right)}^{1} \cdot \color{blue}{{\left(\frac{F}{B}\right)}^{0.5}}\right)}^{0.3333333333333333} \]

      6. pow-prod-up 18.8%

         \[\leadsto -\sqrt{2} \cdot {\color{blue}{\left({\left(\frac{F}{B}\right)}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333} \]

      7. metadata-eval 18.8%

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

   7. Applied egg-rr 18.8%

      \[\leadsto -\sqrt{2} \cdot \color{blue}{{\left({\left(\frac{F}{B}\right)}^{1.5}\right)}^{0.3333333333333333}} \]
   8. Step-by-step derivation

      1. pow-pow 23.6%

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

      2. metadata-eval 23.6%

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

      3. pow1/2 23.6%

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

      4. div-inv 23.6%

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

      5. sqrt-unprod 28.2%

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

      6. *-commutative 28.2%

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

      7. add-sqr-sqrt 28.2%

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

      8. associate-*r* 28.2%

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

      9. inv-pow 28.2%

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

      10. sqrt-pow1 28.3%

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

      11. metadata-eval 28.3%

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

      12. pow1/2 28.3%

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

      13. metadata-eval 28.3%

          \[\leadsto -\sqrt{2} \cdot \left(\left({B}^{-0.5} \cdot \sqrt{{F}^{\color{blue}{\left(1.5 \cdot 0.3333333333333333\right)}}}\right) \cdot \sqrt{\sqrt{F}}\right) \]

      14. sqrt-pow1 28.3%

          \[\leadsto -\sqrt{2} \cdot \left(\left({B}^{-0.5} \cdot \color{blue}{{F}^{\left(\frac{1.5 \cdot 0.3333333333333333}{2}\right)}}\right) \cdot \sqrt{\sqrt{F}}\right) \]

      15. metadata-eval 28.3%

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

      16. metadata-eval 28.3%

          \[\leadsto -\sqrt{2} \cdot \left(\left({B}^{-0.5} \cdot {F}^{\color{blue}{0.25}}\right) \cdot \sqrt{\sqrt{F}}\right) \]

      17. pow1/2 28.3%

          \[\leadsto -\sqrt{2} \cdot \left(\left({B}^{-0.5} \cdot {F}^{0.25}\right) \cdot \sqrt{\color{blue}{{F}^{0.5}}}\right) \]

      18. metadata-eval 28.3%

          \[\leadsto -\sqrt{2} \cdot \left(\left({B}^{-0.5} \cdot {F}^{0.25}\right) \cdot \sqrt{{F}^{\color{blue}{\left(1.5 \cdot 0.3333333333333333\right)}}}\right) \]

      19. sqrt-pow1 28.3%

          \[\leadsto -\sqrt{2} \cdot \left(\left({B}^{-0.5} \cdot {F}^{0.25}\right) \cdot \color{blue}{{F}^{\left(\frac{1.5 \cdot 0.3333333333333333}{2}\right)}}\right) \]

      20. metadata-eval 28.3%

          \[\leadsto -\sqrt{2} \cdot \left(\left({B}^{-0.5} \cdot {F}^{0.25}\right) \cdot {F}^{\left(\frac{\color{blue}{0.5}}{2}\right)}\right) \]

      21. metadata-eval 28.3%

          \[\leadsto -\sqrt{2} \cdot \left(\left({B}^{-0.5} \cdot {F}^{0.25}\right) \cdot {F}^{\color{blue}{0.25}}\right) \]

   9. Applied egg-rr 28.3%

      \[\leadsto -\sqrt{2} \cdot \color{blue}{\left(\left({B}^{-0.5} \cdot {F}^{0.25}\right) \cdot {F}^{0.25}\right)} \]
   10. Step-by-step derivation

       1. *-commutative 28.3%

          \[\leadsto -\sqrt{2} \cdot \color{blue}{\left({F}^{0.25} \cdot \left({B}^{-0.5} \cdot {F}^{0.25}\right)\right)} \]

   11. Simplified 28.3%

       \[\leadsto -\sqrt{2} \cdot \color{blue}{\left({F}^{0.25} \cdot \left({B}^{-0.5} \cdot {F}^{0.25}\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 32.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 2 \cdot 10^{-86}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot \left(4 \cdot C - \frac{{B}^{2}}{A}\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{+202}:\\ \;\;\;\;-\sqrt{2 \cdot \left(F \cdot \frac{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)}{\mathsf{fma}\left(-4, A \cdot C, {B}^{2}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left({F}^{0.25} \cdot \left({F}^{0.25} \cdot \left(-{B}^{-0.5}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 54.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 2 \cdot 10^{-86}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot t\_0\right) \cdot \left(4 \cdot C\right)}}{-t\_0}\\ \mathbf{elif}\;{B\_m}^{2} \leq 2 \cdot 10^{+202}:\\ \;\;\;\;-\sqrt{2 \cdot \left(F \cdot \frac{C + \left(A + \mathsf{hypot}\left(B\_m, A - C\right)\right)}{\mathsf{fma}\left(-4, A \cdot C, {B\_m}^{2}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left({F}^{0.25} \cdot \left({F}^{0.25} \cdot \left(-{B\_m}^{-0.5}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* A (* C -4.0)))))
   (if (<= (pow B_m 2.0) 2e-86)
     (/ (sqrt (* (* F t_0) (* 4.0 C))) (- t_0))
     (if (<= (pow B_m 2.0) 2e+202)
       (-
        (sqrt
         (*
          2.0
          (*
           F
           (/
            (+ C (+ A (hypot B_m (- A C))))
            (fma -4.0 (* A C) (pow B_m 2.0)))))))
       (* (sqrt 2.0) (* (pow F 0.25) (* (pow F 0.25) (- (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 = fma(B_m, B_m, (A * (C * -4.0)));
	double tmp;
	if (pow(B_m, 2.0) <= 2e-86) {
		tmp = sqrt(((F * t_0) * (4.0 * C))) / -t_0;
	} else if (pow(B_m, 2.0) <= 2e+202) {
		tmp = -sqrt((2.0 * (F * ((C + (A + hypot(B_m, (A - C)))) / fma(-4.0, (A * C), pow(B_m, 2.0))))));
	} else {
		tmp = sqrt(2.0) * (pow(F, 0.25) * (pow(F, 0.25) * -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 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 2e-86)
		tmp = Float64(sqrt(Float64(Float64(F * t_0) * Float64(4.0 * C))) / Float64(-t_0));
	elseif ((B_m ^ 2.0) <= 2e+202)
		tmp = Float64(-sqrt(Float64(2.0 * Float64(F * Float64(Float64(C + Float64(A + hypot(B_m, Float64(A - C)))) / fma(-4.0, Float64(A * C), (B_m ^ 2.0)))))));
	else
		tmp = Float64(sqrt(2.0) * Float64((F ^ 0.25) * Float64((F ^ 0.25) * Float64(-(B_m ^ -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[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e-86], N[(N[Sqrt[N[(N[(F * t$95$0), $MachinePrecision] * N[(4.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+202], (-N[Sqrt[N[(2.0 * N[(F * N[(N[(C + N[(A + N[Sqrt[B$95$m ^ 2 + N[(A - C), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(-4.0 * N[(A * C), $MachinePrecision] + N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Power[F, 0.25], $MachinePrecision] * N[(N[Power[F, 0.25], $MachinePrecision] * (-N[Power[B$95$m, -0.5], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 2.00000000000000017e-86

   1. Initial program 21.6%

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

   3. Simplified 29.3%

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

   5. Taylor expanded in A around -inf 21.1%

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

      1. *-commutative 21.1%

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

   7. Simplified 21.1%

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

   if 2.00000000000000017e-86 < (pow.f64 B #s(literal 2 binary64)) < 1.9999999999999998e202

   1. Initial program 40.1%

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

   3. Taylor expanded in F around 0 40.5%

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

   5. Simplified 56.9%

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

      1. neg-sub0 56.9%

         \[\leadsto \color{blue}{0 - \sqrt{2} \cdot \sqrt{F \cdot \frac{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)}{\mathsf{fma}\left(-4, C \cdot A, {B}^{2}\right)}}} \]

      2. sqrt-unprod 57.2%

         \[\leadsto 0 - \color{blue}{\sqrt{2 \cdot \left(F \cdot \frac{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)}{\mathsf{fma}\left(-4, C \cdot A, {B}^{2}\right)}\right)}} \]

      3. associate-*r* 57.2%

         \[\leadsto 0 - \sqrt{\color{blue}{\left(2 \cdot F\right) \cdot \frac{A + \left(C + \mathsf{hypot}\left(B, A - C\right)\right)}{\mathsf{fma}\left(-4, C \cdot A, {B}^{2}\right)}}} \]

      4. +-commutative 57.2%

         \[\leadsto 0 - \sqrt{\left(2 \cdot F\right) \cdot \frac{\color{blue}{\left(C + \mathsf{hypot}\left(B, A - C\right)\right) + A}}{\mathsf{fma}\left(-4, C \cdot A, {B}^{2}\right)}} \]

      5. associate-+l+ 56.9%

         \[\leadsto 0 - \sqrt{\left(2 \cdot F\right) \cdot \frac{\color{blue}{C + \left(\mathsf{hypot}\left(B, A - C\right) + A\right)}}{\mathsf{fma}\left(-4, C \cdot A, {B}^{2}\right)}} \]

      6. *-commutative 56.9%

         \[\leadsto 0 - \sqrt{\left(2 \cdot F\right) \cdot \frac{C + \left(\mathsf{hypot}\left(B, A - C\right) + A\right)}{\mathsf{fma}\left(-4, \color{blue}{A \cdot C}, {B}^{2}\right)}} \]

   7. Applied egg-rr 56.9%

      \[\leadsto \color{blue}{0 - \sqrt{\left(2 \cdot F\right) \cdot \frac{C + \left(\mathsf{hypot}\left(B, A - C\right) + A\right)}{\mathsf{fma}\left(-4, A \cdot C, {B}^{2}\right)}}} \]

   9. Simplified 56.9%

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

   if 1.9999999999999998e202 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 5.5%

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

   3. Taylor expanded in B around inf 23.6%

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

      1. mul-1-neg 23.6%

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

      2. *-commutative 23.6%

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

   5. Simplified 23.6%

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

      1. add-cbrt-cube 19.9%

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

      2. pow1/3 18.8%

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

      3. add-sqr-sqrt 18.8%

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

      4. pow1 18.8%

         \[\leadsto -\sqrt{2} \cdot {\left(\color{blue}{{\left(\frac{F}{B}\right)}^{1}} \cdot \sqrt{\frac{F}{B}}\right)}^{0.3333333333333333} \]

      5. pow1/2 18.8%

         \[\leadsto -\sqrt{2} \cdot {\left({\left(\frac{F}{B}\right)}^{1} \cdot \color{blue}{{\left(\frac{F}{B}\right)}^{0.5}}\right)}^{0.3333333333333333} \]

      6. pow-prod-up 18.8%

         \[\leadsto -\sqrt{2} \cdot {\color{blue}{\left({\left(\frac{F}{B}\right)}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333} \]

      7. metadata-eval 18.8%

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

   7. Applied egg-rr 18.8%

      \[\leadsto -\sqrt{2} \cdot \color{blue}{{\left({\left(\frac{F}{B}\right)}^{1.5}\right)}^{0.3333333333333333}} \]
   8. Step-by-step derivation

      1. pow-pow 23.6%

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

      2. metadata-eval 23.6%

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

      3. pow1/2 23.6%

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

      4. div-inv 23.6%

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

      5. sqrt-unprod 28.2%

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

      6. *-commutative 28.2%

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

      7. add-sqr-sqrt 28.2%

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

      8. associate-*r* 28.2%

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

      9. inv-pow 28.2%

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

      10. sqrt-pow1 28.3%

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

      11. metadata-eval 28.3%

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

      12. pow1/2 28.3%

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

      13. metadata-eval 28.3%

          \[\leadsto -\sqrt{2} \cdot \left(\left({B}^{-0.5} \cdot \sqrt{{F}^{\color{blue}{\left(1.5 \cdot 0.3333333333333333\right)}}}\right) \cdot \sqrt{\sqrt{F}}\right) \]

      14. sqrt-pow1 28.3%

          \[\leadsto -\sqrt{2} \cdot \left(\left({B}^{-0.5} \cdot \color{blue}{{F}^{\left(\frac{1.5 \cdot 0.3333333333333333}{2}\right)}}\right) \cdot \sqrt{\sqrt{F}}\right) \]

      15. metadata-eval 28.3%

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

      16. metadata-eval 28.3%

          \[\leadsto -\sqrt{2} \cdot \left(\left({B}^{-0.5} \cdot {F}^{\color{blue}{0.25}}\right) \cdot \sqrt{\sqrt{F}}\right) \]

      17. pow1/2 28.3%

          \[\leadsto -\sqrt{2} \cdot \left(\left({B}^{-0.5} \cdot {F}^{0.25}\right) \cdot \sqrt{\color{blue}{{F}^{0.5}}}\right) \]

      18. metadata-eval 28.3%

          \[\leadsto -\sqrt{2} \cdot \left(\left({B}^{-0.5} \cdot {F}^{0.25}\right) \cdot \sqrt{{F}^{\color{blue}{\left(1.5 \cdot 0.3333333333333333\right)}}}\right) \]

      19. sqrt-pow1 28.3%

          \[\leadsto -\sqrt{2} \cdot \left(\left({B}^{-0.5} \cdot {F}^{0.25}\right) \cdot \color{blue}{{F}^{\left(\frac{1.5 \cdot 0.3333333333333333}{2}\right)}}\right) \]

      20. metadata-eval 28.3%

          \[\leadsto -\sqrt{2} \cdot \left(\left({B}^{-0.5} \cdot {F}^{0.25}\right) \cdot {F}^{\left(\frac{\color{blue}{0.5}}{2}\right)}\right) \]

      21. metadata-eval 28.3%

          \[\leadsto -\sqrt{2} \cdot \left(\left({B}^{-0.5} \cdot {F}^{0.25}\right) \cdot {F}^{\color{blue}{0.25}}\right) \]

   9. Applied egg-rr 28.3%

      \[\leadsto -\sqrt{2} \cdot \color{blue}{\left(\left({B}^{-0.5} \cdot {F}^{0.25}\right) \cdot {F}^{0.25}\right)} \]
   10. Step-by-step derivation

       1. *-commutative 28.3%

          \[\leadsto -\sqrt{2} \cdot \color{blue}{\left({F}^{0.25} \cdot \left({B}^{-0.5} \cdot {F}^{0.25}\right)\right)} \]

   11. Simplified 28.3%

       \[\leadsto -\sqrt{2} \cdot \color{blue}{\left({F}^{0.25} \cdot \left({B}^{-0.5} \cdot {F}^{0.25}\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 32.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 2 \cdot 10^{-86}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot \left(4 \cdot C\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{+202}:\\ \;\;\;\;-\sqrt{2 \cdot \left(F \cdot \frac{C + \left(A + \mathsf{hypot}\left(B, A - C\right)\right)}{\mathsf{fma}\left(-4, A \cdot C, {B}^{2}\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left({F}^{0.25} \cdot \left({F}^{0.25} \cdot \left(-{B}^{-0.5}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 53.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 4 \cdot 10^{+23}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot t\_0\right) \cdot \left(4 \cdot C\right)}}{-t\_0}\\ \mathbf{elif}\;{B\_m}^{2} \leq 2 \cdot 10^{+202}:\\ \;\;\;\;\left(-\sqrt{2}\right) \cdot \sqrt{F \cdot \frac{C + \mathsf{hypot}\left(B\_m, C\right)}{{B\_m}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left({F}^{0.25} \cdot \left({F}^{0.25} \cdot \left(-{B\_m}^{-0.5}\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* A (* C -4.0)))))
   (if (<= (pow B_m 2.0) 4e+23)
     (/ (sqrt (* (* F t_0) (* 4.0 C))) (- t_0))
     (if (<= (pow B_m 2.0) 2e+202)
       (* (- (sqrt 2.0)) (sqrt (* F (/ (+ C (hypot B_m C)) (pow B_m 2.0)))))
       (* (sqrt 2.0) (* (pow F 0.25) (* (pow F 0.25) (- (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 = fma(B_m, B_m, (A * (C * -4.0)));
	double tmp;
	if (pow(B_m, 2.0) <= 4e+23) {
		tmp = sqrt(((F * t_0) * (4.0 * C))) / -t_0;
	} else if (pow(B_m, 2.0) <= 2e+202) {
		tmp = -sqrt(2.0) * sqrt((F * ((C + hypot(B_m, C)) / pow(B_m, 2.0))));
	} else {
		tmp = sqrt(2.0) * (pow(F, 0.25) * (pow(F, 0.25) * -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 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 4e+23)
		tmp = Float64(sqrt(Float64(Float64(F * t_0) * Float64(4.0 * C))) / Float64(-t_0));
	elseif ((B_m ^ 2.0) <= 2e+202)
		tmp = Float64(Float64(-sqrt(2.0)) * sqrt(Float64(F * Float64(Float64(C + hypot(B_m, C)) / (B_m ^ 2.0)))));
	else
		tmp = Float64(sqrt(2.0) * Float64((F ^ 0.25) * Float64((F ^ 0.25) * Float64(-(B_m ^ -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[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 4e+23], N[(N[Sqrt[N[(N[(F * t$95$0), $MachinePrecision] * N[(4.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 2e+202], N[((-N[Sqrt[2.0], $MachinePrecision]) * N[Sqrt[N[(F * N[(N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision] / N[Power[B$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Power[F, 0.25], $MachinePrecision] * N[(N[Power[F, 0.25], $MachinePrecision] * (-N[Power[B$95$m, -0.5], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 3.9999999999999997e23

   1. Initial program 25.1%

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

   3. Simplified 33.9%

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

   5. Taylor expanded in A around -inf 22.2%

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

      1. *-commutative 22.2%

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

   7. Simplified 22.2%

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

   if 3.9999999999999997e23 < (pow.f64 B #s(literal 2 binary64)) < 1.9999999999999998e202

   1. Initial program 38.8%

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

   3. Taylor expanded in F around 0 38.5%

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

   5. Simplified 59.4%

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

   6. Taylor expanded in A around 0 40.8%

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

      1. unpow2 40.8%

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

      2. unpow2 40.8%

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

      3. hypot-define 46.1%

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

   8. Simplified 46.1%

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

   if 1.9999999999999998e202 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 5.5%

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

   3. Taylor expanded in B around inf 23.6%

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

      1. mul-1-neg 23.6%

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

      2. *-commutative 23.6%

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

   5. Simplified 23.6%

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

      1. add-cbrt-cube 19.9%

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

      2. pow1/3 18.8%

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

      3. add-sqr-sqrt 18.8%

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

      4. pow1 18.8%

         \[\leadsto -\sqrt{2} \cdot {\left(\color{blue}{{\left(\frac{F}{B}\right)}^{1}} \cdot \sqrt{\frac{F}{B}}\right)}^{0.3333333333333333} \]

      5. pow1/2 18.8%

         \[\leadsto -\sqrt{2} \cdot {\left({\left(\frac{F}{B}\right)}^{1} \cdot \color{blue}{{\left(\frac{F}{B}\right)}^{0.5}}\right)}^{0.3333333333333333} \]

      6. pow-prod-up 18.8%

         \[\leadsto -\sqrt{2} \cdot {\color{blue}{\left({\left(\frac{F}{B}\right)}^{\left(1 + 0.5\right)}\right)}}^{0.3333333333333333} \]

      7. metadata-eval 18.8%

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

   7. Applied egg-rr 18.8%

      \[\leadsto -\sqrt{2} \cdot \color{blue}{{\left({\left(\frac{F}{B}\right)}^{1.5}\right)}^{0.3333333333333333}} \]
   8. Step-by-step derivation

      1. pow-pow 23.6%

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

      2. metadata-eval 23.6%

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

      3. pow1/2 23.6%

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

      4. div-inv 23.6%

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

      5. sqrt-unprod 28.2%

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

      6. *-commutative 28.2%

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

      7. add-sqr-sqrt 28.2%

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

      8. associate-*r* 28.2%

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

      9. inv-pow 28.2%

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

      10. sqrt-pow1 28.3%

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

      11. metadata-eval 28.3%

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

      12. pow1/2 28.3%

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

      13. metadata-eval 28.3%

          \[\leadsto -\sqrt{2} \cdot \left(\left({B}^{-0.5} \cdot \sqrt{{F}^{\color{blue}{\left(1.5 \cdot 0.3333333333333333\right)}}}\right) \cdot \sqrt{\sqrt{F}}\right) \]

      14. sqrt-pow1 28.3%

          \[\leadsto -\sqrt{2} \cdot \left(\left({B}^{-0.5} \cdot \color{blue}{{F}^{\left(\frac{1.5 \cdot 0.3333333333333333}{2}\right)}}\right) \cdot \sqrt{\sqrt{F}}\right) \]

      15. metadata-eval 28.3%

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

      16. metadata-eval 28.3%

          \[\leadsto -\sqrt{2} \cdot \left(\left({B}^{-0.5} \cdot {F}^{\color{blue}{0.25}}\right) \cdot \sqrt{\sqrt{F}}\right) \]

      17. pow1/2 28.3%

          \[\leadsto -\sqrt{2} \cdot \left(\left({B}^{-0.5} \cdot {F}^{0.25}\right) \cdot \sqrt{\color{blue}{{F}^{0.5}}}\right) \]

      18. metadata-eval 28.3%

          \[\leadsto -\sqrt{2} \cdot \left(\left({B}^{-0.5} \cdot {F}^{0.25}\right) \cdot \sqrt{{F}^{\color{blue}{\left(1.5 \cdot 0.3333333333333333\right)}}}\right) \]

      19. sqrt-pow1 28.3%

          \[\leadsto -\sqrt{2} \cdot \left(\left({B}^{-0.5} \cdot {F}^{0.25}\right) \cdot \color{blue}{{F}^{\left(\frac{1.5 \cdot 0.3333333333333333}{2}\right)}}\right) \]

      20. metadata-eval 28.3%

          \[\leadsto -\sqrt{2} \cdot \left(\left({B}^{-0.5} \cdot {F}^{0.25}\right) \cdot {F}^{\left(\frac{\color{blue}{0.5}}{2}\right)}\right) \]

      21. metadata-eval 28.3%

          \[\leadsto -\sqrt{2} \cdot \left(\left({B}^{-0.5} \cdot {F}^{0.25}\right) \cdot {F}^{\color{blue}{0.25}}\right) \]

   9. Applied egg-rr 28.3%

      \[\leadsto -\sqrt{2} \cdot \color{blue}{\left(\left({B}^{-0.5} \cdot {F}^{0.25}\right) \cdot {F}^{0.25}\right)} \]
   10. Step-by-step derivation

       1. *-commutative 28.3%

          \[\leadsto -\sqrt{2} \cdot \color{blue}{\left({F}^{0.25} \cdot \left({B}^{-0.5} \cdot {F}^{0.25}\right)\right)} \]

   11. Simplified 28.3%

       \[\leadsto -\sqrt{2} \cdot \color{blue}{\left({F}^{0.25} \cdot \left({B}^{-0.5} \cdot {F}^{0.25}\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 27.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 4 \cdot 10^{+23}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot \left(4 \cdot C\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;{B}^{2} \leq 2 \cdot 10^{+202}:\\ \;\;\;\;\left(-\sqrt{2}\right) \cdot \sqrt{F \cdot \frac{C + \mathsf{hypot}\left(B, C\right)}{{B}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left({F}^{0.25} \cdot \left({F}^{0.25} \cdot \left(-{B}^{-0.5}\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 52.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} t_0 := \mathsf{fma}\left(B\_m, B\_m, A \cdot \left(C \cdot -4\right)\right)\\ \mathbf{if}\;{B\_m}^{2} \leq 4 \cdot 10^{+23}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot t\_0\right) \cdot \left(4 \cdot C\right)}}{-t\_0}\\ \mathbf{elif}\;{B\_m}^{2} \leq 10^{+156}:\\ \;\;\;\;\sqrt{2} \cdot \left(\sqrt{F \cdot \left(C + \mathsf{hypot}\left(B\_m, C\right)\right)} \cdot \frac{-1}{B\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \frac{\sqrt{F}}{-\sqrt{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (let* ((t_0 (fma B_m B_m (* A (* C -4.0)))))
   (if (<= (pow B_m 2.0) 4e+23)
     (/ (sqrt (* (* F t_0) (* 4.0 C))) (- t_0))
     (if (<= (pow B_m 2.0) 1e+156)
       (* (sqrt 2.0) (* (sqrt (* F (+ C (hypot B_m C)))) (/ -1.0 B_m)))
       (* (sqrt 2.0) (/ (sqrt F) (- (sqrt B_m))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double t_0 = fma(B_m, B_m, (A * (C * -4.0)));
	double tmp;
	if (pow(B_m, 2.0) <= 4e+23) {
		tmp = sqrt(((F * t_0) * (4.0 * C))) / -t_0;
	} else if (pow(B_m, 2.0) <= 1e+156) {
		tmp = sqrt(2.0) * (sqrt((F * (C + hypot(B_m, C)))) * (-1.0 / B_m));
	} else {
		tmp = sqrt(2.0) * (sqrt(F) / -sqrt(B_m));
	}
	return tmp;
}

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	t_0 = fma(B_m, B_m, Float64(A * Float64(C * -4.0)))
	tmp = 0.0
	if ((B_m ^ 2.0) <= 4e+23)
		tmp = Float64(sqrt(Float64(Float64(F * t_0) * Float64(4.0 * C))) / Float64(-t_0));
	elseif ((B_m ^ 2.0) <= 1e+156)
		tmp = Float64(sqrt(2.0) * Float64(sqrt(Float64(F * Float64(C + hypot(B_m, C)))) * Float64(-1.0 / B_m)));
	else
		tmp = Float64(sqrt(2.0) * Float64(sqrt(F) / Float64(-sqrt(B_m))));
	end
	return tmp
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := Block[{t$95$0 = N[(B$95$m * B$95$m + N[(A * N[(C * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 4e+23], N[(N[Sqrt[N[(N[(F * t$95$0), $MachinePrecision] * N[(4.0 * C), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t$95$0)), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e+156], N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sqrt[N[(F * N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sqrt[F], $MachinePrecision] / (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 3.9999999999999997e23

   1. Initial program 25.1%

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

   3. Simplified 33.9%

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

   5. Taylor expanded in A around -inf 22.2%

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

      1. *-commutative 22.2%

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

   7. Simplified 22.2%

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

   if 3.9999999999999997e23 < (pow.f64 B #s(literal 2 binary64)) < 9.9999999999999998e155

   1. Initial program 42.4%

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

   3. Taylor expanded in F around 0 42.1%

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

   5. Simplified 57.3%

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

   6. Taylor expanded in A around 0 25.5%

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

      1. unpow2 25.5%

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

      2. unpow2 25.5%

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

      3. hypot-define 29.5%

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

   8. Simplified 29.5%

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

   if 9.9999999999999998e155 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 8.4%

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

   3. Taylor expanded in B around inf 22.9%

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

      1. mul-1-neg 22.9%

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

      2. *-commutative 22.9%

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

   5. Simplified 22.9%

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

      1. sqrt-div 26.9%

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

   7. Applied egg-rr 26.9%

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

4. Final simplification 24.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 4 \cdot 10^{+23}:\\ \;\;\;\;\frac{\sqrt{\left(F \cdot \mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)\right) \cdot \left(4 \cdot C\right)}}{-\mathsf{fma}\left(B, B, A \cdot \left(C \cdot -4\right)\right)}\\ \mathbf{elif}\;{B}^{2} \leq 10^{+156}:\\ \;\;\;\;\sqrt{2} \cdot \left(\sqrt{F \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)} \cdot \frac{-1}{B}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \frac{\sqrt{F}}{-\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 48.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;{B\_m}^{2} \leq 4 \cdot 10^{+23}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{F \cdot \frac{-0.5}{A}}\right)\\ \mathbf{elif}\;{B\_m}^{2} \leq 10^{+156}:\\ \;\;\;\;\sqrt{2} \cdot \left(\sqrt{F \cdot \left(C + \mathsf{hypot}\left(B\_m, C\right)\right)} \cdot \frac{-1}{B\_m}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \frac{\sqrt{F}}{-\sqrt{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= (pow B_m 2.0) 4e+23)
   (* (sqrt 2.0) (- (sqrt (* F (/ -0.5 A)))))
   (if (<= (pow B_m 2.0) 1e+156)
     (* (sqrt 2.0) (* (sqrt (* F (+ C (hypot B_m C)))) (/ -1.0 B_m)))
     (* (sqrt 2.0) (/ (sqrt F) (- (sqrt B_m)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (pow(B_m, 2.0) <= 4e+23) {
		tmp = sqrt(2.0) * -sqrt((F * (-0.5 / A)));
	} else if (pow(B_m, 2.0) <= 1e+156) {
		tmp = sqrt(2.0) * (sqrt((F * (C + hypot(B_m, C)))) * (-1.0 / B_m));
	} else {
		tmp = sqrt(2.0) * (sqrt(F) / -sqrt(B_m));
	}
	return tmp;
}

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (Math.pow(B_m, 2.0) <= 4e+23) {
		tmp = Math.sqrt(2.0) * -Math.sqrt((F * (-0.5 / A)));
	} else if (Math.pow(B_m, 2.0) <= 1e+156) {
		tmp = Math.sqrt(2.0) * (Math.sqrt((F * (C + Math.hypot(B_m, C)))) * (-1.0 / B_m));
	} else {
		tmp = Math.sqrt(2.0) * (Math.sqrt(F) / -Math.sqrt(B_m));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if math.pow(B_m, 2.0) <= 4e+23:
		tmp = math.sqrt(2.0) * -math.sqrt((F * (-0.5 / A)))
	elif math.pow(B_m, 2.0) <= 1e+156:
		tmp = math.sqrt(2.0) * (math.sqrt((F * (C + math.hypot(B_m, C)))) * (-1.0 / B_m))
	else:
		tmp = math.sqrt(2.0) * (math.sqrt(F) / -math.sqrt(B_m))
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if ((B_m ^ 2.0) <= 4e+23)
		tmp = Float64(sqrt(2.0) * Float64(-sqrt(Float64(F * Float64(-0.5 / A)))));
	elseif ((B_m ^ 2.0) <= 1e+156)
		tmp = Float64(sqrt(2.0) * Float64(sqrt(Float64(F * Float64(C + hypot(B_m, C)))) * Float64(-1.0 / B_m)));
	else
		tmp = Float64(sqrt(2.0) * Float64(sqrt(F) / Float64(-sqrt(B_m))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if ((B_m ^ 2.0) <= 4e+23)
		tmp = sqrt(2.0) * -sqrt((F * (-0.5 / A)));
	elseif ((B_m ^ 2.0) <= 1e+156)
		tmp = sqrt(2.0) * (sqrt((F * (C + hypot(B_m, C)))) * (-1.0 / B_m));
	else
		tmp = sqrt(2.0) * (sqrt(F) / -sqrt(B_m));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 4e+23], N[(N[Sqrt[2.0], $MachinePrecision] * (-N[Sqrt[N[(F * N[(-0.5 / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], If[LessEqual[N[Power[B$95$m, 2.0], $MachinePrecision], 1e+156], N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sqrt[N[(F * N[(C + N[Sqrt[B$95$m ^ 2 + C ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-1.0 / B$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sqrt[F], $MachinePrecision] / (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (pow.f64 B #s(literal 2 binary64)) < 3.9999999999999997e23

   1. Initial program 25.1%

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

   3. Taylor expanded in F around 0 21.9%

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

   5. Simplified 28.8%

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

   6. Taylor expanded in A around -inf 16.5%

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

   if 3.9999999999999997e23 < (pow.f64 B #s(literal 2 binary64)) < 9.9999999999999998e155

   1. Initial program 42.4%

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

   3. Taylor expanded in F around 0 42.1%

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

   5. Simplified 57.3%

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

   6. Taylor expanded in A around 0 25.5%

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

      1. unpow2 25.5%

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

      2. unpow2 25.5%

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

      3. hypot-define 29.5%

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

   8. Simplified 29.5%

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

   if 9.9999999999999998e155 < (pow.f64 B #s(literal 2 binary64))

   1. Initial program 8.4%

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

   3. Taylor expanded in B around inf 22.9%

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

      1. mul-1-neg 22.9%

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

      2. *-commutative 22.9%

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

   5. Simplified 22.9%

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

      1. sqrt-div 26.9%

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

   7. Applied egg-rr 26.9%

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

4. Final simplification 21.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;{B}^{2} \leq 4 \cdot 10^{+23}:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{F \cdot \frac{-0.5}{A}}\right)\\ \mathbf{elif}\;{B}^{2} \leq 10^{+156}:\\ \;\;\;\;\sqrt{2} \cdot \left(\sqrt{F \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)} \cdot \frac{-1}{B}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \frac{\sqrt{F}}{-\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 48.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 1900000000000:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{F \cdot \frac{-0.5}{A}}\right)\\ \mathbf{elif}\;B\_m \leq 1.8 \cdot 10^{+79}:\\ \;\;\;\;\sqrt{F \cdot \left(C + \mathsf{hypot}\left(B\_m, C\right)\right)} \cdot \frac{\sqrt{2}}{-B\_m}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\sqrt{F} \cdot \left(-\sqrt{\frac{1}{B\_m}}\right)\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 1900000000000.0)
   (* (sqrt 2.0) (- (sqrt (* F (/ -0.5 A)))))
   (if (<= B_m 1.8e+79)
     (* (sqrt (* F (+ C (hypot B_m C)))) (/ (sqrt 2.0) (- B_m)))
     (* (sqrt 2.0) (* (sqrt F) (- (sqrt (/ 1.0 B_m))))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 1900000000000.0) {
		tmp = sqrt(2.0) * -sqrt((F * (-0.5 / A)));
	} else if (B_m <= 1.8e+79) {
		tmp = sqrt((F * (C + hypot(B_m, C)))) * (sqrt(2.0) / -B_m);
	} else {
		tmp = sqrt(2.0) * (sqrt(F) * -sqrt((1.0 / B_m)));
	}
	return tmp;
}

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 1900000000000.0) {
		tmp = Math.sqrt(2.0) * -Math.sqrt((F * (-0.5 / A)));
	} else if (B_m <= 1.8e+79) {
		tmp = Math.sqrt((F * (C + Math.hypot(B_m, C)))) * (Math.sqrt(2.0) / -B_m);
	} else {
		tmp = Math.sqrt(2.0) * (Math.sqrt(F) * -Math.sqrt((1.0 / B_m)));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 1900000000000.0:
		tmp = math.sqrt(2.0) * -math.sqrt((F * (-0.5 / A)))
	elif B_m <= 1.8e+79:
		tmp = math.sqrt((F * (C + math.hypot(B_m, C)))) * (math.sqrt(2.0) / -B_m)
	else:
		tmp = math.sqrt(2.0) * (math.sqrt(F) * -math.sqrt((1.0 / B_m)))
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 1900000000000.0)
		tmp = Float64(sqrt(2.0) * Float64(-sqrt(Float64(F * Float64(-0.5 / A)))));
	elseif (B_m <= 1.8e+79)
		tmp = Float64(sqrt(Float64(F * Float64(C + hypot(B_m, C)))) * Float64(sqrt(2.0) / Float64(-B_m)));
	else
		tmp = Float64(sqrt(2.0) * Float64(sqrt(F) * Float64(-sqrt(Float64(1.0 / B_m)))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (B_m <= 1900000000000.0)
		tmp = sqrt(2.0) * -sqrt((F * (-0.5 / A)));
	elseif (B_m <= 1.8e+79)
		tmp = sqrt((F * (C + hypot(B_m, C)))) * (sqrt(2.0) / -B_m);
	else
		tmp = sqrt(2.0) * (sqrt(F) * -sqrt((1.0 / B_m)));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 1900000000000.0], N[(N[Sqrt[2.0], $MachinePrecision] * (-N[Sqrt[N[(F * N[(-0.5 / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], If[LessEqual[B$95$m, 1.8e+79], N[(N[Sqrt[N[(F * 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], N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sqrt[F], $MachinePrecision] * (-N[Sqrt[N[(1.0 / B$95$m), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if B < 1.9e12

   1. Initial program 23.0%

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

   3. Taylor expanded in F around 0 19.9%

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

   5. Simplified 29.2%

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

   6. Taylor expanded in A around -inf 13.5%

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

   if 1.9e12 < B < 1.8e79

   1. Initial program 41.2%

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

   3. Taylor expanded in A around 0 47.7%

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

      1. mul-1-neg 47.7%

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

      2. unpow2 47.7%

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

      3. unpow2 47.7%

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

      4. hypot-define 54.9%

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

   5. Simplified 54.9%

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

   if 1.8e79 < B

   1. Initial program 3.3%

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

   3. Taylor expanded in B around inf 52.7%

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

      1. mul-1-neg 52.7%

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

      2. *-commutative 52.7%

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

   5. Simplified 52.7%

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

      1. pow1/2 52.7%

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

      2. div-inv 52.7%

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

      3. unpow-prod-down 65.2%

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

      4. pow1/2 65.2%

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

   7. Applied egg-rr 65.2%

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

      1. unpow1/2 65.2%

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

   9. Simplified 65.2%

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

4. Final simplification 23.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1900000000000:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{F \cdot \frac{-0.5}{A}}\right)\\ \mathbf{elif}\;B \leq 1.8 \cdot 10^{+79}:\\ \;\;\;\;\sqrt{F \cdot \left(C + \mathsf{hypot}\left(B, C\right)\right)} \cdot \frac{\sqrt{2}}{-B}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\sqrt{F} \cdot \left(-\sqrt{\frac{1}{B}}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 47.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 1060000000000:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{F \cdot \frac{-0.5}{A}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\sqrt{F} \cdot \left(-\sqrt{\frac{1}{B\_m}}\right)\right)\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 1060000000000.0)
   (* (sqrt 2.0) (- (sqrt (* F (/ -0.5 A)))))
   (* (sqrt 2.0) (* (sqrt F) (- (sqrt (/ 1.0 B_m)))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 1060000000000.0) {
		tmp = sqrt(2.0) * -sqrt((F * (-0.5 / A)));
	} else {
		tmp = sqrt(2.0) * (sqrt(F) * -sqrt((1.0 / B_m)));
	}
	return tmp;
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (b_m <= 1060000000000.0d0) then
        tmp = sqrt(2.0d0) * -sqrt((f * ((-0.5d0) / a)))
    else
        tmp = sqrt(2.0d0) * (sqrt(f) * -sqrt((1.0d0 / b_m)))
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 1060000000000.0) {
		tmp = Math.sqrt(2.0) * -Math.sqrt((F * (-0.5 / A)));
	} else {
		tmp = Math.sqrt(2.0) * (Math.sqrt(F) * -Math.sqrt((1.0 / B_m)));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 1060000000000.0:
		tmp = math.sqrt(2.0) * -math.sqrt((F * (-0.5 / A)))
	else:
		tmp = math.sqrt(2.0) * (math.sqrt(F) * -math.sqrt((1.0 / B_m)))
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 1060000000000.0)
		tmp = Float64(sqrt(2.0) * Float64(-sqrt(Float64(F * Float64(-0.5 / A)))));
	else
		tmp = Float64(sqrt(2.0) * Float64(sqrt(F) * Float64(-sqrt(Float64(1.0 / B_m)))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (B_m <= 1060000000000.0)
		tmp = sqrt(2.0) * -sqrt((F * (-0.5 / A)));
	else
		tmp = sqrt(2.0) * (sqrt(F) * -sqrt((1.0 / B_m)));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 1060000000000.0], N[(N[Sqrt[2.0], $MachinePrecision] * (-N[Sqrt[N[(F * N[(-0.5 / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sqrt[F], $MachinePrecision] * (-N[Sqrt[N[(1.0 / B$95$m), $MachinePrecision]], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 1.06e12

   1. Initial program 23.0%

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

   3. Taylor expanded in F around 0 19.9%

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

   5. Simplified 29.2%

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

   6. Taylor expanded in A around -inf 13.5%

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

   if 1.06e12 < B

   1. Initial program 14.2%

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

   3. Taylor expanded in B around inf 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%

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

      2. *-commutative 47.1%

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

   5. Simplified 47.1%

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

      1. pow1/2 47.1%

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

      2. div-inv 47.1%

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

      3. unpow-prod-down 57.7%

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

      4. pow1/2 57.7%

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

   7. Applied egg-rr 57.7%

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

      1. unpow1/2 57.7%

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

   9. Simplified 57.7%

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

4. Final simplification 22.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1060000000000:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{F \cdot \frac{-0.5}{A}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \left(\sqrt{F} \cdot \left(-\sqrt{\frac{1}{B}}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 47.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 500000000000:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{F \cdot \frac{-0.5}{A}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \frac{\sqrt{F}}{-\sqrt{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 500000000000.0)
   (* (sqrt 2.0) (- (sqrt (* F (/ -0.5 A)))))
   (* (sqrt 2.0) (/ (sqrt F) (- (sqrt B_m))))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 500000000000.0) {
		tmp = sqrt(2.0) * -sqrt((F * (-0.5 / A)));
	} else {
		tmp = sqrt(2.0) * (sqrt(F) / -sqrt(B_m));
	}
	return tmp;
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (b_m <= 500000000000.0d0) then
        tmp = sqrt(2.0d0) * -sqrt((f * ((-0.5d0) / a)))
    else
        tmp = sqrt(2.0d0) * (sqrt(f) / -sqrt(b_m))
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 500000000000.0) {
		tmp = Math.sqrt(2.0) * -Math.sqrt((F * (-0.5 / A)));
	} else {
		tmp = Math.sqrt(2.0) * (Math.sqrt(F) / -Math.sqrt(B_m));
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 500000000000.0:
		tmp = math.sqrt(2.0) * -math.sqrt((F * (-0.5 / A)))
	else:
		tmp = math.sqrt(2.0) * (math.sqrt(F) / -math.sqrt(B_m))
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 500000000000.0)
		tmp = Float64(sqrt(2.0) * Float64(-sqrt(Float64(F * Float64(-0.5 / A)))));
	else
		tmp = Float64(sqrt(2.0) * Float64(sqrt(F) / Float64(-sqrt(B_m))));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (B_m <= 500000000000.0)
		tmp = sqrt(2.0) * -sqrt((F * (-0.5 / A)));
	else
		tmp = sqrt(2.0) * (sqrt(F) / -sqrt(B_m));
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 500000000000.0], N[(N[Sqrt[2.0], $MachinePrecision] * (-N[Sqrt[N[(F * N[(-0.5 / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[2.0], $MachinePrecision] * N[(N[Sqrt[F], $MachinePrecision] / (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 5e11

   1. Initial program 23.0%

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

   3. Taylor expanded in F around 0 19.9%

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

   5. Simplified 29.2%

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

   6. Taylor expanded in A around -inf 13.5%

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

   if 5e11 < B

   1. Initial program 14.2%

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

   3. Taylor expanded in B around inf 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%

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

      2. *-commutative 47.1%

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

   5. Simplified 47.1%

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

      1. sqrt-div 57.8%

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

   7. Applied egg-rr 57.8%

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

4. Final simplification 22.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 500000000000:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{F \cdot \frac{-0.5}{A}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2} \cdot \frac{\sqrt{F}}{-\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 47.7% accurate, 3.0× speedup

Math

\[\begin{array}{l} B_m = \left|B\right| \\ [A, B_m, C, F] = \mathsf{sort}([A, B_m, C, F])\\ \\ \begin{array}{l} \mathbf{if}\;B\_m \leq 1360000000000:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{F \cdot \frac{-0.5}{A}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-\sqrt{B\_m}}\\ \end{array} \end{array} \]

FPCore

B_m = (fabs.f64 B)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
(FPCore (A B_m C F)
 :precision binary64
 (if (<= B_m 1360000000000.0)
   (* (sqrt 2.0) (- (sqrt (* F (/ -0.5 A)))))
   (/ (sqrt (* 2.0 F)) (- (sqrt B_m)))))

C

B_m = fabs(B);
assert(A < B_m && B_m < C && C < F);
double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 1360000000000.0) {
		tmp = sqrt(2.0) * -sqrt((F * (-0.5 / A)));
	} else {
		tmp = sqrt((2.0 * F)) / -sqrt(B_m);
	}
	return tmp;
}

Fortran

B_m = abs(b)
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
real(8) function code(a, b_m, c, f)
    real(8), intent (in) :: a
    real(8), intent (in) :: b_m
    real(8), intent (in) :: c
    real(8), intent (in) :: f
    real(8) :: tmp
    if (b_m <= 1360000000000.0d0) then
        tmp = sqrt(2.0d0) * -sqrt((f * ((-0.5d0) / a)))
    else
        tmp = sqrt((2.0d0 * f)) / -sqrt(b_m)
    end if
    code = tmp
end function

Java

B_m = Math.abs(B);
assert A < B_m && B_m < C && C < F;
public static double code(double A, double B_m, double C, double F) {
	double tmp;
	if (B_m <= 1360000000000.0) {
		tmp = Math.sqrt(2.0) * -Math.sqrt((F * (-0.5 / A)));
	} else {
		tmp = Math.sqrt((2.0 * F)) / -Math.sqrt(B_m);
	}
	return tmp;
}

Python

B_m = math.fabs(B)
[A, B_m, C, F] = sort([A, B_m, C, F])
def code(A, B_m, C, F):
	tmp = 0
	if B_m <= 1360000000000.0:
		tmp = math.sqrt(2.0) * -math.sqrt((F * (-0.5 / A)))
	else:
		tmp = math.sqrt((2.0 * F)) / -math.sqrt(B_m)
	return tmp

Julia

B_m = abs(B)
A, B_m, C, F = sort([A, B_m, C, F])
function code(A, B_m, C, F)
	tmp = 0.0
	if (B_m <= 1360000000000.0)
		tmp = Float64(sqrt(2.0) * Float64(-sqrt(Float64(F * Float64(-0.5 / A)))));
	else
		tmp = Float64(sqrt(Float64(2.0 * F)) / Float64(-sqrt(B_m)));
	end
	return tmp
end

MATLAB

B_m = abs(B);
A, B_m, C, F = num2cell(sort([A, B_m, C, F])){:}
function tmp_2 = code(A, B_m, C, F)
	tmp = 0.0;
	if (B_m <= 1360000000000.0)
		tmp = sqrt(2.0) * -sqrt((F * (-0.5 / A)));
	else
		tmp = sqrt((2.0 * F)) / -sqrt(B_m);
	end
	tmp_2 = tmp;
end

Wolfram

B_m = N[Abs[B], $MachinePrecision]
NOTE: A, B_m, C, and F should be sorted in increasing order before calling this function.
code[A_, B$95$m_, C_, F_] := If[LessEqual[B$95$m, 1360000000000.0], N[(N[Sqrt[2.0], $MachinePrecision] * (-N[Sqrt[N[(F * N[(-0.5 / A), $MachinePrecision]), $MachinePrecision]], $MachinePrecision])), $MachinePrecision], N[(N[Sqrt[N[(2.0 * F), $MachinePrecision]], $MachinePrecision] / (-N[Sqrt[B$95$m], $MachinePrecision])), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if B < 1.36e12

   1. Initial program 23.0%

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

   3. Taylor expanded in F around 0 19.9%

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

   5. Simplified 29.2%

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

   6. Taylor expanded in A around -inf 13.5%

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

   if 1.36e12 < B

   1. Initial program 14.2%

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

   3. Taylor expanded in B around inf 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%

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

      2. *-commutative 47.1%

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

   5. Simplified 47.1%

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

      1. pow1/2 47.1%

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

      2. div-inv 47.1%

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

      3. unpow-prod-down 57.7%

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

      4. pow1/2 57.7%

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

   7. Applied egg-rr 57.7%

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

      1. unpow1/2 57.7%

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

   9. Simplified 57.7%

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

       1. associate-*r* 57.7%

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

       2. sqrt-div 57.7%

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

       3. metadata-eval 57.7%

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

       4. un-div-inv 57.8%

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

       5. sqrt-unprod 57.9%

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

   11. Applied egg-rr 57.9%

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

       1. *-commutative 57.9%

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

   13. Simplified 57.9%

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

4. Final simplification 22.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;B \leq 1360000000000:\\ \;\;\;\;\sqrt{2} \cdot \left(-\sqrt{F \cdot \frac{-0.5}{A}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2 \cdot F}}{-\sqrt{B}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 35.9% 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 21.2%

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

3. Taylor expanded in B around inf 14.0%

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

   1. mul-1-neg 14.0%

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

   2. *-commutative 14.0%

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

5. Simplified 14.0%

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

   1. pow1/2 14.2%

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

   2. div-inv 14.2%

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

   3. unpow-prod-down 15.7%

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

   4. pow1/2 15.7%

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

7. Applied egg-rr 15.7%

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

   1. unpow1/2 15.7%

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

9. Simplified 15.7%

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

    1. associate-*r* 15.7%

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

    2. sqrt-div 15.7%

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

    3. metadata-eval 15.7%

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

    4. un-div-inv 15.7%

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

    5. sqrt-unprod 15.8%

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

11. Applied egg-rr 15.8%

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

    1. *-commutative 15.8%

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

13. Simplified 15.8%

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

14. Final simplification 15.8%

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

Alternative 13: 27.7% accurate, 5.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 21.2%

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

3. Taylor expanded in B around inf 14.0%

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

   1. mul-1-neg 14.0%

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

   2. *-commutative 14.0%

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

5. Simplified 14.0%

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

   1. *-commutative 14.0%

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

   2. pow1/2 14.2%

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

   3. pow1/2 14.2%

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

   4. pow-prod-down 14.3%

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

7. Applied egg-rr 14.3%

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

8. Final simplification 14.3%

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

Alternative 14: 27.7% accurate, 6.0× speedup

Math

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

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 21.2%

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

3. Taylor expanded in B around inf 14.0%

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

   1. mul-1-neg 14.0%

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

   2. *-commutative 14.0%

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

5. Simplified 14.0%

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

   1. *-commutative 14.0%

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

   2. pow1/2 14.2%

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

   3. pow1/2 14.2%

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

   4. pow-prod-down 14.3%

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

7. Applied egg-rr 14.3%

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

   1. unpow1/2 14.1%

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

9. Simplified 14.1%

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

10. Final simplification 14.1%

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

Alternative 15: 27.7% accurate, 6.0× speedup

Math

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

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 21.2%

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

3. Taylor expanded in B around inf 14.0%

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

   1. mul-1-neg 14.0%

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

   2. *-commutative 14.0%

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

5. Simplified 14.0%

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

   1. *-commutative 14.0%

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

   2. pow1/2 14.2%

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

   3. pow1/2 14.2%

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

   4. pow-prod-down 14.3%

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

7. Applied egg-rr 14.3%

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

   1. unpow1/2 14.1%

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

9. Simplified 14.1%

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

    1. *-commutative 14.1%

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

    2. clear-num 14.1%

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

    3. un-div-inv 14.1%

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

11. Applied egg-rr 14.1%

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

    1. associate-/r/ 14.1%

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

13. Simplified 14.1%

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

14. Final simplification 14.1%

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

Alternative 16: 4.4% accurate, 6.0× speedup

Math

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

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 (* 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 21.2%

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

3. Taylor expanded in B around inf 14.0%

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

   1. mul-1-neg 14.0%

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

   2. *-commutative 14.0%

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

5. Simplified 14.0%

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

   1. *-commutative 14.0%

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

   2. pow1/2 14.2%

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

   3. pow1/2 14.2%

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

   4. pow-prod-down 14.3%

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

7. Applied egg-rr 14.3%

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

   1. unpow1/2 14.1%

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

9. Simplified 14.1%

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

    1. *-commutative 14.1%

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

    2. clear-num 14.1%

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

    3. un-div-inv 14.1%

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

11. Applied egg-rr 14.1%

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

    1. associate-/r/ 14.1%

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

13. Simplified 14.1%

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

    1. div-inv 14.1%

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

    2. inv-pow 14.1%

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

    3. metadata-eval 14.1%

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

    4. pow-prod-up 13.5%

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

    5. pow-sqr 14.1%

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

    6. metadata-eval 14.1%

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

    7. inv-pow 14.1%

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

    8. add-exp-log 12.7%

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

    9. neg-log 12.7%

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

    10. add-sqr-sqrt 3.3%

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

    11. sqrt-unprod 4.4%

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

    12. sqr-neg 4.4%

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

    13. sqrt-unprod 1.0%

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

    14. add-sqr-sqrt 2.3%

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

    15. add-exp-log 2.6%

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

15. Applied egg-rr 2.6%

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

    1. *-commutative 2.6%

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

17. Simplified 2.6%

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

18. Final simplification 2.6%

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

Reproduce

herbie shell --seed 2024118 
(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))))