Cubic critical

Percentage Accurate: 52.8% → 85.5%

Time: 16.0s

Alternatives: 11

Speedup: 11.6×

Specification

Math

\[\begin{array}{l} \\ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-b + sqrt(((b * b) - ((3.0d0 * a) * c)))) / (3.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]

Initial Program: 52.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-b + sqrt(((b * b) - ((3.0d0 * a) * c)))) / (3.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]

Alternative 1: 85.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5.2 \cdot 10^{+94}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a}\\ \mathbf{elif}\;b \leq 10^{-80}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5.2e+94)
   (/ (* b -0.6666666666666666) a)
   (if (<= b 1e-80)
     (/ (- (sqrt (fma b b (* a (* c -3.0)))) b) (* a 3.0))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5.2e+94) {
		tmp = (b * -0.6666666666666666) / a;
	} else if (b <= 1e-80) {
		tmp = (sqrt(fma(b, b, (a * (c * -3.0)))) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5.2e+94)
		tmp = Float64(Float64(b * -0.6666666666666666) / a);
	elseif (b <= 1e-80)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(c * -3.0)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5.2e+94], N[(N[(b * -0.6666666666666666), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 1e-80], N[(N[(N[Sqrt[N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -5.1999999999999998e94

   1. Initial program 54.1%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. fma-neg 54.1%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(3 \cdot a\right) \cdot c\right)}}}{3 \cdot a} \]

      2. associate-*l* 54.1%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, -\color{blue}{3 \cdot \left(a \cdot c\right)}\right)}}{3 \cdot a} \]

      3. distribute-lft-neg-in 54.1%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(-3\right) \cdot \left(a \cdot c\right)}\right)}}{3 \cdot a} \]

      4. metadata-eval 54.1%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \color{blue}{-3} \cdot \left(a \cdot c\right)\right)}}{3 \cdot a} \]

      5. *-commutative 54.1%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(a \cdot c\right) \cdot -3}\right)}}{3 \cdot a} \]

      6. associate-*r* 54.1%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \color{blue}{a \cdot \left(c \cdot -3\right)}\right)}}{3 \cdot a} \]

      7. add-cube-cbrt 54.1%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\left(\sqrt[3]{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} \cdot \sqrt[3]{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}}}}{3 \cdot a} \]

      8. pow3 54.1%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}\right)}^{3}}}}{3 \cdot a} \]

      9. *-commutative 54.1%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\left(\sqrt[3]{\mathsf{fma}\left(b, b, a \cdot \color{blue}{\left(-3 \cdot c\right)}\right)}\right)}^{3}}}{3 \cdot a} \]

      10. associate-*r* 54.1%

          \[\leadsto \frac{\left(-b\right) + \sqrt{{\left(\sqrt[3]{\mathsf{fma}\left(b, b, \color{blue}{\left(a \cdot -3\right) \cdot c}\right)}\right)}^{3}}}{3 \cdot a} \]

      11. metadata-eval 54.1%

          \[\leadsto \frac{\left(-b\right) + \sqrt{{\left(\sqrt[3]{\mathsf{fma}\left(b, b, \left(a \cdot \color{blue}{\left(-3\right)}\right) \cdot c\right)}\right)}^{3}}}{3 \cdot a} \]

      12. distribute-rgt-neg-in 54.1%

          \[\leadsto \frac{\left(-b\right) + \sqrt{{\left(\sqrt[3]{\mathsf{fma}\left(b, b, \color{blue}{\left(-a \cdot 3\right)} \cdot c\right)}\right)}^{3}}}{3 \cdot a} \]

      13. *-commutative 54.1%

          \[\leadsto \frac{\left(-b\right) + \sqrt{{\left(\sqrt[3]{\mathsf{fma}\left(b, b, \left(-\color{blue}{3 \cdot a}\right) \cdot c\right)}\right)}^{3}}}{3 \cdot a} \]

      14. *-commutative 54.1%

          \[\leadsto \frac{\left(-b\right) + \sqrt{{\left(\sqrt[3]{\mathsf{fma}\left(b, b, \color{blue}{c \cdot \left(-3 \cdot a\right)}\right)}\right)}^{3}}}{3 \cdot a} \]

      15. *-commutative 54.1%

          \[\leadsto \frac{\left(-b\right) + \sqrt{{\left(\sqrt[3]{\mathsf{fma}\left(b, b, c \cdot \left(-\color{blue}{a \cdot 3}\right)\right)}\right)}^{3}}}{3 \cdot a} \]

      16. distribute-rgt-neg-in 54.1%

          \[\leadsto \frac{\left(-b\right) + \sqrt{{\left(\sqrt[3]{\mathsf{fma}\left(b, b, c \cdot \color{blue}{\left(a \cdot \left(-3\right)\right)}\right)}\right)}^{3}}}{3 \cdot a} \]

      17. metadata-eval 54.1%

          \[\leadsto \frac{\left(-b\right) + \sqrt{{\left(\sqrt[3]{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot \color{blue}{-3}\right)\right)}\right)}^{3}}}{3 \cdot a} \]

   4. Applied egg-rr 54.1%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)}\right)}^{3}}}}{3 \cdot a} \]

   5. Taylor expanded in b around -inf 93.6%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a}} \]
   6. Step-by-step derivation

      1. associate-*r/ 93.8%

         \[\leadsto \color{blue}{\frac{-0.6666666666666666 \cdot b}{a}} \]

      2. *-commutative 93.8%

         \[\leadsto \frac{\color{blue}{b \cdot -0.6666666666666666}}{a} \]

   7. Simplified 93.8%

      \[\leadsto \color{blue}{\frac{b \cdot -0.6666666666666666}{a}} \]

   if -5.1999999999999998e94 < b < 9.99999999999999961e-81

   1. Initial program 86.4%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Step-by-step derivation

      1. +-commutative 86.4%

         \[\leadsto \frac{\color{blue}{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} + \left(-b\right)}}{3 \cdot a} \]

      2. sqr-neg 86.4%

         \[\leadsto \frac{\sqrt{\color{blue}{\left(-b\right) \cdot \left(-b\right)} - \left(3 \cdot a\right) \cdot c} + \left(-b\right)}{3 \cdot a} \]

      3. unsub-neg 86.4%

         \[\leadsto \frac{\color{blue}{\sqrt{\left(-b\right) \cdot \left(-b\right) - \left(3 \cdot a\right) \cdot c} - b}}{3 \cdot a} \]

      4. div-sub 86.3%

         \[\leadsto \color{blue}{\frac{\sqrt{\left(-b\right) \cdot \left(-b\right) - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} - \frac{b}{3 \cdot a}} \]

      5. --rgt-identity 86.3%

         \[\leadsto \frac{\color{blue}{\sqrt{\left(-b\right) \cdot \left(-b\right) - \left(3 \cdot a\right) \cdot c} - 0}}{3 \cdot a} - \frac{b}{3 \cdot a} \]

      6. div-sub 86.4%

         \[\leadsto \color{blue}{\frac{\left(\sqrt{\left(-b\right) \cdot \left(-b\right) - \left(3 \cdot a\right) \cdot c} - 0\right) - b}{3 \cdot a}} \]

   3. Simplified 86.4%

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{3 \cdot a}} \]
   4. Add Preprocessing

   if 9.99999999999999961e-81 < b

   1. Initial program 15.5%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf 84.7%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
   4. Step-by-step derivation

      1. associate-*r/ 84.7%

         \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b}} \]

   5. Simplified 84.7%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.2 \cdot 10^{+94}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a}\\ \mathbf{elif}\;b \leq 10^{-80}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.2 \cdot 10^{+94}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a}\\ \mathbf{elif}\;b \leq 7 \cdot 10^{-68}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.2e+94)
   (/ (* b -0.6666666666666666) a)
   (if (<= b 7e-68)
     (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.2e+94) {
		tmp = (b * -0.6666666666666666) / a;
	} else if (b <= 7e-68) {
		tmp = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-2.2d+94)) then
        tmp = (b * (-0.6666666666666666d0)) / a
    else if (b <= 7d-68) then
        tmp = (sqrt(((b * b) - (c * (a * 3.0d0)))) - b) / (a * 3.0d0)
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.2e+94) {
		tmp = (b * -0.6666666666666666) / a;
	} else if (b <= 7e-68) {
		tmp = (Math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.2e+94:
		tmp = (b * -0.6666666666666666) / a
	elif b <= 7e-68:
		tmp = (math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.2e+94)
		tmp = Float64(Float64(b * -0.6666666666666666) / a);
	elseif (b <= 7e-68)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.2e+94)
		tmp = (b * -0.6666666666666666) / a;
	elseif (b <= 7e-68)
		tmp = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.2e+94], N[(N[(b * -0.6666666666666666), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 7e-68], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.20000000000000012e94

   1. Initial program 54.1%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. fma-neg 54.1%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(3 \cdot a\right) \cdot c\right)}}}{3 \cdot a} \]

      2. associate-*l* 54.1%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, -\color{blue}{3 \cdot \left(a \cdot c\right)}\right)}}{3 \cdot a} \]

      3. distribute-lft-neg-in 54.1%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(-3\right) \cdot \left(a \cdot c\right)}\right)}}{3 \cdot a} \]

      4. metadata-eval 54.1%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \color{blue}{-3} \cdot \left(a \cdot c\right)\right)}}{3 \cdot a} \]

      5. *-commutative 54.1%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(a \cdot c\right) \cdot -3}\right)}}{3 \cdot a} \]

      6. associate-*r* 54.1%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \color{blue}{a \cdot \left(c \cdot -3\right)}\right)}}{3 \cdot a} \]

      7. add-cube-cbrt 54.1%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\left(\sqrt[3]{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} \cdot \sqrt[3]{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}}}}{3 \cdot a} \]

      8. pow3 54.1%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}\right)}^{3}}}}{3 \cdot a} \]

      9. *-commutative 54.1%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\left(\sqrt[3]{\mathsf{fma}\left(b, b, a \cdot \color{blue}{\left(-3 \cdot c\right)}\right)}\right)}^{3}}}{3 \cdot a} \]

      10. associate-*r* 54.1%

          \[\leadsto \frac{\left(-b\right) + \sqrt{{\left(\sqrt[3]{\mathsf{fma}\left(b, b, \color{blue}{\left(a \cdot -3\right) \cdot c}\right)}\right)}^{3}}}{3 \cdot a} \]

      11. metadata-eval 54.1%

          \[\leadsto \frac{\left(-b\right) + \sqrt{{\left(\sqrt[3]{\mathsf{fma}\left(b, b, \left(a \cdot \color{blue}{\left(-3\right)}\right) \cdot c\right)}\right)}^{3}}}{3 \cdot a} \]

      12. distribute-rgt-neg-in 54.1%

          \[\leadsto \frac{\left(-b\right) + \sqrt{{\left(\sqrt[3]{\mathsf{fma}\left(b, b, \color{blue}{\left(-a \cdot 3\right)} \cdot c\right)}\right)}^{3}}}{3 \cdot a} \]

      13. *-commutative 54.1%

          \[\leadsto \frac{\left(-b\right) + \sqrt{{\left(\sqrt[3]{\mathsf{fma}\left(b, b, \left(-\color{blue}{3 \cdot a}\right) \cdot c\right)}\right)}^{3}}}{3 \cdot a} \]

      14. *-commutative 54.1%

          \[\leadsto \frac{\left(-b\right) + \sqrt{{\left(\sqrt[3]{\mathsf{fma}\left(b, b, \color{blue}{c \cdot \left(-3 \cdot a\right)}\right)}\right)}^{3}}}{3 \cdot a} \]

      15. *-commutative 54.1%

          \[\leadsto \frac{\left(-b\right) + \sqrt{{\left(\sqrt[3]{\mathsf{fma}\left(b, b, c \cdot \left(-\color{blue}{a \cdot 3}\right)\right)}\right)}^{3}}}{3 \cdot a} \]

      16. distribute-rgt-neg-in 54.1%

          \[\leadsto \frac{\left(-b\right) + \sqrt{{\left(\sqrt[3]{\mathsf{fma}\left(b, b, c \cdot \color{blue}{\left(a \cdot \left(-3\right)\right)}\right)}\right)}^{3}}}{3 \cdot a} \]

      17. metadata-eval 54.1%

          \[\leadsto \frac{\left(-b\right) + \sqrt{{\left(\sqrt[3]{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot \color{blue}{-3}\right)\right)}\right)}^{3}}}{3 \cdot a} \]

   4. Applied egg-rr 54.1%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)}\right)}^{3}}}}{3 \cdot a} \]

   5. Taylor expanded in b around -inf 93.6%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a}} \]
   6. Step-by-step derivation

      1. associate-*r/ 93.8%

         \[\leadsto \color{blue}{\frac{-0.6666666666666666 \cdot b}{a}} \]

      2. *-commutative 93.8%

         \[\leadsto \frac{\color{blue}{b \cdot -0.6666666666666666}}{a} \]

   7. Simplified 93.8%

      \[\leadsto \color{blue}{\frac{b \cdot -0.6666666666666666}{a}} \]

   if -2.20000000000000012e94 < b < 7.00000000000000026e-68

   1. Initial program 84.9%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing

   if 7.00000000000000026e-68 < b

   1. Initial program 14.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf 86.3%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
   4. Step-by-step derivation

      1. associate-*r/ 86.3%

         \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b}} \]

   5. Simplified 86.3%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.2 \cdot 10^{+94}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a}\\ \mathbf{elif}\;b \leq 7 \cdot 10^{-68}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5.2 \cdot 10^{-16}:\\ \;\;\;\;\frac{\left(a \cdot \left(c \cdot \frac{1.5}{b}\right) - b\right) - b}{a \cdot 3}\\ \mathbf{elif}\;b \leq 6.8 \cdot 10^{-89}:\\ \;\;\;\;\frac{\sqrt{-3 \cdot \left(a \cdot c\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5.2e-16)
   (/ (- (- (* a (* c (/ 1.5 b))) b) b) (* a 3.0))
   (if (<= b 6.8e-89)
     (/ (- (sqrt (* -3.0 (* a c))) b) (* a 3.0))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5.2e-16) {
		tmp = (((a * (c * (1.5 / b))) - b) - b) / (a * 3.0);
	} else if (b <= 6.8e-89) {
		tmp = (sqrt((-3.0 * (a * c))) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-5.2d-16)) then
        tmp = (((a * (c * (1.5d0 / b))) - b) - b) / (a * 3.0d0)
    else if (b <= 6.8d-89) then
        tmp = (sqrt(((-3.0d0) * (a * c))) - b) / (a * 3.0d0)
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5.2e-16) {
		tmp = (((a * (c * (1.5 / b))) - b) - b) / (a * 3.0);
	} else if (b <= 6.8e-89) {
		tmp = (Math.sqrt((-3.0 * (a * c))) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5.2e-16:
		tmp = (((a * (c * (1.5 / b))) - b) - b) / (a * 3.0)
	elif b <= 6.8e-89:
		tmp = (math.sqrt((-3.0 * (a * c))) - b) / (a * 3.0)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5.2e-16)
		tmp = Float64(Float64(Float64(Float64(a * Float64(c * Float64(1.5 / b))) - b) - b) / Float64(a * 3.0));
	elseif (b <= 6.8e-89)
		tmp = Float64(Float64(sqrt(Float64(-3.0 * Float64(a * c))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5.2e-16)
		tmp = (((a * (c * (1.5 / b))) - b) - b) / (a * 3.0);
	elseif (b <= 6.8e-89)
		tmp = (sqrt((-3.0 * (a * c))) - b) / (a * 3.0);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5.2e-16], N[(N[(N[(N[(a * N[(c * N[(1.5 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 6.8e-89], N[(N[(N[Sqrt[N[(-3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -5.1999999999999997e-16

   1. Initial program 66.7%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. fma-neg 66.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(3 \cdot a\right) \cdot c\right)}}}{3 \cdot a} \]

      2. associate-*l* 66.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, -\color{blue}{3 \cdot \left(a \cdot c\right)}\right)}}{3 \cdot a} \]

      3. distribute-lft-neg-in 66.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(-3\right) \cdot \left(a \cdot c\right)}\right)}}{3 \cdot a} \]

      4. metadata-eval 66.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \color{blue}{-3} \cdot \left(a \cdot c\right)\right)}}{3 \cdot a} \]

      5. *-commutative 66.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(a \cdot c\right) \cdot -3}\right)}}{3 \cdot a} \]

      6. associate-*r* 66.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \color{blue}{a \cdot \left(c \cdot -3\right)}\right)}}{3 \cdot a} \]

      7. add-cube-cbrt 66.6%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\left(\sqrt[3]{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} \cdot \sqrt[3]{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}}}}{3 \cdot a} \]

      8. pow3 66.6%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}\right)}^{3}}}}{3 \cdot a} \]

      9. *-commutative 66.6%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\left(\sqrt[3]{\mathsf{fma}\left(b, b, a \cdot \color{blue}{\left(-3 \cdot c\right)}\right)}\right)}^{3}}}{3 \cdot a} \]

      10. associate-*r* 66.6%

          \[\leadsto \frac{\left(-b\right) + \sqrt{{\left(\sqrt[3]{\mathsf{fma}\left(b, b, \color{blue}{\left(a \cdot -3\right) \cdot c}\right)}\right)}^{3}}}{3 \cdot a} \]

      11. metadata-eval 66.6%

          \[\leadsto \frac{\left(-b\right) + \sqrt{{\left(\sqrt[3]{\mathsf{fma}\left(b, b, \left(a \cdot \color{blue}{\left(-3\right)}\right) \cdot c\right)}\right)}^{3}}}{3 \cdot a} \]

      12. distribute-rgt-neg-in 66.6%

          \[\leadsto \frac{\left(-b\right) + \sqrt{{\left(\sqrt[3]{\mathsf{fma}\left(b, b, \color{blue}{\left(-a \cdot 3\right)} \cdot c\right)}\right)}^{3}}}{3 \cdot a} \]

      13. *-commutative 66.6%

          \[\leadsto \frac{\left(-b\right) + \sqrt{{\left(\sqrt[3]{\mathsf{fma}\left(b, b, \left(-\color{blue}{3 \cdot a}\right) \cdot c\right)}\right)}^{3}}}{3 \cdot a} \]

      14. *-commutative 66.6%

          \[\leadsto \frac{\left(-b\right) + \sqrt{{\left(\sqrt[3]{\mathsf{fma}\left(b, b, \color{blue}{c \cdot \left(-3 \cdot a\right)}\right)}\right)}^{3}}}{3 \cdot a} \]

      15. *-commutative 66.6%

          \[\leadsto \frac{\left(-b\right) + \sqrt{{\left(\sqrt[3]{\mathsf{fma}\left(b, b, c \cdot \left(-\color{blue}{a \cdot 3}\right)\right)}\right)}^{3}}}{3 \cdot a} \]

      16. distribute-rgt-neg-in 66.6%

          \[\leadsto \frac{\left(-b\right) + \sqrt{{\left(\sqrt[3]{\mathsf{fma}\left(b, b, c \cdot \color{blue}{\left(a \cdot \left(-3\right)\right)}\right)}\right)}^{3}}}{3 \cdot a} \]

      17. metadata-eval 66.6%

          \[\leadsto \frac{\left(-b\right) + \sqrt{{\left(\sqrt[3]{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot \color{blue}{-3}\right)\right)}\right)}^{3}}}{3 \cdot a} \]

   4. Applied egg-rr 66.6%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)}\right)}^{3}}}}{3 \cdot a} \]

   5. Taylor expanded in b around -inf 91.5%

      \[\leadsto \frac{\left(-b\right) + \color{blue}{\left(-1 \cdot b + 1.5 \cdot \frac{a \cdot c}{b}\right)}}{3 \cdot a} \]
   6. Step-by-step derivation

      1. +-commutative 91.5%

         \[\leadsto \frac{\left(-b\right) + \color{blue}{\left(1.5 \cdot \frac{a \cdot c}{b} + -1 \cdot b\right)}}{3 \cdot a} \]

      2. mul-1-neg 91.5%

         \[\leadsto \frac{\left(-b\right) + \left(1.5 \cdot \frac{a \cdot c}{b} + \color{blue}{\left(-b\right)}\right)}{3 \cdot a} \]

      3. unsub-neg 91.5%

         \[\leadsto \frac{\left(-b\right) + \color{blue}{\left(1.5 \cdot \frac{a \cdot c}{b} - b\right)}}{3 \cdot a} \]

      4. associate-/l* 92.8%

         \[\leadsto \frac{\left(-b\right) + \left(1.5 \cdot \color{blue}{\frac{a}{\frac{b}{c}}} - b\right)}{3 \cdot a} \]

      5. associate-*r/ 92.8%

         \[\leadsto \frac{\left(-b\right) + \left(\color{blue}{\frac{1.5 \cdot a}{\frac{b}{c}}} - b\right)}{3 \cdot a} \]

      6. *-commutative 92.8%

         \[\leadsto \frac{\left(-b\right) + \left(\frac{\color{blue}{a \cdot 1.5}}{\frac{b}{c}} - b\right)}{3 \cdot a} \]

      7. associate-*r/ 92.8%

         \[\leadsto \frac{\left(-b\right) + \left(\color{blue}{a \cdot \frac{1.5}{\frac{b}{c}}} - b\right)}{3 \cdot a} \]

      8. associate-/r/ 92.8%

         \[\leadsto \frac{\left(-b\right) + \left(a \cdot \color{blue}{\left(\frac{1.5}{b} \cdot c\right)} - b\right)}{3 \cdot a} \]

   7. Simplified 92.8%

      \[\leadsto \frac{\left(-b\right) + \color{blue}{\left(a \cdot \left(\frac{1.5}{b} \cdot c\right) - b\right)}}{3 \cdot a} \]

   if -5.1999999999999997e-16 < b < 6.8000000000000001e-89

   1. Initial program 82.2%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around 0 69.1%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{-3 \cdot \left(a \cdot c\right)}}}{3 \cdot a} \]

   if 6.8000000000000001e-89 < b

   1. Initial program 15.5%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf 84.7%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
   4. Step-by-step derivation

      1. associate-*r/ 84.7%

         \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b}} \]

   5. Simplified 84.7%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.2 \cdot 10^{-16}:\\ \;\;\;\;\frac{\left(a \cdot \left(c \cdot \frac{1.5}{b}\right) - b\right) - b}{a \cdot 3}\\ \mathbf{elif}\;b \leq 6.8 \cdot 10^{-89}:\\ \;\;\;\;\frac{\sqrt{-3 \cdot \left(a \cdot c\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-14}:\\ \;\;\;\;\frac{\left(a \cdot \left(c \cdot \frac{1.5}{b}\right) - b\right) - b}{a \cdot 3}\\ \mathbf{elif}\;b \leq 3.7 \cdot 10^{-85}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2e-14)
   (/ (- (- (* a (* c (/ 1.5 b))) b) b) (* a 3.0))
   (if (<= b 3.7e-85)
     (/ (- (sqrt (* a (* c -3.0))) b) (* a 3.0))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e-14) {
		tmp = (((a * (c * (1.5 / b))) - b) - b) / (a * 3.0);
	} else if (b <= 3.7e-85) {
		tmp = (sqrt((a * (c * -3.0))) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-2d-14)) then
        tmp = (((a * (c * (1.5d0 / b))) - b) - b) / (a * 3.0d0)
    else if (b <= 3.7d-85) then
        tmp = (sqrt((a * (c * (-3.0d0)))) - b) / (a * 3.0d0)
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e-14) {
		tmp = (((a * (c * (1.5 / b))) - b) - b) / (a * 3.0);
	} else if (b <= 3.7e-85) {
		tmp = (Math.sqrt((a * (c * -3.0))) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2e-14:
		tmp = (((a * (c * (1.5 / b))) - b) - b) / (a * 3.0)
	elif b <= 3.7e-85:
		tmp = (math.sqrt((a * (c * -3.0))) - b) / (a * 3.0)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2e-14)
		tmp = Float64(Float64(Float64(Float64(a * Float64(c * Float64(1.5 / b))) - b) - b) / Float64(a * 3.0));
	elseif (b <= 3.7e-85)
		tmp = Float64(Float64(sqrt(Float64(a * Float64(c * -3.0))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2e-14)
		tmp = (((a * (c * (1.5 / b))) - b) - b) / (a * 3.0);
	elseif (b <= 3.7e-85)
		tmp = (sqrt((a * (c * -3.0))) - b) / (a * 3.0);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2e-14], N[(N[(N[(N[(a * N[(c * N[(1.5 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 3.7e-85], N[(N[(N[Sqrt[N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2e-14

   1. Initial program 66.7%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. fma-neg 66.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(3 \cdot a\right) \cdot c\right)}}}{3 \cdot a} \]

      2. associate-*l* 66.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, -\color{blue}{3 \cdot \left(a \cdot c\right)}\right)}}{3 \cdot a} \]

      3. distribute-lft-neg-in 66.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(-3\right) \cdot \left(a \cdot c\right)}\right)}}{3 \cdot a} \]

      4. metadata-eval 66.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \color{blue}{-3} \cdot \left(a \cdot c\right)\right)}}{3 \cdot a} \]

      5. *-commutative 66.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(a \cdot c\right) \cdot -3}\right)}}{3 \cdot a} \]

      6. associate-*r* 66.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \color{blue}{a \cdot \left(c \cdot -3\right)}\right)}}{3 \cdot a} \]

      7. add-cube-cbrt 66.6%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\left(\sqrt[3]{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} \cdot \sqrt[3]{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}}}}{3 \cdot a} \]

      8. pow3 66.6%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}\right)}^{3}}}}{3 \cdot a} \]

      9. *-commutative 66.6%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\left(\sqrt[3]{\mathsf{fma}\left(b, b, a \cdot \color{blue}{\left(-3 \cdot c\right)}\right)}\right)}^{3}}}{3 \cdot a} \]

      10. associate-*r* 66.6%

          \[\leadsto \frac{\left(-b\right) + \sqrt{{\left(\sqrt[3]{\mathsf{fma}\left(b, b, \color{blue}{\left(a \cdot -3\right) \cdot c}\right)}\right)}^{3}}}{3 \cdot a} \]

      11. metadata-eval 66.6%

          \[\leadsto \frac{\left(-b\right) + \sqrt{{\left(\sqrt[3]{\mathsf{fma}\left(b, b, \left(a \cdot \color{blue}{\left(-3\right)}\right) \cdot c\right)}\right)}^{3}}}{3 \cdot a} \]

      12. distribute-rgt-neg-in 66.6%

          \[\leadsto \frac{\left(-b\right) + \sqrt{{\left(\sqrt[3]{\mathsf{fma}\left(b, b, \color{blue}{\left(-a \cdot 3\right)} \cdot c\right)}\right)}^{3}}}{3 \cdot a} \]

      13. *-commutative 66.6%

          \[\leadsto \frac{\left(-b\right) + \sqrt{{\left(\sqrt[3]{\mathsf{fma}\left(b, b, \left(-\color{blue}{3 \cdot a}\right) \cdot c\right)}\right)}^{3}}}{3 \cdot a} \]

      14. *-commutative 66.6%

          \[\leadsto \frac{\left(-b\right) + \sqrt{{\left(\sqrt[3]{\mathsf{fma}\left(b, b, \color{blue}{c \cdot \left(-3 \cdot a\right)}\right)}\right)}^{3}}}{3 \cdot a} \]

      15. *-commutative 66.6%

          \[\leadsto \frac{\left(-b\right) + \sqrt{{\left(\sqrt[3]{\mathsf{fma}\left(b, b, c \cdot \left(-\color{blue}{a \cdot 3}\right)\right)}\right)}^{3}}}{3 \cdot a} \]

      16. distribute-rgt-neg-in 66.6%

          \[\leadsto \frac{\left(-b\right) + \sqrt{{\left(\sqrt[3]{\mathsf{fma}\left(b, b, c \cdot \color{blue}{\left(a \cdot \left(-3\right)\right)}\right)}\right)}^{3}}}{3 \cdot a} \]

      17. metadata-eval 66.6%

          \[\leadsto \frac{\left(-b\right) + \sqrt{{\left(\sqrt[3]{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot \color{blue}{-3}\right)\right)}\right)}^{3}}}{3 \cdot a} \]

   4. Applied egg-rr 66.6%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)}\right)}^{3}}}}{3 \cdot a} \]

   5. Taylor expanded in b around -inf 91.5%

      \[\leadsto \frac{\left(-b\right) + \color{blue}{\left(-1 \cdot b + 1.5 \cdot \frac{a \cdot c}{b}\right)}}{3 \cdot a} \]
   6. Step-by-step derivation

      1. +-commutative 91.5%

         \[\leadsto \frac{\left(-b\right) + \color{blue}{\left(1.5 \cdot \frac{a \cdot c}{b} + -1 \cdot b\right)}}{3 \cdot a} \]

      2. mul-1-neg 91.5%

         \[\leadsto \frac{\left(-b\right) + \left(1.5 \cdot \frac{a \cdot c}{b} + \color{blue}{\left(-b\right)}\right)}{3 \cdot a} \]

      3. unsub-neg 91.5%

         \[\leadsto \frac{\left(-b\right) + \color{blue}{\left(1.5 \cdot \frac{a \cdot c}{b} - b\right)}}{3 \cdot a} \]

      4. associate-/l* 92.8%

         \[\leadsto \frac{\left(-b\right) + \left(1.5 \cdot \color{blue}{\frac{a}{\frac{b}{c}}} - b\right)}{3 \cdot a} \]

      5. associate-*r/ 92.8%

         \[\leadsto \frac{\left(-b\right) + \left(\color{blue}{\frac{1.5 \cdot a}{\frac{b}{c}}} - b\right)}{3 \cdot a} \]

      6. *-commutative 92.8%

         \[\leadsto \frac{\left(-b\right) + \left(\frac{\color{blue}{a \cdot 1.5}}{\frac{b}{c}} - b\right)}{3 \cdot a} \]

      7. associate-*r/ 92.8%

         \[\leadsto \frac{\left(-b\right) + \left(\color{blue}{a \cdot \frac{1.5}{\frac{b}{c}}} - b\right)}{3 \cdot a} \]

      8. associate-/r/ 92.8%

         \[\leadsto \frac{\left(-b\right) + \left(a \cdot \color{blue}{\left(\frac{1.5}{b} \cdot c\right)} - b\right)}{3 \cdot a} \]

   7. Simplified 92.8%

      \[\leadsto \frac{\left(-b\right) + \color{blue}{\left(a \cdot \left(\frac{1.5}{b} \cdot c\right) - b\right)}}{3 \cdot a} \]

   if -2e-14 < b < 3.69999999999999983e-85

   1. Initial program 82.2%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around 0 69.1%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{-3 \cdot \left(a \cdot c\right)}}}{3 \cdot a} \]
   4. Step-by-step derivation

      1. *-commutative 69.1%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\left(a \cdot c\right) \cdot -3}}}{3 \cdot a} \]

      2. associate-*l* 69.2%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{a \cdot \left(c \cdot -3\right)}}}{3 \cdot a} \]

   5. Simplified 69.2%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{a \cdot \left(c \cdot -3\right)}}}{3 \cdot a} \]

   if 3.69999999999999983e-85 < b

   1. Initial program 15.5%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf 84.7%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
   4. Step-by-step derivation

      1. associate-*r/ 84.7%

         \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b}} \]

   5. Simplified 84.7%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-14}:\\ \;\;\;\;\frac{\left(a \cdot \left(c \cdot \frac{1.5}{b}\right) - b\right) - b}{a \cdot 3}\\ \mathbf{elif}\;b \leq 3.7 \cdot 10^{-85}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.0% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{\left(a \cdot \left(c \cdot \frac{1.5}{b}\right) - b\right) - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e-310)
   (/ (- (- (* a (* c (/ 1.5 b))) b) b) (* a 3.0))
   (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = (((a * (c * (1.5 / b))) - b) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-5d-310)) then
        tmp = (((a * (c * (1.5d0 / b))) - b) - b) / (a * 3.0d0)
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = (((a * (c * (1.5 / b))) - b) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e-310:
		tmp = (((a * (c * (1.5 / b))) - b) - b) / (a * 3.0)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e-310)
		tmp = Float64(Float64(Float64(Float64(a * Float64(c * Float64(1.5 / b))) - b) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5e-310)
		tmp = (((a * (c * (1.5 / b))) - b) - b) / (a * 3.0);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-310], N[(N[(N[(N[(a * N[(c * N[(1.5 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 74.3%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. fma-neg 74.3%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(3 \cdot a\right) \cdot c\right)}}}{3 \cdot a} \]

      2. associate-*l* 74.3%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, -\color{blue}{3 \cdot \left(a \cdot c\right)}\right)}}{3 \cdot a} \]

      3. distribute-lft-neg-in 74.3%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(-3\right) \cdot \left(a \cdot c\right)}\right)}}{3 \cdot a} \]

      4. metadata-eval 74.3%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \color{blue}{-3} \cdot \left(a \cdot c\right)\right)}}{3 \cdot a} \]

      5. *-commutative 74.3%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(a \cdot c\right) \cdot -3}\right)}}{3 \cdot a} \]

      6. associate-*r* 74.3%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \color{blue}{a \cdot \left(c \cdot -3\right)}\right)}}{3 \cdot a} \]

      7. add-cube-cbrt 74.1%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\left(\sqrt[3]{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} \cdot \sqrt[3]{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}}}}{3 \cdot a} \]

      8. pow3 74.1%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}\right)}^{3}}}}{3 \cdot a} \]

      9. *-commutative 74.1%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\left(\sqrt[3]{\mathsf{fma}\left(b, b, a \cdot \color{blue}{\left(-3 \cdot c\right)}\right)}\right)}^{3}}}{3 \cdot a} \]

      10. associate-*r* 74.1%

          \[\leadsto \frac{\left(-b\right) + \sqrt{{\left(\sqrt[3]{\mathsf{fma}\left(b, b, \color{blue}{\left(a \cdot -3\right) \cdot c}\right)}\right)}^{3}}}{3 \cdot a} \]

      11. metadata-eval 74.1%

          \[\leadsto \frac{\left(-b\right) + \sqrt{{\left(\sqrt[3]{\mathsf{fma}\left(b, b, \left(a \cdot \color{blue}{\left(-3\right)}\right) \cdot c\right)}\right)}^{3}}}{3 \cdot a} \]

      12. distribute-rgt-neg-in 74.1%

          \[\leadsto \frac{\left(-b\right) + \sqrt{{\left(\sqrt[3]{\mathsf{fma}\left(b, b, \color{blue}{\left(-a \cdot 3\right)} \cdot c\right)}\right)}^{3}}}{3 \cdot a} \]

      13. *-commutative 74.1%

          \[\leadsto \frac{\left(-b\right) + \sqrt{{\left(\sqrt[3]{\mathsf{fma}\left(b, b, \left(-\color{blue}{3 \cdot a}\right) \cdot c\right)}\right)}^{3}}}{3 \cdot a} \]

      14. *-commutative 74.1%

          \[\leadsto \frac{\left(-b\right) + \sqrt{{\left(\sqrt[3]{\mathsf{fma}\left(b, b, \color{blue}{c \cdot \left(-3 \cdot a\right)}\right)}\right)}^{3}}}{3 \cdot a} \]

      15. *-commutative 74.1%

          \[\leadsto \frac{\left(-b\right) + \sqrt{{\left(\sqrt[3]{\mathsf{fma}\left(b, b, c \cdot \left(-\color{blue}{a \cdot 3}\right)\right)}\right)}^{3}}}{3 \cdot a} \]

      16. distribute-rgt-neg-in 74.1%

          \[\leadsto \frac{\left(-b\right) + \sqrt{{\left(\sqrt[3]{\mathsf{fma}\left(b, b, c \cdot \color{blue}{\left(a \cdot \left(-3\right)\right)}\right)}\right)}^{3}}}{3 \cdot a} \]

      17. metadata-eval 74.1%

          \[\leadsto \frac{\left(-b\right) + \sqrt{{\left(\sqrt[3]{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot \color{blue}{-3}\right)\right)}\right)}^{3}}}{3 \cdot a} \]

   4. Applied egg-rr 74.1%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)}\right)}^{3}}}}{3 \cdot a} \]

   5. Taylor expanded in b around -inf 69.4%

      \[\leadsto \frac{\left(-b\right) + \color{blue}{\left(-1 \cdot b + 1.5 \cdot \frac{a \cdot c}{b}\right)}}{3 \cdot a} \]
   6. Step-by-step derivation

      1. +-commutative 69.4%

         \[\leadsto \frac{\left(-b\right) + \color{blue}{\left(1.5 \cdot \frac{a \cdot c}{b} + -1 \cdot b\right)}}{3 \cdot a} \]

      2. mul-1-neg 69.4%

         \[\leadsto \frac{\left(-b\right) + \left(1.5 \cdot \frac{a \cdot c}{b} + \color{blue}{\left(-b\right)}\right)}{3 \cdot a} \]

      3. unsub-neg 69.4%

         \[\leadsto \frac{\left(-b\right) + \color{blue}{\left(1.5 \cdot \frac{a \cdot c}{b} - b\right)}}{3 \cdot a} \]

      4. associate-/l* 70.3%

         \[\leadsto \frac{\left(-b\right) + \left(1.5 \cdot \color{blue}{\frac{a}{\frac{b}{c}}} - b\right)}{3 \cdot a} \]

      5. associate-*r/ 70.3%

         \[\leadsto \frac{\left(-b\right) + \left(\color{blue}{\frac{1.5 \cdot a}{\frac{b}{c}}} - b\right)}{3 \cdot a} \]

      6. *-commutative 70.3%

         \[\leadsto \frac{\left(-b\right) + \left(\frac{\color{blue}{a \cdot 1.5}}{\frac{b}{c}} - b\right)}{3 \cdot a} \]

      7. associate-*r/ 70.3%

         \[\leadsto \frac{\left(-b\right) + \left(\color{blue}{a \cdot \frac{1.5}{\frac{b}{c}}} - b\right)}{3 \cdot a} \]

      8. associate-/r/ 70.3%

         \[\leadsto \frac{\left(-b\right) + \left(a \cdot \color{blue}{\left(\frac{1.5}{b} \cdot c\right)} - b\right)}{3 \cdot a} \]

   7. Simplified 70.3%

      \[\leadsto \frac{\left(-b\right) + \color{blue}{\left(a \cdot \left(\frac{1.5}{b} \cdot c\right) - b\right)}}{3 \cdot a} \]

   if -4.999999999999985e-310 < b

   1. Initial program 27.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf 69.8%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
   4. Step-by-step derivation

      1. associate-*r/ 69.8%

         \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b}} \]

   5. Simplified 69.8%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{\left(a \cdot \left(c \cdot \frac{1.5}{b}\right) - b\right) - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.0% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e-310)
   (+ (* -0.6666666666666666 (/ b a)) (* 0.5 (/ c b)))
   (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-5d-310)) then
        tmp = ((-0.6666666666666666d0) * (b / a)) + (0.5d0 * (c / b))
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e-310:
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b))
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e-310)
		tmp = Float64(Float64(-0.6666666666666666 * Float64(b / a)) + Float64(0.5 * Float64(c / b)));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5e-310)
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-310], N[(N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 74.3%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around -inf 70.2%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}} \]

   if -4.999999999999985e-310 < b

   1. Initial program 27.8%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf 69.8%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
   4. Step-by-step derivation

      1. associate-*r/ 69.8%

         \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b}} \]

   5. Simplified 69.8%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 66.4% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 9 \cdot 10^{-278}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{\frac{b}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 9e-278) (* b (/ -0.6666666666666666 a)) (/ -0.5 (/ b c))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 9e-278) {
		tmp = b * (-0.6666666666666666 / a);
	} else {
		tmp = -0.5 / (b / c);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 9d-278) then
        tmp = b * ((-0.6666666666666666d0) / a)
    else
        tmp = (-0.5d0) / (b / c)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 9e-278) {
		tmp = b * (-0.6666666666666666 / a);
	} else {
		tmp = -0.5 / (b / c);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 9e-278:
		tmp = b * (-0.6666666666666666 / a)
	else:
		tmp = -0.5 / (b / c)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 9e-278)
		tmp = Float64(b * Float64(-0.6666666666666666 / a));
	else
		tmp = Float64(-0.5 / Float64(b / c));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 9e-278)
		tmp = b * (-0.6666666666666666 / a);
	else
		tmp = -0.5 / (b / c);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 9e-278], N[(b * N[(-0.6666666666666666 / a), $MachinePrecision]), $MachinePrecision], N[(-0.5 / N[(b / c), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 8.9999999999999996e-278

   1. Initial program 73.3%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around -inf 68.7%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a}} \]
   4. Step-by-step derivation

      1. *-commutative 68.7%

         \[\leadsto \color{blue}{\frac{b}{a} \cdot -0.6666666666666666} \]

   5. Simplified 68.7%

      \[\leadsto \color{blue}{\frac{b}{a} \cdot -0.6666666666666666} \]
   6. Step-by-step derivation

      1. associate-*l/ 68.8%

         \[\leadsto \color{blue}{\frac{b \cdot -0.6666666666666666}{a}} \]

      2. clear-num 68.7%

         \[\leadsto \color{blue}{\frac{1}{\frac{a}{b \cdot -0.6666666666666666}}} \]

   7. Applied egg-rr 68.7%

      \[\leadsto \color{blue}{\frac{1}{\frac{a}{b \cdot -0.6666666666666666}}} \]

   8. Taylor expanded in a around 0 68.7%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a}} \]
   9. Step-by-step derivation

      1. associate-*r/ 68.8%

         \[\leadsto \color{blue}{\frac{-0.6666666666666666 \cdot b}{a}} \]

      2. *-commutative 68.8%

         \[\leadsto \frac{\color{blue}{b \cdot -0.6666666666666666}}{a} \]

      3. associate-*r/ 68.8%

         \[\leadsto \color{blue}{b \cdot \frac{-0.6666666666666666}{a}} \]

   10. Simplified 68.8%

       \[\leadsto \color{blue}{b \cdot \frac{-0.6666666666666666}{a}} \]

   if 8.9999999999999996e-278 < b

   1. Initial program 28.2%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf 71.0%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
   4. Step-by-step derivation

      1. associate-*r/ 71.0%

         \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b}} \]

      2. associate-/l* 69.7%

         \[\leadsto \color{blue}{\frac{-0.5}{\frac{b}{c}}} \]

   5. Simplified 69.7%

      \[\leadsto \color{blue}{\frac{-0.5}{\frac{b}{c}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 9 \cdot 10^{-278}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{\frac{b}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 66.5% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 9 \cdot 10^{-278}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{\frac{b}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 9e-278) (/ b (* a -1.5)) (/ -0.5 (/ b c))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 9e-278) {
		tmp = b / (a * -1.5);
	} else {
		tmp = -0.5 / (b / c);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 9d-278) then
        tmp = b / (a * (-1.5d0))
    else
        tmp = (-0.5d0) / (b / c)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 9e-278) {
		tmp = b / (a * -1.5);
	} else {
		tmp = -0.5 / (b / c);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 9e-278:
		tmp = b / (a * -1.5)
	else:
		tmp = -0.5 / (b / c)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 9e-278)
		tmp = Float64(b / Float64(a * -1.5));
	else
		tmp = Float64(-0.5 / Float64(b / c));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 9e-278)
		tmp = b / (a * -1.5);
	else
		tmp = -0.5 / (b / c);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 9e-278], N[(b / N[(a * -1.5), $MachinePrecision]), $MachinePrecision], N[(-0.5 / N[(b / c), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 8.9999999999999996e-278

   1. Initial program 73.3%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around -inf 68.7%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a}} \]
   4. Step-by-step derivation

      1. *-commutative 68.7%

         \[\leadsto \color{blue}{\frac{b}{a} \cdot -0.6666666666666666} \]

   5. Simplified 68.7%

      \[\leadsto \color{blue}{\frac{b}{a} \cdot -0.6666666666666666} \]
   6. Step-by-step derivation

      1. associate-*l/ 68.8%

         \[\leadsto \color{blue}{\frac{b \cdot -0.6666666666666666}{a}} \]

      2. clear-num 68.7%

         \[\leadsto \color{blue}{\frac{1}{\frac{a}{b \cdot -0.6666666666666666}}} \]

   7. Applied egg-rr 68.7%

      \[\leadsto \color{blue}{\frac{1}{\frac{a}{b \cdot -0.6666666666666666}}} \]

   8. Taylor expanded in a around 0 68.7%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a}} \]
   9. Step-by-step derivation

      1. associate-*r/ 68.8%

         \[\leadsto \color{blue}{\frac{-0.6666666666666666 \cdot b}{a}} \]

      2. *-commutative 68.8%

         \[\leadsto \frac{\color{blue}{b \cdot -0.6666666666666666}}{a} \]

      3. associate-*r/ 68.8%

         \[\leadsto \color{blue}{b \cdot \frac{-0.6666666666666666}{a}} \]

   10. Simplified 68.8%

       \[\leadsto \color{blue}{b \cdot \frac{-0.6666666666666666}{a}} \]
   11. Step-by-step derivation

       1. clear-num 68.7%

          \[\leadsto b \cdot \color{blue}{\frac{1}{\frac{a}{-0.6666666666666666}}} \]

       2. un-div-inv 68.7%

          \[\leadsto \color{blue}{\frac{b}{\frac{a}{-0.6666666666666666}}} \]

       3. div-inv 68.8%

          \[\leadsto \frac{b}{\color{blue}{a \cdot \frac{1}{-0.6666666666666666}}} \]

       4. metadata-eval 68.8%

          \[\leadsto \frac{b}{a \cdot \color{blue}{-1.5}} \]

   12. Applied egg-rr 68.8%

       \[\leadsto \color{blue}{\frac{b}{a \cdot -1.5}} \]

   if 8.9999999999999996e-278 < b

   1. Initial program 28.2%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf 71.0%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
   4. Step-by-step derivation

      1. associate-*r/ 71.0%

         \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b}} \]

      2. associate-/l* 69.7%

         \[\leadsto \color{blue}{\frac{-0.5}{\frac{b}{c}}} \]

   5. Simplified 69.7%

      \[\leadsto \color{blue}{\frac{-0.5}{\frac{b}{c}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 9 \cdot 10^{-278}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{\frac{b}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 66.8% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 9 \cdot 10^{-278}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 9e-278) (/ b (* a -1.5)) (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 9e-278) {
		tmp = b / (a * -1.5);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 9d-278) then
        tmp = b / (a * (-1.5d0))
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 9e-278) {
		tmp = b / (a * -1.5);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 9e-278:
		tmp = b / (a * -1.5)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 9e-278)
		tmp = Float64(b / Float64(a * -1.5));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 9e-278)
		tmp = b / (a * -1.5);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 9e-278], N[(b / N[(a * -1.5), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 8.9999999999999996e-278

   1. Initial program 73.3%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around -inf 68.7%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a}} \]
   4. Step-by-step derivation

      1. *-commutative 68.7%

         \[\leadsto \color{blue}{\frac{b}{a} \cdot -0.6666666666666666} \]

   5. Simplified 68.7%

      \[\leadsto \color{blue}{\frac{b}{a} \cdot -0.6666666666666666} \]
   6. Step-by-step derivation

      1. associate-*l/ 68.8%

         \[\leadsto \color{blue}{\frac{b \cdot -0.6666666666666666}{a}} \]

      2. clear-num 68.7%

         \[\leadsto \color{blue}{\frac{1}{\frac{a}{b \cdot -0.6666666666666666}}} \]

   7. Applied egg-rr 68.7%

      \[\leadsto \color{blue}{\frac{1}{\frac{a}{b \cdot -0.6666666666666666}}} \]

   8. Taylor expanded in a around 0 68.7%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a}} \]
   9. Step-by-step derivation

      1. associate-*r/ 68.8%

         \[\leadsto \color{blue}{\frac{-0.6666666666666666 \cdot b}{a}} \]

      2. *-commutative 68.8%

         \[\leadsto \frac{\color{blue}{b \cdot -0.6666666666666666}}{a} \]

      3. associate-*r/ 68.8%

         \[\leadsto \color{blue}{b \cdot \frac{-0.6666666666666666}{a}} \]

   10. Simplified 68.8%

       \[\leadsto \color{blue}{b \cdot \frac{-0.6666666666666666}{a}} \]
   11. Step-by-step derivation

       1. clear-num 68.7%

          \[\leadsto b \cdot \color{blue}{\frac{1}{\frac{a}{-0.6666666666666666}}} \]

       2. un-div-inv 68.7%

          \[\leadsto \color{blue}{\frac{b}{\frac{a}{-0.6666666666666666}}} \]

       3. div-inv 68.8%

          \[\leadsto \frac{b}{\color{blue}{a \cdot \frac{1}{-0.6666666666666666}}} \]

       4. metadata-eval 68.8%

          \[\leadsto \frac{b}{a \cdot \color{blue}{-1.5}} \]

   12. Applied egg-rr 68.8%

       \[\leadsto \color{blue}{\frac{b}{a \cdot -1.5}} \]

   if 8.9999999999999996e-278 < b

   1. Initial program 28.2%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf 71.0%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
   4. Step-by-step derivation

      1. associate-*r/ 71.0%

         \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b}} \]

   5. Simplified 71.0%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 9 \cdot 10^{-278}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 66.8% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 9 \cdot 10^{-278}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 9e-278) (/ (* b -0.6666666666666666) a) (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 9e-278) {
		tmp = (b * -0.6666666666666666) / a;
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 9d-278) then
        tmp = (b * (-0.6666666666666666d0)) / a
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 9e-278) {
		tmp = (b * -0.6666666666666666) / a;
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 9e-278:
		tmp = (b * -0.6666666666666666) / a
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 9e-278)
		tmp = Float64(Float64(b * -0.6666666666666666) / a);
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 9e-278)
		tmp = (b * -0.6666666666666666) / a;
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 9e-278], N[(N[(b * -0.6666666666666666), $MachinePrecision] / a), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 8.9999999999999996e-278

   1. Initial program 73.3%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. fma-neg 73.3%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(3 \cdot a\right) \cdot c\right)}}}{3 \cdot a} \]

      2. associate-*l* 73.3%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, -\color{blue}{3 \cdot \left(a \cdot c\right)}\right)}}{3 \cdot a} \]

      3. distribute-lft-neg-in 73.3%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(-3\right) \cdot \left(a \cdot c\right)}\right)}}{3 \cdot a} \]

      4. metadata-eval 73.3%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \color{blue}{-3} \cdot \left(a \cdot c\right)\right)}}{3 \cdot a} \]

      5. *-commutative 73.3%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(a \cdot c\right) \cdot -3}\right)}}{3 \cdot a} \]

      6. associate-*r* 73.3%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\mathsf{fma}\left(b, b, \color{blue}{a \cdot \left(c \cdot -3\right)}\right)}}{3 \cdot a} \]

      7. add-cube-cbrt 73.1%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\left(\sqrt[3]{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} \cdot \sqrt[3]{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}\right) \cdot \sqrt[3]{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}}}}{3 \cdot a} \]

      8. pow3 73.0%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)}\right)}^{3}}}}{3 \cdot a} \]

      9. *-commutative 73.0%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\left(\sqrt[3]{\mathsf{fma}\left(b, b, a \cdot \color{blue}{\left(-3 \cdot c\right)}\right)}\right)}^{3}}}{3 \cdot a} \]

      10. associate-*r* 73.0%

          \[\leadsto \frac{\left(-b\right) + \sqrt{{\left(\sqrt[3]{\mathsf{fma}\left(b, b, \color{blue}{\left(a \cdot -3\right) \cdot c}\right)}\right)}^{3}}}{3 \cdot a} \]

      11. metadata-eval 73.0%

          \[\leadsto \frac{\left(-b\right) + \sqrt{{\left(\sqrt[3]{\mathsf{fma}\left(b, b, \left(a \cdot \color{blue}{\left(-3\right)}\right) \cdot c\right)}\right)}^{3}}}{3 \cdot a} \]

      12. distribute-rgt-neg-in 73.0%

          \[\leadsto \frac{\left(-b\right) + \sqrt{{\left(\sqrt[3]{\mathsf{fma}\left(b, b, \color{blue}{\left(-a \cdot 3\right)} \cdot c\right)}\right)}^{3}}}{3 \cdot a} \]

      13. *-commutative 73.0%

          \[\leadsto \frac{\left(-b\right) + \sqrt{{\left(\sqrt[3]{\mathsf{fma}\left(b, b, \left(-\color{blue}{3 \cdot a}\right) \cdot c\right)}\right)}^{3}}}{3 \cdot a} \]

      14. *-commutative 73.0%

          \[\leadsto \frac{\left(-b\right) + \sqrt{{\left(\sqrt[3]{\mathsf{fma}\left(b, b, \color{blue}{c \cdot \left(-3 \cdot a\right)}\right)}\right)}^{3}}}{3 \cdot a} \]

      15. *-commutative 73.0%

          \[\leadsto \frac{\left(-b\right) + \sqrt{{\left(\sqrt[3]{\mathsf{fma}\left(b, b, c \cdot \left(-\color{blue}{a \cdot 3}\right)\right)}\right)}^{3}}}{3 \cdot a} \]

      16. distribute-rgt-neg-in 73.0%

          \[\leadsto \frac{\left(-b\right) + \sqrt{{\left(\sqrt[3]{\mathsf{fma}\left(b, b, c \cdot \color{blue}{\left(a \cdot \left(-3\right)\right)}\right)}\right)}^{3}}}{3 \cdot a} \]

      17. metadata-eval 73.0%

          \[\leadsto \frac{\left(-b\right) + \sqrt{{\left(\sqrt[3]{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot \color{blue}{-3}\right)\right)}\right)}^{3}}}{3 \cdot a} \]

   4. Applied egg-rr 73.0%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)}\right)}^{3}}}}{3 \cdot a} \]

   5. Taylor expanded in b around -inf 68.7%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a}} \]
   6. Step-by-step derivation

      1. associate-*r/ 68.8%

         \[\leadsto \color{blue}{\frac{-0.6666666666666666 \cdot b}{a}} \]

      2. *-commutative 68.8%

         \[\leadsto \frac{\color{blue}{b \cdot -0.6666666666666666}}{a} \]

   7. Simplified 68.8%

      \[\leadsto \color{blue}{\frac{b \cdot -0.6666666666666666}{a}} \]

   if 8.9999999999999996e-278 < b

   1. Initial program 28.2%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Add Preprocessing

   3. Taylor expanded in b around inf 71.0%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
   4. Step-by-step derivation

      1. associate-*r/ 71.0%

         \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b}} \]

   5. Simplified 71.0%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 9 \cdot 10^{-278}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 34.3% accurate, 23.2× speedup

Math

\[\begin{array}{l} \\ b \cdot \frac{-0.6666666666666666}{a} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (* b (/ -0.6666666666666666 a)))

C

double code(double a, double b, double c) {
	return b * (-0.6666666666666666 / a);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = b * ((-0.6666666666666666d0) / a)
end function

Java

public static double code(double a, double b, double c) {
	return b * (-0.6666666666666666 / a);
}

Python

def code(a, b, c):
	return b * (-0.6666666666666666 / a)

Julia

function code(a, b, c)
	return Float64(b * Float64(-0.6666666666666666 / a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = b * (-0.6666666666666666 / a);
end

Wolfram

code[a_, b_, c_] := N[(b * N[(-0.6666666666666666 / a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.5%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
2. Add Preprocessing

3. Taylor expanded in b around -inf 38.3%

   \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a}} \]
4. Step-by-step derivation

   1. *-commutative 38.3%

      \[\leadsto \color{blue}{\frac{b}{a} \cdot -0.6666666666666666} \]

5. Simplified 38.3%

   \[\leadsto \color{blue}{\frac{b}{a} \cdot -0.6666666666666666} \]
6. Step-by-step derivation

   1. associate-*l/ 38.3%

      \[\leadsto \color{blue}{\frac{b \cdot -0.6666666666666666}{a}} \]

   2. clear-num 38.2%

      \[\leadsto \color{blue}{\frac{1}{\frac{a}{b \cdot -0.6666666666666666}}} \]

7. Applied egg-rr 38.2%

   \[\leadsto \color{blue}{\frac{1}{\frac{a}{b \cdot -0.6666666666666666}}} \]

8. Taylor expanded in a around 0 38.3%

   \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a}} \]
9. Step-by-step derivation

   1. associate-*r/ 38.3%

      \[\leadsto \color{blue}{\frac{-0.6666666666666666 \cdot b}{a}} \]

   2. *-commutative 38.3%

      \[\leadsto \frac{\color{blue}{b \cdot -0.6666666666666666}}{a} \]

   3. associate-*r/ 38.3%

      \[\leadsto \color{blue}{b \cdot \frac{-0.6666666666666666}{a}} \]

10. Simplified 38.3%

    \[\leadsto \color{blue}{b \cdot \frac{-0.6666666666666666}{a}} \]

11. Final simplification 38.3%

    \[\leadsto b \cdot \frac{-0.6666666666666666}{a} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024020 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))