Cubic critical

Percentage Accurate: 51.6% → 85.4%

Time: 15.4s

Alternatives: 14

Speedup: 16.4×

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: 51.6% 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: 85.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.2 \cdot 10^{+143}:\\ \;\;\;\;\frac{-2}{\frac{a \cdot 3}{b}}\\ \mathbf{elif}\;b \leq 1.95 \cdot 10^{-62}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 3\right) \cdot c} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.2e+143)
   (/ -2.0 (/ (* a 3.0) b))
   (if (<= b 1.95e-62)
     (/ (- (sqrt (- (* b b) (* (* a 3.0) c))) b) (* a 3.0))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.2e+143) {
		tmp = -2.0 / ((a * 3.0) / b);
	} else if (b <= 1.95e-62) {
		tmp = (sqrt(((b * b) - ((a * 3.0) * 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 <= (-3.2d+143)) then
        tmp = (-2.0d0) / ((a * 3.0d0) / b)
    else if (b <= 1.95d-62) then
        tmp = (sqrt(((b * b) - ((a * 3.0d0) * 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 <= -3.2e+143) {
		tmp = -2.0 / ((a * 3.0) / b);
	} else if (b <= 1.95e-62) {
		tmp = (Math.sqrt(((b * b) - ((a * 3.0) * c))) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -3.2e+143:
		tmp = -2.0 / ((a * 3.0) / b)
	elif b <= 1.95e-62:
		tmp = (math.sqrt(((b * b) - ((a * 3.0) * c))) - b) / (a * 3.0)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.2e+143)
		tmp = Float64(-2.0 / Float64(Float64(a * 3.0) / b));
	elseif (b <= 1.95e-62)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(a * 3.0) * 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 <= -3.2e+143)
		tmp = -2.0 / ((a * 3.0) / b);
	elseif (b <= 1.95e-62)
		tmp = (sqrt(((b * b) - ((a * 3.0) * c))) - b) / (a * 3.0);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.2e+143], N[(-2.0 / N[(N[(a * 3.0), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.95e-62], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(a * 3.0), $MachinePrecision] * 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 < -3.20000000000000016e143

   1. Initial program 48.7%

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

   4. Applied egg-rr 56.9%

      \[\leadsto \color{blue}{\left(b - \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -3\right)}\right)\right) \cdot \frac{1}{a \cdot -3}} \]
   5. Step-by-step derivation

      1. associate-/r* 57.0%

         \[\leadsto \left(b - \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -3\right)}\right)\right) \cdot \color{blue}{\frac{\frac{1}{a}}{-3}} \]

   6. Simplified 57.0%

      \[\leadsto \color{blue}{\left(b - \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -3\right)}\right)\right) \cdot \frac{\frac{1}{a}}{-3}} \]

   7. Taylor expanded in b around -inf 94.8%

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

      1. add-exp-log 46.0%

         \[\leadsto \color{blue}{e^{\log \left(\left(2 \cdot b\right) \cdot \frac{\frac{1}{a}}{-3}\right)}} \]

      2. *-commutative 46.0%

         \[\leadsto e^{\log \left(\color{blue}{\left(b \cdot 2\right)} \cdot \frac{\frac{1}{a}}{-3}\right)} \]

      3. div-inv 46.0%

         \[\leadsto e^{\log \left(\left(b \cdot 2\right) \cdot \color{blue}{\left(\frac{1}{a} \cdot \frac{1}{-3}\right)}\right)} \]

      4. metadata-eval 46.0%

         \[\leadsto e^{\log \left(\left(b \cdot 2\right) \cdot \left(\frac{1}{a} \cdot \color{blue}{-0.3333333333333333}\right)\right)} \]

   9. Applied egg-rr 46.0%

      \[\leadsto \color{blue}{e^{\log \left(\left(b \cdot 2\right) \cdot \left(\frac{1}{a} \cdot -0.3333333333333333\right)\right)}} \]
   10. Step-by-step derivation

       1. rem-exp-log 94.5%

          \[\leadsto \color{blue}{\left(b \cdot 2\right) \cdot \left(\frac{1}{a} \cdot -0.3333333333333333\right)} \]

       2. associate-*r* 94.4%

          \[\leadsto \color{blue}{\left(\left(b \cdot 2\right) \cdot \frac{1}{a}\right) \cdot -0.3333333333333333} \]

       3. metadata-eval 94.4%

          \[\leadsto \left(\left(b \cdot \color{blue}{\left(--2\right)}\right) \cdot \frac{1}{a}\right) \cdot -0.3333333333333333 \]

       4. distribute-rgt-neg-in 94.4%

          \[\leadsto \left(\color{blue}{\left(-b \cdot -2\right)} \cdot \frac{1}{a}\right) \cdot -0.3333333333333333 \]

       5. distribute-lft-neg-in 94.4%

          \[\leadsto \color{blue}{\left(-\left(b \cdot -2\right) \cdot \frac{1}{a}\right)} \cdot -0.3333333333333333 \]

       6. div-inv 94.6%

          \[\leadsto \left(-\color{blue}{\frac{b \cdot -2}{a}}\right) \cdot -0.3333333333333333 \]

       7. metadata-eval 94.6%

          \[\leadsto \left(-\frac{b \cdot -2}{a}\right) \cdot \color{blue}{\frac{1}{-3}} \]

       8. div-inv 94.8%

          \[\leadsto \color{blue}{\frac{-\frac{b \cdot -2}{a}}{-3}} \]

       9. metadata-eval 94.8%

          \[\leadsto \frac{-\frac{b \cdot -2}{a}}{\color{blue}{-3}} \]

       10. frac-2neg 94.8%

           \[\leadsto \color{blue}{\frac{\frac{b \cdot -2}{a}}{3}} \]

       11. associate-/l/ 94.8%

           \[\leadsto \color{blue}{\frac{b \cdot -2}{3 \cdot a}} \]

       12. *-commutative 94.8%

           \[\leadsto \frac{\color{blue}{-2 \cdot b}}{3 \cdot a} \]

       13. associate-/l* 94.8%

           \[\leadsto \color{blue}{\frac{-2}{\frac{3 \cdot a}{b}}} \]

       14. *-commutative 94.8%

           \[\leadsto \frac{-2}{\frac{\color{blue}{a \cdot 3}}{b}} \]

   11. Applied egg-rr 94.8%

       \[\leadsto \color{blue}{\frac{-2}{\frac{a \cdot 3}{b}}} \]

   if -3.20000000000000016e143 < b < 1.9500000000000002e-62

   1. Initial program 85.5%

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

   if 1.9500000000000002e-62 < b

   1. Initial program 20.1%

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

      1. fma-neg 20.1%

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

      2. associate-*l* 20.1%

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

      3. distribute-lft-neg-in 20.1%

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

      4. metadata-eval 20.1%

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

      5. *-commutative 20.1%

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

      6. associate-*r* 20.1%

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

      7. add-cube-cbrt 14.7%

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

      8. pow3 15.7%

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

      9. associate-*r* 15.7%

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

      10. *-commutative 15.7%

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

      11. metadata-eval 15.7%

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

      12. distribute-lft-neg-in 15.7%

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

      13. associate-*l* 15.7%

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

      14. *-commutative 15.7%

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

      15. distribute-rgt-neg-in 15.7%

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

      16. *-commutative 15.7%

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

      17. distribute-rgt-neg-in 15.7%

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

      18. metadata-eval 15.7%

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

   4. Applied egg-rr 15.7%

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

   5. Taylor expanded in b around inf 83.0%

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

      1. associate-*r/ 83.0%

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

   7. Simplified 83.0%

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

4. Final simplification 85.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.2 \cdot 10^{+143}:\\ \;\;\;\;\frac{-2}{\frac{a \cdot 3}{b}}\\ \mathbf{elif}\;b \leq 1.95 \cdot 10^{-62}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 3\right) \cdot c} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 80.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.9 \cdot 10^{-58}:\\ \;\;\;\;\frac{\left(\frac{a}{\frac{b}{c}} \cdot 1.5 - b\right) - b}{a \cdot 3}\\ \mathbf{elif}\;b \leq 3.6 \cdot 10^{-62}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{b + \sqrt{-3 \cdot \left(a \cdot c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.9e-58)
   (/ (- (- (* (/ a (/ b c)) 1.5) b) b) (* a 3.0))
   (if (<= b 3.6e-62)
     (* 0.3333333333333333 (/ (+ b (sqrt (* -3.0 (* a c)))) a))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.9e-58) {
		tmp = ((((a / (b / c)) * 1.5) - b) - b) / (a * 3.0);
	} else if (b <= 3.6e-62) {
		tmp = 0.3333333333333333 * ((b + sqrt((-3.0 * (a * c)))) / 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 <= (-2.9d-58)) then
        tmp = ((((a / (b / c)) * 1.5d0) - b) - b) / (a * 3.0d0)
    else if (b <= 3.6d-62) then
        tmp = 0.3333333333333333d0 * ((b + sqrt(((-3.0d0) * (a * c)))) / 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 <= -2.9e-58) {
		tmp = ((((a / (b / c)) * 1.5) - b) - b) / (a * 3.0);
	} else if (b <= 3.6e-62) {
		tmp = 0.3333333333333333 * ((b + Math.sqrt((-3.0 * (a * c)))) / a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.9e-58:
		tmp = ((((a / (b / c)) * 1.5) - b) - b) / (a * 3.0)
	elif b <= 3.6e-62:
		tmp = 0.3333333333333333 * ((b + math.sqrt((-3.0 * (a * c)))) / a)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.9e-58)
		tmp = Float64(Float64(Float64(Float64(Float64(a / Float64(b / c)) * 1.5) - b) - b) / Float64(a * 3.0));
	elseif (b <= 3.6e-62)
		tmp = Float64(0.3333333333333333 * Float64(Float64(b + sqrt(Float64(-3.0 * Float64(a * c)))) / 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 <= -2.9e-58)
		tmp = ((((a / (b / c)) * 1.5) - b) - b) / (a * 3.0);
	elseif (b <= 3.6e-62)
		tmp = 0.3333333333333333 * ((b + sqrt((-3.0 * (a * c)))) / a);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.9e-58], N[(N[(N[(N[(N[(a / N[(b / c), $MachinePrecision]), $MachinePrecision] * 1.5), $MachinePrecision] - b), $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 3.6e-62], N[(0.3333333333333333 * N[(N[(b + N[Sqrt[N[(-3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.8999999999999999e-58

   1. Initial program 73.5%

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

      1. fma-neg 73.5%

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

      2. associate-*l* 73.5%

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

      3. distribute-lft-neg-in 73.5%

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

      4. metadata-eval 73.5%

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

      5. *-commutative 73.5%

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

      6. associate-*r* 73.5%

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

      7. add-cube-cbrt 73.3%

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

      8. pow3 73.4%

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

      9. associate-*r* 73.4%

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

      10. *-commutative 73.4%

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

      11. metadata-eval 73.4%

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

      12. distribute-lft-neg-in 73.4%

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

      13. associate-*l* 73.4%

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

      14. *-commutative 73.4%

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

      15. distribute-rgt-neg-in 73.4%

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

      16. *-commutative 73.4%

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

      17. distribute-rgt-neg-in 73.4%

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

      18. metadata-eval 73.4%

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

   4. Applied egg-rr 73.4%

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

   5. Taylor expanded in b around -inf 84.6%

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

      1. +-commutative 84.6%

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

      2. mul-1-neg 84.6%

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

      3. unsub-neg 84.6%

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

      4. *-commutative 84.6%

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

      5. associate-/l* 85.8%

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

   7. Simplified 85.8%

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

   if -2.8999999999999999e-58 < b < 3.6e-62

   1. Initial program 80.9%

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

   3. Taylor expanded in b around 0 72.1%

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

      1. associate-*r* 72.3%

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

      2. *-commutative 72.3%

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

      3. *-commutative 72.3%

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

   5. Simplified 72.3%

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

      1. sqrt-prod 37.8%

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

   7. Applied egg-rr 37.8%

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

      1. *-commutative 37.8%

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

   9. Simplified 37.8%

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

       1. expm1-log1p-u 21.6%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(-b\right) + \sqrt{a \cdot -3} \cdot \sqrt{c}}{3 \cdot a}\right)\right)} \]

       2. expm1-udef 1.0%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\left(-b\right) + \sqrt{a \cdot -3} \cdot \sqrt{c}}{3 \cdot a}\right)} - 1} \]

       3. add-sqr-sqrt 0.5%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{-b} \cdot \sqrt{-b}} + \sqrt{a \cdot -3} \cdot \sqrt{c}}{3 \cdot a}\right)} - 1 \]

       4. sqrt-unprod 1.0%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{\left(-b\right) \cdot \left(-b\right)}} + \sqrt{a \cdot -3} \cdot \sqrt{c}}{3 \cdot a}\right)} - 1 \]

       5. sqr-neg 1.0%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{\sqrt{\color{blue}{b \cdot b}} + \sqrt{a \cdot -3} \cdot \sqrt{c}}{3 \cdot a}\right)} - 1 \]

       6. sqrt-unprod 0.5%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{b} \cdot \sqrt{b}} + \sqrt{a \cdot -3} \cdot \sqrt{c}}{3 \cdot a}\right)} - 1 \]

       7. add-sqr-sqrt 1.0%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{b} + \sqrt{a \cdot -3} \cdot \sqrt{c}}{3 \cdot a}\right)} - 1 \]

       8. sqrt-unprod 19.3%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{b + \color{blue}{\sqrt{\left(a \cdot -3\right) \cdot c}}}{3 \cdot a}\right)} - 1 \]

       9. *-commutative 19.3%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{b + \sqrt{\left(a \cdot -3\right) \cdot c}}{\color{blue}{a \cdot 3}}\right)} - 1 \]

   11. Applied egg-rr 19.3%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{b + \sqrt{\left(a \cdot -3\right) \cdot c}}{a \cdot 3}\right)} - 1} \]
   12. Step-by-step derivation

       1. expm1-def 56.9%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{b + \sqrt{\left(a \cdot -3\right) \cdot c}}{a \cdot 3}\right)\right)} \]

       2. expm1-log1p 71.0%

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

       3. *-rgt-identity 71.0%

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

       4. times-frac 70.7%

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

       5. metadata-eval 70.7%

          \[\leadsto \frac{b + \sqrt{\left(a \cdot -3\right) \cdot c}}{a} \cdot \color{blue}{0.3333333333333333} \]

       6. *-commutative 70.7%

          \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{b + \sqrt{\left(a \cdot -3\right) \cdot c}}{a}} \]

       7. associate-*l* 70.7%

          \[\leadsto 0.3333333333333333 \cdot \frac{b + \sqrt{\color{blue}{a \cdot \left(-3 \cdot c\right)}}}{a} \]

       8. *-commutative 70.7%

          \[\leadsto 0.3333333333333333 \cdot \frac{b + \sqrt{a \cdot \color{blue}{\left(c \cdot -3\right)}}}{a} \]

   13. Simplified 70.7%

       \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{b + \sqrt{a \cdot \left(c \cdot -3\right)}}{a}} \]

   14. Taylor expanded in a around 0 70.6%

       \[\leadsto 0.3333333333333333 \cdot \frac{b + \sqrt{\color{blue}{-3 \cdot \left(a \cdot c\right)}}}{a} \]

   if 3.6e-62 < b

   1. Initial program 20.1%

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

      1. fma-neg 20.1%

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

      2. associate-*l* 20.1%

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

      3. distribute-lft-neg-in 20.1%

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

      4. metadata-eval 20.1%

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

      5. *-commutative 20.1%

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

      6. associate-*r* 20.1%

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

      7. add-cube-cbrt 14.7%

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

      8. pow3 15.7%

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

      9. associate-*r* 15.7%

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

      10. *-commutative 15.7%

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

      11. metadata-eval 15.7%

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

      12. distribute-lft-neg-in 15.7%

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

      13. associate-*l* 15.7%

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

      14. *-commutative 15.7%

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

      15. distribute-rgt-neg-in 15.7%

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

      16. *-commutative 15.7%

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

      17. distribute-rgt-neg-in 15.7%

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

      18. metadata-eval 15.7%

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

   4. Applied egg-rr 15.7%

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

   5. Taylor expanded in b around inf 83.0%

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

      1. associate-*r/ 83.0%

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

   7. Simplified 83.0%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.9 \cdot 10^{-58}:\\ \;\;\;\;\frac{\left(\frac{a}{\frac{b}{c}} \cdot 1.5 - b\right) - b}{a \cdot 3}\\ \mathbf{elif}\;b \leq 3.6 \cdot 10^{-62}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{b + \sqrt{-3 \cdot \left(a \cdot c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.6 \cdot 10^{-58}:\\ \;\;\;\;\frac{\left(\frac{a}{\frac{b}{c}} \cdot 1.5 - b\right) - b}{a \cdot 3}\\ \mathbf{elif}\;b \leq 10^{-59}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{b + \sqrt{a \cdot \left(c \cdot -3\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.6e-58)
   (/ (- (- (* (/ a (/ b c)) 1.5) b) b) (* a 3.0))
   (if (<= b 1e-59)
     (* 0.3333333333333333 (/ (+ b (sqrt (* a (* c -3.0)))) a))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.6e-58) {
		tmp = ((((a / (b / c)) * 1.5) - b) - b) / (a * 3.0);
	} else if (b <= 1e-59) {
		tmp = 0.3333333333333333 * ((b + sqrt((a * (c * -3.0)))) / 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 <= (-2.6d-58)) then
        tmp = ((((a / (b / c)) * 1.5d0) - b) - b) / (a * 3.0d0)
    else if (b <= 1d-59) then
        tmp = 0.3333333333333333d0 * ((b + sqrt((a * (c * (-3.0d0))))) / 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 <= -2.6e-58) {
		tmp = ((((a / (b / c)) * 1.5) - b) - b) / (a * 3.0);
	} else if (b <= 1e-59) {
		tmp = 0.3333333333333333 * ((b + Math.sqrt((a * (c * -3.0)))) / a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.6e-58:
		tmp = ((((a / (b / c)) * 1.5) - b) - b) / (a * 3.0)
	elif b <= 1e-59:
		tmp = 0.3333333333333333 * ((b + math.sqrt((a * (c * -3.0)))) / a)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.6e-58)
		tmp = Float64(Float64(Float64(Float64(Float64(a / Float64(b / c)) * 1.5) - b) - b) / Float64(a * 3.0));
	elseif (b <= 1e-59)
		tmp = Float64(0.3333333333333333 * Float64(Float64(b + sqrt(Float64(a * Float64(c * -3.0)))) / 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 <= -2.6e-58)
		tmp = ((((a / (b / c)) * 1.5) - b) - b) / (a * 3.0);
	elseif (b <= 1e-59)
		tmp = 0.3333333333333333 * ((b + sqrt((a * (c * -3.0)))) / a);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.6e-58], N[(N[(N[(N[(N[(a / N[(b / c), $MachinePrecision]), $MachinePrecision] * 1.5), $MachinePrecision] - b), $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1e-59], N[(0.3333333333333333 * N[(N[(b + N[Sqrt[N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.60000000000000007e-58

   1. Initial program 73.5%

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

      1. fma-neg 73.5%

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

      2. associate-*l* 73.5%

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

      3. distribute-lft-neg-in 73.5%

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

      4. metadata-eval 73.5%

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

      5. *-commutative 73.5%

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

      6. associate-*r* 73.5%

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

      7. add-cube-cbrt 73.3%

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

      8. pow3 73.4%

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

      9. associate-*r* 73.4%

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

      10. *-commutative 73.4%

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

      11. metadata-eval 73.4%

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

      12. distribute-lft-neg-in 73.4%

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

      13. associate-*l* 73.4%

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

      14. *-commutative 73.4%

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

      15. distribute-rgt-neg-in 73.4%

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

      16. *-commutative 73.4%

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

      17. distribute-rgt-neg-in 73.4%

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

      18. metadata-eval 73.4%

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

   4. Applied egg-rr 73.4%

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

   5. Taylor expanded in b around -inf 84.6%

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

      1. +-commutative 84.6%

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

      2. mul-1-neg 84.6%

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

      3. unsub-neg 84.6%

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

      4. *-commutative 84.6%

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

      5. associate-/l* 85.8%

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

   7. Simplified 85.8%

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

   if -2.60000000000000007e-58 < b < 1e-59

   1. Initial program 80.9%

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

   3. Taylor expanded in b around 0 72.1%

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

      1. associate-*r* 72.3%

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

      2. *-commutative 72.3%

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

      3. *-commutative 72.3%

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

   5. Simplified 72.3%

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

      1. sqrt-prod 37.8%

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

   7. Applied egg-rr 37.8%

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

      1. *-commutative 37.8%

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

   9. Simplified 37.8%

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

       1. expm1-log1p-u 21.6%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(-b\right) + \sqrt{a \cdot -3} \cdot \sqrt{c}}{3 \cdot a}\right)\right)} \]

       2. expm1-udef 1.0%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\left(-b\right) + \sqrt{a \cdot -3} \cdot \sqrt{c}}{3 \cdot a}\right)} - 1} \]

       3. add-sqr-sqrt 0.5%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{-b} \cdot \sqrt{-b}} + \sqrt{a \cdot -3} \cdot \sqrt{c}}{3 \cdot a}\right)} - 1 \]

       4. sqrt-unprod 1.0%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{\left(-b\right) \cdot \left(-b\right)}} + \sqrt{a \cdot -3} \cdot \sqrt{c}}{3 \cdot a}\right)} - 1 \]

       5. sqr-neg 1.0%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{\sqrt{\color{blue}{b \cdot b}} + \sqrt{a \cdot -3} \cdot \sqrt{c}}{3 \cdot a}\right)} - 1 \]

       6. sqrt-unprod 0.5%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{b} \cdot \sqrt{b}} + \sqrt{a \cdot -3} \cdot \sqrt{c}}{3 \cdot a}\right)} - 1 \]

       7. add-sqr-sqrt 1.0%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{b} + \sqrt{a \cdot -3} \cdot \sqrt{c}}{3 \cdot a}\right)} - 1 \]

       8. sqrt-unprod 19.3%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{b + \color{blue}{\sqrt{\left(a \cdot -3\right) \cdot c}}}{3 \cdot a}\right)} - 1 \]

       9. *-commutative 19.3%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{b + \sqrt{\left(a \cdot -3\right) \cdot c}}{\color{blue}{a \cdot 3}}\right)} - 1 \]

   11. Applied egg-rr 19.3%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{b + \sqrt{\left(a \cdot -3\right) \cdot c}}{a \cdot 3}\right)} - 1} \]
   12. Step-by-step derivation

       1. expm1-def 56.9%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{b + \sqrt{\left(a \cdot -3\right) \cdot c}}{a \cdot 3}\right)\right)} \]

       2. expm1-log1p 71.0%

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

       3. *-rgt-identity 71.0%

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

       4. times-frac 70.7%

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

       5. metadata-eval 70.7%

          \[\leadsto \frac{b + \sqrt{\left(a \cdot -3\right) \cdot c}}{a} \cdot \color{blue}{0.3333333333333333} \]

       6. *-commutative 70.7%

          \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{b + \sqrt{\left(a \cdot -3\right) \cdot c}}{a}} \]

       7. associate-*l* 70.7%

          \[\leadsto 0.3333333333333333 \cdot \frac{b + \sqrt{\color{blue}{a \cdot \left(-3 \cdot c\right)}}}{a} \]

       8. *-commutative 70.7%

          \[\leadsto 0.3333333333333333 \cdot \frac{b + \sqrt{a \cdot \color{blue}{\left(c \cdot -3\right)}}}{a} \]

   13. Simplified 70.7%

       \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{b + \sqrt{a \cdot \left(c \cdot -3\right)}}{a}} \]

   if 1e-59 < b

   1. Initial program 20.1%

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

      1. fma-neg 20.1%

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

      2. associate-*l* 20.1%

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

      3. distribute-lft-neg-in 20.1%

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

      4. metadata-eval 20.1%

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

      5. *-commutative 20.1%

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

      6. associate-*r* 20.1%

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

      7. add-cube-cbrt 14.7%

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

      8. pow3 15.7%

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

      9. associate-*r* 15.7%

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

      10. *-commutative 15.7%

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

      11. metadata-eval 15.7%

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

      12. distribute-lft-neg-in 15.7%

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

      13. associate-*l* 15.7%

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

      14. *-commutative 15.7%

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

      15. distribute-rgt-neg-in 15.7%

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

      16. *-commutative 15.7%

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

      17. distribute-rgt-neg-in 15.7%

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

      18. metadata-eval 15.7%

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

   4. Applied egg-rr 15.7%

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

   5. Taylor expanded in b around inf 83.0%

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

      1. associate-*r/ 83.0%

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

   7. Simplified 83.0%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.6 \cdot 10^{-58}:\\ \;\;\;\;\frac{\left(\frac{a}{\frac{b}{c}} \cdot 1.5 - b\right) - b}{a \cdot 3}\\ \mathbf{elif}\;b \leq 10^{-59}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{b + \sqrt{a \cdot \left(c \cdot -3\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.8 \cdot 10^{-58}:\\ \;\;\;\;\frac{\left(\frac{a}{\frac{b}{c}} \cdot 1.5 - b\right) - b}{a \cdot 3}\\ \mathbf{elif}\;b \leq 7 \cdot 10^{-60}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{b + \sqrt{c \cdot \left(a \cdot -3\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.8e-58)
   (/ (- (- (* (/ a (/ b c)) 1.5) b) b) (* a 3.0))
   (if (<= b 7e-60)
     (* 0.3333333333333333 (/ (+ b (sqrt (* c (* a -3.0)))) a))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.8e-58) {
		tmp = ((((a / (b / c)) * 1.5) - b) - b) / (a * 3.0);
	} else if (b <= 7e-60) {
		tmp = 0.3333333333333333 * ((b + sqrt((c * (a * -3.0)))) / 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 <= (-2.8d-58)) then
        tmp = ((((a / (b / c)) * 1.5d0) - b) - b) / (a * 3.0d0)
    else if (b <= 7d-60) then
        tmp = 0.3333333333333333d0 * ((b + sqrt((c * (a * (-3.0d0))))) / 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 <= -2.8e-58) {
		tmp = ((((a / (b / c)) * 1.5) - b) - b) / (a * 3.0);
	} else if (b <= 7e-60) {
		tmp = 0.3333333333333333 * ((b + Math.sqrt((c * (a * -3.0)))) / a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.8e-58:
		tmp = ((((a / (b / c)) * 1.5) - b) - b) / (a * 3.0)
	elif b <= 7e-60:
		tmp = 0.3333333333333333 * ((b + math.sqrt((c * (a * -3.0)))) / a)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.8e-58)
		tmp = Float64(Float64(Float64(Float64(Float64(a / Float64(b / c)) * 1.5) - b) - b) / Float64(a * 3.0));
	elseif (b <= 7e-60)
		tmp = Float64(0.3333333333333333 * Float64(Float64(b + sqrt(Float64(c * Float64(a * -3.0)))) / 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 <= -2.8e-58)
		tmp = ((((a / (b / c)) * 1.5) - b) - b) / (a * 3.0);
	elseif (b <= 7e-60)
		tmp = 0.3333333333333333 * ((b + sqrt((c * (a * -3.0)))) / a);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.8e-58], N[(N[(N[(N[(N[(a / N[(b / c), $MachinePrecision]), $MachinePrecision] * 1.5), $MachinePrecision] - b), $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 7e-60], N[(0.3333333333333333 * N[(N[(b + N[Sqrt[N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.8000000000000001e-58

   1. Initial program 73.5%

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

      1. fma-neg 73.5%

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

      2. associate-*l* 73.5%

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

      3. distribute-lft-neg-in 73.5%

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

      4. metadata-eval 73.5%

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

      5. *-commutative 73.5%

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

      6. associate-*r* 73.5%

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

      7. add-cube-cbrt 73.3%

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

      8. pow3 73.4%

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

      9. associate-*r* 73.4%

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

      10. *-commutative 73.4%

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

      11. metadata-eval 73.4%

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

      12. distribute-lft-neg-in 73.4%

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

      13. associate-*l* 73.4%

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

      14. *-commutative 73.4%

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

      15. distribute-rgt-neg-in 73.4%

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

      16. *-commutative 73.4%

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

      17. distribute-rgt-neg-in 73.4%

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

      18. metadata-eval 73.4%

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

   4. Applied egg-rr 73.4%

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

   5. Taylor expanded in b around -inf 84.6%

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

      1. +-commutative 84.6%

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

      2. mul-1-neg 84.6%

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

      3. unsub-neg 84.6%

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

      4. *-commutative 84.6%

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

      5. associate-/l* 85.8%

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

   7. Simplified 85.8%

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

   if -2.8000000000000001e-58 < b < 6.99999999999999952e-60

   1. Initial program 80.9%

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

   3. Taylor expanded in b around 0 72.1%

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

      1. associate-*r* 72.3%

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

      2. *-commutative 72.3%

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

      3. *-commutative 72.3%

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

   5. Simplified 72.3%

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

      1. sqrt-prod 37.8%

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

   7. Applied egg-rr 37.8%

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

      1. *-commutative 37.8%

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

   9. Simplified 37.8%

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

       1. *-un-lft-identity 37.8%

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

       2. times-frac 37.6%

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

       3. metadata-eval 37.6%

          \[\leadsto \color{blue}{0.3333333333333333} \cdot \frac{\left(-b\right) + \sqrt{a \cdot -3} \cdot \sqrt{c}}{a} \]

       4. add-sqr-sqrt 20.4%

          \[\leadsto 0.3333333333333333 \cdot \frac{\color{blue}{\sqrt{-b} \cdot \sqrt{-b}} + \sqrt{a \cdot -3} \cdot \sqrt{c}}{a} \]

       5. sqrt-unprod 37.8%

          \[\leadsto 0.3333333333333333 \cdot \frac{\color{blue}{\sqrt{\left(-b\right) \cdot \left(-b\right)}} + \sqrt{a \cdot -3} \cdot \sqrt{c}}{a} \]

       6. sqr-neg 37.8%

          \[\leadsto 0.3333333333333333 \cdot \frac{\sqrt{\color{blue}{b \cdot b}} + \sqrt{a \cdot -3} \cdot \sqrt{c}}{a} \]

       7. sqrt-unprod 17.5%

          \[\leadsto 0.3333333333333333 \cdot \frac{\color{blue}{\sqrt{b} \cdot \sqrt{b}} + \sqrt{a \cdot -3} \cdot \sqrt{c}}{a} \]

       8. add-sqr-sqrt 37.6%

          \[\leadsto 0.3333333333333333 \cdot \frac{\color{blue}{b} + \sqrt{a \cdot -3} \cdot \sqrt{c}}{a} \]

       9. sqrt-unprod 70.7%

          \[\leadsto 0.3333333333333333 \cdot \frac{b + \color{blue}{\sqrt{\left(a \cdot -3\right) \cdot c}}}{a} \]

   11. Applied egg-rr 70.7%

       \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{b + \sqrt{\left(a \cdot -3\right) \cdot c}}{a}} \]

   if 6.99999999999999952e-60 < b

   1. Initial program 20.1%

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

      1. fma-neg 20.1%

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

      2. associate-*l* 20.1%

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

      3. distribute-lft-neg-in 20.1%

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

      4. metadata-eval 20.1%

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

      5. *-commutative 20.1%

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

      6. associate-*r* 20.1%

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

      7. add-cube-cbrt 14.7%

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

      8. pow3 15.7%

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

      9. associate-*r* 15.7%

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

      10. *-commutative 15.7%

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

      11. metadata-eval 15.7%

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

      12. distribute-lft-neg-in 15.7%

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

      13. associate-*l* 15.7%

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

      14. *-commutative 15.7%

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

      15. distribute-rgt-neg-in 15.7%

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

      16. *-commutative 15.7%

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

      17. distribute-rgt-neg-in 15.7%

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

      18. metadata-eval 15.7%

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

   4. Applied egg-rr 15.7%

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

   5. Taylor expanded in b around inf 83.0%

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

      1. associate-*r/ 83.0%

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

   7. Simplified 83.0%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.8 \cdot 10^{-58}:\\ \;\;\;\;\frac{\left(\frac{a}{\frac{b}{c}} \cdot 1.5 - b\right) - b}{a \cdot 3}\\ \mathbf{elif}\;b \leq 7 \cdot 10^{-60}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{b + \sqrt{c \cdot \left(a \cdot -3\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.9 \cdot 10^{-58}:\\ \;\;\;\;\frac{\left(\frac{a}{\frac{b}{c}} \cdot 1.5 - b\right) - b}{a \cdot 3}\\ \mathbf{elif}\;b \leq 5.5 \cdot 10^{-61}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.9e-58)
   (/ (- (- (* (/ a (/ b c)) 1.5) b) b) (* a 3.0))
   (if (<= b 5.5e-61)
     (/ (- (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 <= -2.9e-58) {
		tmp = ((((a / (b / c)) * 1.5) - b) - b) / (a * 3.0);
	} else if (b <= 5.5e-61) {
		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 <= (-2.9d-58)) then
        tmp = ((((a / (b / c)) * 1.5d0) - b) - b) / (a * 3.0d0)
    else if (b <= 5.5d-61) 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 <= -2.9e-58) {
		tmp = ((((a / (b / c)) * 1.5) - b) - b) / (a * 3.0);
	} else if (b <= 5.5e-61) {
		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 <= -2.9e-58:
		tmp = ((((a / (b / c)) * 1.5) - b) - b) / (a * 3.0)
	elif b <= 5.5e-61:
		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 <= -2.9e-58)
		tmp = Float64(Float64(Float64(Float64(Float64(a / Float64(b / c)) * 1.5) - b) - b) / Float64(a * 3.0));
	elseif (b <= 5.5e-61)
		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 <= -2.9e-58)
		tmp = ((((a / (b / c)) * 1.5) - b) - b) / (a * 3.0);
	elseif (b <= 5.5e-61)
		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, -2.9e-58], N[(N[(N[(N[(N[(a / N[(b / c), $MachinePrecision]), $MachinePrecision] * 1.5), $MachinePrecision] - b), $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5.5e-61], 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 < -2.8999999999999999e-58

   1. Initial program 73.5%

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

      1. fma-neg 73.5%

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

      2. associate-*l* 73.5%

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

      3. distribute-lft-neg-in 73.5%

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

      4. metadata-eval 73.5%

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

      5. *-commutative 73.5%

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

      6. associate-*r* 73.5%

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

      7. add-cube-cbrt 73.3%

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

      8. pow3 73.4%

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

      9. associate-*r* 73.4%

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

      10. *-commutative 73.4%

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

      11. metadata-eval 73.4%

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

      12. distribute-lft-neg-in 73.4%

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

      13. associate-*l* 73.4%

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

      14. *-commutative 73.4%

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

      15. distribute-rgt-neg-in 73.4%

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

      16. *-commutative 73.4%

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

      17. distribute-rgt-neg-in 73.4%

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

      18. metadata-eval 73.4%

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

   4. Applied egg-rr 73.4%

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

   5. Taylor expanded in b around -inf 84.6%

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

      1. +-commutative 84.6%

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

      2. mul-1-neg 84.6%

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

      3. unsub-neg 84.6%

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

      4. *-commutative 84.6%

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

      5. associate-/l* 85.8%

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

   7. Simplified 85.8%

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

   if -2.8999999999999999e-58 < b < 5.4999999999999997e-61

   1. Initial program 80.9%

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

   3. Taylor expanded in b around 0 72.1%

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

      1. associate-*r* 72.3%

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

      2. *-commutative 72.3%

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

      3. *-commutative 72.3%

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

   5. Simplified 72.3%

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

      1. +-commutative 72.3%

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

      2. unsub-neg 72.3%

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

   7. Applied egg-rr 72.3%

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

      1. associate-*r* 72.1%

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

      2. *-commutative 72.1%

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

      3. rem-square-sqrt 0.0%

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

      4. unpow2 0.0%

         \[\leadsto \frac{\sqrt{\left(a \cdot c\right) \cdot \color{blue}{{\left(\sqrt{-3}\right)}^{2}}} - b}{3 \cdot a} \]

      5. associate-*r* 0.0%

         \[\leadsto \frac{\sqrt{\color{blue}{a \cdot \left(c \cdot {\left(\sqrt{-3}\right)}^{2}\right)}} - b}{3 \cdot a} \]

      6. unpow2 0.0%

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

      7. rem-square-sqrt 72.2%

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

   9. Simplified 72.2%

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

   if 5.4999999999999997e-61 < b

   1. Initial program 20.1%

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

      1. fma-neg 20.1%

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

      2. associate-*l* 20.1%

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

      3. distribute-lft-neg-in 20.1%

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

      4. metadata-eval 20.1%

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

      5. *-commutative 20.1%

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

      6. associate-*r* 20.1%

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

      7. add-cube-cbrt 14.7%

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

      8. pow3 15.7%

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

      9. associate-*r* 15.7%

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

      10. *-commutative 15.7%

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

      11. metadata-eval 15.7%

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

      12. distribute-lft-neg-in 15.7%

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

      13. associate-*l* 15.7%

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

      14. *-commutative 15.7%

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

      15. distribute-rgt-neg-in 15.7%

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

      16. *-commutative 15.7%

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

      17. distribute-rgt-neg-in 15.7%

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

      18. metadata-eval 15.7%

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

   4. Applied egg-rr 15.7%

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

   5. Taylor expanded in b around inf 83.0%

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

      1. associate-*r/ 83.0%

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

   7. Simplified 83.0%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.9 \cdot 10^{-58}:\\ \;\;\;\;\frac{\left(\frac{a}{\frac{b}{c}} \cdot 1.5 - b\right) - b}{a \cdot 3}\\ \mathbf{elif}\;b \leq 5.5 \cdot 10^{-61}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.4 \cdot 10^{-58}:\\ \;\;\;\;\frac{\left(\frac{a}{\frac{b}{c}} \cdot 1.5 - b\right) - b}{a \cdot 3}\\ \mathbf{elif}\;b \leq 7.8 \cdot 10^{-62}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.4e-58)
   (/ (- (- (* (/ a (/ b c)) 1.5) b) b) (* a 3.0))
   (if (<= b 7.8e-62)
     (/ (- (sqrt (* 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 <= -2.4e-58) {
		tmp = ((((a / (b / c)) * 1.5) - b) - b) / (a * 3.0);
	} else if (b <= 7.8e-62) {
		tmp = (sqrt((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 <= (-2.4d-58)) then
        tmp = ((((a / (b / c)) * 1.5d0) - b) - b) / (a * 3.0d0)
    else if (b <= 7.8d-62) then
        tmp = (sqrt((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 <= -2.4e-58) {
		tmp = ((((a / (b / c)) * 1.5) - b) - b) / (a * 3.0);
	} else if (b <= 7.8e-62) {
		tmp = (Math.sqrt((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 <= -2.4e-58:
		tmp = ((((a / (b / c)) * 1.5) - b) - b) / (a * 3.0)
	elif b <= 7.8e-62:
		tmp = (math.sqrt((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 <= -2.4e-58)
		tmp = Float64(Float64(Float64(Float64(Float64(a / Float64(b / c)) * 1.5) - b) - b) / Float64(a * 3.0));
	elseif (b <= 7.8e-62)
		tmp = Float64(Float64(sqrt(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 <= -2.4e-58)
		tmp = ((((a / (b / c)) * 1.5) - b) - b) / (a * 3.0);
	elseif (b <= 7.8e-62)
		tmp = (sqrt((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, -2.4e-58], N[(N[(N[(N[(N[(a / N[(b / c), $MachinePrecision]), $MachinePrecision] * 1.5), $MachinePrecision] - b), $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 7.8e-62], N[(N[(N[Sqrt[N[(c * N[(a * -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 < -2.4000000000000001e-58

   1. Initial program 73.5%

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

      1. fma-neg 73.5%

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

      2. associate-*l* 73.5%

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

      3. distribute-lft-neg-in 73.5%

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

      4. metadata-eval 73.5%

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

      5. *-commutative 73.5%

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

      6. associate-*r* 73.5%

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

      7. add-cube-cbrt 73.3%

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

      8. pow3 73.4%

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

      9. associate-*r* 73.4%

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

      10. *-commutative 73.4%

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

      11. metadata-eval 73.4%

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

      12. distribute-lft-neg-in 73.4%

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

      13. associate-*l* 73.4%

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

      14. *-commutative 73.4%

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

      15. distribute-rgt-neg-in 73.4%

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

      16. *-commutative 73.4%

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

      17. distribute-rgt-neg-in 73.4%

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

      18. metadata-eval 73.4%

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

   4. Applied egg-rr 73.4%

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

   5. Taylor expanded in b around -inf 84.6%

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

      1. +-commutative 84.6%

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

      2. mul-1-neg 84.6%

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

      3. unsub-neg 84.6%

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

      4. *-commutative 84.6%

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

      5. associate-/l* 85.8%

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

   7. Simplified 85.8%

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

   if -2.4000000000000001e-58 < b < 7.8000000000000007e-62

   1. Initial program 80.9%

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

   3. Taylor expanded in b around 0 72.1%

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

      1. associate-*r* 72.3%

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

      2. *-commutative 72.3%

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

      3. *-commutative 72.3%

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

   5. Simplified 72.3%

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

      1. +-commutative 72.3%

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

      2. unsub-neg 72.3%

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

   7. Applied egg-rr 72.3%

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

   if 7.8000000000000007e-62 < b

   1. Initial program 20.1%

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

      1. fma-neg 20.1%

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

      2. associate-*l* 20.1%

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

      3. distribute-lft-neg-in 20.1%

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

      4. metadata-eval 20.1%

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

      5. *-commutative 20.1%

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

      6. associate-*r* 20.1%

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

      7. add-cube-cbrt 14.7%

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

      8. pow3 15.7%

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

      9. associate-*r* 15.7%

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

      10. *-commutative 15.7%

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

      11. metadata-eval 15.7%

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

      12. distribute-lft-neg-in 15.7%

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

      13. associate-*l* 15.7%

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

      14. *-commutative 15.7%

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

      15. distribute-rgt-neg-in 15.7%

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

      16. *-commutative 15.7%

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

      17. distribute-rgt-neg-in 15.7%

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

      18. metadata-eval 15.7%

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

   4. Applied egg-rr 15.7%

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

   5. Taylor expanded in b around inf 83.0%

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

      1. associate-*r/ 83.0%

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

   7. Simplified 83.0%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.4 \cdot 10^{-58}:\\ \;\;\;\;\frac{\left(\frac{a}{\frac{b}{c}} \cdot 1.5 - b\right) - b}{a \cdot 3}\\ \mathbf{elif}\;b \leq 7.8 \cdot 10^{-62}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.2% accurate, 12.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{-2}{\frac{a \cdot 3}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e-310) (/ -2.0 (/ (* a 3.0) b)) (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = -2.0 / ((a * 3.0) / 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 = (-2.0d0) / ((a * 3.0d0) / 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 = -2.0 / ((a * 3.0) / b);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e-310:
		tmp = -2.0 / ((a * 3.0) / b)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e-310)
		tmp = Float64(-2.0 / Float64(Float64(a * 3.0) / 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 = -2.0 / ((a * 3.0) / b);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-310], N[(-2.0 / N[(N[(a * 3.0), $MachinePrecision] / b), $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 78.4%

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

   4. Applied egg-rr 64.8%

      \[\leadsto \color{blue}{\left(b - \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -3\right)}\right)\right) \cdot \frac{1}{a \cdot -3}} \]
   5. Step-by-step derivation

      1. associate-/r* 64.8%

         \[\leadsto \left(b - \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -3\right)}\right)\right) \cdot \color{blue}{\frac{\frac{1}{a}}{-3}} \]

   6. Simplified 64.8%

      \[\leadsto \color{blue}{\left(b - \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -3\right)}\right)\right) \cdot \frac{\frac{1}{a}}{-3}} \]

   7. Taylor expanded in b around -inf 66.3%

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

      1. add-exp-log 28.0%

         \[\leadsto \color{blue}{e^{\log \left(\left(2 \cdot b\right) \cdot \frac{\frac{1}{a}}{-3}\right)}} \]

      2. *-commutative 28.0%

         \[\leadsto e^{\log \left(\color{blue}{\left(b \cdot 2\right)} \cdot \frac{\frac{1}{a}}{-3}\right)} \]

      3. div-inv 28.0%

         \[\leadsto e^{\log \left(\left(b \cdot 2\right) \cdot \color{blue}{\left(\frac{1}{a} \cdot \frac{1}{-3}\right)}\right)} \]

      4. metadata-eval 28.0%

         \[\leadsto e^{\log \left(\left(b \cdot 2\right) \cdot \left(\frac{1}{a} \cdot \color{blue}{-0.3333333333333333}\right)\right)} \]

   9. Applied egg-rr 28.0%

      \[\leadsto \color{blue}{e^{\log \left(\left(b \cdot 2\right) \cdot \left(\frac{1}{a} \cdot -0.3333333333333333\right)\right)}} \]
   10. Step-by-step derivation

       1. rem-exp-log 66.2%

          \[\leadsto \color{blue}{\left(b \cdot 2\right) \cdot \left(\frac{1}{a} \cdot -0.3333333333333333\right)} \]

       2. associate-*r* 66.2%

          \[\leadsto \color{blue}{\left(\left(b \cdot 2\right) \cdot \frac{1}{a}\right) \cdot -0.3333333333333333} \]

       3. metadata-eval 66.2%

          \[\leadsto \left(\left(b \cdot \color{blue}{\left(--2\right)}\right) \cdot \frac{1}{a}\right) \cdot -0.3333333333333333 \]

       4. distribute-rgt-neg-in 66.2%

          \[\leadsto \left(\color{blue}{\left(-b \cdot -2\right)} \cdot \frac{1}{a}\right) \cdot -0.3333333333333333 \]

       5. distribute-lft-neg-in 66.2%

          \[\leadsto \color{blue}{\left(-\left(b \cdot -2\right) \cdot \frac{1}{a}\right)} \cdot -0.3333333333333333 \]

       6. div-inv 66.2%

          \[\leadsto \left(-\color{blue}{\frac{b \cdot -2}{a}}\right) \cdot -0.3333333333333333 \]

       7. metadata-eval 66.2%

          \[\leadsto \left(-\frac{b \cdot -2}{a}\right) \cdot \color{blue}{\frac{1}{-3}} \]

       8. div-inv 66.4%

          \[\leadsto \color{blue}{\frac{-\frac{b \cdot -2}{a}}{-3}} \]

       9. metadata-eval 66.4%

          \[\leadsto \frac{-\frac{b \cdot -2}{a}}{\color{blue}{-3}} \]

       10. frac-2neg 66.4%

           \[\leadsto \color{blue}{\frac{\frac{b \cdot -2}{a}}{3}} \]

       11. associate-/l/ 66.4%

           \[\leadsto \color{blue}{\frac{b \cdot -2}{3 \cdot a}} \]

       12. *-commutative 66.4%

           \[\leadsto \frac{\color{blue}{-2 \cdot b}}{3 \cdot a} \]

       13. associate-/l* 66.4%

           \[\leadsto \color{blue}{\frac{-2}{\frac{3 \cdot a}{b}}} \]

       14. *-commutative 66.4%

           \[\leadsto \frac{-2}{\frac{\color{blue}{a \cdot 3}}{b}} \]

   11. Applied egg-rr 66.4%

       \[\leadsto \color{blue}{\frac{-2}{\frac{a \cdot 3}{b}}} \]

   if -4.999999999999985e-310 < b

   1. Initial program 31.9%

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

      1. fma-neg 31.9%

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

      2. associate-*l* 31.9%

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

      3. distribute-lft-neg-in 31.9%

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

      4. metadata-eval 31.9%

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

      5. *-commutative 31.9%

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

      6. associate-*r* 31.9%

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

      7. add-cube-cbrt 27.6%

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

      8. pow3 28.4%

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

      9. associate-*r* 28.3%

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

      10. *-commutative 28.3%

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

      11. metadata-eval 28.3%

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

      12. distribute-lft-neg-in 28.3%

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

      13. associate-*l* 28.4%

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

      14. *-commutative 28.4%

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

      15. distribute-rgt-neg-in 28.4%

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

      16. *-commutative 28.4%

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

      17. distribute-rgt-neg-in 28.4%

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

      18. metadata-eval 28.4%

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

   4. Applied egg-rr 28.4%

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

   5. Taylor expanded in b around inf 68.0%

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

      1. associate-*r/ 68.0%

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

   7. Simplified 68.0%

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

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{-2}{\frac{a \cdot 3}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 68.2% accurate, 12.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2 \cdot 10^{-309}:\\ \;\;\;\;\frac{b \cdot -2}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 2e-309) (/ (* b -2.0) (* a 3.0)) (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 2e-309) {
		tmp = (b * -2.0) / (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-309) then
        tmp = (b * (-2.0d0)) / (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-309) {
		tmp = (b * -2.0) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 2e-309:
		tmp = (b * -2.0) / (a * 3.0)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 2e-309)
		tmp = Float64(Float64(b * -2.0) / 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-309)
		tmp = (b * -2.0) / (a * 3.0);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 2e-309], N[(N[(b * -2.0), $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 < 1.9999999999999988e-309

   1. Initial program 78.4%

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

   3. Taylor expanded in b around -inf 66.4%

      \[\leadsto \frac{\color{blue}{-2 \cdot b}}{3 \cdot a} \]
   4. Step-by-step derivation

      1. *-commutative 66.4%

         \[\leadsto \frac{\color{blue}{b \cdot -2}}{3 \cdot a} \]

   5. Simplified 66.4%

      \[\leadsto \frac{\color{blue}{b \cdot -2}}{3 \cdot a} \]

   if 1.9999999999999988e-309 < b

   1. Initial program 31.9%

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

      1. fma-neg 31.9%

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

      2. associate-*l* 31.9%

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

      3. distribute-lft-neg-in 31.9%

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

      4. metadata-eval 31.9%

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

      5. *-commutative 31.9%

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

      6. associate-*r* 31.9%

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

      7. add-cube-cbrt 27.6%

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

      8. pow3 28.4%

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

      9. associate-*r* 28.3%

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

      10. *-commutative 28.3%

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

      11. metadata-eval 28.3%

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

      12. distribute-lft-neg-in 28.3%

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

      13. associate-*l* 28.4%

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

      14. *-commutative 28.4%

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

      15. distribute-rgt-neg-in 28.4%

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

      16. *-commutative 28.4%

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

      17. distribute-rgt-neg-in 28.4%

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

      18. metadata-eval 28.4%

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

   4. Applied egg-rr 28.4%

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

   5. Taylor expanded in b around inf 68.0%

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

      1. associate-*r/ 68.0%

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

   7. Simplified 68.0%

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

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2 \cdot 10^{-309}:\\ \;\;\;\;\frac{b \cdot -2}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 48.2% accurate, 16.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{b}{a} \cdot -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e-310) (* (/ b a) -0.3333333333333333) (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = (b / a) * -0.3333333333333333;
	} else {
		tmp = -0.5 * (c / 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 = (b / a) * (-0.3333333333333333d0)
    else
        tmp = (-0.5d0) * (c / 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 = (b / a) * -0.3333333333333333;
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e-310:
		tmp = (b / a) * -0.3333333333333333
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e-310)
		tmp = Float64(Float64(b / a) * -0.3333333333333333);
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5e-310)
		tmp = (b / a) * -0.3333333333333333;
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-310], N[(N[(b / a), $MachinePrecision] * -0.3333333333333333), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 78.4%

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

   3. Taylor expanded in b around 0 44.1%

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

      1. associate-*r* 44.2%

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

      2. *-commutative 44.2%

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

      3. *-commutative 44.2%

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

   5. Simplified 44.2%

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

   6. Taylor expanded in b around inf 27.7%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{b}{a}} \]

   if -4.999999999999985e-310 < b

   1. Initial program 31.9%

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

   3. Taylor expanded in b around inf 68.0%

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

4. Final simplification 48.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{b}{a} \cdot -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 68.2% accurate, 16.4× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = b * (-0.6666666666666666 / a);
	} else {
		tmp = -0.5 * (c / 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 = b * ((-0.6666666666666666d0) / a)
    else
        tmp = (-0.5d0) * (c / 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 = b * (-0.6666666666666666 / a);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 78.4%

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

   4. Applied egg-rr 64.8%

      \[\leadsto \color{blue}{\left(b - \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -3\right)}\right)\right) \cdot \frac{1}{a \cdot -3}} \]
   5. Step-by-step derivation

      1. associate-/r* 64.8%

         \[\leadsto \left(b - \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -3\right)}\right)\right) \cdot \color{blue}{\frac{\frac{1}{a}}{-3}} \]

   6. Simplified 64.8%

      \[\leadsto \color{blue}{\left(b - \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -3\right)}\right)\right) \cdot \frac{\frac{1}{a}}{-3}} \]

   7. Taylor expanded in b around -inf 66.3%

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

      1. expm1-log1p-u 35.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(2 \cdot b\right) \cdot \frac{\frac{1}{a}}{-3}\right)\right)} \]

      2. expm1-udef 26.7%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(2 \cdot b\right) \cdot \frac{\frac{1}{a}}{-3}\right)} - 1} \]

      3. *-commutative 26.7%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\left(b \cdot 2\right)} \cdot \frac{\frac{1}{a}}{-3}\right)} - 1 \]

      4. div-inv 26.7%

         \[\leadsto e^{\mathsf{log1p}\left(\left(b \cdot 2\right) \cdot \color{blue}{\left(\frac{1}{a} \cdot \frac{1}{-3}\right)}\right)} - 1 \]

      5. metadata-eval 26.7%

         \[\leadsto e^{\mathsf{log1p}\left(\left(b \cdot 2\right) \cdot \left(\frac{1}{a} \cdot \color{blue}{-0.3333333333333333}\right)\right)} - 1 \]

   9. Applied egg-rr 26.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(b \cdot 2\right) \cdot \left(\frac{1}{a} \cdot -0.3333333333333333\right)\right)} - 1} \]
   10. Step-by-step derivation

       1. expm1-def 35.0%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(b \cdot 2\right) \cdot \left(\frac{1}{a} \cdot -0.3333333333333333\right)\right)\right)} \]

       2. expm1-log1p 66.2%

          \[\leadsto \color{blue}{\left(b \cdot 2\right) \cdot \left(\frac{1}{a} \cdot -0.3333333333333333\right)} \]

       3. associate-*l* 66.2%

          \[\leadsto \color{blue}{b \cdot \left(2 \cdot \left(\frac{1}{a} \cdot -0.3333333333333333\right)\right)} \]

       4. associate-*l/ 66.2%

          \[\leadsto b \cdot \left(2 \cdot \color{blue}{\frac{1 \cdot -0.3333333333333333}{a}}\right) \]

       5. metadata-eval 66.2%

          \[\leadsto b \cdot \left(2 \cdot \frac{\color{blue}{-0.3333333333333333}}{a}\right) \]

       6. associate-*r/ 66.2%

          \[\leadsto b \cdot \color{blue}{\frac{2 \cdot -0.3333333333333333}{a}} \]

       7. metadata-eval 66.2%

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

   11. Simplified 66.2%

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

   if -4.999999999999985e-310 < b

   1. Initial program 31.9%

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

   3. Taylor expanded in b around inf 68.0%

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

4. Final simplification 67.1%

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

Alternative 11: 68.2% accurate, 16.4× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = -0.6666666666666666 * (b / a);
	} else {
		tmp = -0.5 * (c / 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)
    else
        tmp = (-0.5d0) * (c / 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);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

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

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e-310)
		tmp = Float64(-0.6666666666666666 * Float64(b / a));
	else
		tmp = Float64(-0.5 * Float64(c / 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);
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 78.4%

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

   3. Taylor expanded in b around -inf 66.2%

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

      1. *-commutative 66.2%

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

   5. Simplified 66.2%

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

   if -4.999999999999985e-310 < b

   1. Initial program 31.9%

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

   3. Taylor expanded in b around inf 68.0%

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

4. Final simplification 67.1%

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

Alternative 12: 68.3% accurate, 16.4× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = -0.6666666666666666 * (b / 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 <= (-5d-310)) then
        tmp = (-0.6666666666666666d0) * (b / 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 <= -5e-310) {
		tmp = -0.6666666666666666 * (b / a);
	} 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)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e-310)
		tmp = Float64(-0.6666666666666666 * Float64(b / 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 <= -5e-310)
		tmp = -0.6666666666666666 * (b / a);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-310], N[(-0.6666666666666666 * N[(b / a), $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 78.4%

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

   3. Taylor expanded in b around -inf 66.2%

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

      1. *-commutative 66.2%

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

   5. Simplified 66.2%

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

   if -4.999999999999985e-310 < b

   1. Initial program 31.9%

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

      1. fma-neg 31.9%

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

      2. associate-*l* 31.9%

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

      3. distribute-lft-neg-in 31.9%

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

      4. metadata-eval 31.9%

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

      5. *-commutative 31.9%

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

      6. associate-*r* 31.9%

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

      7. add-cube-cbrt 27.6%

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

      8. pow3 28.4%

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

      9. associate-*r* 28.3%

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

      10. *-commutative 28.3%

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

      11. metadata-eval 28.3%

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

      12. distribute-lft-neg-in 28.3%

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

      13. associate-*l* 28.4%

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

      14. *-commutative 28.4%

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

      15. distribute-rgt-neg-in 28.4%

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

      16. *-commutative 28.4%

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

      17. distribute-rgt-neg-in 28.4%

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

      18. metadata-eval 28.4%

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

   4. Applied egg-rr 28.4%

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

   5. Taylor expanded in b around inf 68.0%

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

      1. associate-*r/ 68.0%

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

   7. Simplified 68.0%

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

4. Final simplification 67.1%

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

Alternative 13: 68.3% accurate, 16.4× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

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

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-310], 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 < -4.999999999999985e-310

   1. Initial program 78.4%

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

   3. Taylor expanded in b around -inf 66.2%

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

      1. *-commutative 66.2%

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

      2. associate-*l/ 66.3%

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

   5. Simplified 66.3%

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

   if -4.999999999999985e-310 < b

   1. Initial program 31.9%

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

      1. fma-neg 31.9%

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

      2. associate-*l* 31.9%

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

      3. distribute-lft-neg-in 31.9%

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

      4. metadata-eval 31.9%

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

      5. *-commutative 31.9%

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

      6. associate-*r* 31.9%

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

      7. add-cube-cbrt 27.6%

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

      8. pow3 28.4%

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

      9. associate-*r* 28.3%

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

      10. *-commutative 28.3%

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

      11. metadata-eval 28.3%

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

      12. distribute-lft-neg-in 28.3%

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

      13. associate-*l* 28.4%

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

      14. *-commutative 28.4%

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

      15. distribute-rgt-neg-in 28.4%

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

      16. *-commutative 28.4%

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

      17. distribute-rgt-neg-in 28.4%

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

      18. metadata-eval 28.4%

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

   4. Applied egg-rr 28.4%

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

   5. Taylor expanded in b around inf 68.0%

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

      1. associate-*r/ 68.0%

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

   7. Simplified 68.0%

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

4. Final simplification 67.2%

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

Alternative 14: 35.2% accurate, 23.2× speedup

Math

\[\begin{array}{l} \\ -0.5 \cdot \frac{c}{b} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (* -0.5 (/ c b)))

C

double code(double a, double b, double c) {
	return -0.5 * (c / b);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-0.5d0) * (c / b)
end function

Java

public static double code(double a, double b, double c) {
	return -0.5 * (c / b);
}

Python

def code(a, b, c):
	return -0.5 * (c / b)

Julia

function code(a, b, c)
	return Float64(-0.5 * Float64(c / b))
end

MATLAB

function tmp = code(a, b, c)
	tmp = -0.5 * (c / b);
end

Wolfram

code[a_, b_, c_] := N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 54.6%

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

3. Taylor expanded in b around inf 35.9%

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

4. Final simplification 35.9%

   \[\leadsto -0.5 \cdot \frac{c}{b} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024010 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))