Cubic critical

Percentage Accurate: 53.0% → 85.3%

Time: 12.9s

Alternatives: 11

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: 53.0% 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: 85.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5.8 \cdot 10^{+134}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 3.5 \cdot 10^{-116}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 3\right) \cdot c} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5.8e+134)
   (/ b (* a -1.5))
   (if (<= b 3.5e-116)
     (/ (- (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 <= -5.8e+134) {
		tmp = b / (a * -1.5);
	} else if (b <= 3.5e-116) {
		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 <= (-5.8d+134)) then
        tmp = b / (a * (-1.5d0))
    else if (b <= 3.5d-116) 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 <= -5.8e+134) {
		tmp = b / (a * -1.5);
	} else if (b <= 3.5e-116) {
		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 <= -5.8e+134:
		tmp = b / (a * -1.5)
	elif b <= 3.5e-116:
		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 <= -5.8e+134)
		tmp = Float64(b / Float64(a * -1.5));
	elseif (b <= 3.5e-116)
		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 <= -5.8e+134)
		tmp = b / (a * -1.5);
	elseif (b <= 3.5e-116)
		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, -5.8e+134], N[(b / N[(a * -1.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 3.5e-116], 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 < -5.80000000000000023e134

   1. Initial program 52.4%

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

   2. Taylor expanded in b around -inf 92.5%

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

      1. *-commutative 92.5%

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

   4. Simplified 92.5%

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

      1. metadata-eval 92.5%

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

      2. times-frac 92.9%

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

      3. *-commutative 92.9%

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

      4. associate-/l* 92.9%

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

      5. frac-2neg 92.9%

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

      6. add-sqr-sqrt 92.5%

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

      7. sqrt-unprod 52.6%

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

      8. sqr-neg 52.6%

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

      9. sqrt-unprod 0.0%

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

      10. add-sqr-sqrt 0.5%

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

      11. div-inv 0.5%

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

      12. add-sqr-sqrt 0.2%

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

      13. sqrt-unprod 38.2%

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

      14. *-commutative 38.2%

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

      15. *-commutative 38.2%

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

      16. swap-sqr 38.2%

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

      17. metadata-eval 38.2%

          \[\leadsto \frac{b}{-\sqrt{\left(a \cdot a\right) \cdot \color{blue}{9}} \cdot \frac{1}{-2}} \]

      18. metadata-eval 38.2%

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

      19. swap-sqr 38.2%

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

      20. sqrt-unprod 45.1%

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

      21. add-sqr-sqrt 92.9%

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

      22. metadata-eval 92.9%

          \[\leadsto \frac{b}{-\left(a \cdot -3\right) \cdot \color{blue}{-0.5}} \]

   6. Applied egg-rr 92.9%

      \[\leadsto \color{blue}{\frac{b}{-\left(a \cdot -3\right) \cdot -0.5}} \]
   7. Step-by-step derivation

      1. associate-*l* 92.9%

         \[\leadsto \frac{b}{-\color{blue}{a \cdot \left(-3 \cdot -0.5\right)}} \]

      2. metadata-eval 92.9%

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

      3. distribute-rgt-neg-in 92.9%

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

      4. metadata-eval 92.9%

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

   8. Simplified 92.9%

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

   if -5.80000000000000023e134 < b < 3.49999999999999984e-116

   1. Initial program 88.8%

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

   if 3.49999999999999984e-116 < b

   1. Initial program 17.9%

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

      1. fma-neg 17.9%

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

      2. associate-*l* 17.9%

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

      3. distribute-lft-neg-in 17.9%

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

      4. metadata-eval 17.9%

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

      5. *-commutative 17.9%

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

      6. associate-*r* 17.9%

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

      7. add-cube-cbrt 15.1%

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

      8. sqrt-prod 15.2%

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

   3. Applied egg-rr 15.2%

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

   4. Taylor expanded in b around inf 84.3%

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

      1. associate-*r/ 84.3%

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

      2. *-commutative 84.3%

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

   6. Simplified 84.3%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.8 \cdot 10^{+134}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 3.5 \cdot 10^{-116}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 3\right) \cdot c} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 2: 80.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.15 \cdot 10^{-69}:\\ \;\;\;\;\frac{\frac{b \cdot 2}{-3}}{a}\\ \mathbf{elif}\;b \leq 3.5 \cdot 10^{-116}:\\ \;\;\;\;\left(b + \sqrt{a \cdot \left(c \cdot -3\right)}\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.15e-69)
   (/ (/ (* b 2.0) -3.0) a)
   (if (<= b 3.5e-116)
     (* (+ b (sqrt (* a (* c -3.0)))) (/ 0.3333333333333333 a))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.15e-69) {
		tmp = ((b * 2.0) / -3.0) / a;
	} else if (b <= 3.5e-116) {
		tmp = (b + sqrt((a * (c * -3.0)))) * (0.3333333333333333 / 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 <= (-1.15d-69)) then
        tmp = ((b * 2.0d0) / (-3.0d0)) / a
    else if (b <= 3.5d-116) then
        tmp = (b + sqrt((a * (c * (-3.0d0))))) * (0.3333333333333333d0 / 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 <= -1.15e-69) {
		tmp = ((b * 2.0) / -3.0) / a;
	} else if (b <= 3.5e-116) {
		tmp = (b + Math.sqrt((a * (c * -3.0)))) * (0.3333333333333333 / a);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.15e-69:
		tmp = ((b * 2.0) / -3.0) / a
	elif b <= 3.5e-116:
		tmp = (b + math.sqrt((a * (c * -3.0)))) * (0.3333333333333333 / a)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.15e-69)
		tmp = Float64(Float64(Float64(b * 2.0) / -3.0) / a);
	elseif (b <= 3.5e-116)
		tmp = Float64(Float64(b + sqrt(Float64(a * Float64(c * -3.0)))) * Float64(0.3333333333333333 / 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 <= -1.15e-69)
		tmp = ((b * 2.0) / -3.0) / a;
	elseif (b <= 3.5e-116)
		tmp = (b + sqrt((a * (c * -3.0)))) * (0.3333333333333333 / a);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.15e-69], N[(N[(N[(b * 2.0), $MachinePrecision] / -3.0), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 3.5e-116], N[(N[(b + N[Sqrt[N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(0.3333333333333333 / a), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.15e-69

   1. Initial program 78.9%

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

   3. Applied egg-rr 8.1%

      \[\leadsto \color{blue}{\frac{1}{a} \cdot \left(\left(b + \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -3\right)}\right)\right) \cdot 0.3333333333333333\right)} \]

   5. Applied egg-rr 62.7%

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

   6. Taylor expanded in b around -inf 89.6%

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

      1. *-commutative 89.6%

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

   8. Simplified 89.6%

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

   if -1.15e-69 < b < 3.49999999999999984e-116

   1. Initial program 79.8%

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

   2. Taylor expanded in b around 0 77.7%

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

      1. associate-*r* 77.8%

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

      2. *-commutative 77.8%

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

      3. *-commutative 77.8%

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

   4. Simplified 77.8%

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

      1. add-cbrt-cube 57.2%

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

      2. pow3 57.2%

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

      3. sqrt-pow2 57.2%

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

      4. metadata-eval 57.2%

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

   6. Applied egg-rr 57.2%

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

      1. expm1-log1p-u 41.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(-b\right) + \sqrt[3]{{\left(c \cdot \left(a \cdot -3\right)\right)}^{1.5}}}{3 \cdot a}\right)\right)} \]

      2. expm1-udef 16.9%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\left(-b\right) + \sqrt[3]{{\left(c \cdot \left(a \cdot -3\right)\right)}^{1.5}}}{3 \cdot a}\right)} - 1} \]

   8. Applied egg-rr 23.2%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(b + \sqrt{c \cdot \left(a \cdot -3\right)}\right) \cdot \frac{0.3333333333333333}{a}\right)} - 1} \]
   9. Step-by-step derivation

      1. expm1-def 59.2%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(b + \sqrt{c \cdot \left(a \cdot -3\right)}\right) \cdot \frac{0.3333333333333333}{a}\right)\right)} \]

      2. expm1-log1p 77.1%

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

      3. *-commutative 77.1%

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

      4. *-commutative 77.1%

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

      5. associate-*l* 77.3%

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

      6. *-commutative 77.3%

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

   10. Simplified 77.3%

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

   if 3.49999999999999984e-116 < b

   1. Initial program 17.9%

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

      1. fma-neg 17.9%

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

      2. associate-*l* 17.9%

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

      3. distribute-lft-neg-in 17.9%

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

      4. metadata-eval 17.9%

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

      5. *-commutative 17.9%

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

      6. associate-*r* 17.9%

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

      7. add-cube-cbrt 15.1%

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

      8. sqrt-prod 15.2%

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

   3. Applied egg-rr 15.2%

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

   4. Taylor expanded in b around inf 84.3%

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

      1. associate-*r/ 84.3%

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

      2. *-commutative 84.3%

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

   6. Simplified 84.3%

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

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.15 \cdot 10^{-69}:\\ \;\;\;\;\frac{\frac{b \cdot 2}{-3}}{a}\\ \mathbf{elif}\;b \leq 3.5 \cdot 10^{-116}:\\ \;\;\;\;\left(b + \sqrt{a \cdot \left(c \cdot -3\right)}\right) \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 3: 80.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.7 \cdot 10^{-67}:\\ \;\;\;\;\frac{\frac{b \cdot 2}{-3}}{a}\\ \mathbf{elif}\;b \leq 3.5 \cdot 10^{-116}:\\ \;\;\;\;\frac{\frac{b + \sqrt{a \cdot \left(c \cdot -3\right)}}{a}}{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.7e-67)
   (/ (/ (* b 2.0) -3.0) a)
   (if (<= b 3.5e-116)
     (/ (/ (+ b (sqrt (* a (* c -3.0)))) a) 3.0)
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.7e-67) {
		tmp = ((b * 2.0) / -3.0) / a;
	} else if (b <= 3.5e-116) {
		tmp = ((b + sqrt((a * (c * -3.0)))) / 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 <= (-2.7d-67)) then
        tmp = ((b * 2.0d0) / (-3.0d0)) / a
    else if (b <= 3.5d-116) then
        tmp = ((b + sqrt((a * (c * (-3.0d0))))) / 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 <= -2.7e-67) {
		tmp = ((b * 2.0) / -3.0) / a;
	} else if (b <= 3.5e-116) {
		tmp = ((b + Math.sqrt((a * (c * -3.0)))) / a) / 3.0;
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.7e-67:
		tmp = ((b * 2.0) / -3.0) / a
	elif b <= 3.5e-116:
		tmp = ((b + math.sqrt((a * (c * -3.0)))) / a) / 3.0
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.7e-67)
		tmp = Float64(Float64(Float64(b * 2.0) / -3.0) / a);
	elseif (b <= 3.5e-116)
		tmp = Float64(Float64(Float64(b + sqrt(Float64(a * Float64(c * -3.0)))) / 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 <= -2.7e-67)
		tmp = ((b * 2.0) / -3.0) / a;
	elseif (b <= 3.5e-116)
		tmp = ((b + sqrt((a * (c * -3.0)))) / a) / 3.0;
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.7e-67], N[(N[(N[(b * 2.0), $MachinePrecision] / -3.0), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 3.5e-116], N[(N[(N[(b + N[Sqrt[N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] / 3.0), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.70000000000000016e-67

   1. Initial program 78.9%

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

   3. Applied egg-rr 8.1%

      \[\leadsto \color{blue}{\frac{1}{a} \cdot \left(\left(b + \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -3\right)}\right)\right) \cdot 0.3333333333333333\right)} \]

   5. Applied egg-rr 62.7%

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

   6. Taylor expanded in b around -inf 89.6%

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

      1. *-commutative 89.6%

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

   8. Simplified 89.6%

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

   if -2.70000000000000016e-67 < b < 3.49999999999999984e-116

   1. Initial program 79.8%

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

   2. Taylor expanded in b around 0 77.7%

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

      1. associate-*r* 77.8%

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

      2. *-commutative 77.8%

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

      3. *-commutative 77.8%

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

   4. Simplified 77.8%

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

      1. add-cbrt-cube 57.2%

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

      2. pow3 57.2%

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

      3. sqrt-pow2 57.2%

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

      4. metadata-eval 57.2%

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

   6. Applied egg-rr 57.2%

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

      1. add-cube-cbrt 56.6%

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

      2. *-commutative 56.6%

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

      3. times-frac 56.6%

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

   8. Applied egg-rr 76.1%

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

      1. associate-*l/ 76.1%

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

      2. associate-*r/ 76.0%

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

      3. unpow2 76.0%

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

      4. rem-3cbrt-lft 77.3%

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

      5. rem-square-sqrt 77.0%

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

      6. associate-*r/ 77.2%

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

      7. associate-*l/ 77.0%

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

      8. *-commutative 77.0%

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

      9. associate-*l/ 77.1%

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

   10. Simplified 77.4%

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

   if 3.49999999999999984e-116 < b

   1. Initial program 17.9%

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

      1. fma-neg 17.9%

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

      2. associate-*l* 17.9%

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

      3. distribute-lft-neg-in 17.9%

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

      4. metadata-eval 17.9%

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

      5. *-commutative 17.9%

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

      6. associate-*r* 17.9%

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

      7. add-cube-cbrt 15.1%

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

      8. sqrt-prod 15.2%

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

   3. Applied egg-rr 15.2%

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

   4. Taylor expanded in b around inf 84.3%

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

      1. associate-*r/ 84.3%

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

      2. *-commutative 84.3%

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

   6. Simplified 84.3%

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

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.7 \cdot 10^{-67}:\\ \;\;\;\;\frac{\frac{b \cdot 2}{-3}}{a}\\ \mathbf{elif}\;b \leq 3.5 \cdot 10^{-116}:\\ \;\;\;\;\frac{\frac{b + \sqrt{a \cdot \left(c \cdot -3\right)}}{a}}{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 4: 80.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.3 \cdot 10^{-53}:\\ \;\;\;\;\frac{\frac{b \cdot 2}{-3}}{a}\\ \mathbf{elif}\;b \leq 2.8 \cdot 10^{-116}:\\ \;\;\;\;\frac{\sqrt{-3 \cdot \left(a \cdot c\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.3e-53)
   (/ (/ (* b 2.0) -3.0) a)
   (if (<= b 2.8e-116)
     (/ (- (sqrt (* -3.0 (* a c))) b) (* a 3.0))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.3e-53) {
		tmp = ((b * 2.0) / -3.0) / a;
	} else if (b <= 2.8e-116) {
		tmp = (sqrt((-3.0 * (a * c))) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-3.3d-53)) then
        tmp = ((b * 2.0d0) / (-3.0d0)) / a
    else if (b <= 2.8d-116) then
        tmp = (sqrt(((-3.0d0) * (a * c))) - b) / (a * 3.0d0)
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.3e-53) {
		tmp = ((b * 2.0) / -3.0) / a;
	} else if (b <= 2.8e-116) {
		tmp = (Math.sqrt((-3.0 * (a * c))) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -3.3e-53:
		tmp = ((b * 2.0) / -3.0) / a
	elif b <= 2.8e-116:
		tmp = (math.sqrt((-3.0 * (a * c))) - b) / (a * 3.0)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.3e-53)
		tmp = Float64(Float64(Float64(b * 2.0) / -3.0) / a);
	elseif (b <= 2.8e-116)
		tmp = Float64(Float64(sqrt(Float64(-3.0 * Float64(a * c))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -3.3e-53)
		tmp = ((b * 2.0) / -3.0) / a;
	elseif (b <= 2.8e-116)
		tmp = (sqrt((-3.0 * (a * c))) - b) / (a * 3.0);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.3e-53], N[(N[(N[(b * 2.0), $MachinePrecision] / -3.0), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 2.8e-116], N[(N[(N[Sqrt[N[(-3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.30000000000000004e-53

   1. Initial program 78.4%

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

   3. Applied egg-rr 7.1%

      \[\leadsto \color{blue}{\frac{1}{a} \cdot \left(\left(b + \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -3\right)}\right)\right) \cdot 0.3333333333333333\right)} \]

   5. Applied egg-rr 61.8%

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

   6. Taylor expanded in b around -inf 90.4%

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

      1. *-commutative 90.4%

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

   8. Simplified 90.4%

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

   if -3.30000000000000004e-53 < b < 2.7999999999999999e-116

   1. Initial program 80.4%

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

   2. Taylor expanded in b around 0 77.1%

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

   if 2.7999999999999999e-116 < b

   1. Initial program 17.9%

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

      1. fma-neg 17.9%

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

      2. associate-*l* 17.9%

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

      3. distribute-lft-neg-in 17.9%

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

      4. metadata-eval 17.9%

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

      5. *-commutative 17.9%

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

      6. associate-*r* 17.9%

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

      7. add-cube-cbrt 15.1%

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

      8. sqrt-prod 15.2%

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

   3. Applied egg-rr 15.2%

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

   4. Taylor expanded in b around inf 84.3%

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

      1. associate-*r/ 84.3%

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

      2. *-commutative 84.3%

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

   6. Simplified 84.3%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.3 \cdot 10^{-53}:\\ \;\;\;\;\frac{\frac{b \cdot 2}{-3}}{a}\\ \mathbf{elif}\;b \leq 2.8 \cdot 10^{-116}:\\ \;\;\;\;\frac{\sqrt{-3 \cdot \left(a \cdot c\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 5: 80.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -6.4 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{b \cdot 2}{-3}}{a}\\ \mathbf{elif}\;b \leq 3.5 \cdot 10^{-116}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -6.4e-52)
   (/ (/ (* b 2.0) -3.0) a)
   (if (<= b 3.5e-116)
     (/ (- (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 <= -6.4e-52) {
		tmp = ((b * 2.0) / -3.0) / a;
	} else if (b <= 3.5e-116) {
		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 <= (-6.4d-52)) then
        tmp = ((b * 2.0d0) / (-3.0d0)) / a
    else if (b <= 3.5d-116) 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 <= -6.4e-52) {
		tmp = ((b * 2.0) / -3.0) / a;
	} else if (b <= 3.5e-116) {
		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 <= -6.4e-52:
		tmp = ((b * 2.0) / -3.0) / a
	elif b <= 3.5e-116:
		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 <= -6.4e-52)
		tmp = Float64(Float64(Float64(b * 2.0) / -3.0) / a);
	elseif (b <= 3.5e-116)
		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 <= -6.4e-52)
		tmp = ((b * 2.0) / -3.0) / a;
	elseif (b <= 3.5e-116)
		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, -6.4e-52], N[(N[(N[(b * 2.0), $MachinePrecision] / -3.0), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 3.5e-116], 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 < -6.4000000000000002e-52

   1. Initial program 78.4%

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

   3. Applied egg-rr 7.1%

      \[\leadsto \color{blue}{\frac{1}{a} \cdot \left(\left(b + \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -3\right)}\right)\right) \cdot 0.3333333333333333\right)} \]

   5. Applied egg-rr 61.8%

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

   6. Taylor expanded in b around -inf 90.4%

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

      1. *-commutative 90.4%

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

   8. Simplified 90.4%

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

   if -6.4000000000000002e-52 < b < 3.49999999999999984e-116

   1. Initial program 80.4%

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

   2. Taylor expanded in b around 0 77.1%

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

      1. associate-*r* 77.2%

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

      2. *-commutative 77.2%

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

      3. *-commutative 77.2%

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

   4. Simplified 77.2%

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

   if 3.49999999999999984e-116 < b

   1. Initial program 17.9%

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

      1. fma-neg 17.9%

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

      2. associate-*l* 17.9%

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

      3. distribute-lft-neg-in 17.9%

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

      4. metadata-eval 17.9%

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

      5. *-commutative 17.9%

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

      6. associate-*r* 17.9%

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

      7. add-cube-cbrt 15.1%

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

      8. sqrt-prod 15.2%

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

   3. Applied egg-rr 15.2%

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

   4. Taylor expanded in b around inf 84.3%

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

      1. associate-*r/ 84.3%

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

      2. *-commutative 84.3%

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

   6. Simplified 84.3%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6.4 \cdot 10^{-52}:\\ \;\;\;\;\frac{\frac{b \cdot 2}{-3}}{a}\\ \mathbf{elif}\;b \leq 3.5 \cdot 10^{-116}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 6: 67.1% accurate, 12.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.6 \cdot 10^{-287}:\\ \;\;\;\;\frac{\frac{b \cdot 2}{-3}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 2.6e-287) (/ (/ (* b 2.0) -3.0) a) (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.6e-287) {
		tmp = ((b * 2.0) / -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-287) then
        tmp = ((b * 2.0d0) / (-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-287) {
		tmp = ((b * 2.0) / -3.0) / a;
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 2.6e-287:
		tmp = ((b * 2.0) / -3.0) / a
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 2.6e-287)
		tmp = Float64(Float64(Float64(b * 2.0) / -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-287)
		tmp = ((b * 2.0) / -3.0) / a;
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 2.6e-287], N[(N[(N[(b * 2.0), $MachinePrecision] / -3.0), $MachinePrecision] / a), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.6e-287

   1. Initial program 80.4%

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

   3. Applied egg-rr 29.1%

      \[\leadsto \color{blue}{\frac{1}{a} \cdot \left(\left(b + \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -3\right)}\right)\right) \cdot 0.3333333333333333\right)} \]

   5. Applied egg-rr 69.6%

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

   6. Taylor expanded in b around -inf 66.5%

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

      1. *-commutative 66.5%

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

   8. Simplified 66.5%

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

   if 2.6e-287 < b

   1. Initial program 28.2%

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

      1. fma-neg 28.2%

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

      2. associate-*l* 28.2%

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

      3. distribute-lft-neg-in 28.2%

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

      4. metadata-eval 28.2%

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

      5. *-commutative 28.2%

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

      6. associate-*r* 28.2%

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

      7. add-cube-cbrt 25.9%

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

      8. sqrt-prod 25.9%

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

   3. Applied egg-rr 25.9%

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

   4. Taylor expanded in b around inf 71.5%

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

      1. associate-*r/ 71.5%

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

      2. *-commutative 71.5%

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

   6. Simplified 71.5%

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

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.6 \cdot 10^{-287}:\\ \;\;\;\;\frac{\frac{b \cdot 2}{-3}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 7: 67.1% accurate, 16.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.6 \cdot 10^{-287}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 2.6e-287) (* b (/ -0.6666666666666666 a)) (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.6e-287) {
		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 <= 2.6d-287) 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 <= 2.6e-287) {
		tmp = b * (-0.6666666666666666 / a);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 2.6e-287:
		tmp = b * (-0.6666666666666666 / a)
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 2.6e-287)
		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 <= 2.6e-287)
		tmp = b * (-0.6666666666666666 / a);
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 2.6e-287], 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 < 2.6e-287

   1. Initial program 80.4%

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

      1. fma-neg 80.4%

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

      2. associate-*l* 80.4%

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

      3. distribute-lft-neg-in 80.4%

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

      4. metadata-eval 80.4%

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

      5. *-commutative 80.4%

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

      6. associate-*r* 80.4%

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

      7. add-cube-cbrt 80.1%

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

      8. sqrt-prod 80.1%

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

   3. Applied egg-rr 80.2%

      \[\leadsto \frac{\left(-b\right) + \color{blue}{\sqrt{{\left(\sqrt[3]{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)}\right)}^{2}} \cdot \sqrt{\sqrt[3]{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)}}}}{3 \cdot a} \]
   4. Step-by-step derivation

      1. sqrt-pow1 80.2%

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

      2. metadata-eval 80.2%

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

      3. pow1 80.2%

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

   5. Applied egg-rr 80.2%

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

   6. Taylor expanded in b around -inf 66.3%

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

      1. *-commutative 66.3%

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

      2. associate-*l/ 66.4%

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

      3. associate-*r/ 66.3%

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

   8. Simplified 66.3%

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

   if 2.6e-287 < b

   1. Initial program 28.2%

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

   2. Taylor expanded in b around inf 71.5%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.6 \cdot 10^{-287}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 8: 67.1% accurate, 16.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.6 \cdot 10^{-287}:\\ \;\;\;\;\frac{-0.6666666666666666}{\frac{a}{b}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 2.6e-287) (/ -0.6666666666666666 (/ a b)) (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.6e-287) {
		tmp = -0.6666666666666666 / (a / b);
	} 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 <= 2.6d-287) then
        tmp = (-0.6666666666666666d0) / (a / b)
    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 <= 2.6e-287) {
		tmp = -0.6666666666666666 / (a / b);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 2.6e-287:
		tmp = -0.6666666666666666 / (a / b)
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 2.6e-287)
		tmp = Float64(-0.6666666666666666 / Float64(a / b));
	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 <= 2.6e-287)
		tmp = -0.6666666666666666 / (a / b);
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 2.6e-287], N[(-0.6666666666666666 / N[(a / b), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.6e-287

   1. Initial program 80.4%

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

   2. Taylor expanded in b around -inf 66.3%

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

      1. *-commutative 66.3%

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

   4. Simplified 66.3%

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

      1. *-commutative 66.3%

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

      2. clear-num 66.2%

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

      3. un-div-inv 66.3%

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

   6. Applied egg-rr 66.3%

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

   if 2.6e-287 < b

   1. Initial program 28.2%

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

   2. Taylor expanded in b around inf 71.5%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.6 \cdot 10^{-287}:\\ \;\;\;\;\frac{-0.6666666666666666}{\frac{a}{b}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 9: 67.1% accurate, 16.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.6 \cdot 10^{-287}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 2.6e-287) (/ b (* a -1.5)) (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.6e-287) {
		tmp = b / (a * -1.5);
	} 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 <= 2.6d-287) then
        tmp = b / (a * (-1.5d0))
    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 <= 2.6e-287) {
		tmp = b / (a * -1.5);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 2.6e-287:
		tmp = b / (a * -1.5)
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 2.6e-287)
		tmp = Float64(b / Float64(a * -1.5));
	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 <= 2.6e-287)
		tmp = b / (a * -1.5);
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 2.6e-287], N[(b / N[(a * -1.5), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.6e-287

   1. Initial program 80.4%

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

   2. Taylor expanded in b around -inf 66.3%

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

      1. *-commutative 66.3%

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

   4. Simplified 66.3%

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

      1. metadata-eval 66.3%

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

      2. times-frac 66.4%

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

      3. *-commutative 66.4%

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

      4. associate-/l* 66.4%

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

      5. frac-2neg 66.4%

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

      6. add-sqr-sqrt 66.2%

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

      7. sqrt-unprod 52.9%

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

      8. sqr-neg 52.9%

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

      9. sqrt-unprod 0.1%

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

      10. add-sqr-sqrt 1.4%

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

      11. div-inv 1.4%

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

      12. add-sqr-sqrt 0.7%

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

      13. sqrt-unprod 29.1%

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

      14. *-commutative 29.1%

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

      15. *-commutative 29.1%

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

      16. swap-sqr 29.1%

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

      17. metadata-eval 29.1%

          \[\leadsto \frac{b}{-\sqrt{\left(a \cdot a\right) \cdot \color{blue}{9}} \cdot \frac{1}{-2}} \]

      18. metadata-eval 29.1%

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

      19. swap-sqr 29.1%

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

      20. sqrt-unprod 32.6%

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

      21. add-sqr-sqrt 66.4%

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

      22. metadata-eval 66.4%

          \[\leadsto \frac{b}{-\left(a \cdot -3\right) \cdot \color{blue}{-0.5}} \]

   6. Applied egg-rr 66.4%

      \[\leadsto \color{blue}{\frac{b}{-\left(a \cdot -3\right) \cdot -0.5}} \]
   7. Step-by-step derivation

      1. associate-*l* 66.4%

         \[\leadsto \frac{b}{-\color{blue}{a \cdot \left(-3 \cdot -0.5\right)}} \]

      2. metadata-eval 66.4%

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

      3. distribute-rgt-neg-in 66.4%

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

      4. metadata-eval 66.4%

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

   8. Simplified 66.4%

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

   if 2.6e-287 < b

   1. Initial program 28.2%

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

   2. Taylor expanded in b around inf 71.5%

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

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.6 \cdot 10^{-287}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 10: 67.1% accurate, 16.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 2.6 \cdot 10^{-287}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 2.6e-287) (/ b (* a -1.5)) (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.6e-287) {
		tmp = b / (a * -1.5);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 2.6d-287) then
        tmp = b / (a * (-1.5d0))
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 2.6e-287) {
		tmp = b / (a * -1.5);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 2.6e-287:
		tmp = b / (a * -1.5)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 2.6e-287)
		tmp = Float64(b / Float64(a * -1.5));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 2.6e-287)
		tmp = b / (a * -1.5);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 2.6e-287], N[(b / N[(a * -1.5), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 2.6e-287

   1. Initial program 80.4%

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

   2. Taylor expanded in b around -inf 66.3%

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

      1. *-commutative 66.3%

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

   4. Simplified 66.3%

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

      1. metadata-eval 66.3%

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

      2. times-frac 66.4%

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

      3. *-commutative 66.4%

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

      4. associate-/l* 66.4%

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

      5. frac-2neg 66.4%

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

      6. add-sqr-sqrt 66.2%

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

      7. sqrt-unprod 52.9%

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

      8. sqr-neg 52.9%

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

      9. sqrt-unprod 0.1%

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

      10. add-sqr-sqrt 1.4%

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

      11. div-inv 1.4%

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

      12. add-sqr-sqrt 0.7%

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

      13. sqrt-unprod 29.1%

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

      14. *-commutative 29.1%

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

      15. *-commutative 29.1%

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

      16. swap-sqr 29.1%

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

      17. metadata-eval 29.1%

          \[\leadsto \frac{b}{-\sqrt{\left(a \cdot a\right) \cdot \color{blue}{9}} \cdot \frac{1}{-2}} \]

      18. metadata-eval 29.1%

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

      19. swap-sqr 29.1%

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

      20. sqrt-unprod 32.6%

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

      21. add-sqr-sqrt 66.4%

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

      22. metadata-eval 66.4%

          \[\leadsto \frac{b}{-\left(a \cdot -3\right) \cdot \color{blue}{-0.5}} \]

   6. Applied egg-rr 66.4%

      \[\leadsto \color{blue}{\frac{b}{-\left(a \cdot -3\right) \cdot -0.5}} \]
   7. Step-by-step derivation

      1. associate-*l* 66.4%

         \[\leadsto \frac{b}{-\color{blue}{a \cdot \left(-3 \cdot -0.5\right)}} \]

      2. metadata-eval 66.4%

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

      3. distribute-rgt-neg-in 66.4%

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

      4. metadata-eval 66.4%

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

   8. Simplified 66.4%

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

   if 2.6e-287 < b

   1. Initial program 28.2%

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

      1. fma-neg 28.2%

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

      2. associate-*l* 28.2%

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

      3. distribute-lft-neg-in 28.2%

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

      4. metadata-eval 28.2%

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

      5. *-commutative 28.2%

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

      6. associate-*r* 28.2%

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

      7. add-cube-cbrt 25.9%

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

      8. sqrt-prod 25.9%

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

   3. Applied egg-rr 25.9%

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

   4. Taylor expanded in b around inf 71.5%

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

      1. associate-*r/ 71.5%

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

      2. *-commutative 71.5%

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

   6. Simplified 71.5%

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

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 2.6 \cdot 10^{-287}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 11: 34.6% 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 54.3%

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

2. Taylor expanded in b around inf 36.7%

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

3. Final simplification 36.7%

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

Reproduce

herbie shell --seed 2023321 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))