Cubic critical

Percentage Accurate: 51.9% → 85.7%

Time: 13.9s

Alternatives: 10

Speedup: 11.6×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 51.9% 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: 85.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -7.2 \cdot 10^{+63}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 2.3 \cdot 10^{-78}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 3\right) \cdot c} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -7.2e+63)
   (/ b (* a -1.5))
   (if (<= b 2.3e-78)
     (/ (- (sqrt (- (* b b) (* (* a 3.0) c))) b) (* a 3.0))
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -7.2e+63) {
		tmp = b / (a * -1.5);
	} else if (b <= 2.3e-78) {
		tmp = (sqrt(((b * b) - ((a * 3.0) * c))) - b) / (a * 3.0);
	} else {
		tmp = (c / b) * -0.5;
	}
	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 <= (-7.2d+63)) then
        tmp = b / (a * (-1.5d0))
    else if (b <= 2.3d-78) then
        tmp = (sqrt(((b * b) - ((a * 3.0d0) * c))) - b) / (a * 3.0d0)
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -7.2e+63) {
		tmp = b / (a * -1.5);
	} else if (b <= 2.3e-78) {
		tmp = (Math.sqrt(((b * b) - ((a * 3.0) * c))) - b) / (a * 3.0);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -7.2e+63:
		tmp = b / (a * -1.5)
	elif b <= 2.3e-78:
		tmp = (math.sqrt(((b * b) - ((a * 3.0) * c))) - b) / (a * 3.0)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -7.2e+63)
		tmp = Float64(b / Float64(a * -1.5));
	elseif (b <= 2.3e-78)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(a * 3.0) * c))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -7.2e+63)
		tmp = b / (a * -1.5);
	elseif (b <= 2.3e-78)
		tmp = (sqrt(((b * b) - ((a * 3.0) * c))) - b) / (a * 3.0);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -7.2e+63], N[(b / N[(a * -1.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.3e-78], 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 / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -7.19999999999999998e63

   1. Initial program 64.8%

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

      1. add-cube-cbrt 64.4%

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

      2. pow3 64.4%

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

   4. Applied egg-rr 64.4%

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

   5. Taylor expanded in b around -inf 95.9%

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

      1. associate-*r/ 95.9%

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

      2. rem-cube-cbrt 96.6%

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

      3. *-commutative 96.6%

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

      4. times-frac 96.4%

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

      5. metadata-eval 96.4%

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

      6. associate-*r/ 96.5%

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

      7. *-commutative 96.5%

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

      8. associate-/l* 96.3%

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

   7. Simplified 96.3%

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

      1. clear-num 96.3%

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

      2. un-div-inv 96.5%

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

      3. div-inv 96.6%

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

      4. metadata-eval 96.6%

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

   9. Applied egg-rr 96.6%

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

   if -7.19999999999999998e63 < b < 2.3000000000000002e-78

   1. Initial program 82.4%

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

   if 2.3000000000000002e-78 < b

   1. Initial program 18.6%

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

      1. sqr-neg 18.6%

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

      2. sqr-neg 18.6%

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

      3. associate-*l* 18.6%

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

   3. Simplified 18.6%

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

   5. Taylor expanded in b around inf 86.4%

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

      1. *-commutative 86.4%

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

   7. Simplified 86.4%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -7.2 \cdot 10^{+63}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 2.3 \cdot 10^{-78}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 3\right) \cdot c} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.06 \cdot 10^{+63}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 2.2 \cdot 10^{-78}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.06e+63)
   (/ b (* a -1.5))
   (if (<= b 2.2e-78)
     (/ (- (sqrt (- (* b b) (* 3.0 (* a c)))) b) (* a 3.0))
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.06e+63) {
		tmp = b / (a * -1.5);
	} else if (b <= 2.2e-78) {
		tmp = (sqrt(((b * b) - (3.0 * (a * c)))) - b) / (a * 3.0);
	} else {
		tmp = (c / b) * -0.5;
	}
	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 <= (-1.06d+63)) then
        tmp = b / (a * (-1.5d0))
    else if (b <= 2.2d-78) then
        tmp = (sqrt(((b * b) - (3.0d0 * (a * c)))) - b) / (a * 3.0d0)
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.06e+63) {
		tmp = b / (a * -1.5);
	} else if (b <= 2.2e-78) {
		tmp = (Math.sqrt(((b * b) - (3.0 * (a * c)))) - b) / (a * 3.0);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.06e+63:
		tmp = b / (a * -1.5)
	elif b <= 2.2e-78:
		tmp = (math.sqrt(((b * b) - (3.0 * (a * c)))) - b) / (a * 3.0)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.06e+63)
		tmp = Float64(b / Float64(a * -1.5));
	elseif (b <= 2.2e-78)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(3.0 * Float64(a * c)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.06e+63)
		tmp = b / (a * -1.5);
	elseif (b <= 2.2e-78)
		tmp = (sqrt(((b * b) - (3.0 * (a * c)))) - b) / (a * 3.0);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.06e+63], N[(b / N[(a * -1.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.2e-78], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.0600000000000001e63

   1. Initial program 64.8%

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

      1. add-cube-cbrt 64.4%

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

      2. pow3 64.4%

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

   4. Applied egg-rr 64.4%

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

   5. Taylor expanded in b around -inf 95.9%

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

      1. associate-*r/ 95.9%

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

      2. rem-cube-cbrt 96.6%

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

      3. *-commutative 96.6%

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

      4. times-frac 96.4%

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

      5. metadata-eval 96.4%

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

      6. associate-*r/ 96.5%

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

      7. *-commutative 96.5%

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

      8. associate-/l* 96.3%

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

   7. Simplified 96.3%

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

      1. clear-num 96.3%

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

      2. un-div-inv 96.5%

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

      3. div-inv 96.6%

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

      4. metadata-eval 96.6%

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

   9. Applied egg-rr 96.6%

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

   if -1.0600000000000001e63 < b < 2.1999999999999999e-78

   1. Initial program 82.4%

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

      1. sqr-neg 82.4%

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

      2. sqr-neg 82.4%

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

      3. associate-*l* 82.3%

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

   3. Simplified 82.3%

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

   if 2.1999999999999999e-78 < b

   1. Initial program 18.6%

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

      1. sqr-neg 18.6%

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

      2. sqr-neg 18.6%

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

      3. associate-*l* 18.6%

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

   3. Simplified 18.6%

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

   5. Taylor expanded in b around inf 86.4%

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

      1. *-commutative 86.4%

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

   7. Simplified 86.4%

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

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.06 \cdot 10^{+63}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 2.2 \cdot 10^{-78}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.55 \cdot 10^{-27}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 1.65 \cdot 10^{-85}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.55e-27)
   (/ b (* a -1.5))
   (if (<= b 1.65e-85)
     (/ (- (sqrt (* c (* a -3.0))) b) (* a 3.0))
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.55e-27) {
		tmp = b / (a * -1.5);
	} else if (b <= 1.65e-85) {
		tmp = (sqrt((c * (a * -3.0))) - b) / (a * 3.0);
	} else {
		tmp = (c / b) * -0.5;
	}
	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 <= (-1.55d-27)) then
        tmp = b / (a * (-1.5d0))
    else if (b <= 1.65d-85) then
        tmp = (sqrt((c * (a * (-3.0d0)))) - b) / (a * 3.0d0)
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.55e-27) {
		tmp = b / (a * -1.5);
	} else if (b <= 1.65e-85) {
		tmp = (Math.sqrt((c * (a * -3.0))) - b) / (a * 3.0);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.55e-27:
		tmp = b / (a * -1.5)
	elif b <= 1.65e-85:
		tmp = (math.sqrt((c * (a * -3.0))) - b) / (a * 3.0)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.55e-27)
		tmp = Float64(b / Float64(a * -1.5));
	elseif (b <= 1.65e-85)
		tmp = Float64(Float64(sqrt(Float64(c * Float64(a * -3.0))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.55e-27)
		tmp = b / (a * -1.5);
	elseif (b <= 1.65e-85)
		tmp = (sqrt((c * (a * -3.0))) - b) / (a * 3.0);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.55e-27], N[(b / N[(a * -1.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.65e-85], N[(N[(N[Sqrt[N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.5499999999999999e-27

   1. Initial program 72.6%

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

      1. add-cube-cbrt 72.2%

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

      2. pow3 72.2%

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

   4. Applied egg-rr 72.2%

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

   5. Taylor expanded in b around -inf 94.6%

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

      1. associate-*r/ 94.6%

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

      2. rem-cube-cbrt 95.2%

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

      3. *-commutative 95.2%

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

      4. times-frac 95.0%

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

      5. metadata-eval 95.0%

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

      6. associate-*r/ 95.1%

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

      7. *-commutative 95.1%

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

      8. associate-/l* 95.0%

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

   7. Simplified 95.0%

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

      1. clear-num 95.0%

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

      2. un-div-inv 95.1%

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

      3. div-inv 95.2%

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

      4. metadata-eval 95.2%

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

   9. Applied egg-rr 95.2%

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

   if -1.5499999999999999e-27 < b < 1.64999999999999986e-85

   1. Initial program 78.1%

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

      1. sqr-neg 78.1%

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

      2. sqr-neg 78.1%

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

      3. associate-*l* 78.0%

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

   3. Simplified 78.0%

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

   5. Taylor expanded in b around 0 68.6%

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

      1. associate-*r* 68.7%

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

      2. *-commutative 68.7%

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

   7. Simplified 68.7%

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

      1. +-commutative 68.7%

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

      2. unsub-neg 68.7%

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

      3. associate-*l* 68.6%

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

      4. *-commutative 68.6%

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

   9. Applied egg-rr 68.6%

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

       1. associate-*r* 68.6%

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

       2. *-commutative 68.6%

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

       3. associate-*r* 68.7%

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

       4. *-commutative 68.7%

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

   11. Simplified 68.7%

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

   if 1.64999999999999986e-85 < b

   1. Initial program 19.2%

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

      1. sqr-neg 19.2%

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

      2. sqr-neg 19.2%

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

      3. associate-*l* 19.2%

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

   3. Simplified 19.2%

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

   5. Taylor expanded in b around inf 86.0%

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

      1. *-commutative 86.0%

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

   7. Simplified 86.0%

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

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.55 \cdot 10^{-27}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 1.65 \cdot 10^{-85}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.55 \cdot 10^{-27}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 1.42 \cdot 10^{-81}:\\ \;\;\;\;\frac{0.3333333333333333 \cdot \left(\sqrt{a \cdot \left(c \cdot -3\right)} - b\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.55e-27)
   (/ b (* a -1.5))
   (if (<= b 1.42e-81)
     (/ (* 0.3333333333333333 (- (sqrt (* a (* c -3.0))) b)) a)
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.55e-27) {
		tmp = b / (a * -1.5);
	} else if (b <= 1.42e-81) {
		tmp = (0.3333333333333333 * (sqrt((a * (c * -3.0))) - b)) / a;
	} else {
		tmp = (c / b) * -0.5;
	}
	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 <= (-1.55d-27)) then
        tmp = b / (a * (-1.5d0))
    else if (b <= 1.42d-81) then
        tmp = (0.3333333333333333d0 * (sqrt((a * (c * (-3.0d0)))) - b)) / a
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.55e-27) {
		tmp = b / (a * -1.5);
	} else if (b <= 1.42e-81) {
		tmp = (0.3333333333333333 * (Math.sqrt((a * (c * -3.0))) - b)) / a;
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.55e-27:
		tmp = b / (a * -1.5)
	elif b <= 1.42e-81:
		tmp = (0.3333333333333333 * (math.sqrt((a * (c * -3.0))) - b)) / a
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.55e-27)
		tmp = Float64(b / Float64(a * -1.5));
	elseif (b <= 1.42e-81)
		tmp = Float64(Float64(0.3333333333333333 * Float64(sqrt(Float64(a * Float64(c * -3.0))) - b)) / a);
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.55e-27)
		tmp = b / (a * -1.5);
	elseif (b <= 1.42e-81)
		tmp = (0.3333333333333333 * (sqrt((a * (c * -3.0))) - b)) / a;
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.55e-27], N[(b / N[(a * -1.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.42e-81], N[(N[(0.3333333333333333 * N[(N[Sqrt[N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.5499999999999999e-27

   1. Initial program 72.6%

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

      1. add-cube-cbrt 72.2%

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

      2. pow3 72.2%

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

   4. Applied egg-rr 72.2%

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

   5. Taylor expanded in b around -inf 94.6%

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

      1. associate-*r/ 94.6%

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

      2. rem-cube-cbrt 95.2%

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

      3. *-commutative 95.2%

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

      4. times-frac 95.0%

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

      5. metadata-eval 95.0%

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

      6. associate-*r/ 95.1%

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

      7. *-commutative 95.1%

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

      8. associate-/l* 95.0%

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

   7. Simplified 95.0%

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

      1. clear-num 95.0%

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

      2. un-div-inv 95.1%

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

      3. div-inv 95.2%

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

      4. metadata-eval 95.2%

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

   9. Applied egg-rr 95.2%

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

   if -1.5499999999999999e-27 < b < 1.42000000000000009e-81

   1. Initial program 78.1%

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

      1. sqr-neg 78.1%

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

      2. sqr-neg 78.1%

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

      3. associate-*l* 78.0%

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

   3. Simplified 78.0%

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

   5. Taylor expanded in b around 0 68.6%

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

      1. associate-*r* 68.7%

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

      2. *-commutative 68.7%

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

   7. Simplified 68.7%

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

      1. +-commutative 68.7%

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

      2. unsub-neg 68.7%

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

      3. associate-*l* 68.6%

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

      4. *-commutative 68.6%

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

   9. Applied egg-rr 68.6%

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

       1. associate-*r* 68.6%

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

       2. *-commutative 68.6%

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

       3. associate-*r* 68.7%

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

       4. *-commutative 68.7%

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

   11. Simplified 68.7%

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

       1. div-sub 68.7%

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

       2. sub-neg 68.7%

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

       3. *-un-lft-identity 68.7%

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

       4. times-frac 68.5%

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

       5. metadata-eval 68.5%

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

       6. associate-*l* 68.5%

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

       7. *-commutative 68.5%

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

       8. *-un-lft-identity 68.5%

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

       9. times-frac 68.5%

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

       10. metadata-eval 68.5%

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

   13. Applied egg-rr 68.5%

       \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{\sqrt{a \cdot \left(c \cdot -3\right)}}{a} + \left(-0.3333333333333333 \cdot \frac{b}{a}\right)} \]
   14. Step-by-step derivation

       1. sub-neg 68.5%

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

       2. distribute-lft-out-- 68.5%

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

       3. div-sub 68.5%

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

       4. associate-*r/ 68.5%

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

   15. Simplified 68.5%

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

   if 1.42000000000000009e-81 < b

   1. Initial program 19.2%

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

      1. sqr-neg 19.2%

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

      2. sqr-neg 19.2%

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

      3. associate-*l* 19.2%

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

   3. Simplified 19.2%

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

   5. Taylor expanded in b around inf 86.0%

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

      1. *-commutative 86.0%

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

   7. Simplified 86.0%

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

Alternative 5: 80.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.8 \cdot 10^{-27}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 1.16 \cdot 10^{-80}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\sqrt{a \cdot \left(c \cdot -3\right)} - b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.8e-27)
   (/ b (* a -1.5))
   (if (<= b 1.16e-80)
     (* 0.3333333333333333 (/ (- (sqrt (* a (* c -3.0))) b) a))
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.8e-27) {
		tmp = b / (a * -1.5);
	} else if (b <= 1.16e-80) {
		tmp = 0.3333333333333333 * ((sqrt((a * (c * -3.0))) - b) / a);
	} else {
		tmp = (c / b) * -0.5;
	}
	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 <= (-3.8d-27)) then
        tmp = b / (a * (-1.5d0))
    else if (b <= 1.16d-80) then
        tmp = 0.3333333333333333d0 * ((sqrt((a * (c * (-3.0d0)))) - b) / a)
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.8e-27) {
		tmp = b / (a * -1.5);
	} else if (b <= 1.16e-80) {
		tmp = 0.3333333333333333 * ((Math.sqrt((a * (c * -3.0))) - b) / a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -3.8e-27:
		tmp = b / (a * -1.5)
	elif b <= 1.16e-80:
		tmp = 0.3333333333333333 * ((math.sqrt((a * (c * -3.0))) - b) / a)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.8e-27)
		tmp = Float64(b / Float64(a * -1.5));
	elseif (b <= 1.16e-80)
		tmp = Float64(0.3333333333333333 * Float64(Float64(sqrt(Float64(a * Float64(c * -3.0))) - b) / a));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -3.8e-27)
		tmp = b / (a * -1.5);
	elseif (b <= 1.16e-80)
		tmp = 0.3333333333333333 * ((sqrt((a * (c * -3.0))) - b) / a);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.8e-27], N[(b / N[(a * -1.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.16e-80], N[(0.3333333333333333 * N[(N[(N[Sqrt[N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.8e-27

   1. Initial program 72.6%

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

      1. add-cube-cbrt 72.2%

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

      2. pow3 72.2%

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

   4. Applied egg-rr 72.2%

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

   5. Taylor expanded in b around -inf 94.6%

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

      1. associate-*r/ 94.6%

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

      2. rem-cube-cbrt 95.2%

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

      3. *-commutative 95.2%

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

      4. times-frac 95.0%

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

      5. metadata-eval 95.0%

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

      6. associate-*r/ 95.1%

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

      7. *-commutative 95.1%

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

      8. associate-/l* 95.0%

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

   7. Simplified 95.0%

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

      1. clear-num 95.0%

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

      2. un-div-inv 95.1%

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

      3. div-inv 95.2%

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

      4. metadata-eval 95.2%

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

   9. Applied egg-rr 95.2%

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

   if -3.8e-27 < b < 1.15999999999999996e-80

   1. Initial program 78.1%

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

      1. sqr-neg 78.1%

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

      2. sqr-neg 78.1%

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

      3. associate-*l* 78.0%

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

   3. Simplified 78.0%

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

   5. Taylor expanded in b around 0 68.6%

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

      1. associate-*r* 68.7%

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

      2. *-commutative 68.7%

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

   7. Simplified 68.7%

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

      1. +-commutative 68.7%

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

      2. unsub-neg 68.7%

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

      3. associate-*l* 68.6%

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

      4. *-commutative 68.6%

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

   9. Applied egg-rr 68.6%

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

       1. associate-*r* 68.6%

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

       2. *-commutative 68.6%

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

       3. associate-*r* 68.7%

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

       4. *-commutative 68.7%

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

   11. Simplified 68.7%

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

       1. *-un-lft-identity 68.7%

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

       2. times-frac 68.5%

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

       3. metadata-eval 68.5%

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

       4. associate-*l* 68.5%

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

       5. *-commutative 68.5%

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

   13. Applied egg-rr 68.5%

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

   if 1.15999999999999996e-80 < b

   1. Initial program 19.2%

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

      1. sqr-neg 19.2%

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

      2. sqr-neg 19.2%

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

      3. associate-*l* 19.2%

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

   3. Simplified 19.2%

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

   5. Taylor expanded in b around inf 86.0%

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

      1. *-commutative 86.0%

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

   7. Simplified 86.0%

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

Alternative 6: 79.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.8 \cdot 10^{-27}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 6.3 \cdot 10^{-81}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{b + \sqrt{c \cdot \left(a \cdot -3\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.8e-27)
   (/ b (* a -1.5))
   (if (<= b 6.3e-81)
     (* 0.3333333333333333 (/ (+ b (sqrt (* c (* a -3.0)))) a))
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.8e-27) {
		tmp = b / (a * -1.5);
	} else if (b <= 6.3e-81) {
		tmp = 0.3333333333333333 * ((b + sqrt((c * (a * -3.0)))) / a);
	} else {
		tmp = (c / b) * -0.5;
	}
	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 <= (-3.8d-27)) then
        tmp = b / (a * (-1.5d0))
    else if (b <= 6.3d-81) then
        tmp = 0.3333333333333333d0 * ((b + sqrt((c * (a * (-3.0d0))))) / a)
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.8e-27) {
		tmp = b / (a * -1.5);
	} else if (b <= 6.3e-81) {
		tmp = 0.3333333333333333 * ((b + Math.sqrt((c * (a * -3.0)))) / a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -3.8e-27:
		tmp = b / (a * -1.5)
	elif b <= 6.3e-81:
		tmp = 0.3333333333333333 * ((b + math.sqrt((c * (a * -3.0)))) / a)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.8e-27)
		tmp = Float64(b / Float64(a * -1.5));
	elseif (b <= 6.3e-81)
		tmp = Float64(0.3333333333333333 * Float64(Float64(b + sqrt(Float64(c * Float64(a * -3.0)))) / a));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -3.8e-27)
		tmp = b / (a * -1.5);
	elseif (b <= 6.3e-81)
		tmp = 0.3333333333333333 * ((b + sqrt((c * (a * -3.0)))) / a);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.8e-27], N[(b / N[(a * -1.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 6.3e-81], N[(0.3333333333333333 * N[(N[(b + N[Sqrt[N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.8e-27

   1. Initial program 72.6%

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

      1. add-cube-cbrt 72.2%

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

      2. pow3 72.2%

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

   4. Applied egg-rr 72.2%

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

   5. Taylor expanded in b around -inf 94.6%

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

      1. associate-*r/ 94.6%

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

      2. rem-cube-cbrt 95.2%

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

      3. *-commutative 95.2%

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

      4. times-frac 95.0%

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

      5. metadata-eval 95.0%

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

      6. associate-*r/ 95.1%

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

      7. *-commutative 95.1%

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

      8. associate-/l* 95.0%

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

   7. Simplified 95.0%

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

      1. clear-num 95.0%

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

      2. un-div-inv 95.1%

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

      3. div-inv 95.2%

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

      4. metadata-eval 95.2%

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

   9. Applied egg-rr 95.2%

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

   if -3.8e-27 < b < 6.30000000000000023e-81

   1. Initial program 78.1%

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

      1. sqr-neg 78.1%

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

      2. sqr-neg 78.1%

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

      3. associate-*l* 78.0%

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

   3. Simplified 78.0%

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

   5. Taylor expanded in b around 0 68.6%

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

      1. associate-*r* 68.7%

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

      2. *-commutative 68.7%

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

   7. Simplified 68.7%

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

      1. *-un-lft-identity 68.7%

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

      2. times-frac 68.5%

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

      3. metadata-eval 68.5%

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

      4. add-sqr-sqrt 42.5%

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

      5. sqrt-unprod 68.1%

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

      6. sqr-neg 68.1%

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

      7. sqrt-unprod 25.8%

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

      8. add-sqr-sqrt 67.0%

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

      9. associate-*l* 67.0%

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

      10. *-commutative 67.0%

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

   9. Applied egg-rr 67.0%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{b + \sqrt{a \cdot \left(c \cdot -3\right)}}{a}} \]
   10. Step-by-step derivation

       1. associate-*r* 67.0%

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

       2. *-commutative 67.0%

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

       3. associate-*r* 67.0%

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

       4. *-commutative 67.0%

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

   11. Simplified 67.0%

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

   if 6.30000000000000023e-81 < b

   1. Initial program 19.2%

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

      1. sqr-neg 19.2%

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

      2. sqr-neg 19.2%

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

      3. associate-*l* 19.2%

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

   3. Simplified 19.2%

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

   5. Taylor expanded in b around inf 86.0%

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

      1. *-commutative 86.0%

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

   7. Simplified 86.0%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.8 \cdot 10^{-27}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 6.3 \cdot 10^{-81}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{b + \sqrt{c \cdot \left(a \cdot -3\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 69.0% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.5 \cdot 10^{-302}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 3.5e-302) (/ b (* a -1.5)) (* (/ c b) -0.5)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 3.5e-302) {
		tmp = b / (a * -1.5);
	} else {
		tmp = (c / b) * -0.5;
	}
	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 <= 3.5d-302) then
        tmp = b / (a * (-1.5d0))
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 3.5e-302) {
		tmp = b / (a * -1.5);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 3.5e-302:
		tmp = b / (a * -1.5)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 3.5e-302)
		tmp = Float64(b / Float64(a * -1.5));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 3.5e-302)
		tmp = b / (a * -1.5);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 3.5e-302], N[(b / N[(a * -1.5), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.5000000000000001e-302

   1. Initial program 76.4%

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

      1. add-cube-cbrt 75.7%

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

      2. pow3 75.7%

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

   4. Applied egg-rr 75.7%

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

   5. Taylor expanded in b around -inf 67.5%

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

      1. associate-*r/ 67.5%

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

      2. rem-cube-cbrt 68.0%

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

      3. *-commutative 68.0%

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

      4. times-frac 67.8%

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

      5. metadata-eval 67.8%

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

      6. associate-*r/ 67.9%

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

      7. *-commutative 67.9%

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

      8. associate-/l* 67.8%

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

   7. Simplified 67.8%

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

      1. clear-num 67.8%

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

      2. un-div-inv 67.9%

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

      3. div-inv 68.0%

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

      4. metadata-eval 68.0%

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

   9. Applied egg-rr 68.0%

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

   if 3.5000000000000001e-302 < b

   1. Initial program 31.0%

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

      1. sqr-neg 31.0%

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

      2. sqr-neg 31.0%

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

      3. associate-*l* 30.9%

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

   3. Simplified 30.9%

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

   5. Taylor expanded in b around inf 68.8%

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

      1. *-commutative 68.8%

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

   7. Simplified 68.8%

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

Alternative 8: 69.0% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.5 \cdot 10^{-302}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 3.5e-302) (* b (/ -0.6666666666666666 a)) (* (/ c b) -0.5)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 3.5e-302) {
		tmp = b * (-0.6666666666666666 / a);
	} else {
		tmp = (c / b) * -0.5;
	}
	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 <= 3.5d-302) then
        tmp = b * ((-0.6666666666666666d0) / a)
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 3.5e-302) {
		tmp = b * (-0.6666666666666666 / a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 3.5e-302:
		tmp = b * (-0.6666666666666666 / a)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 3.5e-302)
		tmp = Float64(b * Float64(-0.6666666666666666 / a));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 3.5e-302)
		tmp = b * (-0.6666666666666666 / a);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 3.5e-302], N[(b * N[(-0.6666666666666666 / a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.5000000000000001e-302

   1. Initial program 76.4%

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

      1. add-cube-cbrt 75.7%

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

      2. pow3 75.7%

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

   4. Applied egg-rr 75.7%

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

   5. Taylor expanded in b around -inf 67.5%

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

      1. associate-*r/ 67.5%

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

      2. rem-cube-cbrt 68.0%

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

      3. *-commutative 68.0%

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

      4. times-frac 67.8%

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

      5. metadata-eval 67.8%

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

      6. associate-*r/ 67.9%

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

      7. *-commutative 67.9%

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

      8. associate-/l* 67.8%

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

   7. Simplified 67.8%

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

   if 3.5000000000000001e-302 < b

   1. Initial program 31.0%

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

      1. sqr-neg 31.0%

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

      2. sqr-neg 31.0%

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

      3. associate-*l* 30.9%

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

   3. Simplified 30.9%

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

   5. Taylor expanded in b around inf 68.8%

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

      1. *-commutative 68.8%

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

   7. Simplified 68.8%

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

Alternative 9: 44.4% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 8 \cdot 10^{+26}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 8e+26) (* b (/ -0.6666666666666666 a)) (* c (/ 0.5 b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 8e+26) {
		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 <= 8d+26) 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 <= 8e+26) {
		tmp = b * (-0.6666666666666666 / a);
	} else {
		tmp = c * (0.5 / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 8e+26:
		tmp = b * (-0.6666666666666666 / a)
	else:
		tmp = c * (0.5 / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 8e+26)
		tmp = Float64(b * Float64(-0.6666666666666666 / a));
	else
		tmp = Float64(c * Float64(0.5 / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 8e+26)
		tmp = b * (-0.6666666666666666 / a);
	else
		tmp = c * (0.5 / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 8e+26], N[(b * N[(-0.6666666666666666 / a), $MachinePrecision]), $MachinePrecision], N[(c * N[(0.5 / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 8.00000000000000038e26

   1. Initial program 71.4%

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

      1. add-cube-cbrt 70.5%

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

      2. pow3 70.5%

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

   4. Applied egg-rr 70.5%

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

   5. Taylor expanded in b around -inf 49.0%

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

      1. associate-*r/ 49.0%

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

      2. rem-cube-cbrt 49.3%

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

      3. *-commutative 49.3%

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

      4. times-frac 49.2%

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

      5. metadata-eval 49.2%

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

      6. associate-*r/ 49.3%

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

      7. *-commutative 49.3%

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

      8. associate-/l* 49.2%

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

   7. Simplified 49.2%

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

   if 8.00000000000000038e26 < b

   1. Initial program 11.3%

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

      1. add-cube-cbrt 11.2%

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

      2. pow3 11.2%

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

   4. Applied egg-rr 11.2%

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

   5. Taylor expanded in b around inf 94.2%

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

      1. associate-*r/ 94.4%

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

      2. rem-cube-cbrt 94.8%

         \[\leadsto \frac{-1.5 \cdot c}{b \cdot \color{blue}{3}} \]

   7. Simplified 94.8%

      \[\leadsto \color{blue}{\frac{-1.5 \cdot c}{b \cdot 3}} \]
   8. Step-by-step derivation

      1. div-inv 94.7%

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

      2. *-commutative 94.7%

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

      3. *-commutative 94.7%

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

      4. add-sqr-sqrt 94.6%

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

      5. sqrt-unprod 63.6%

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

      6. sqr-neg 63.6%

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

      7. sqrt-unprod 0.0%

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

      8. add-sqr-sqrt 33.1%

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

      9. distribute-rgt-neg-in 33.1%

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

      10. distribute-lft-neg-in 33.1%

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

      11. metadata-eval 33.1%

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

   9. Applied egg-rr 33.1%

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

       1. associate-*r/ 33.1%

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

       2. *-rgt-identity 33.1%

          \[\leadsto \frac{\color{blue}{c \cdot -1.5}}{-3 \cdot b} \]

       3. associate-*r/ 33.1%

          \[\leadsto \color{blue}{c \cdot \frac{-1.5}{-3 \cdot b}} \]

       4. associate-/r* 33.1%

          \[\leadsto c \cdot \color{blue}{\frac{\frac{-1.5}{-3}}{b}} \]

       5. metadata-eval 33.1%

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

   11. Simplified 33.1%

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

Alternative 10: 36.3% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.2%

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

   1. add-cube-cbrt 53.6%

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

   2. pow3 53.6%

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

4. Applied egg-rr 53.6%

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

5. Taylor expanded in b around -inf 35.9%

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

   1. associate-*r/ 35.9%

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

   2. rem-cube-cbrt 36.1%

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

   3. *-commutative 36.1%

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

   4. times-frac 36.0%

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

   5. metadata-eval 36.0%

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

   6. associate-*r/ 36.0%

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

   7. *-commutative 36.0%

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

   8. associate-/l* 36.0%

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

7. Simplified 36.0%

   \[\leadsto \color{blue}{b \cdot \frac{-0.6666666666666666}{a}} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024121 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))