Cubic critical

Percentage Accurate: 51.8% → 86.5%

Time: 13.3s

Alternatives: 9

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: 51.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 86.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.6 \cdot 10^{+135}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 8.2 \cdot 10^{-73}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.6e+135)
   (+ (* -0.6666666666666666 (/ b a)) (* 0.5 (/ c b)))
   (if (<= b 8.2e-73)
     (/ (- (sqrt (- (* b b) (* 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 <= -1.6e+135) {
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	} else if (b <= 8.2e-73) {
		tmp = (sqrt(((b * b) - (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 <= (-1.6d+135)) then
        tmp = ((-0.6666666666666666d0) * (b / a)) + (0.5d0 * (c / b))
    else if (b <= 8.2d-73) then
        tmp = (sqrt(((b * b) - (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 <= -1.6e+135) {
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	} else if (b <= 8.2e-73) {
		tmp = (Math.sqrt(((b * b) - (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 <= -1.6e+135:
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b))
	elif b <= 8.2e-73:
		tmp = (math.sqrt(((b * b) - (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 <= -1.6e+135)
		tmp = Float64(Float64(-0.6666666666666666 * Float64(b / a)) + Float64(0.5 * Float64(c / b)));
	elseif (b <= 8.2e-73)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - 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 <= -1.6e+135)
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	elseif (b <= 8.2e-73)
		tmp = (sqrt(((b * b) - (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, -1.6e+135], N[(N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 8.2e-73], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.59999999999999987e135

   1. Initial program 47.3%

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

      1. sqr-neg 47.3%

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

      2. sqr-neg 47.3%

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

      3. associate-*l* 47.3%

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

   3. Simplified 47.3%

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

   5. Taylor expanded in b around -inf 89.4%

      \[\leadsto \color{blue}{-1 \cdot \left(b \cdot \left(-0.5 \cdot \frac{c}{{b}^{2}} + 0.6666666666666666 \cdot \frac{1}{a}\right)\right)} \]

   6. Taylor expanded in c around 0 90.4%

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

   if -1.59999999999999987e135 < b < 8.20000000000000032e-73

   1. Initial program 81.8%

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

   if 8.20000000000000032e-73 < b

   1. Initial program 18.0%

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

      1. sqr-neg 18.0%

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

      2. sqr-neg 18.0%

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

      3. associate-*l* 18.0%

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

   3. Simplified 18.0%

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

   5. Taylor expanded in b around inf 85.1%

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

      1. *-commutative 85.1%

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

   7. Simplified 85.1%

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

4. Final simplification 84.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.6 \cdot 10^{+135}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 8.2 \cdot 10^{-73}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 86.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.7 \cdot 10^{+135}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 2.1 \cdot 10^{-71}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4.7e+135)
   (+ (* -0.6666666666666666 (/ b a)) (* 0.5 (/ c b)))
   (if (<= b 2.1e-71)
     (/ (- (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 <= -4.7e+135) {
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	} else if (b <= 2.1e-71) {
		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 <= (-4.7d+135)) then
        tmp = ((-0.6666666666666666d0) * (b / a)) + (0.5d0 * (c / b))
    else if (b <= 2.1d-71) 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 <= -4.7e+135) {
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	} else if (b <= 2.1e-71) {
		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 <= -4.7e+135:
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b))
	elif b <= 2.1e-71:
		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 <= -4.7e+135)
		tmp = Float64(Float64(-0.6666666666666666 * Float64(b / a)) + Float64(0.5 * Float64(c / b)));
	elseif (b <= 2.1e-71)
		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 <= -4.7e+135)
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	elseif (b <= 2.1e-71)
		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, -4.7e+135], N[(N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.1e-71], 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 < -4.6999999999999998e135

   1. Initial program 47.3%

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

      1. sqr-neg 47.3%

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

      2. sqr-neg 47.3%

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

      3. associate-*l* 47.3%

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

   3. Simplified 47.3%

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

   5. Taylor expanded in b around -inf 89.4%

      \[\leadsto \color{blue}{-1 \cdot \left(b \cdot \left(-0.5 \cdot \frac{c}{{b}^{2}} + 0.6666666666666666 \cdot \frac{1}{a}\right)\right)} \]

   6. Taylor expanded in c around 0 90.4%

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

   if -4.6999999999999998e135 < b < 2.1000000000000001e-71

   1. Initial program 81.8%

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

      1. sqr-neg 81.8%

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

      2. sqr-neg 81.8%

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

      3. associate-*l* 81.6%

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

   3. Simplified 81.6%

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

   if 2.1000000000000001e-71 < b

   1. Initial program 18.0%

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

      1. sqr-neg 18.0%

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

      2. sqr-neg 18.0%

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

      3. associate-*l* 18.0%

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

   3. Simplified 18.0%

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

   5. Taylor expanded in b around inf 85.1%

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

      1. *-commutative 85.1%

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

   7. Simplified 85.1%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.7 \cdot 10^{+135}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 2.1 \cdot 10^{-71}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.2 \cdot 10^{-68}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 2.1 \cdot 10^{-70}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4.2e-68)
   (+ (* -0.6666666666666666 (/ b a)) (* 0.5 (/ c b)))
   (if (<= b 2.1e-70)
     (/ (- (sqrt (* a (* c -3.0))) b) (* a 3.0))
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.2e-68) {
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	} else if (b <= 2.1e-70) {
		tmp = (sqrt((a * (c * -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 <= (-4.2d-68)) then
        tmp = ((-0.6666666666666666d0) * (b / a)) + (0.5d0 * (c / b))
    else if (b <= 2.1d-70) then
        tmp = (sqrt((a * (c * (-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 <= -4.2e-68) {
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	} else if (b <= 2.1e-70) {
		tmp = (Math.sqrt((a * (c * -3.0))) - b) / (a * 3.0);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4.2e-68:
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b))
	elif b <= 2.1e-70:
		tmp = (math.sqrt((a * (c * -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 <= -4.2e-68)
		tmp = Float64(Float64(-0.6666666666666666 * Float64(b / a)) + Float64(0.5 * Float64(c / b)));
	elseif (b <= 2.1e-70)
		tmp = Float64(Float64(sqrt(Float64(a * Float64(c * -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 <= -4.2e-68)
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	elseif (b <= 2.1e-70)
		tmp = (sqrt((a * (c * -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, -4.2e-68], N[(N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.1e-70], N[(N[(N[Sqrt[N[(a * N[(c * -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 < -4.20000000000000016e-68

   1. Initial program 63.3%

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

      1. sqr-neg 63.3%

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

      2. sqr-neg 63.3%

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

      3. associate-*l* 63.3%

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

   3. Simplified 63.3%

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

   5. Taylor expanded in b around -inf 80.7%

      \[\leadsto \color{blue}{-1 \cdot \left(b \cdot \left(-0.5 \cdot \frac{c}{{b}^{2}} + 0.6666666666666666 \cdot \frac{1}{a}\right)\right)} \]

   6. Taylor expanded in c around 0 81.4%

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

   if -4.20000000000000016e-68 < b < 2.1000000000000001e-70

   1. Initial program 80.9%

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

      1. sqr-neg 80.9%

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

      2. sqr-neg 80.9%

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

      3. associate-*l* 80.6%

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

   3. Simplified 80.6%

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

   5. Taylor expanded in b around 0 74.5%

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

      1. *-commutative 74.5%

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

      2. associate-*l* 74.7%

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

      3. *-commutative 74.7%

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

      4. *-commutative 74.7%

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

   7. Simplified 74.7%

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

   if 2.1000000000000001e-70 < b

   1. Initial program 18.0%

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

      1. sqr-neg 18.0%

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

      2. sqr-neg 18.0%

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

      3. associate-*l* 18.0%

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

   3. Simplified 18.0%

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

   5. Taylor expanded in b around inf 85.1%

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

      1. *-commutative 85.1%

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

   7. Simplified 85.1%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.2 \cdot 10^{-68}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 2.1 \cdot 10^{-70}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.4 \cdot 10^{-75}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 1.8 \cdot 10^{-70}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{b - \sqrt{\left(a \cdot c\right) \cdot -3}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.4e-75)
   (+ (* -0.6666666666666666 (/ b a)) (* 0.5 (/ c b)))
   (if (<= b 1.8e-70)
     (* -0.3333333333333333 (/ (- b (sqrt (* (* a c) -3.0))) a))
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.4e-75) {
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	} else if (b <= 1.8e-70) {
		tmp = -0.3333333333333333 * ((b - sqrt(((a * c) * -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 <= (-2.4d-75)) then
        tmp = ((-0.6666666666666666d0) * (b / a)) + (0.5d0 * (c / b))
    else if (b <= 1.8d-70) then
        tmp = (-0.3333333333333333d0) * ((b - sqrt(((a * c) * (-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 <= -2.4e-75) {
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	} else if (b <= 1.8e-70) {
		tmp = -0.3333333333333333 * ((b - Math.sqrt(((a * c) * -3.0))) / a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.4e-75:
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b))
	elif b <= 1.8e-70:
		tmp = -0.3333333333333333 * ((b - math.sqrt(((a * c) * -3.0))) / a)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.4e-75)
		tmp = Float64(Float64(-0.6666666666666666 * Float64(b / a)) + Float64(0.5 * Float64(c / b)));
	elseif (b <= 1.8e-70)
		tmp = Float64(-0.3333333333333333 * Float64(Float64(b - sqrt(Float64(Float64(a * c) * -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 <= -2.4e-75)
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	elseif (b <= 1.8e-70)
		tmp = -0.3333333333333333 * ((b - sqrt(((a * c) * -3.0))) / a);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.4e-75], N[(N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.8e-70], N[(-0.3333333333333333 * N[(N[(b - N[Sqrt[N[(N[(a * c), $MachinePrecision] * -3.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.40000000000000019e-75

   1. Initial program 63.3%

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

      1. sqr-neg 63.3%

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

      2. sqr-neg 63.3%

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

      3. associate-*l* 63.3%

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

   3. Simplified 63.3%

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

   5. Taylor expanded in b around -inf 80.7%

      \[\leadsto \color{blue}{-1 \cdot \left(b \cdot \left(-0.5 \cdot \frac{c}{{b}^{2}} + 0.6666666666666666 \cdot \frac{1}{a}\right)\right)} \]

   6. Taylor expanded in c around 0 81.4%

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

   if -2.40000000000000019e-75 < b < 1.8000000000000001e-70

   1. Initial program 80.9%

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

      1. sqr-neg 80.9%

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

      2. sqr-neg 80.9%

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

      3. associate-*l* 80.6%

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

   3. Simplified 80.6%

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

   5. Taylor expanded in b around 0 74.5%

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

      1. *-un-lft-identity 74.5%

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

      2. frac-2neg 74.5%

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

      3. distribute-neg-in 74.5%

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

      4. add-sqr-sqrt 37.9%

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

      5. sqrt-unprod 73.8%

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

      6. sqr-neg 73.8%

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

      7. sqrt-unprod 36.7%

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

      8. add-sqr-sqrt 73.1%

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

      9. sub-neg 73.1%

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

      10. add-sqr-sqrt 36.5%

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

      11. sqrt-unprod 73.2%

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

      12. sqr-neg 73.2%

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

      13. sqrt-unprod 36.7%

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

      14. add-sqr-sqrt 74.5%

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

      15. distribute-lft-neg-in 74.5%

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

      16. metadata-eval 74.5%

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

   7. Applied egg-rr 74.5%

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

      1. associate-*r/ 74.5%

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

      2. times-frac 74.7%

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

      3. metadata-eval 74.7%

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

      4. *-commutative 74.7%

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

   9. Simplified 74.7%

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

   if 1.8000000000000001e-70 < b

   1. Initial program 18.0%

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

      1. sqr-neg 18.0%

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

      2. sqr-neg 18.0%

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

      3. associate-*l* 18.0%

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

   3. Simplified 18.0%

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

   5. Taylor expanded in b around inf 85.1%

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

      1. *-commutative 85.1%

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

   7. Simplified 85.1%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.4 \cdot 10^{-75}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 1.8 \cdot 10^{-70}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{b - \sqrt{\left(a \cdot c\right) \cdot -3}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 68.3% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{-310}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4e-310)
   (+ (* -0.6666666666666666 (/ b a)) (* 0.5 (/ c b)))
   (* (/ c b) -0.5)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4e-310) {
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	} 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 <= (-4d-310)) then
        tmp = ((-0.6666666666666666d0) * (b / a)) + (0.5d0 * (c / b))
    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 <= -4e-310) {
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4e-310:
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b))
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4e-310)
		tmp = Float64(Float64(-0.6666666666666666 * Float64(b / a)) + Float64(0.5 * Float64(c / b)));
	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 <= -4e-310)
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4e-310], N[(N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -3.999999999999988e-310

   1. Initial program 69.7%

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

      1. sqr-neg 69.7%

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

      2. sqr-neg 69.7%

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

      3. associate-*l* 69.6%

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

   3. Simplified 69.6%

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

   5. Taylor expanded in b around -inf 62.5%

      \[\leadsto \color{blue}{-1 \cdot \left(b \cdot \left(-0.5 \cdot \frac{c}{{b}^{2}} + 0.6666666666666666 \cdot \frac{1}{a}\right)\right)} \]

   6. Taylor expanded in c around 0 65.5%

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

   if -3.999999999999988e-310 < b

   1. Initial program 33.9%

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

      1. sqr-neg 33.9%

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

      2. sqr-neg 33.9%

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

      3. associate-*l* 33.8%

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

   3. Simplified 33.8%

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

   5. Taylor expanded in b around inf 66.2%

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

      1. *-commutative 66.2%

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

   7. Simplified 66.2%

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

Alternative 6: 68.2% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 10^{-309}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1e-309) (* -0.6666666666666666 (/ b a)) (* (/ c b) -0.5)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1e-309) {
		tmp = -0.6666666666666666 * (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 <= 1d-309) then
        tmp = (-0.6666666666666666d0) * (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 <= 1e-309) {
		tmp = -0.6666666666666666 * (b / a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 1e-309:
		tmp = -0.6666666666666666 * (b / a)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1e-309)
		tmp = Float64(-0.6666666666666666 * Float64(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 <= 1e-309)
		tmp = -0.6666666666666666 * (b / a);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1e-309], N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.000000000000002e-309

   1. Initial program 69.7%

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

      1. sqr-neg 69.7%

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

      2. sqr-neg 69.7%

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

      3. associate-*l* 69.6%

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

   3. Simplified 69.6%

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

   5. Taylor expanded in b around -inf 64.9%

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

      1. *-commutative 64.9%

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

   7. Simplified 64.9%

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

   if 1.000000000000002e-309 < b

   1. Initial program 33.9%

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

      1. sqr-neg 33.9%

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

      2. sqr-neg 33.9%

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

      3. associate-*l* 33.8%

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

   3. Simplified 33.8%

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

   5. Taylor expanded in b around inf 66.2%

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

      1. *-commutative 66.2%

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

   7. Simplified 66.2%

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

4. Final simplification 65.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 10^{-309}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.1% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{-310}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4e-310) (* -0.6666666666666666 (/ b a)) (* c (/ -0.5 b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4e-310) {
		tmp = -0.6666666666666666 * (b / a);
	} else {
		tmp = c * (-0.5 / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-4d-310)) then
        tmp = (-0.6666666666666666d0) * (b / a)
    else
        tmp = c * ((-0.5d0) / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -4e-310) {
		tmp = -0.6666666666666666 * (b / a);
	} else {
		tmp = c * (-0.5 / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4e-310:
		tmp = -0.6666666666666666 * (b / a)
	else:
		tmp = c * (-0.5 / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4e-310)
		tmp = Float64(-0.6666666666666666 * Float64(b / 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 <= -4e-310)
		tmp = -0.6666666666666666 * (b / a);
	else
		tmp = c * (-0.5 / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4e-310], N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision], N[(c * N[(-0.5 / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -3.999999999999988e-310

   1. Initial program 69.7%

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

      1. sqr-neg 69.7%

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

      2. sqr-neg 69.7%

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

      3. associate-*l* 69.6%

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

   3. Simplified 69.6%

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

   5. Taylor expanded in b around -inf 64.9%

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

      1. *-commutative 64.9%

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

   7. Simplified 64.9%

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

   if -3.999999999999988e-310 < b

   1. Initial program 33.9%

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

      1. sqr-neg 33.9%

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

      2. sqr-neg 33.9%

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

      3. associate-*l* 33.8%

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

   3. Simplified 33.8%

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

   5. Taylor expanded in b around inf 66.2%

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

      1. *-commutative 66.2%

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

   7. Simplified 66.2%

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

      1. *-commutative 66.2%

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

      2. clear-num 65.3%

         \[\leadsto -0.5 \cdot \color{blue}{\frac{1}{\frac{b}{c}}} \]

      3. un-div-inv 65.3%

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

   9. Applied egg-rr 65.3%

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

       1. associate-/r/ 66.1%

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

   11. Simplified 66.1%

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

4. Final simplification 65.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{-310}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 68.1% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{-310}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4e-310) (* b (/ -0.6666666666666666 a)) (* c (/ -0.5 b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4e-310) {
		tmp = b * (-0.6666666666666666 / a);
	} else {
		tmp = c * (-0.5 / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-4d-310)) then
        tmp = b * ((-0.6666666666666666d0) / a)
    else
        tmp = c * ((-0.5d0) / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -4e-310) {
		tmp = b * (-0.6666666666666666 / a);
	} else {
		tmp = c * (-0.5 / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4e-310:
		tmp = b * (-0.6666666666666666 / a)
	else:
		tmp = c * (-0.5 / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4e-310)
		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 <= -4e-310)
		tmp = b * (-0.6666666666666666 / a);
	else
		tmp = c * (-0.5 / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4e-310], 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 < -3.999999999999988e-310

   1. Initial program 69.7%

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

      1. sqr-neg 69.7%

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

      2. sqr-neg 69.7%

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

      3. associate-*l* 69.6%

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

   3. Simplified 69.6%

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

      1. frac-2neg 69.6%

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

      2. div-inv 69.6%

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

   6. Applied egg-rr 69.7%

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

      1. fma-undefine 69.7%

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

      2. unpow2 69.7%

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

      3. associate-*l* 69.6%

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

      4. *-commutative 69.6%

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

      5. +-commutative 69.6%

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

      6. fma-define 69.7%

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

      7. associate-/r* 69.6%

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

   8. Simplified 69.6%

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

   9. Taylor expanded in b around -inf 64.9%

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

       1. *-commutative 64.9%

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

       2. associate-*l/ 64.9%

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

       3. associate-*r/ 64.9%

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

   11. Simplified 64.9%

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

   if -3.999999999999988e-310 < b

   1. Initial program 33.9%

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

      1. sqr-neg 33.9%

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

      2. sqr-neg 33.9%

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

      3. associate-*l* 33.8%

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

   3. Simplified 33.8%

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

   5. Taylor expanded in b around inf 66.2%

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

      1. *-commutative 66.2%

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

   7. Simplified 66.2%

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

      1. *-commutative 66.2%

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

      2. clear-num 65.3%

         \[\leadsto -0.5 \cdot \color{blue}{\frac{1}{\frac{b}{c}}} \]

      3. un-div-inv 65.3%

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

   9. Applied egg-rr 65.3%

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

       1. associate-/r/ 66.1%

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

   11. Simplified 66.1%

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

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{-310}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 35.0% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.2%

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

   1. sqr-neg 52.2%

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

   2. sqr-neg 52.2%

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

   3. associate-*l* 52.1%

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

3. Simplified 52.1%

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

   1. frac-2neg 52.1%

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

   2. div-inv 52.1%

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

6. Applied egg-rr 52.2%

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

   1. fma-undefine 52.2%

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

   2. unpow2 52.2%

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

   3. associate-*l* 52.1%

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

   4. *-commutative 52.1%

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

   5. +-commutative 52.1%

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

   6. fma-define 52.1%

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

   7. associate-/r* 52.2%

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

8. Simplified 52.2%

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

9. Taylor expanded in b around -inf 34.5%

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

    1. *-commutative 34.5%

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

    2. associate-*l/ 34.5%

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

    3. associate-*r/ 34.5%

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

11. Simplified 34.5%

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

Reproduce

herbie shell --seed 2024135 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))