Cubic critical

Percentage Accurate: 51.3% → 84.6%

Time: 14.0s

Alternatives: 11

Speedup: 11.6×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 51.3% 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: 84.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -6.5 \cdot 10^{+31}:\\ \;\;\;\;\frac{\frac{b \cdot 2}{a}}{-3}\\ \mathbf{elif}\;b \leq 2.2 \cdot 10^{-50}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 3\right) \cdot c} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -6.5e+31)
   (/ (/ (* b 2.0) a) -3.0)
   (if (<= b 2.2e-50)
     (/ (- (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 <= -6.5e+31) {
		tmp = ((b * 2.0) / a) / -3.0;
	} else if (b <= 2.2e-50) {
		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 <= (-6.5d+31)) then
        tmp = ((b * 2.0d0) / a) / (-3.0d0)
    else if (b <= 2.2d-50) 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 <= -6.5e+31) {
		tmp = ((b * 2.0) / a) / -3.0;
	} else if (b <= 2.2e-50) {
		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 <= -6.5e+31:
		tmp = ((b * 2.0) / a) / -3.0
	elif b <= 2.2e-50:
		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 <= -6.5e+31)
		tmp = Float64(Float64(Float64(b * 2.0) / a) / -3.0);
	elseif (b <= 2.2e-50)
		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 <= -6.5e+31)
		tmp = ((b * 2.0) / a) / -3.0;
	elseif (b <= 2.2e-50)
		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, -6.5e+31], N[(N[(N[(b * 2.0), $MachinePrecision] / a), $MachinePrecision] / -3.0), $MachinePrecision], If[LessEqual[b, 2.2e-50], 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 < -6.5000000000000004e31

   1. Initial program 78.5%

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

      1. sqr-neg 78.5%

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

      2. sqr-neg 78.5%

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

      3. associate-*l* 78.5%

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

   3. Simplified 78.5%

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

   6. Applied egg-rr 57.7%

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

      1. associate-*l/ 57.7%

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

      2. un-div-inv 57.7%

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

   8. Applied egg-rr 57.7%

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

   9. Taylor expanded in b around -inf 98.5%

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

       1. *-commutative 98.5%

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

   11. Simplified 98.5%

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

   if -6.5000000000000004e31 < b < 2.1999999999999999e-50

   1. Initial program 74.2%

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

   if 2.1999999999999999e-50 < b

   1. Initial program 15.7%

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

      1. sqr-neg 15.7%

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

      2. sqr-neg 15.7%

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

      3. associate-*l* 15.7%

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

   3. Simplified 15.7%

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

   5. Taylor expanded in b around inf 84.4%

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

      1. *-commutative 84.4%

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

   7. Simplified 84.4%

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

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6.5 \cdot 10^{+31}:\\ \;\;\;\;\frac{\frac{b \cdot 2}{a}}{-3}\\ \mathbf{elif}\;b \leq 2.2 \cdot 10^{-50}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 3\right) \cdot c} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 84.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.7 \cdot 10^{+31}:\\ \;\;\;\;\frac{\frac{b \cdot 2}{a}}{-3}\\ \mathbf{elif}\;b \leq 5.1 \cdot 10^{-49}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.7e+31)
   (/ (/ (* b 2.0) a) -3.0)
   (if (<= b 5.1e-49)
     (/ (- (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 <= -2.7e+31) {
		tmp = ((b * 2.0) / a) / -3.0;
	} else if (b <= 5.1e-49) {
		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 <= (-2.7d+31)) then
        tmp = ((b * 2.0d0) / a) / (-3.0d0)
    else if (b <= 5.1d-49) 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 <= -2.7e+31) {
		tmp = ((b * 2.0) / a) / -3.0;
	} else if (b <= 5.1e-49) {
		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 <= -2.7e+31:
		tmp = ((b * 2.0) / a) / -3.0
	elif b <= 5.1e-49:
		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 <= -2.7e+31)
		tmp = Float64(Float64(Float64(b * 2.0) / a) / -3.0);
	elseif (b <= 5.1e-49)
		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 <= -2.7e+31)
		tmp = ((b * 2.0) / a) / -3.0;
	elseif (b <= 5.1e-49)
		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, -2.7e+31], N[(N[(N[(b * 2.0), $MachinePrecision] / a), $MachinePrecision] / -3.0), $MachinePrecision], If[LessEqual[b, 5.1e-49], 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 < -2.69999999999999986e31

   1. Initial program 78.5%

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

      1. sqr-neg 78.5%

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

      2. sqr-neg 78.5%

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

      3. associate-*l* 78.5%

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

   3. Simplified 78.5%

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

   6. Applied egg-rr 57.7%

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

      1. associate-*l/ 57.7%

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

      2. un-div-inv 57.7%

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

   8. Applied egg-rr 57.7%

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

   9. Taylor expanded in b around -inf 98.5%

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

       1. *-commutative 98.5%

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

   11. Simplified 98.5%

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

   if -2.69999999999999986e31 < b < 5.10000000000000026e-49

   1. Initial program 74.2%

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

      1. sqr-neg 74.2%

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

      2. sqr-neg 74.2%

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

      3. associate-*l* 74.2%

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

   3. Simplified 74.2%

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

   if 5.10000000000000026e-49 < b

   1. Initial program 15.7%

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

      1. sqr-neg 15.7%

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

      2. sqr-neg 15.7%

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

      3. associate-*l* 15.7%

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

   3. Simplified 15.7%

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

   5. Taylor expanded in b around inf 84.4%

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

      1. *-commutative 84.4%

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

   7. Simplified 84.4%

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

4. Final simplification 84.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.7 \cdot 10^{+31}:\\ \;\;\;\;\frac{\frac{b \cdot 2}{a}}{-3}\\ \mathbf{elif}\;b \leq 5.1 \cdot 10^{-49}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.45 \cdot 10^{+16}:\\ \;\;\;\;\frac{\frac{b \cdot 2}{a}}{-3}\\ \mathbf{elif}\;b \leq 2.6 \cdot 10^{-50}:\\ \;\;\;\;\frac{\frac{b \cdot 2 - \sqrt{a \cdot \left(-3 \cdot c\right)}}{a}}{-3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.45e+16)
   (/ (/ (* b 2.0) a) -3.0)
   (if (<= b 2.6e-50)
     (/ (/ (- (* b 2.0) (sqrt (* a (* -3.0 c)))) a) -3.0)
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.45e+16) {
		tmp = ((b * 2.0) / a) / -3.0;
	} else if (b <= 2.6e-50) {
		tmp = (((b * 2.0) - sqrt((a * (-3.0 * c)))) / 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 <= (-1.45d+16)) then
        tmp = ((b * 2.0d0) / a) / (-3.0d0)
    else if (b <= 2.6d-50) then
        tmp = (((b * 2.0d0) - sqrt((a * ((-3.0d0) * c)))) / 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 <= -1.45e+16) {
		tmp = ((b * 2.0) / a) / -3.0;
	} else if (b <= 2.6e-50) {
		tmp = (((b * 2.0) - Math.sqrt((a * (-3.0 * c)))) / a) / -3.0;
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.45e+16:
		tmp = ((b * 2.0) / a) / -3.0
	elif b <= 2.6e-50:
		tmp = (((b * 2.0) - math.sqrt((a * (-3.0 * c)))) / a) / -3.0
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.45e+16)
		tmp = Float64(Float64(Float64(b * 2.0) / a) / -3.0);
	elseif (b <= 2.6e-50)
		tmp = Float64(Float64(Float64(Float64(b * 2.0) - sqrt(Float64(a * Float64(-3.0 * c)))) / 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 <= -1.45e+16)
		tmp = ((b * 2.0) / a) / -3.0;
	elseif (b <= 2.6e-50)
		tmp = (((b * 2.0) - sqrt((a * (-3.0 * c)))) / a) / -3.0;
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.45e+16], N[(N[(N[(b * 2.0), $MachinePrecision] / a), $MachinePrecision] / -3.0), $MachinePrecision], If[LessEqual[b, 2.6e-50], N[(N[(N[(N[(b * 2.0), $MachinePrecision] - N[Sqrt[N[(a * N[(-3.0 * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] / -3.0), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.45e16

   1. Initial program 79.6%

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

      1. sqr-neg 79.6%

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

      2. sqr-neg 79.6%

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

      3. associate-*l* 79.6%

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

   3. Simplified 79.6%

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

   6. Applied egg-rr 57.3%

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

      1. associate-*l/ 57.3%

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

      2. un-div-inv 57.3%

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

   8. Applied egg-rr 57.3%

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

   9. Taylor expanded in b around -inf 97.3%

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

       1. *-commutative 97.3%

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

   11. Simplified 97.3%

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

   if -1.45e16 < b < 2.6000000000000001e-50

   1. Initial program 73.1%

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

      1. sqr-neg 73.1%

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

      2. sqr-neg 73.1%

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

      3. associate-*l* 73.1%

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

   3. Simplified 73.1%

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

   6. Applied egg-rr 71.5%

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

      1. associate-*l/ 71.4%

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

      2. un-div-inv 71.5%

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

   8. Applied egg-rr 71.5%

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

      1. div-inv 71.4%

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

   10. Applied egg-rr 70.2%

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

       1. associate-*r/ 70.3%

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

       2. *-rgt-identity 70.3%

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

       3. associate-+r- 70.3%

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

       4. count-2 70.3%

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

   12. Simplified 70.3%

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

   if 2.6000000000000001e-50 < b

   1. Initial program 15.7%

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

      1. sqr-neg 15.7%

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

      2. sqr-neg 15.7%

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

      3. associate-*l* 15.7%

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

   3. Simplified 15.7%

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

   5. Taylor expanded in b around inf 84.4%

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

      1. *-commutative 84.4%

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

   7. Simplified 84.4%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.45 \cdot 10^{+16}:\\ \;\;\;\;\frac{\frac{b \cdot 2}{a}}{-3}\\ \mathbf{elif}\;b \leq 2.6 \cdot 10^{-50}:\\ \;\;\;\;\frac{\frac{b \cdot 2 - \sqrt{a \cdot \left(-3 \cdot c\right)}}{a}}{-3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.05 \cdot 10^{-58}:\\ \;\;\;\;\frac{\frac{b \cdot 2}{a}}{-3}\\ \mathbf{elif}\;b \leq 4.4 \cdot 10^{-51}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.05e-58)
   (/ (/ (* b 2.0) a) -3.0)
   (if (<= b 4.4e-51)
     (/ (- (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 <= -2.05e-58) {
		tmp = ((b * 2.0) / a) / -3.0;
	} else if (b <= 4.4e-51) {
		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 <= (-2.05d-58)) then
        tmp = ((b * 2.0d0) / a) / (-3.0d0)
    else if (b <= 4.4d-51) 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 <= -2.05e-58) {
		tmp = ((b * 2.0) / a) / -3.0;
	} else if (b <= 4.4e-51) {
		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 <= -2.05e-58:
		tmp = ((b * 2.0) / a) / -3.0
	elif b <= 4.4e-51:
		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 <= -2.05e-58)
		tmp = Float64(Float64(Float64(b * 2.0) / a) / -3.0);
	elseif (b <= 4.4e-51)
		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 <= -2.05e-58)
		tmp = ((b * 2.0) / a) / -3.0;
	elseif (b <= 4.4e-51)
		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, -2.05e-58], N[(N[(N[(b * 2.0), $MachinePrecision] / a), $MachinePrecision] / -3.0), $MachinePrecision], If[LessEqual[b, 4.4e-51], 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 < -2.05000000000000014e-58

   1. Initial program 78.5%

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

      1. sqr-neg 78.5%

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

      2. sqr-neg 78.5%

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

      3. associate-*l* 78.6%

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

   3. Simplified 78.6%

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

   6. Applied egg-rr 58.6%

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

      1. associate-*l/ 58.5%

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

      2. un-div-inv 58.6%

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

   8. Applied egg-rr 58.6%

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

   9. Taylor expanded in b around -inf 86.8%

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

       1. *-commutative 86.8%

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

   11. Simplified 86.8%

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

   if -2.05000000000000014e-58 < b < 4.4e-51

   1. Initial program 72.8%

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

      1. sqr-neg 72.8%

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

      2. sqr-neg 72.8%

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

      3. associate-*l* 72.8%

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

   3. Simplified 72.8%

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

   5. Taylor expanded in b around 0 68.0%

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

      1. associate-*r* 68.1%

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

      2. *-commutative 68.1%

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

   7. Simplified 68.1%

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

   if 4.4e-51 < b

   1. Initial program 15.7%

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

      1. sqr-neg 15.7%

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

      2. sqr-neg 15.7%

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

      3. associate-*l* 15.7%

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

   3. Simplified 15.7%

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

   5. Taylor expanded in b around inf 84.4%

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

      1. *-commutative 84.4%

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

   7. Simplified 84.4%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.05 \cdot 10^{-58}:\\ \;\;\;\;\frac{\frac{b \cdot 2}{a}}{-3}\\ \mathbf{elif}\;b \leq 4.4 \cdot 10^{-51}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 80.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5.3 \cdot 10^{-64}:\\ \;\;\;\;\frac{\frac{b \cdot 2}{a}}{-3}\\ \mathbf{elif}\;b \leq 4.6 \cdot 10^{-49}:\\ \;\;\;\;\frac{\sqrt{-3 \cdot \left(a \cdot c\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5.3e-64)
   (/ (/ (* b 2.0) a) -3.0)
   (if (<= b 4.6e-49)
     (/ (- (sqrt (* -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 <= -5.3e-64) {
		tmp = ((b * 2.0) / a) / -3.0;
	} else if (b <= 4.6e-49) {
		tmp = (sqrt((-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 <= (-5.3d-64)) then
        tmp = ((b * 2.0d0) / a) / (-3.0d0)
    else if (b <= 4.6d-49) then
        tmp = (sqrt(((-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 <= -5.3e-64) {
		tmp = ((b * 2.0) / a) / -3.0;
	} else if (b <= 4.6e-49) {
		tmp = (Math.sqrt((-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 <= -5.3e-64:
		tmp = ((b * 2.0) / a) / -3.0
	elif b <= 4.6e-49:
		tmp = (math.sqrt((-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 <= -5.3e-64)
		tmp = Float64(Float64(Float64(b * 2.0) / a) / -3.0);
	elseif (b <= 4.6e-49)
		tmp = Float64(Float64(sqrt(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 <= -5.3e-64)
		tmp = ((b * 2.0) / a) / -3.0;
	elseif (b <= 4.6e-49)
		tmp = (sqrt((-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, -5.3e-64], N[(N[(N[(b * 2.0), $MachinePrecision] / a), $MachinePrecision] / -3.0), $MachinePrecision], If[LessEqual[b, 4.6e-49], N[(N[(N[Sqrt[N[(-3.0 * N[(a * 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 < -5.3000000000000002e-64

   1. Initial program 78.5%

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

      1. sqr-neg 78.5%

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

      2. sqr-neg 78.5%

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

      3. associate-*l* 78.6%

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

   3. Simplified 78.6%

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

   6. Applied egg-rr 58.6%

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

      1. associate-*l/ 58.5%

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

      2. un-div-inv 58.6%

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

   8. Applied egg-rr 58.6%

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

   9. Taylor expanded in b around -inf 86.8%

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

       1. *-commutative 86.8%

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

   11. Simplified 86.8%

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

   if -5.3000000000000002e-64 < b < 4.5999999999999998e-49

   1. Initial program 72.8%

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

      1. sqr-neg 72.8%

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

      2. sqr-neg 72.8%

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

      3. associate-*l* 72.8%

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

   3. Simplified 72.8%

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

   5. Taylor expanded in b around 0 68.0%

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

      1. associate-*r* 68.1%

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

      2. *-commutative 68.1%

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

   7. Simplified 68.1%

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

      1. +-commutative 68.1%

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

      2. associate-*r* 68.0%

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

      3. sqrt-prod 41.6%

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

      4. *-commutative 41.6%

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

      5. sqrt-prod 68.0%

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

      6. unsub-neg 68.0%

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

   9. Applied egg-rr 68.0%

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

   10. Taylor expanded in a around 0 68.0%

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

   if 4.5999999999999998e-49 < b

   1. Initial program 15.7%

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

      1. sqr-neg 15.7%

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

      2. sqr-neg 15.7%

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

      3. associate-*l* 15.7%

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

   3. Simplified 15.7%

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

   5. Taylor expanded in b around inf 84.4%

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

      1. *-commutative 84.4%

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

   7. Simplified 84.4%

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

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.3 \cdot 10^{-64}:\\ \;\;\;\;\frac{\frac{b \cdot 2}{a}}{-3}\\ \mathbf{elif}\;b \leq 4.6 \cdot 10^{-49}:\\ \;\;\;\;\frac{\sqrt{-3 \cdot \left(a \cdot c\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -7.2 \cdot 10^{-62}:\\ \;\;\;\;\frac{\frac{b \cdot 2}{a}}{-3}\\ \mathbf{elif}\;b \leq 1.28 \cdot 10^{-49}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{b - \sqrt{a \cdot \left(-3 \cdot c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -7.2e-62)
   (/ (/ (* b 2.0) a) -3.0)
   (if (<= b 1.28e-49)
     (* -0.3333333333333333 (/ (- b (sqrt (* a (* -3.0 c)))) a))
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -7.2e-62) {
		tmp = ((b * 2.0) / a) / -3.0;
	} else if (b <= 1.28e-49) {
		tmp = -0.3333333333333333 * ((b - sqrt((a * (-3.0 * c)))) / 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 <= (-7.2d-62)) then
        tmp = ((b * 2.0d0) / a) / (-3.0d0)
    else if (b <= 1.28d-49) then
        tmp = (-0.3333333333333333d0) * ((b - sqrt((a * ((-3.0d0) * c)))) / 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 <= -7.2e-62) {
		tmp = ((b * 2.0) / a) / -3.0;
	} else if (b <= 1.28e-49) {
		tmp = -0.3333333333333333 * ((b - Math.sqrt((a * (-3.0 * c)))) / a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -7.2e-62:
		tmp = ((b * 2.0) / a) / -3.0
	elif b <= 1.28e-49:
		tmp = -0.3333333333333333 * ((b - math.sqrt((a * (-3.0 * c)))) / a)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -7.2e-62)
		tmp = Float64(Float64(Float64(b * 2.0) / a) / -3.0);
	elseif (b <= 1.28e-49)
		tmp = Float64(-0.3333333333333333 * Float64(Float64(b - sqrt(Float64(a * Float64(-3.0 * c)))) / 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 <= -7.2e-62)
		tmp = ((b * 2.0) / a) / -3.0;
	elseif (b <= 1.28e-49)
		tmp = -0.3333333333333333 * ((b - sqrt((a * (-3.0 * c)))) / a);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -7.2e-62], N[(N[(N[(b * 2.0), $MachinePrecision] / a), $MachinePrecision] / -3.0), $MachinePrecision], If[LessEqual[b, 1.28e-49], N[(-0.3333333333333333 * N[(N[(b - N[Sqrt[N[(a * N[(-3.0 * c), $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 < -7.1999999999999999e-62

   1. Initial program 78.5%

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

      1. sqr-neg 78.5%

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

      2. sqr-neg 78.5%

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

      3. associate-*l* 78.6%

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

   3. Simplified 78.6%

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

   6. Applied egg-rr 58.6%

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

      1. associate-*l/ 58.5%

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

      2. un-div-inv 58.6%

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

   8. Applied egg-rr 58.6%

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

   9. Taylor expanded in b around -inf 86.8%

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

       1. *-commutative 86.8%

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

   11. Simplified 86.8%

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

   if -7.1999999999999999e-62 < b < 1.28e-49

   1. Initial program 72.8%

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

      1. sqr-neg 72.8%

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

      2. sqr-neg 72.8%

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

      3. associate-*l* 72.8%

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

   3. Simplified 72.8%

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

   5. Taylor expanded in b around 0 68.0%

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

      1. associate-*r* 68.1%

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

      2. *-commutative 68.1%

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

   7. Simplified 68.1%

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

      1. +-commutative 68.1%

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

      2. associate-*r* 68.0%

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

      3. sqrt-prod 41.6%

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

      4. *-commutative 41.6%

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

      5. sqrt-prod 68.0%

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

      6. unsub-neg 68.0%

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

   9. Applied egg-rr 68.0%

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

       1. frac-2neg 68.0%

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

       2. div-inv 67.9%

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

       3. sub-neg 67.9%

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

       4. +-commutative 67.9%

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

       5. distribute-neg-in 67.9%

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

       6. add-sqr-sqrt 37.9%

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

       7. sqrt-unprod 67.3%

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

       8. sqr-neg 67.3%

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

       9. sqrt-unprod 30.1%

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

       10. add-sqr-sqrt 66.6%

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

       11. sub-neg 66.6%

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

       12. add-sqr-sqrt 36.5%

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

       13. sqrt-unprod 66.7%

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

       14. sqr-neg 66.7%

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

       15. sqrt-unprod 30.0%

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

       16. add-sqr-sqrt 67.9%

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

       17. *-commutative 67.9%

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

       18. distribute-rgt-neg-in 67.9%

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

       19. metadata-eval 67.9%

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

   11. Applied egg-rr 67.9%

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

       1. associate-*r/ 68.0%

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

       2. *-rgt-identity 68.0%

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

       3. metadata-eval 68.0%

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

       4. distribute-rgt-neg-in 68.0%

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

       5. distribute-neg-frac2 68.0%

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

       6. distribute-neg-frac 68.0%

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

       7. neg-mul-1 68.0%

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

       8. *-commutative 68.0%

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

       9. times-frac 67.9%

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

       10. metadata-eval 67.9%

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

   13. Simplified 67.9%

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

   if 1.28e-49 < b

   1. Initial program 15.7%

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

      1. sqr-neg 15.7%

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

      2. sqr-neg 15.7%

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

      3. associate-*l* 15.7%

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

   3. Simplified 15.7%

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

   5. Taylor expanded in b around inf 84.4%

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

      1. *-commutative 84.4%

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

   7. Simplified 84.4%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -7.2 \cdot 10^{-62}:\\ \;\;\;\;\frac{\frac{b \cdot 2}{a}}{-3}\\ \mathbf{elif}\;b \leq 1.28 \cdot 10^{-49}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{b - \sqrt{a \cdot \left(-3 \cdot c\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 68.0% accurate, 9.7× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2e-310) (/ (/ (* b 2.0) a) -3.0) (* (/ c b) -0.5)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e-310) {
		tmp = ((b * 2.0) / 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 <= (-2d-310)) then
        tmp = ((b * 2.0d0) / 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 <= -2e-310) {
		tmp = ((b * 2.0) / a) / -3.0;
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2e-310:
		tmp = ((b * 2.0) / a) / -3.0
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2e-310)
		tmp = Float64(Float64(Float64(b * 2.0) / 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 <= -2e-310)
		tmp = ((b * 2.0) / a) / -3.0;
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2e-310], N[(N[(N[(b * 2.0), $MachinePrecision] / a), $MachinePrecision] / -3.0), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.999999999999994e-310

   1. Initial program 79.8%

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

      1. sqr-neg 79.8%

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

      2. sqr-neg 79.8%

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

      3. associate-*l* 79.8%

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

   3. Simplified 79.8%

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

   6. Applied egg-rr 66.0%

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

      1. associate-*l/ 65.9%

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

      2. un-div-inv 66.0%

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

   8. Applied egg-rr 66.0%

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

   9. Taylor expanded in b around -inf 67.5%

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

       1. *-commutative 67.5%

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

   11. Simplified 67.5%

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

   if -1.999999999999994e-310 < b

   1. Initial program 29.1%

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

      1. sqr-neg 29.1%

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

      2. sqr-neg 29.1%

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

      3. associate-*l* 29.1%

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

   3. Simplified 29.1%

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

   5. Taylor expanded in b around inf 67.5%

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

      1. *-commutative 67.5%

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

   7. Simplified 67.5%

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

Alternative 8: 68.0% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3 \cdot 10^{-309}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 3e-309) (/ b (* a -1.5)) (* (/ c b) -0.5)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 3e-309) {
		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 <= 3d-309) 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 <= 3e-309) {
		tmp = b / (a * -1.5);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 3e-309:
		tmp = b / (a * -1.5)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 3e-309)
		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 <= 3e-309)
		tmp = b / (a * -1.5);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 3e-309], 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 < 3.000000000000001e-309

   1. Initial program 79.8%

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

      1. add-cube-cbrt 78.9%

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

      2. pow3 79.0%

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

   4. Applied egg-rr 79.0%

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

   5. Taylor expanded in b around -inf 67.1%

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

      1. associate-*r/ 67.1%

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

      2. *-commutative 67.1%

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

      3. rem-cube-cbrt 67.5%

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

      4. times-frac 67.4%

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

      5. metadata-eval 67.4%

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

      6. associate-*l/ 67.3%

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

      7. associate-/l* 67.4%

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

   7. Simplified 67.4%

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

      1. clear-num 67.2%

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

      2. un-div-inv 67.3%

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

      3. div-inv 67.5%

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

      4. metadata-eval 67.5%

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

   9. Applied egg-rr 67.5%

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

   if 3.000000000000001e-309 < b

   1. Initial program 29.1%

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

      1. sqr-neg 29.1%

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

      2. sqr-neg 29.1%

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

      3. associate-*l* 29.1%

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

   3. Simplified 29.1%

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

   5. Taylor expanded in b around inf 67.5%

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

      1. *-commutative 67.5%

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

   7. Simplified 67.5%

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

Alternative 9: 68.0% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2e-310) (* b (/ -0.6666666666666666 a)) (* (/ c b) -0.5)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e-310) {
		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 <= (-2d-310)) 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 <= -2e-310) {
		tmp = b * (-0.6666666666666666 / a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2e-310:
		tmp = b * (-0.6666666666666666 / a)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2e-310)
		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 <= -2e-310)
		tmp = b * (-0.6666666666666666 / a);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2e-310], 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 < -1.999999999999994e-310

   1. Initial program 79.8%

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

      1. add-cube-cbrt 78.9%

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

      2. pow3 79.0%

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

   4. Applied egg-rr 79.0%

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

   5. Taylor expanded in b around -inf 67.1%

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

      1. associate-*r/ 67.1%

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

      2. *-commutative 67.1%

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

      3. rem-cube-cbrt 67.5%

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

      4. times-frac 67.4%

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

      5. metadata-eval 67.4%

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

      6. associate-*l/ 67.3%

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

      7. associate-/l* 67.4%

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

   7. Simplified 67.4%

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

   if -1.999999999999994e-310 < b

   1. Initial program 29.1%

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

      1. sqr-neg 29.1%

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

      2. sqr-neg 29.1%

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

      3. associate-*l* 29.1%

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

   3. Simplified 29.1%

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

   5. Taylor expanded in b around inf 67.5%

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

      1. *-commutative 67.5%

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

   7. Simplified 67.5%

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

Alternative 10: 67.9% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-310}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{-0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2e-310) (* b (/ -0.6666666666666666 a)) (* c (/ -0.5 b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e-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 <= (-2d-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 <= -2e-310) {
		tmp = b * (-0.6666666666666666 / a);
	} else {
		tmp = c * (-0.5 / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2e-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 <= -2e-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 <= -2e-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, -2e-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 < -1.999999999999994e-310

   1. Initial program 79.8%

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

      1. add-cube-cbrt 78.9%

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

      2. pow3 79.0%

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

   4. Applied egg-rr 79.0%

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

   5. Taylor expanded in b around -inf 67.1%

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

      1. associate-*r/ 67.1%

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

      2. *-commutative 67.1%

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

      3. rem-cube-cbrt 67.5%

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

      4. times-frac 67.4%

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

      5. metadata-eval 67.4%

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

      6. associate-*l/ 67.3%

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

      7. associate-/l* 67.4%

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

   7. Simplified 67.4%

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

   if -1.999999999999994e-310 < b

   1. Initial program 29.1%

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

      1. sqr-neg 29.1%

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

      2. sqr-neg 29.1%

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

      3. associate-*l* 29.1%

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

   3. Simplified 29.1%

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

   5. Taylor expanded in b around inf 67.5%

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

      1. *-commutative 67.5%

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

   7. Simplified 67.5%

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

   8. Taylor expanded in c around 0 67.5%

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

      1. *-commutative 67.5%

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

      2. associate-*l/ 67.5%

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

      3. associate-*r/ 67.3%

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

   10. Simplified 67.3%

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

Alternative 11: 34.6% 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 55.3%

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

   1. add-cube-cbrt 54.6%

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

   2. pow3 54.7%

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

4. Applied egg-rr 54.7%

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

5. Taylor expanded in b around -inf 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. *-commutative 35.9%

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

   3. rem-cube-cbrt 36.1%

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

   4. times-frac 36.1%

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

   5. metadata-eval 36.1%

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

   6. associate-*l/ 36.1%

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

   7. associate-/l* 36.1%

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

7. Simplified 36.1%

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

Reproduce

herbie shell --seed 2024119 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))