Cubic critical

Percentage Accurate: 52.4% → 85.4%

Time: 15.5s

Alternatives: 12

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: 52.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{+126}:\\ \;\;\;\;\frac{\left(\frac{1.5 \cdot a}{\frac{b}{c}} - b\right) - b}{a \cdot 3}\\ \mathbf{elif}\;b \leq 2.6 \cdot 10^{-69}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4e+126)
   (/ (- (- (/ (* 1.5 a) (/ b c)) b) b) (* a 3.0))
   (if (<= b 2.6e-69)
     (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4e+126) {
		tmp = ((((1.5 * a) / (b / c)) - b) - b) / (a * 3.0);
	} else if (b <= 2.6e-69) {
		tmp = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-4d+126)) then
        tmp = ((((1.5d0 * a) / (b / c)) - b) - b) / (a * 3.0d0)
    else if (b <= 2.6d-69) then
        tmp = (sqrt(((b * b) - (c * (a * 3.0d0)))) - b) / (a * 3.0d0)
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -4e+126) {
		tmp = ((((1.5 * a) / (b / c)) - b) - b) / (a * 3.0);
	} else if (b <= 2.6e-69) {
		tmp = (Math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4e+126:
		tmp = ((((1.5 * a) / (b / c)) - b) - b) / (a * 3.0)
	elif b <= 2.6e-69:
		tmp = (math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4e+126)
		tmp = Float64(Float64(Float64(Float64(Float64(1.5 * a) / Float64(b / c)) - b) - b) / Float64(a * 3.0));
	elseif (b <= 2.6e-69)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -4e+126)
		tmp = ((((1.5 * a) / (b / c)) - b) - b) / (a * 3.0);
	elseif (b <= 2.6e-69)
		tmp = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4e+126], N[(N[(N[(N[(N[(1.5 * a), $MachinePrecision] / N[(b / c), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.6e-69], 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 * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.9999999999999997e126

   1. Initial program 53.2%

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

   3. Taylor expanded in b around -inf 91.4%

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

      1. +-commutative 91.4%

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

      2. mul-1-neg 91.4%

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

      3. unsub-neg 91.4%

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

      4. associate-/l* 97.1%

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

      5. associate-*r/ 97.1%

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

   5. Simplified 97.1%

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

   if -3.9999999999999997e126 < b < 2.6000000000000002e-69

   1. Initial program 84.2%

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

   if 2.6000000000000002e-69 < b

   1. Initial program 16.3%

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

   3. Taylor expanded in b around inf 88.5%

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

      1. associate-*r/ 88.5%

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

   5. Simplified 88.5%

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

4. Final simplification 89.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4 \cdot 10^{+126}:\\ \;\;\;\;\frac{\left(\frac{1.5 \cdot a}{\frac{b}{c}} - b\right) - b}{a \cdot 3}\\ \mathbf{elif}\;b \leq 2.6 \cdot 10^{-69}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 78.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -8.5 \cdot 10^{-177}:\\ \;\;\;\;\frac{\left(\frac{1.5 \cdot a}{\frac{b}{c}} - b\right) - b}{a \cdot 3}\\ \mathbf{elif}\;b \leq 2.6 \cdot 10^{-69}:\\ \;\;\;\;\frac{\sqrt{-3 \cdot \left(a \cdot c\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -8.5e-177)
   (/ (- (- (/ (* 1.5 a) (/ b c)) b) b) (* a 3.0))
   (if (<= b 2.6e-69)
     (/ (- (sqrt (* -3.0 (* a c))) b) (* a 3.0))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -8.5e-177) {
		tmp = ((((1.5 * a) / (b / c)) - b) - b) / (a * 3.0);
	} else if (b <= 2.6e-69) {
		tmp = (sqrt((-3.0 * (a * c))) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-8.5d-177)) then
        tmp = ((((1.5d0 * a) / (b / c)) - b) - b) / (a * 3.0d0)
    else if (b <= 2.6d-69) then
        tmp = (sqrt(((-3.0d0) * (a * c))) - b) / (a * 3.0d0)
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -8.5e-177) {
		tmp = ((((1.5 * a) / (b / c)) - b) - b) / (a * 3.0);
	} else if (b <= 2.6e-69) {
		tmp = (Math.sqrt((-3.0 * (a * c))) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -8.5e-177:
		tmp = ((((1.5 * a) / (b / c)) - b) - b) / (a * 3.0)
	elif b <= 2.6e-69:
		tmp = (math.sqrt((-3.0 * (a * c))) - b) / (a * 3.0)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -8.5e-177)
		tmp = Float64(Float64(Float64(Float64(Float64(1.5 * a) / Float64(b / c)) - b) - b) / Float64(a * 3.0));
	elseif (b <= 2.6e-69)
		tmp = Float64(Float64(sqrt(Float64(-3.0 * Float64(a * c))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -8.5e-177)
		tmp = ((((1.5 * a) / (b / c)) - b) - b) / (a * 3.0);
	elseif (b <= 2.6e-69)
		tmp = (sqrt((-3.0 * (a * c))) - b) / (a * 3.0);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -8.5e-177], N[(N[(N[(N[(N[(1.5 * a), $MachinePrecision] / N[(b / c), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.6e-69], N[(N[(N[Sqrt[N[(-3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -8.4999999999999993e-177

   1. Initial program 69.5%

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

   3. Taylor expanded in b around -inf 79.9%

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

      1. +-commutative 79.9%

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

      2. mul-1-neg 79.9%

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

      3. unsub-neg 79.9%

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

      4. associate-/l* 83.2%

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

      5. associate-*r/ 83.2%

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

   5. Simplified 83.2%

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

   if -8.4999999999999993e-177 < b < 2.6000000000000002e-69

   1. Initial program 78.3%

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

   3. Taylor expanded in b around 0 77.1%

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

   if 2.6000000000000002e-69 < b

   1. Initial program 16.3%

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

   3. Taylor expanded in b around inf 88.5%

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

      1. associate-*r/ 88.5%

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

   5. Simplified 88.5%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -8.5 \cdot 10^{-177}:\\ \;\;\;\;\frac{\left(\frac{1.5 \cdot a}{\frac{b}{c}} - b\right) - b}{a \cdot 3}\\ \mathbf{elif}\;b \leq 2.6 \cdot 10^{-69}:\\ \;\;\;\;\frac{\sqrt{-3 \cdot \left(a \cdot c\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 78.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -8.5 \cdot 10^{-177}:\\ \;\;\;\;\frac{\left(\frac{1.5 \cdot a}{\frac{b}{c}} - b\right) - b}{a \cdot 3}\\ \mathbf{elif}\;b \leq 3.9 \cdot 10^{-70}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -8.5e-177)
   (/ (- (- (/ (* 1.5 a) (/ b c)) b) b) (* a 3.0))
   (if (<= b 3.9e-70)
     (/ (- (sqrt (* c (* a -3.0))) b) (* a 3.0))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -8.5e-177) {
		tmp = ((((1.5 * a) / (b / c)) - b) - b) / (a * 3.0);
	} else if (b <= 3.9e-70) {
		tmp = (sqrt((c * (a * -3.0))) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-8.5d-177)) then
        tmp = ((((1.5d0 * a) / (b / c)) - b) - b) / (a * 3.0d0)
    else if (b <= 3.9d-70) then
        tmp = (sqrt((c * (a * (-3.0d0)))) - b) / (a * 3.0d0)
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -8.5e-177) {
		tmp = ((((1.5 * a) / (b / c)) - b) - b) / (a * 3.0);
	} else if (b <= 3.9e-70) {
		tmp = (Math.sqrt((c * (a * -3.0))) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -8.5e-177:
		tmp = ((((1.5 * a) / (b / c)) - b) - b) / (a * 3.0)
	elif b <= 3.9e-70:
		tmp = (math.sqrt((c * (a * -3.0))) - b) / (a * 3.0)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -8.5e-177)
		tmp = Float64(Float64(Float64(Float64(Float64(1.5 * a) / Float64(b / c)) - b) - b) / Float64(a * 3.0));
	elseif (b <= 3.9e-70)
		tmp = Float64(Float64(sqrt(Float64(c * Float64(a * -3.0))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -8.5e-177)
		tmp = ((((1.5 * a) / (b / c)) - b) - b) / (a * 3.0);
	elseif (b <= 3.9e-70)
		tmp = (sqrt((c * (a * -3.0))) - b) / (a * 3.0);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -8.5e-177], N[(N[(N[(N[(N[(1.5 * a), $MachinePrecision] / N[(b / c), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 3.9e-70], N[(N[(N[Sqrt[N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -8.4999999999999993e-177

   1. Initial program 69.5%

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

   3. Taylor expanded in b around -inf 79.9%

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

      1. +-commutative 79.9%

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

      2. mul-1-neg 79.9%

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

      3. unsub-neg 79.9%

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

      4. associate-/l* 83.2%

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

      5. associate-*r/ 83.2%

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

   5. Simplified 83.2%

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

   if -8.4999999999999993e-177 < b < 3.90000000000000019e-70

   1. Initial program 78.3%

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

   3. Taylor expanded in b around 0 77.1%

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

      1. associate-*r* 77.2%

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

      2. *-commutative 77.2%

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

      3. *-commutative 77.2%

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

   5. Simplified 77.2%

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

   if 3.90000000000000019e-70 < b

   1. Initial program 16.3%

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

   3. Taylor expanded in b around inf 88.5%

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

      1. associate-*r/ 88.5%

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

   5. Simplified 88.5%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -8.5 \cdot 10^{-177}:\\ \;\;\;\;\frac{\left(\frac{1.5 \cdot a}{\frac{b}{c}} - b\right) - b}{a \cdot 3}\\ \mathbf{elif}\;b \leq 3.9 \cdot 10^{-70}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 67.0% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{-0.3333333333333333}{a} \cdot \left(a \cdot \frac{c \cdot -1.5}{b} + b \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e-310)
   (* (/ -0.3333333333333333 a) (+ (* a (/ (* c -1.5) b)) (* b 2.0)))
   (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = (-0.3333333333333333 / a) * ((a * ((c * -1.5) / b)) + (b * 2.0));
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-5d-310)) then
        tmp = ((-0.3333333333333333d0) / a) * ((a * ((c * (-1.5d0)) / b)) + (b * 2.0d0))
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = (-0.3333333333333333 / a) * ((a * ((c * -1.5) / b)) + (b * 2.0));
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e-310:
		tmp = (-0.3333333333333333 / a) * ((a * ((c * -1.5) / b)) + (b * 2.0))
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e-310)
		tmp = Float64(Float64(-0.3333333333333333 / a) * Float64(Float64(a * Float64(Float64(c * -1.5) / b)) + Float64(b * 2.0)));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5e-310)
		tmp = (-0.3333333333333333 / a) * ((a * ((c * -1.5) / b)) + (b * 2.0));
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-310], N[(N[(-0.3333333333333333 / a), $MachinePrecision] * N[(N[(a * N[(N[(c * -1.5), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision] + N[(b * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 70.4%

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

   3. Taylor expanded in b around inf 2.0%

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

      1. associate-*r/ 2.0%

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

      2. clear-num 2.0%

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

   5. Applied egg-rr 2.0%

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

      1. associate-/r/ 2.0%

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

      2. *-commutative 2.0%

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

      3. associate-*l* 2.0%

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

   7. Simplified 2.0%

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

      1. div-inv 2.0%

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

      2. associate-+r+ 2.1%

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

      3. add-sqr-sqrt 10.9%

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

      4. sqrt-unprod 21.4%

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

      5. sqr-neg 21.4%

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

      6. sqrt-unprod 0.0%

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

      7. add-sqr-sqrt 0.9%

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

      8. associate-*l/ 0.9%

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

      9. *-un-lft-identity 0.9%

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

      10. *-un-lft-identity 0.9%

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

      11. times-frac 0.8%

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

      12. /-rgt-identity 0.8%

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

      13. add-sqr-sqrt 0.4%

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

      14. sqrt-unprod 24.0%

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

      15. *-commutative 24.0%

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

      16. *-commutative 24.0%

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

      17. swap-sqr 24.0%

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

      18. metadata-eval 24.0%

          \[\leadsto \left(\left(b + b\right) + a \cdot \frac{c \cdot -1.5}{b}\right) \cdot \frac{1}{\sqrt{\left(a \cdot a\right) \cdot \color{blue}{9}}} \]

      19. metadata-eval 24.0%

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

      20. swap-sqr 24.0%

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

      21. sqrt-unprod 30.5%

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

   9. Applied egg-rr 68.8%

      \[\leadsto \color{blue}{\left(\left(b + b\right) + a \cdot \frac{c \cdot -1.5}{b}\right) \cdot \left(\frac{1}{a} \cdot -0.3333333333333333\right)} \]
   10. Step-by-step derivation

       1. *-commutative 68.8%

          \[\leadsto \color{blue}{\left(\frac{1}{a} \cdot -0.3333333333333333\right) \cdot \left(\left(b + b\right) + a \cdot \frac{c \cdot -1.5}{b}\right)} \]

       2. associate-*l/ 68.8%

          \[\leadsto \color{blue}{\frac{1 \cdot -0.3333333333333333}{a}} \cdot \left(\left(b + b\right) + a \cdot \frac{c \cdot -1.5}{b}\right) \]

       3. metadata-eval 68.8%

          \[\leadsto \frac{\color{blue}{-0.3333333333333333}}{a} \cdot \left(\left(b + b\right) + a \cdot \frac{c \cdot -1.5}{b}\right) \]

       4. +-commutative 68.8%

          \[\leadsto \frac{-0.3333333333333333}{a} \cdot \color{blue}{\left(a \cdot \frac{c \cdot -1.5}{b} + \left(b + b\right)\right)} \]

       5. count-2 68.8%

          \[\leadsto \frac{-0.3333333333333333}{a} \cdot \left(a \cdot \frac{c \cdot -1.5}{b} + \color{blue}{2 \cdot b}\right) \]

       6. *-commutative 68.8%

          \[\leadsto \frac{-0.3333333333333333}{a} \cdot \left(a \cdot \frac{c \cdot -1.5}{b} + \color{blue}{b \cdot 2}\right) \]

   11. Simplified 68.8%

       \[\leadsto \color{blue}{\frac{-0.3333333333333333}{a} \cdot \left(a \cdot \frac{c \cdot -1.5}{b} + b \cdot 2\right)} \]

   if -4.999999999999985e-310 < b

   1. Initial program 34.2%

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

   3. Taylor expanded in b around inf 68.9%

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

      1. associate-*r/ 68.9%

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

   5. Simplified 68.9%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{-0.3333333333333333}{a} \cdot \left(a \cdot \frac{c \cdot -1.5}{b} + b \cdot 2\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 67.0% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{\left(\frac{1.5 \cdot a}{\frac{b}{c}} - b\right) - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e-310)
   (/ (- (- (/ (* 1.5 a) (/ b c)) b) b) (* a 3.0))
   (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = ((((1.5 * a) / (b / c)) - b) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-5d-310)) then
        tmp = ((((1.5d0 * a) / (b / c)) - b) - b) / (a * 3.0d0)
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = ((((1.5 * a) / (b / c)) - b) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e-310:
		tmp = ((((1.5 * a) / (b / c)) - b) - b) / (a * 3.0)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e-310)
		tmp = Float64(Float64(Float64(Float64(Float64(1.5 * a) / Float64(b / c)) - b) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5e-310)
		tmp = ((((1.5 * a) / (b / c)) - b) - b) / (a * 3.0);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-310], N[(N[(N[(N[(N[(1.5 * a), $MachinePrecision] / N[(b / c), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 70.4%

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

   3. Taylor expanded in b around -inf 66.2%

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

      1. +-commutative 66.2%

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

      2. mul-1-neg 66.2%

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

      3. unsub-neg 66.2%

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

      4. associate-/l* 68.9%

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

      5. associate-*r/ 68.9%

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

   5. Simplified 68.9%

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

   if -4.999999999999985e-310 < b

   1. Initial program 34.2%

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

   3. Taylor expanded in b around inf 68.9%

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

      1. associate-*r/ 68.9%

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

   5. Simplified 68.9%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{\left(\frac{1.5 \cdot a}{\frac{b}{c}} - b\right) - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 67.1% accurate, 7.2× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-5d-310)) then
        tmp = ((-0.6666666666666666d0) * (b / a)) + (0.5d0 * (c / b))
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 70.4%

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

   3. Taylor expanded in b around -inf 68.8%

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

   if -4.999999999999985e-310 < b

   1. Initial program 34.2%

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

   3. Taylor expanded in b around inf 68.9%

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

      1. associate-*r/ 68.9%

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

   5. Simplified 68.9%

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

4. Final simplification 68.9%

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

Alternative 7: 66.9% accurate, 8.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(b \cdot 2\right) \cdot \frac{\frac{1}{a}}{-3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e-310) (* (* b 2.0) (/ (/ 1.0 a) -3.0)) (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = (b * 2.0) * ((1.0 / a) / -3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-5d-310)) then
        tmp = (b * 2.0d0) * ((1.0d0 / a) / (-3.0d0))
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = (b * 2.0) * ((1.0 / a) / -3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e-310:
		tmp = (b * 2.0) * ((1.0 / a) / -3.0)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e-310)
		tmp = Float64(Float64(b * 2.0) * Float64(Float64(1.0 / a) / -3.0));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5e-310)
		tmp = (b * 2.0) * ((1.0 / a) / -3.0);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-310], N[(N[(b * 2.0), $MachinePrecision] * N[(N[(1.0 / a), $MachinePrecision] / -3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 70.4%

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

   4. Applied egg-rr 66.0%

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

      1. associate-/r* 65.9%

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

   6. Simplified 65.9%

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

   7. Taylor expanded in b around -inf 68.7%

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

      1. *-commutative 68.7%

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

   9. Simplified 68.7%

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

   if -4.999999999999985e-310 < b

   1. Initial program 34.2%

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

   3. Taylor expanded in b around inf 68.9%

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

      1. associate-*r/ 68.9%

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

   5. Simplified 68.9%

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

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\left(b \cdot 2\right) \cdot \frac{\frac{1}{a}}{-3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 43.0% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5.5:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 5.5) (* b (/ -0.6666666666666666 a)) (* c (/ 0.5 b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 5.5) {
		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 <= 5.5d0) 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 <= 5.5) {
		tmp = b * (-0.6666666666666666 / a);
	} else {
		tmp = c * (0.5 / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 5.5:
		tmp = b * (-0.6666666666666666 / a)
	else:
		tmp = c * (0.5 / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 5.5)
		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 <= 5.5)
		tmp = b * (-0.6666666666666666 / a);
	else
		tmp = c * (0.5 / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 5.5], 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 < 5.5

   1. Initial program 70.0%

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

   4. Applied egg-rr 66.0%

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

      1. sub-neg 66.0%

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

      2. distribute-rgt-out-- 66.0%

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

      3. *-commutative 66.0%

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

      4. associate-*r* 66.0%

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

      5. *-commutative 66.0%

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

      6. associate-*l* 66.0%

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

   6. Simplified 66.0%

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

   7. Taylor expanded in b around -inf 54.7%

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

      1. associate-*r/ 54.7%

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

      2. associate-/l* 54.7%

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

      3. associate-/r/ 54.7%

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

   9. Simplified 54.7%

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

   if 5.5 < b

   1. Initial program 11.8%

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

   3. Taylor expanded in b around -inf 2.4%

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

   4. Taylor expanded in b around 0 35.2%

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

      1. metadata-eval 35.2%

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

      2. times-frac 35.2%

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

      3. *-commutative 35.2%

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

      4. times-frac 35.2%

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

      5. /-rgt-identity 35.2%

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

   6. Simplified 35.2%

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

4. Final simplification 49.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5.5:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \frac{0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 66.4% accurate, 11.6× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-5d-310)) then
        tmp = b * ((-0.6666666666666666d0) / a)
    else
        tmp = (-0.5d0) / (b / c)
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 70.4%

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

   4. Applied egg-rr 65.9%

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

      1. sub-neg 65.9%

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

      2. distribute-rgt-out-- 65.9%

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

      3. *-commutative 65.9%

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

      4. associate-*r* 66.0%

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

      5. *-commutative 66.0%

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

      6. associate-*l* 66.0%

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

   6. Simplified 66.0%

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

   7. Taylor expanded in b around -inf 68.7%

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

      1. associate-*r/ 68.7%

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

      2. associate-/l* 68.7%

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

      3. associate-/r/ 68.7%

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

   9. Simplified 68.7%

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

   if -4.999999999999985e-310 < b

   1. Initial program 34.2%

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

   3. Taylor expanded in b around inf 68.9%

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

      1. associate-*r/ 68.9%

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

      2. associate-/l* 68.5%

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

   5. Simplified 68.5%

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

4. Final simplification 68.6%

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

Alternative 10: 66.5% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{\frac{b}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e-310) (/ b (* a -1.5)) (/ -0.5 (/ b c))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = b / (a * -1.5);
	} else {
		tmp = -0.5 / (b / c);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-5d-310)) then
        tmp = b / (a * (-1.5d0))
    else
        tmp = (-0.5d0) / (b / c)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = b / (a * -1.5);
	} else {
		tmp = -0.5 / (b / c);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e-310:
		tmp = b / (a * -1.5)
	else:
		tmp = -0.5 / (b / c)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e-310)
		tmp = Float64(b / Float64(a * -1.5));
	else
		tmp = Float64(-0.5 / Float64(b / c));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5e-310)
		tmp = b / (a * -1.5);
	else
		tmp = -0.5 / (b / c);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-310], N[(b / N[(a * -1.5), $MachinePrecision]), $MachinePrecision], N[(-0.5 / N[(b / c), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 70.4%

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

   4. Applied egg-rr 65.9%

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

      1. sub-neg 65.9%

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

      2. distribute-rgt-out-- 65.9%

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

      3. *-commutative 65.9%

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

      4. associate-*r* 66.0%

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

      5. *-commutative 66.0%

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

      6. associate-*l* 66.0%

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

   6. Simplified 66.0%

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

   7. Taylor expanded in b around -inf 68.7%

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

      1. associate-*r/ 68.7%

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

      2. associate-/l* 68.7%

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

      3. associate-/r/ 68.7%

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

   9. Simplified 68.7%

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

       1. *-commutative 68.7%

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

       2. clear-num 68.6%

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

       3. un-div-inv 68.6%

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

       4. div-inv 68.7%

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

       5. metadata-eval 68.7%

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

   11. Applied egg-rr 68.7%

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

   if -4.999999999999985e-310 < b

   1. Initial program 34.2%

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

   3. Taylor expanded in b around inf 68.9%

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

      1. associate-*r/ 68.9%

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

      2. associate-/l* 68.5%

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

   5. Simplified 68.5%

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

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{\frac{b}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 66.9% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e-310) (/ b (* a -1.5)) (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = b / (a * -1.5);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-5d-310)) then
        tmp = b / (a * (-1.5d0))
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = b / (a * -1.5);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e-310:
		tmp = b / (a * -1.5)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e-310)
		tmp = Float64(b / Float64(a * -1.5));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5e-310)
		tmp = b / (a * -1.5);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-310], N[(b / N[(a * -1.5), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 70.4%

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

   4. Applied egg-rr 65.9%

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

      1. sub-neg 65.9%

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

      2. distribute-rgt-out-- 65.9%

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

      3. *-commutative 65.9%

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

      4. associate-*r* 66.0%

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

      5. *-commutative 66.0%

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

      6. associate-*l* 66.0%

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

   6. Simplified 66.0%

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

   7. Taylor expanded in b around -inf 68.7%

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

      1. associate-*r/ 68.7%

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

      2. associate-/l* 68.7%

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

      3. associate-/r/ 68.7%

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

   9. Simplified 68.7%

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

       1. *-commutative 68.7%

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

       2. clear-num 68.6%

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

       3. un-div-inv 68.6%

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

       4. div-inv 68.7%

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

       5. metadata-eval 68.7%

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

   11. Applied egg-rr 68.7%

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

   if -4.999999999999985e-310 < b

   1. Initial program 34.2%

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

   3. Taylor expanded in b around inf 68.9%

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

      1. associate-*r/ 68.9%

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

   5. Simplified 68.9%

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

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 11.1% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.7%

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

3. Taylor expanded in b around -inf 41.8%

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

4. Taylor expanded in b around 0 11.1%

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

   1. metadata-eval 11.1%

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

   2. times-frac 11.1%

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

   3. *-commutative 11.1%

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

   4. times-frac 11.1%

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

   5. /-rgt-identity 11.1%

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

6. Simplified 11.1%

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

7. Final simplification 11.1%

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

Reproduce

herbie shell --seed 2024016 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))