Cubic critical

Percentage Accurate: 52.4% → 85.8%

Time: 14.6s

Alternatives: 11

Speedup: 16.4×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 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: 85.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{+166}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, -2, \left(\frac{a}{b} \cdot c\right) \cdot 1.5\right)}{a \cdot 3}\\ \mathbf{elif}\;b \leq 3.55 \cdot 10^{-66}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e+166)
   (/ (fma b -2.0 (* (* (/ a b) c) 1.5)) (* a 3.0))
   (if (<= b 3.55e-66)
     (/ (- (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 <= -5e+166) {
		tmp = fma(b, -2.0, (((a / b) * c) * 1.5)) / (a * 3.0);
	} else if (b <= 3.55e-66) {
		tmp = (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 <= -5e+166)
		tmp = Float64(fma(b, -2.0, Float64(Float64(Float64(a / b) * c) * 1.5)) / Float64(a * 3.0));
	elseif (b <= 3.55e-66)
		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

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e+166], N[(N[(b * -2.0 + N[(N[(N[(a / b), $MachinePrecision] * c), $MachinePrecision] * 1.5), $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 3.55e-66], 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 < -5.0000000000000002e166

   1. Initial program 37.5%

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

   2. Taylor expanded in b around -inf 90.2%

      \[\leadsto \frac{\color{blue}{-2 \cdot b + 1.5 \cdot \frac{a \cdot c}{b}}}{3 \cdot a} \]
   3. Step-by-step derivation

      1. *-commutative 90.2%

         \[\leadsto \frac{\color{blue}{b \cdot -2} + 1.5 \cdot \frac{a \cdot c}{b}}{3 \cdot a} \]

      2. fma-def 90.2%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(b, -2, 1.5 \cdot \frac{a \cdot c}{b}\right)}}{3 \cdot a} \]

      3. *-commutative 90.2%

         \[\leadsto \frac{\mathsf{fma}\left(b, -2, \color{blue}{\frac{a \cdot c}{b} \cdot 1.5}\right)}{3 \cdot a} \]

      4. associate-/l* 97.5%

         \[\leadsto \frac{\mathsf{fma}\left(b, -2, \color{blue}{\frac{a}{\frac{b}{c}}} \cdot 1.5\right)}{3 \cdot a} \]

      5. associate-/r/ 97.5%

         \[\leadsto \frac{\mathsf{fma}\left(b, -2, \color{blue}{\left(\frac{a}{b} \cdot c\right)} \cdot 1.5\right)}{3 \cdot a} \]

   4. Simplified 97.5%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(b, -2, \left(\frac{a}{b} \cdot c\right) \cdot 1.5\right)}}{3 \cdot a} \]

   if -5.0000000000000002e166 < b < 3.54999999999999982e-66

   1. Initial program 77.7%

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

   if 3.54999999999999982e-66 < b

   1. Initial program 9.8%

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

   2. Taylor expanded in b around inf 93.1%

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

      1. *-commutative 93.1%

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

      2. associate-*l/ 93.1%

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

   4. Simplified 93.1%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{+166}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, -2, \left(\frac{a}{b} \cdot c\right) \cdot 1.5\right)}{a \cdot 3}\\ \mathbf{elif}\;b \leq 3.55 \cdot 10^{-66}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 2: 81.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.05 \cdot 10^{-73}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 4.5 \cdot 10^{-69}:\\ \;\;\;\;\frac{\sqrt{-3 \cdot \left(a \cdot c\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.05e-73)
   (+ (* b (/ -0.6666666666666666 a)) (* 0.5 (/ c b)))
   (if (<= b 4.5e-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 <= -2.05e-73) {
		tmp = (b * (-0.6666666666666666 / a)) + (0.5 * (c / b));
	} else if (b <= 4.5e-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 <= (-2.05d-73)) then
        tmp = (b * ((-0.6666666666666666d0) / a)) + (0.5d0 * (c / b))
    else if (b <= 4.5d-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 <= -2.05e-73) {
		tmp = (b * (-0.6666666666666666 / a)) + (0.5 * (c / b));
	} else if (b <= 4.5e-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 <= -2.05e-73:
		tmp = (b * (-0.6666666666666666 / a)) + (0.5 * (c / b))
	elif b <= 4.5e-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 <= -2.05e-73)
		tmp = Float64(Float64(b * Float64(-0.6666666666666666 / a)) + Float64(0.5 * Float64(c / b)));
	elseif (b <= 4.5e-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 <= -2.05e-73)
		tmp = (b * (-0.6666666666666666 / a)) + (0.5 * (c / b));
	elseif (b <= 4.5e-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, -2.05e-73], N[(N[(b * N[(-0.6666666666666666 / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 4.5e-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 < -2.05000000000000008e-73

   1. Initial program 65.2%

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

   2. Taylor expanded in b around -inf 84.7%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}} \]
   3. Step-by-step derivation

      1. fma-def 84.7%

         \[\leadsto \color{blue}{\mathsf{fma}\left(-0.6666666666666666, \frac{b}{a}, 0.5 \cdot \frac{c}{b}\right)} \]

      2. associate-*r/ 84.7%

         \[\leadsto \mathsf{fma}\left(-0.6666666666666666, \frac{b}{a}, \color{blue}{\frac{0.5 \cdot c}{b}}\right) \]

   4. Simplified 84.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.6666666666666666, \frac{b}{a}, \frac{0.5 \cdot c}{b}\right)} \]
   5. Step-by-step derivation

      1. fma-udef 84.7%

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

      2. *-commutative 84.7%

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

      3. *-un-lft-identity 84.7%

         \[\leadsto \frac{b}{a} \cdot -0.6666666666666666 + \frac{0.5 \cdot c}{\color{blue}{1 \cdot b}} \]

      4. times-frac 84.7%

         \[\leadsto \frac{b}{a} \cdot -0.6666666666666666 + \color{blue}{\frac{0.5}{1} \cdot \frac{c}{b}} \]

      5. metadata-eval 84.7%

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

   6. Applied egg-rr 84.7%

      \[\leadsto \color{blue}{\frac{b}{a} \cdot -0.6666666666666666 + 0.5 \cdot \frac{c}{b}} \]
   7. Step-by-step derivation

      1. *-commutative 83.9%

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

      2. clear-num 84.0%

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

      3. un-div-inv 84.0%

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

   8. Applied egg-rr 84.7%

      \[\leadsto \color{blue}{\frac{-0.6666666666666666}{\frac{a}{b}}} + 0.5 \cdot \frac{c}{b} \]
   9. Step-by-step derivation

      1. associate-/r/ 84.1%

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

   10. Simplified 84.9%

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

   if -2.05000000000000008e-73 < b < 4.50000000000000009e-69

   1. Initial program 70.1%

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

   2. Taylor expanded in b around 0 62.6%

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

   if 4.50000000000000009e-69 < b

   1. Initial program 9.8%

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

   2. Taylor expanded in b around inf 93.1%

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

      1. *-commutative 93.1%

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

      2. associate-*l/ 93.1%

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

   4. Simplified 93.1%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.05 \cdot 10^{-73}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 4.5 \cdot 10^{-69}:\\ \;\;\;\;\frac{\sqrt{-3 \cdot \left(a \cdot c\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 3: 81.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.8 \cdot 10^{-74}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 5.5 \cdot 10^{-76}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.8e-74)
   (+ (* b (/ -0.6666666666666666 a)) (* 0.5 (/ c b)))
   (if (<= b 5.5e-76)
     (/ (- (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 <= -2.8e-74) {
		tmp = (b * (-0.6666666666666666 / a)) + (0.5 * (c / b));
	} else if (b <= 5.5e-76) {
		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 <= (-2.8d-74)) then
        tmp = (b * ((-0.6666666666666666d0) / a)) + (0.5d0 * (c / b))
    else if (b <= 5.5d-76) 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 <= -2.8e-74) {
		tmp = (b * (-0.6666666666666666 / a)) + (0.5 * (c / b));
	} else if (b <= 5.5e-76) {
		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 <= -2.8e-74:
		tmp = (b * (-0.6666666666666666 / a)) + (0.5 * (c / b))
	elif b <= 5.5e-76:
		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 <= -2.8e-74)
		tmp = Float64(Float64(b * Float64(-0.6666666666666666 / a)) + Float64(0.5 * Float64(c / b)));
	elseif (b <= 5.5e-76)
		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 <= -2.8e-74)
		tmp = (b * (-0.6666666666666666 / a)) + (0.5 * (c / b));
	elseif (b <= 5.5e-76)
		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, -2.8e-74], N[(N[(b * N[(-0.6666666666666666 / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5.5e-76], 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 < -2.79999999999999988e-74

   1. Initial program 65.2%

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

   2. Taylor expanded in b around -inf 84.7%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}} \]
   3. Step-by-step derivation

      1. fma-def 84.7%

         \[\leadsto \color{blue}{\mathsf{fma}\left(-0.6666666666666666, \frac{b}{a}, 0.5 \cdot \frac{c}{b}\right)} \]

      2. associate-*r/ 84.7%

         \[\leadsto \mathsf{fma}\left(-0.6666666666666666, \frac{b}{a}, \color{blue}{\frac{0.5 \cdot c}{b}}\right) \]

   4. Simplified 84.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.6666666666666666, \frac{b}{a}, \frac{0.5 \cdot c}{b}\right)} \]
   5. Step-by-step derivation

      1. fma-udef 84.7%

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

      2. *-commutative 84.7%

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

      3. *-un-lft-identity 84.7%

         \[\leadsto \frac{b}{a} \cdot -0.6666666666666666 + \frac{0.5 \cdot c}{\color{blue}{1 \cdot b}} \]

      4. times-frac 84.7%

         \[\leadsto \frac{b}{a} \cdot -0.6666666666666666 + \color{blue}{\frac{0.5}{1} \cdot \frac{c}{b}} \]

      5. metadata-eval 84.7%

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

   6. Applied egg-rr 84.7%

      \[\leadsto \color{blue}{\frac{b}{a} \cdot -0.6666666666666666 + 0.5 \cdot \frac{c}{b}} \]
   7. Step-by-step derivation

      1. *-commutative 83.9%

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

      2. clear-num 84.0%

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

      3. un-div-inv 84.0%

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

   8. Applied egg-rr 84.7%

      \[\leadsto \color{blue}{\frac{-0.6666666666666666}{\frac{a}{b}}} + 0.5 \cdot \frac{c}{b} \]
   9. Step-by-step derivation

      1. associate-/r/ 84.1%

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

   10. Simplified 84.9%

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

   if -2.79999999999999988e-74 < b < 5.50000000000000014e-76

   1. Initial program 70.1%

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

   2. Taylor expanded in b around 0 62.6%

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

      1. associate-*r* 62.7%

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

      2. *-commutative 62.7%

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

      3. *-commutative 62.7%

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

   4. Simplified 62.7%

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

   if 5.50000000000000014e-76 < b

   1. Initial program 9.8%

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

   2. Taylor expanded in b around inf 93.1%

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

      1. *-commutative 93.1%

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

      2. associate-*l/ 93.1%

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

   4. Simplified 93.1%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.8 \cdot 10^{-74}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 5.5 \cdot 10^{-76}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 4: 81.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.4 \cdot 10^{-73}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, -2, \left(\frac{a}{b} \cdot c\right) \cdot 1.5\right)}{a \cdot 3}\\ \mathbf{elif}\;b \leq 5.7 \cdot 10^{-68}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.4e-73)
   (/ (fma b -2.0 (* (* (/ a b) c) 1.5)) (* a 3.0))
   (if (<= b 5.7e-68)
     (/ (- (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 <= -2.4e-73) {
		tmp = fma(b, -2.0, (((a / b) * c) * 1.5)) / (a * 3.0);
	} else if (b <= 5.7e-68) {
		tmp = (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 <= -2.4e-73)
		tmp = Float64(fma(b, -2.0, Float64(Float64(Float64(a / b) * c) * 1.5)) / Float64(a * 3.0));
	elseif (b <= 5.7e-68)
		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

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.4e-73], N[(N[(b * -2.0 + N[(N[(N[(a / b), $MachinePrecision] * c), $MachinePrecision] * 1.5), $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5.7e-68], 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 < -2.40000000000000006e-73

   1. Initial program 65.2%

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

   2. Taylor expanded in b around -inf 81.5%

      \[\leadsto \frac{\color{blue}{-2 \cdot b + 1.5 \cdot \frac{a \cdot c}{b}}}{3 \cdot a} \]
   3. Step-by-step derivation

      1. *-commutative 81.5%

         \[\leadsto \frac{\color{blue}{b \cdot -2} + 1.5 \cdot \frac{a \cdot c}{b}}{3 \cdot a} \]

      2. fma-def 81.5%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(b, -2, 1.5 \cdot \frac{a \cdot c}{b}\right)}}{3 \cdot a} \]

      3. *-commutative 81.5%

         \[\leadsto \frac{\mathsf{fma}\left(b, -2, \color{blue}{\frac{a \cdot c}{b} \cdot 1.5}\right)}{3 \cdot a} \]

      4. associate-/l* 84.9%

         \[\leadsto \frac{\mathsf{fma}\left(b, -2, \color{blue}{\frac{a}{\frac{b}{c}}} \cdot 1.5\right)}{3 \cdot a} \]

      5. associate-/r/ 84.9%

         \[\leadsto \frac{\mathsf{fma}\left(b, -2, \color{blue}{\left(\frac{a}{b} \cdot c\right)} \cdot 1.5\right)}{3 \cdot a} \]

   4. Simplified 84.9%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(b, -2, \left(\frac{a}{b} \cdot c\right) \cdot 1.5\right)}}{3 \cdot a} \]

   if -2.40000000000000006e-73 < b < 5.7000000000000002e-68

   1. Initial program 70.1%

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

   2. Taylor expanded in b around 0 62.6%

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

      1. associate-*r* 62.7%

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

      2. *-commutative 62.7%

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

      3. *-commutative 62.7%

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

   4. Simplified 62.7%

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

   if 5.7000000000000002e-68 < b

   1. Initial program 9.8%

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

   2. Taylor expanded in b around inf 93.1%

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

      1. *-commutative 93.1%

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

      2. associate-*l/ 93.1%

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

   4. Simplified 93.1%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.4 \cdot 10^{-73}:\\ \;\;\;\;\frac{\mathsf{fma}\left(b, -2, \left(\frac{a}{b} \cdot c\right) \cdot 1.5\right)}{a \cdot 3}\\ \mathbf{elif}\;b \leq 5.7 \cdot 10^{-68}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 5: 68.0% accurate, 8.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-310], N[(N[(0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(-0.6666666666666666 * N[(b / a), $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 68.5%

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

   2. Taylor expanded in b around -inf 70.4%

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

   if -4.999999999999985e-310 < b

   1. Initial program 26.2%

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

   2. Taylor expanded in b around inf 70.9%

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

      1. *-commutative 70.9%

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

      2. associate-*l/ 70.9%

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

   4. Simplified 70.9%

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

4. Final simplification 70.7%

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

Alternative 6: 68.0% accurate, 8.9× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e-310)
   (+ (* b (/ -0.6666666666666666 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 = (b * (-0.6666666666666666 / 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 = (b * ((-0.6666666666666666d0) / 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 = (b * (-0.6666666666666666 / 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 = (b * (-0.6666666666666666 / 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(b * Float64(-0.6666666666666666 / 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 = (b * (-0.6666666666666666 / 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[(b * N[(-0.6666666666666666 / 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 68.5%

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

   2. Taylor expanded in b around -inf 70.4%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}} \]
   3. Step-by-step derivation

      1. fma-def 70.4%

         \[\leadsto \color{blue}{\mathsf{fma}\left(-0.6666666666666666, \frac{b}{a}, 0.5 \cdot \frac{c}{b}\right)} \]

      2. associate-*r/ 70.4%

         \[\leadsto \mathsf{fma}\left(-0.6666666666666666, \frac{b}{a}, \color{blue}{\frac{0.5 \cdot c}{b}}\right) \]

   4. Simplified 70.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.6666666666666666, \frac{b}{a}, \frac{0.5 \cdot c}{b}\right)} \]
   5. Step-by-step derivation

      1. fma-udef 70.4%

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

      2. *-commutative 70.4%

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

      3. *-un-lft-identity 70.4%

         \[\leadsto \frac{b}{a} \cdot -0.6666666666666666 + \frac{0.5 \cdot c}{\color{blue}{1 \cdot b}} \]

      4. times-frac 70.4%

         \[\leadsto \frac{b}{a} \cdot -0.6666666666666666 + \color{blue}{\frac{0.5}{1} \cdot \frac{c}{b}} \]

      5. metadata-eval 70.4%

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

   6. Applied egg-rr 70.4%

      \[\leadsto \color{blue}{\frac{b}{a} \cdot -0.6666666666666666 + 0.5 \cdot \frac{c}{b}} \]
   7. Step-by-step derivation

      1. *-commutative 69.6%

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

      2. clear-num 69.6%

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

      3. un-div-inv 69.6%

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

   8. Applied egg-rr 70.4%

      \[\leadsto \color{blue}{\frac{-0.6666666666666666}{\frac{a}{b}}} + 0.5 \cdot \frac{c}{b} \]
   9. Step-by-step derivation

      1. associate-/r/ 69.7%

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

   10. Simplified 70.5%

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

   if -4.999999999999985e-310 < b

   1. Initial program 26.2%

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

   2. Taylor expanded in b around inf 70.9%

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

      1. *-commutative 70.9%

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

      2. associate-*l/ 70.9%

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

   4. Simplified 70.9%

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

4. Final simplification 70.7%

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

Alternative 7: 43.2% accurate, 16.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 8.8 \cdot 10^{+74}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 8.8e+74) (* b (/ -0.6666666666666666 a)) (* 0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 8.8e+74) {
		tmp = b * (-0.6666666666666666 / a);
	} else {
		tmp = 0.5 * (c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 8.8d+74) then
        tmp = b * ((-0.6666666666666666d0) / a)
    else
        tmp = 0.5d0 * (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 8.8e+74) {
		tmp = b * (-0.6666666666666666 / a);
	} else {
		tmp = 0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 8.8e+74:
		tmp = b * (-0.6666666666666666 / a)
	else:
		tmp = 0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 8.8e+74)
		tmp = Float64(b * Float64(-0.6666666666666666 / a));
	else
		tmp = Float64(0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 8.8e+74)
		tmp = b * (-0.6666666666666666 / a);
	else
		tmp = 0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 8.8e+74], N[(b * N[(-0.6666666666666666 / a), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 8.8000000000000005e74

   1. Initial program 58.5%

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

   2. Taylor expanded in b around -inf 44.9%

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

      1. *-commutative 44.9%

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

   4. Simplified 44.9%

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

      1. *-commutative 44.9%

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

      2. clear-num 44.9%

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

      3. un-div-inv 44.9%

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

   6. Applied egg-rr 44.9%

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

      1. associate-/r/ 44.9%

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

   8. Simplified 44.9%

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

   if 8.8000000000000005e74 < b

   1. Initial program 9.4%

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

   2. Taylor expanded in b around -inf 2.1%

      \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}} \]
   3. Step-by-step derivation

      1. fma-def 2.1%

         \[\leadsto \color{blue}{\mathsf{fma}\left(-0.6666666666666666, \frac{b}{a}, 0.5 \cdot \frac{c}{b}\right)} \]

      2. associate-*r/ 2.1%

         \[\leadsto \mathsf{fma}\left(-0.6666666666666666, \frac{b}{a}, \color{blue}{\frac{0.5 \cdot c}{b}}\right) \]

   4. Simplified 2.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.6666666666666666, \frac{b}{a}, \frac{0.5 \cdot c}{b}\right)} \]

   5. Taylor expanded in b around 0 33.6%

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

4. Final simplification 42.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 8.8 \cdot 10^{+74}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 8: 67.4% accurate, 16.4× 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 68.5%

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

   2. Taylor expanded in b around -inf 69.6%

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

      1. *-commutative 69.6%

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

   4. Simplified 69.6%

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

      1. *-commutative 69.6%

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

      2. clear-num 69.6%

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

      3. un-div-inv 69.6%

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

   6. Applied egg-rr 69.6%

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

      1. associate-/r/ 69.7%

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

   8. Simplified 69.7%

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

   if -4.999999999999985e-310 < b

   1. Initial program 26.2%

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

   2. Taylor expanded in b around inf 57.3%

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

      1. associate-/l* 61.1%

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

      2. associate-/r/ 53.2%

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

   4. Simplified 53.2%

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

      1. log1p-expm1-u 44.4%

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

      2. log1p-udef 20.1%

         \[\leadsto \color{blue}{\log \left(1 + \mathsf{expm1}\left(\frac{-1.5 \cdot \left(\frac{a}{b} \cdot c\right)}{3 \cdot a}\right)\right)} \]

      3. times-frac 20.1%

         \[\leadsto \log \left(1 + \mathsf{expm1}\left(\color{blue}{\frac{-1.5}{3} \cdot \frac{\frac{a}{b} \cdot c}{a}}\right)\right) \]

      4. metadata-eval 20.1%

         \[\leadsto \log \left(1 + \mathsf{expm1}\left(\color{blue}{-0.5} \cdot \frac{\frac{a}{b} \cdot c}{a}\right)\right) \]

      5. associate-*l/ 19.5%

         \[\leadsto \log \left(1 + \mathsf{expm1}\left(-0.5 \cdot \frac{\color{blue}{\frac{a \cdot c}{b}}}{a}\right)\right) \]

      6. associate-/l* 20.8%

         \[\leadsto \log \left(1 + \mathsf{expm1}\left(-0.5 \cdot \frac{\color{blue}{\frac{a}{\frac{b}{c}}}}{a}\right)\right) \]

   6. Applied egg-rr 20.8%

      \[\leadsto \color{blue}{\log \left(1 + \mathsf{expm1}\left(-0.5 \cdot \frac{\frac{a}{\frac{b}{c}}}{a}\right)\right)} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 20.8%

         \[\leadsto \log \color{blue}{\left(1 \cdot \left(1 + \mathsf{expm1}\left(-0.5 \cdot \frac{\frac{a}{\frac{b}{c}}}{a}\right)\right)\right)} \]

      2. log-prod 20.8%

         \[\leadsto \color{blue}{\log 1 + \log \left(1 + \mathsf{expm1}\left(-0.5 \cdot \frac{\frac{a}{\frac{b}{c}}}{a}\right)\right)} \]

      3. metadata-eval 20.8%

         \[\leadsto \color{blue}{0} + \log \left(1 + \mathsf{expm1}\left(-0.5 \cdot \frac{\frac{a}{\frac{b}{c}}}{a}\right)\right) \]

      4. log1p-def 49.5%

         \[\leadsto 0 + \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(-0.5 \cdot \frac{\frac{a}{\frac{b}{c}}}{a}\right)\right)} \]

      5. log1p-expm1-u 61.0%

         \[\leadsto 0 + \color{blue}{-0.5 \cdot \frac{\frac{a}{\frac{b}{c}}}{a}} \]

      6. associate-/l/ 60.6%

         \[\leadsto 0 + -0.5 \cdot \color{blue}{\frac{a}{a \cdot \frac{b}{c}}} \]

   8. Applied egg-rr 60.6%

      \[\leadsto \color{blue}{0 + -0.5 \cdot \frac{a}{a \cdot \frac{b}{c}}} \]
   9. Step-by-step derivation

      1. +-lft-identity 60.6%

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

      2. associate-/r* 70.3%

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

      3. *-inverses 70.3%

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

      4. associate-*r/ 70.3%

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

      5. metadata-eval 70.3%

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

   10. Simplified 70.3%

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

4. Final simplification 70.1%

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

Alternative 9: 67.4% accurate, 16.4× 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 68.5%

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

   2. Taylor expanded in b around -inf 69.6%

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

      1. *-commutative 69.6%

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

   4. Simplified 69.6%

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

      1. *-commutative 69.6%

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

      2. clear-num 69.6%

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

      3. un-div-inv 69.6%

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

   6. Applied egg-rr 69.6%

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

      1. associate-/r/ 69.7%

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

   8. Simplified 69.7%

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

      1. *-commutative 69.7%

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

      2. clear-num 69.6%

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

      3. un-div-inv 69.6%

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

      4. div-inv 69.8%

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

      5. metadata-eval 69.8%

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

   10. Applied egg-rr 69.8%

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

   if -4.999999999999985e-310 < b

   1. Initial program 26.2%

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

   2. Taylor expanded in b around inf 57.3%

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

      1. associate-/l* 61.1%

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

      2. associate-/r/ 53.2%

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

   4. Simplified 53.2%

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

      1. log1p-expm1-u 44.4%

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

      2. log1p-udef 20.1%

         \[\leadsto \color{blue}{\log \left(1 + \mathsf{expm1}\left(\frac{-1.5 \cdot \left(\frac{a}{b} \cdot c\right)}{3 \cdot a}\right)\right)} \]

      3. times-frac 20.1%

         \[\leadsto \log \left(1 + \mathsf{expm1}\left(\color{blue}{\frac{-1.5}{3} \cdot \frac{\frac{a}{b} \cdot c}{a}}\right)\right) \]

      4. metadata-eval 20.1%

         \[\leadsto \log \left(1 + \mathsf{expm1}\left(\color{blue}{-0.5} \cdot \frac{\frac{a}{b} \cdot c}{a}\right)\right) \]

      5. associate-*l/ 19.5%

         \[\leadsto \log \left(1 + \mathsf{expm1}\left(-0.5 \cdot \frac{\color{blue}{\frac{a \cdot c}{b}}}{a}\right)\right) \]

      6. associate-/l* 20.8%

         \[\leadsto \log \left(1 + \mathsf{expm1}\left(-0.5 \cdot \frac{\color{blue}{\frac{a}{\frac{b}{c}}}}{a}\right)\right) \]

   6. Applied egg-rr 20.8%

      \[\leadsto \color{blue}{\log \left(1 + \mathsf{expm1}\left(-0.5 \cdot \frac{\frac{a}{\frac{b}{c}}}{a}\right)\right)} \]
   7. Step-by-step derivation

      1. *-un-lft-identity 20.8%

         \[\leadsto \log \color{blue}{\left(1 \cdot \left(1 + \mathsf{expm1}\left(-0.5 \cdot \frac{\frac{a}{\frac{b}{c}}}{a}\right)\right)\right)} \]

      2. log-prod 20.8%

         \[\leadsto \color{blue}{\log 1 + \log \left(1 + \mathsf{expm1}\left(-0.5 \cdot \frac{\frac{a}{\frac{b}{c}}}{a}\right)\right)} \]

      3. metadata-eval 20.8%

         \[\leadsto \color{blue}{0} + \log \left(1 + \mathsf{expm1}\left(-0.5 \cdot \frac{\frac{a}{\frac{b}{c}}}{a}\right)\right) \]

      4. log1p-def 49.5%

         \[\leadsto 0 + \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(-0.5 \cdot \frac{\frac{a}{\frac{b}{c}}}{a}\right)\right)} \]

      5. log1p-expm1-u 61.0%

         \[\leadsto 0 + \color{blue}{-0.5 \cdot \frac{\frac{a}{\frac{b}{c}}}{a}} \]

      6. associate-/l/ 60.6%

         \[\leadsto 0 + -0.5 \cdot \color{blue}{\frac{a}{a \cdot \frac{b}{c}}} \]

   8. Applied egg-rr 60.6%

      \[\leadsto \color{blue}{0 + -0.5 \cdot \frac{a}{a \cdot \frac{b}{c}}} \]
   9. Step-by-step derivation

      1. +-lft-identity 60.6%

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

      2. associate-/r* 70.3%

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

      3. *-inverses 70.3%

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

      4. associate-*r/ 70.3%

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

      5. metadata-eval 70.3%

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

   10. Simplified 70.3%

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

4. Final simplification 70.1%

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

Alternative 10: 67.8% accurate, 16.4× 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 68.5%

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

   2. Taylor expanded in b around -inf 69.6%

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

      1. *-commutative 69.6%

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

   4. Simplified 69.6%

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

      1. *-commutative 69.6%

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

      2. clear-num 69.6%

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

      3. un-div-inv 69.6%

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

   6. Applied egg-rr 69.6%

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

      1. associate-/r/ 69.7%

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

   8. Simplified 69.7%

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

      1. *-commutative 69.7%

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

      2. clear-num 69.6%

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

      3. un-div-inv 69.6%

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

      4. div-inv 69.8%

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

      5. metadata-eval 69.8%

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

   10. Applied egg-rr 69.8%

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

   if -4.999999999999985e-310 < b

   1. Initial program 26.2%

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

   2. Taylor expanded in b around inf 70.9%

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

      1. *-commutative 70.9%

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

      2. associate-*l/ 70.9%

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

   4. Simplified 70.9%

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

4. Final simplification 70.4%

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

Alternative 11: 11.3% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 46.0%

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

2. Taylor expanded in b around -inf 34.2%

   \[\leadsto \color{blue}{-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}} \]
3. Step-by-step derivation

   1. fma-def 34.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.6666666666666666, \frac{b}{a}, 0.5 \cdot \frac{c}{b}\right)} \]

   2. associate-*r/ 34.2%

      \[\leadsto \mathsf{fma}\left(-0.6666666666666666, \frac{b}{a}, \color{blue}{\frac{0.5 \cdot c}{b}}\right) \]

4. Simplified 34.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-0.6666666666666666, \frac{b}{a}, \frac{0.5 \cdot c}{b}\right)} \]

5. Taylor expanded in b around 0 10.8%

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

6. Final simplification 10.8%

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

Reproduce

herbie shell --seed 2023301 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))