Cubic critical

Percentage Accurate: 53.0% → 85.3%

Time: 12.7s

Alternatives: 8

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: 53.0% 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: 85.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{+130}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 1.2 \cdot 10^{-111}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e+130)
   (+ (* -0.6666666666666666 (/ b a)) (* 0.5 (/ c b)))
   (if (<= b 1.2e-111)
     (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0))
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e+130) {
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	} else if (b <= 1.2e-111) {
		tmp = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-5d+130)) then
        tmp = ((-0.6666666666666666d0) * (b / a)) + (0.5d0 * (c / b))
    else if (b <= 1.2d-111) then
        tmp = (sqrt(((b * b) - (c * (a * 3.0d0)))) - b) / (a * 3.0d0)
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e+130) {
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	} else if (b <= 1.2e-111) {
		tmp = (Math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e+130:
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b))
	elif b <= 1.2e-111:
		tmp = (math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e+130)
		tmp = Float64(Float64(-0.6666666666666666 * Float64(b / a)) + Float64(0.5 * Float64(c / b)));
	elseif (b <= 1.2e-111)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5e+130)
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	elseif (b <= 1.2e-111)
		tmp = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e+130], N[(N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.2e-111], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.9999999999999996e130

   1. Initial program 41.9%

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

   2. Taylor expanded in b around -inf 93.5%

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

   if -4.9999999999999996e130 < b < 1.2e-111

   1. Initial program 80.4%

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

   if 1.2e-111 < b

   1. Initial program 16.6%

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

   2. Taylor expanded in b around inf 85.1%

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

4. Final simplification 84.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{+130}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 1.2 \cdot 10^{-111}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]

Alternative 2: 80.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -3.1 \cdot 10^{-106}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 7.5 \cdot 10^{-112}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\sqrt{-3 \cdot \left(a \cdot c\right)} - b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -3.1e-106)
   (+ (* -0.6666666666666666 (/ b a)) (* 0.5 (/ c b)))
   (if (<= b 7.5e-112)
     (* 0.3333333333333333 (/ (- (sqrt (* -3.0 (* a c))) b) a))
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.1e-106) {
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	} else if (b <= 7.5e-112) {
		tmp = 0.3333333333333333 * ((sqrt((-3.0 * (a * c))) - b) / a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-3.1d-106)) then
        tmp = ((-0.6666666666666666d0) * (b / a)) + (0.5d0 * (c / b))
    else if (b <= 7.5d-112) then
        tmp = 0.3333333333333333d0 * ((sqrt(((-3.0d0) * (a * c))) - b) / a)
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -3.1e-106) {
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	} else if (b <= 7.5e-112) {
		tmp = 0.3333333333333333 * ((Math.sqrt((-3.0 * (a * c))) - b) / a);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -3.1e-106:
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b))
	elif b <= 7.5e-112:
		tmp = 0.3333333333333333 * ((math.sqrt((-3.0 * (a * c))) - b) / a)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -3.1e-106)
		tmp = Float64(Float64(-0.6666666666666666 * Float64(b / a)) + Float64(0.5 * Float64(c / b)));
	elseif (b <= 7.5e-112)
		tmp = Float64(0.3333333333333333 * Float64(Float64(sqrt(Float64(-3.0 * Float64(a * c))) - b) / a));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -3.1e-106)
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	elseif (b <= 7.5e-112)
		tmp = 0.3333333333333333 * ((sqrt((-3.0 * (a * c))) - b) / a);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -3.1e-106], N[(N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 7.5e-112], N[(0.3333333333333333 * N[(N[(N[Sqrt[N[(-3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -3.09999999999999985e-106

   1. Initial program 64.9%

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

   2. Taylor expanded in b around -inf 83.8%

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

   if -3.09999999999999985e-106 < b < 7.5000000000000002e-112

   1. Initial program 71.7%

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

      1. add-sqr-sqrt 71.2%

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

      2. pow2 71.2%

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

      3. pow1/2 71.2%

         \[\leadsto \frac{\left(-b\right) + {\left(\sqrt{\color{blue}{{\left(b \cdot b - \left(3 \cdot a\right) \cdot c\right)}^{0.5}}}\right)}^{2}}{3 \cdot a} \]

      4. sqrt-pow1 71.2%

         \[\leadsto \frac{\left(-b\right) + {\color{blue}{\left({\left(b \cdot b - \left(3 \cdot a\right) \cdot c\right)}^{\left(\frac{0.5}{2}\right)}\right)}}^{2}}{3 \cdot a} \]

      5. sub-neg 71.2%

         \[\leadsto \frac{\left(-b\right) + {\left({\color{blue}{\left(b \cdot b + \left(-\left(3 \cdot a\right) \cdot c\right)\right)}}^{\left(\frac{0.5}{2}\right)}\right)}^{2}}{3 \cdot a} \]

      6. +-commutative 71.2%

         \[\leadsto \frac{\left(-b\right) + {\left({\color{blue}{\left(\left(-\left(3 \cdot a\right) \cdot c\right) + b \cdot b\right)}}^{\left(\frac{0.5}{2}\right)}\right)}^{2}}{3 \cdot a} \]

      7. *-commutative 71.2%

         \[\leadsto \frac{\left(-b\right) + {\left({\left(\left(-\color{blue}{c \cdot \left(3 \cdot a\right)}\right) + b \cdot b\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2}}{3 \cdot a} \]

      8. distribute-rgt-neg-in 71.2%

         \[\leadsto \frac{\left(-b\right) + {\left({\left(\color{blue}{c \cdot \left(-3 \cdot a\right)} + b \cdot b\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2}}{3 \cdot a} \]

      9. distribute-lft-neg-in 71.2%

         \[\leadsto \frac{\left(-b\right) + {\left({\left(c \cdot \color{blue}{\left(\left(-3\right) \cdot a\right)} + b \cdot b\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2}}{3 \cdot a} \]

      10. metadata-eval 71.2%

          \[\leadsto \frac{\left(-b\right) + {\left({\left(c \cdot \left(\color{blue}{-3} \cdot a\right) + b \cdot b\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2}}{3 \cdot a} \]

      11. associate-*r* 71.1%

          \[\leadsto \frac{\left(-b\right) + {\left({\left(\color{blue}{\left(c \cdot -3\right) \cdot a} + b \cdot b\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2}}{3 \cdot a} \]

      12. *-commutative 71.1%

          \[\leadsto \frac{\left(-b\right) + {\left({\left(\color{blue}{a \cdot \left(c \cdot -3\right)} + b \cdot b\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2}}{3 \cdot a} \]

      13. fma-udef 71.1%

          \[\leadsto \frac{\left(-b\right) + {\left({\color{blue}{\left(\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)\right)}}^{\left(\frac{0.5}{2}\right)}\right)}^{2}}{3 \cdot a} \]

      14. pow2 71.1%

          \[\leadsto \frac{\left(-b\right) + {\left({\left(\mathsf{fma}\left(a, c \cdot -3, \color{blue}{{b}^{2}}\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2}}{3 \cdot a} \]

      15. metadata-eval 71.1%

          \[\leadsto \frac{\left(-b\right) + {\left({\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{\color{blue}{0.25}}\right)}^{2}}{3 \cdot a} \]

   3. Applied egg-rr 71.1%

      \[\leadsto \frac{\left(-b\right) + \color{blue}{{\left({\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{0.25}\right)}^{2}}}{3 \cdot a} \]

   4. Taylor expanded in a around inf 30.2%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{{\left(e^{0.25 \cdot \left(\log \left(-3 \cdot c\right) + -1 \cdot \log \left(\frac{1}{a}\right)\right)}\right)}^{2} - b}{a}} \]
   5. Step-by-step derivation

   6. Simplified 67.2%

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

   7. Taylor expanded in a around 0 67.2%

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

   if 7.5000000000000002e-112 < b

   1. Initial program 16.6%

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

   2. Taylor expanded in b around inf 85.1%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -3.1 \cdot 10^{-106}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 7.5 \cdot 10^{-112}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\sqrt{-3 \cdot \left(a \cdot c\right)} - b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]

Alternative 3: 80.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -8.2 \cdot 10^{-106}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 1.1 \cdot 10^{-111}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -8.2e-106)
   (+ (* -0.6666666666666666 (/ b a)) (* 0.5 (/ c b)))
   (if (<= b 1.1e-111)
     (/ (- (sqrt (* a (* c -3.0))) b) (* a 3.0))
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -8.2e-106) {
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	} else if (b <= 1.1e-111) {
		tmp = (sqrt((a * (c * -3.0))) - b) / (a * 3.0);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-8.2d-106)) then
        tmp = ((-0.6666666666666666d0) * (b / a)) + (0.5d0 * (c / b))
    else if (b <= 1.1d-111) then
        tmp = (sqrt((a * (c * (-3.0d0)))) - b) / (a * 3.0d0)
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -8.2e-106) {
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	} else if (b <= 1.1e-111) {
		tmp = (Math.sqrt((a * (c * -3.0))) - b) / (a * 3.0);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -8.2e-106:
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b))
	elif b <= 1.1e-111:
		tmp = (math.sqrt((a * (c * -3.0))) - b) / (a * 3.0)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -8.2e-106)
		tmp = Float64(Float64(-0.6666666666666666 * Float64(b / a)) + Float64(0.5 * Float64(c / b)));
	elseif (b <= 1.1e-111)
		tmp = Float64(Float64(sqrt(Float64(a * Float64(c * -3.0))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -8.2e-106)
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	elseif (b <= 1.1e-111)
		tmp = (sqrt((a * (c * -3.0))) - b) / (a * 3.0);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -8.2e-106], N[(N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.1e-111], N[(N[(N[Sqrt[N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -8.1999999999999998e-106

   1. Initial program 64.9%

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

   2. Taylor expanded in b around -inf 83.8%

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

   if -8.1999999999999998e-106 < b < 1.1e-111

   1. Initial program 71.7%

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

      1. add-sqr-sqrt 71.2%

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

      2. pow2 71.2%

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

      3. pow1/2 71.2%

         \[\leadsto \frac{\left(-b\right) + {\left(\sqrt{\color{blue}{{\left(b \cdot b - \left(3 \cdot a\right) \cdot c\right)}^{0.5}}}\right)}^{2}}{3 \cdot a} \]

      4. sqrt-pow1 71.2%

         \[\leadsto \frac{\left(-b\right) + {\color{blue}{\left({\left(b \cdot b - \left(3 \cdot a\right) \cdot c\right)}^{\left(\frac{0.5}{2}\right)}\right)}}^{2}}{3 \cdot a} \]

      5. sub-neg 71.2%

         \[\leadsto \frac{\left(-b\right) + {\left({\color{blue}{\left(b \cdot b + \left(-\left(3 \cdot a\right) \cdot c\right)\right)}}^{\left(\frac{0.5}{2}\right)}\right)}^{2}}{3 \cdot a} \]

      6. +-commutative 71.2%

         \[\leadsto \frac{\left(-b\right) + {\left({\color{blue}{\left(\left(-\left(3 \cdot a\right) \cdot c\right) + b \cdot b\right)}}^{\left(\frac{0.5}{2}\right)}\right)}^{2}}{3 \cdot a} \]

      7. *-commutative 71.2%

         \[\leadsto \frac{\left(-b\right) + {\left({\left(\left(-\color{blue}{c \cdot \left(3 \cdot a\right)}\right) + b \cdot b\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2}}{3 \cdot a} \]

      8. distribute-rgt-neg-in 71.2%

         \[\leadsto \frac{\left(-b\right) + {\left({\left(\color{blue}{c \cdot \left(-3 \cdot a\right)} + b \cdot b\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2}}{3 \cdot a} \]

      9. distribute-lft-neg-in 71.2%

         \[\leadsto \frac{\left(-b\right) + {\left({\left(c \cdot \color{blue}{\left(\left(-3\right) \cdot a\right)} + b \cdot b\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2}}{3 \cdot a} \]

      10. metadata-eval 71.2%

          \[\leadsto \frac{\left(-b\right) + {\left({\left(c \cdot \left(\color{blue}{-3} \cdot a\right) + b \cdot b\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2}}{3 \cdot a} \]

      11. associate-*r* 71.1%

          \[\leadsto \frac{\left(-b\right) + {\left({\left(\color{blue}{\left(c \cdot -3\right) \cdot a} + b \cdot b\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2}}{3 \cdot a} \]

      12. *-commutative 71.1%

          \[\leadsto \frac{\left(-b\right) + {\left({\left(\color{blue}{a \cdot \left(c \cdot -3\right)} + b \cdot b\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2}}{3 \cdot a} \]

      13. fma-udef 71.1%

          \[\leadsto \frac{\left(-b\right) + {\left({\color{blue}{\left(\mathsf{fma}\left(a, c \cdot -3, b \cdot b\right)\right)}}^{\left(\frac{0.5}{2}\right)}\right)}^{2}}{3 \cdot a} \]

      14. pow2 71.1%

          \[\leadsto \frac{\left(-b\right) + {\left({\left(\mathsf{fma}\left(a, c \cdot -3, \color{blue}{{b}^{2}}\right)\right)}^{\left(\frac{0.5}{2}\right)}\right)}^{2}}{3 \cdot a} \]

      15. metadata-eval 71.1%

          \[\leadsto \frac{\left(-b\right) + {\left({\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{\color{blue}{0.25}}\right)}^{2}}{3 \cdot a} \]

   3. Applied egg-rr 71.1%

      \[\leadsto \frac{\left(-b\right) + \color{blue}{{\left({\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{0.25}\right)}^{2}}}{3 \cdot a} \]

   4. Taylor expanded in a around inf 30.2%

      \[\leadsto \frac{\color{blue}{{\left(e^{0.25 \cdot \left(\log \left(-3 \cdot c\right) + -1 \cdot \log \left(\frac{1}{a}\right)\right)}\right)}^{2} - b}}{3 \cdot a} \]
   5. Step-by-step derivation

      1. unpow2 30.2%

         \[\leadsto \frac{\color{blue}{e^{0.25 \cdot \left(\log \left(-3 \cdot c\right) + -1 \cdot \log \left(\frac{1}{a}\right)\right)} \cdot e^{0.25 \cdot \left(\log \left(-3 \cdot c\right) + -1 \cdot \log \left(\frac{1}{a}\right)\right)}} - b}{3 \cdot a} \]

      2. fma-neg 30.2%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(e^{0.25 \cdot \left(\log \left(-3 \cdot c\right) + -1 \cdot \log \left(\frac{1}{a}\right)\right)}, e^{0.25 \cdot \left(\log \left(-3 \cdot c\right) + -1 \cdot \log \left(\frac{1}{a}\right)\right)}, -b\right)}}{3 \cdot a} \]

   6. Simplified 67.3%

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

   if 1.1e-111 < b

   1. Initial program 16.6%

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

   2. Taylor expanded in b around inf 85.1%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -8.2 \cdot 10^{-106}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 1.1 \cdot 10^{-111}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]

Alternative 4: 80.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -6.6 \cdot 10^{-106}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 4 \cdot 10^{-112}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -6.6e-106)
   (+ (* -0.6666666666666666 (/ b a)) (* 0.5 (/ c b)))
   (if (<= b 4e-112)
     (/ (- (sqrt (* c (* a -3.0))) b) (* a 3.0))
     (* (/ c b) -0.5))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -6.6e-106) {
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	} else if (b <= 4e-112) {
		tmp = (sqrt((c * (a * -3.0))) - b) / (a * 3.0);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-6.6d-106)) then
        tmp = ((-0.6666666666666666d0) * (b / a)) + (0.5d0 * (c / b))
    else if (b <= 4d-112) then
        tmp = (sqrt((c * (a * (-3.0d0)))) - b) / (a * 3.0d0)
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -6.6e-106) {
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	} else if (b <= 4e-112) {
		tmp = (Math.sqrt((c * (a * -3.0))) - b) / (a * 3.0);
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -6.6e-106:
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b))
	elif b <= 4e-112:
		tmp = (math.sqrt((c * (a * -3.0))) - b) / (a * 3.0)
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -6.6e-106)
		tmp = Float64(Float64(-0.6666666666666666 * Float64(b / a)) + Float64(0.5 * Float64(c / b)));
	elseif (b <= 4e-112)
		tmp = Float64(Float64(sqrt(Float64(c * Float64(a * -3.0))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -6.6e-106)
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	elseif (b <= 4e-112)
		tmp = (sqrt((c * (a * -3.0))) - b) / (a * 3.0);
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -6.6e-106], N[(N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 4e-112], N[(N[(N[Sqrt[N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -6.60000000000000031e-106

   1. Initial program 64.9%

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

   2. Taylor expanded in b around -inf 83.8%

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

   if -6.60000000000000031e-106 < b < 3.9999999999999998e-112

   1. Initial program 71.7%

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

   2. Taylor expanded in b around 0 67.4%

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

      1. *-commutative 67.4%

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

   4. Simplified 67.4%

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

      1. associate-*l* 67.3%

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

      2. sqrt-prod 32.9%

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

   6. Applied egg-rr 32.9%

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

      1. sqrt-prod 67.3%

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

      2. +-commutative 67.3%

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

      3. sub-neg 67.3%

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

   8. Applied egg-rr 67.3%

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

      1. *-commutative 67.3%

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

      2. associate-*l* 67.5%

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

      3. *-commutative 67.5%

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

   10. Simplified 67.5%

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

   if 3.9999999999999998e-112 < b

   1. Initial program 16.6%

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

   2. Taylor expanded in b around inf 85.1%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -6.6 \cdot 10^{-106}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 4 \cdot 10^{-112}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]

Alternative 5: 68.0% accurate, 8.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 67.8%

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

   2. Taylor expanded in b around -inf 71.2%

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

   if -4.999999999999985e-310 < b

   1. Initial program 30.7%

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

   2. Taylor expanded in b around inf 67.0%

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

4. Final simplification 69.3%

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

Alternative 6: 67.8% accurate, 16.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 67.8%

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

   2. Taylor expanded in b around -inf 70.6%

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

      1. *-commutative 70.6%

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

   4. Simplified 70.6%

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

   if -4.999999999999985e-310 < b

   1. Initial program 30.7%

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

   2. Taylor expanded in b around inf 67.0%

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

4. Final simplification 68.9%

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

Alternative 7: 67.9% accurate, 16.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 8.6 \cdot 10^{-308}:\\ \;\;\;\;\frac{\frac{b}{-1.5}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 8.6e-308) (/ (/ b -1.5) a) (* (/ c b) -0.5)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 8.6e-308) {
		tmp = (b / -1.5) / a;
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 8.6d-308) then
        tmp = (b / (-1.5d0)) / a
    else
        tmp = (c / b) * (-0.5d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 8.6e-308) {
		tmp = (b / -1.5) / a;
	} else {
		tmp = (c / b) * -0.5;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 8.6e-308:
		tmp = (b / -1.5) / a
	else:
		tmp = (c / b) * -0.5
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 8.6e-308)
		tmp = Float64(Float64(b / -1.5) / a);
	else
		tmp = Float64(Float64(c / b) * -0.5);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 8.6e-308)
		tmp = (b / -1.5) / a;
	else
		tmp = (c / b) * -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 8.6e-308], N[(N[(b / -1.5), $MachinePrecision] / a), $MachinePrecision], N[(N[(c / b), $MachinePrecision] * -0.5), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 8.60000000000000041e-308

   1. Initial program 67.8%

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

   2. Taylor expanded in b around -inf 70.6%

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

      1. *-commutative 70.6%

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

   4. Simplified 70.6%

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

      1. *-commutative 70.6%

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

      2. metadata-eval 70.6%

         \[\leadsto \color{blue}{\frac{1}{-1.5}} \cdot \frac{b}{a} \]

      3. times-frac 70.5%

         \[\leadsto \color{blue}{\frac{1 \cdot b}{-1.5 \cdot a}} \]

      4. *-un-lft-identity 70.5%

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

      5. associate-/r* 70.6%

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

   6. Applied egg-rr 70.6%

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

   if 8.60000000000000041e-308 < b

   1. Initial program 30.7%

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

   2. Taylor expanded in b around inf 67.0%

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

4. Final simplification 69.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 8.6 \cdot 10^{-308}:\\ \;\;\;\;\frac{\frac{b}{-1.5}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c}{b} \cdot -0.5\\ \end{array} \]

Alternative 8: 34.7% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

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 / b) * (-0.5d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 50.9%

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

2. Taylor expanded in b around inf 31.9%

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

3. Final simplification 31.9%

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

Reproduce

herbie shell --seed 2023333 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))