Cubic critical

Percentage Accurate: 52.1% → 86.1%

Time: 15.2s

Alternatives: 12

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: 52.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 86.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.95 \cdot 10^{+156}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 2.4 \cdot 10^{-103}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 3\right) \cdot c} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.95e+156)
   (/ b (* a -1.5))
   (if (<= b 2.4e-103)
     (/ (- (sqrt (- (* b b) (* (* a 3.0) c))) b) (* a 3.0))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.95e+156) {
		tmp = b / (a * -1.5);
	} else if (b <= 2.4e-103) {
		tmp = (sqrt(((b * b) - ((a * 3.0) * 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 <= (-2.95d+156)) then
        tmp = b / (a * (-1.5d0))
    else if (b <= 2.4d-103) then
        tmp = (sqrt(((b * b) - ((a * 3.0d0) * 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 <= -2.95e+156) {
		tmp = b / (a * -1.5);
	} else if (b <= 2.4e-103) {
		tmp = (Math.sqrt(((b * b) - ((a * 3.0) * c))) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.95e+156:
		tmp = b / (a * -1.5)
	elif b <= 2.4e-103:
		tmp = (math.sqrt(((b * b) - ((a * 3.0) * c))) - b) / (a * 3.0)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.95e+156)
		tmp = Float64(b / Float64(a * -1.5));
	elseif (b <= 2.4e-103)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(a * 3.0) * 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 <= -2.95e+156)
		tmp = b / (a * -1.5);
	elseif (b <= 2.4e-103)
		tmp = (sqrt(((b * b) - ((a * 3.0) * c))) - b) / (a * 3.0);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.95e+156], N[(b / N[(a * -1.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2.4e-103], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(a * 3.0), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.9499999999999998e156

   1. Initial program 35.8%

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

      1. add-sqr-sqrt 35.8%

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

      2. pow2 35.8%

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

      3. pow1/2 35.8%

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

      4. sqrt-pow1 35.8%

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

      5. sub-neg 35.8%

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

      6. +-commutative 35.8%

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

      7. *-commutative 35.8%

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

      8. distribute-rgt-neg-in 35.8%

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

      9. fma-def 36.0%

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

      10. *-commutative 36.0%

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

      11. distribute-rgt-neg-in 36.0%

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

      12. metadata-eval 36.0%

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

      13. pow2 36.0%

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

      14. metadata-eval 36.0%

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

   3. Applied egg-rr 36.0%

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

   4. Taylor expanded in b around -inf 99.7%

      \[\leadsto \color{blue}{2 \cdot \left(-0.3333333333333333 \cdot \frac{b}{a}\right)} \]
   5. Step-by-step derivation

      1. associate-*r* 99.7%

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

      2. metadata-eval 99.7%

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

      3. *-commutative 99.7%

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

      4. associate-*l/ 99.5%

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

      5. associate-/l* 99.7%

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

   6. Simplified 99.7%

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

   7. Taylor expanded in a around 0 99.9%

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

      1. *-commutative 99.9%

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

   9. Simplified 99.9%

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

   if -2.9499999999999998e156 < b < 2.4000000000000002e-103

   1. Initial program 84.3%

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

   if 2.4000000000000002e-103 < b

   1. Initial program 15.5%

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

   2. Taylor expanded in b around inf 89.7%

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

      1. *-commutative 89.7%

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

      2. associate-*l/ 89.7%

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

   4. Simplified 89.7%

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

4. Final simplification 89.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.95 \cdot 10^{+156}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{elif}\;b \leq 2.4 \cdot 10^{-103}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 3\right) \cdot c} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 2: 81.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.6 \cdot 10^{-15}:\\ \;\;\;\;\frac{\left(1.5 \cdot \left(c \cdot \frac{a}{b}\right) - b\right) - b}{a \cdot 3}\\ \mathbf{elif}\;b \leq 5.8 \cdot 10^{-103}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -3\right)} - b}{a} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4.6e-15)
   (/ (- (- (* 1.5 (* c (/ a b))) b) b) (* a 3.0))
   (if (<= b 5.8e-103)
     (* (/ (- (sqrt (* a (* c -3.0))) b) a) 0.3333333333333333)
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.6e-15) {
		tmp = (((1.5 * (c * (a / b))) - b) - b) / (a * 3.0);
	} else if (b <= 5.8e-103) {
		tmp = ((sqrt((a * (c * -3.0))) - b) / a) * 0.3333333333333333;
	} 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 <= (-4.6d-15)) then
        tmp = (((1.5d0 * (c * (a / b))) - b) - b) / (a * 3.0d0)
    else if (b <= 5.8d-103) then
        tmp = ((sqrt((a * (c * (-3.0d0)))) - b) / a) * 0.3333333333333333d0
    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 <= -4.6e-15) {
		tmp = (((1.5 * (c * (a / b))) - b) - b) / (a * 3.0);
	} else if (b <= 5.8e-103) {
		tmp = ((Math.sqrt((a * (c * -3.0))) - b) / a) * 0.3333333333333333;
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4.6e-15:
		tmp = (((1.5 * (c * (a / b))) - b) - b) / (a * 3.0)
	elif b <= 5.8e-103:
		tmp = ((math.sqrt((a * (c * -3.0))) - b) / a) * 0.3333333333333333
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4.6e-15)
		tmp = Float64(Float64(Float64(Float64(1.5 * Float64(c * Float64(a / b))) - b) - b) / Float64(a * 3.0));
	elseif (b <= 5.8e-103)
		tmp = Float64(Float64(Float64(sqrt(Float64(a * Float64(c * -3.0))) - b) / a) * 0.3333333333333333);
	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 <= -4.6e-15)
		tmp = (((1.5 * (c * (a / b))) - b) - b) / (a * 3.0);
	elseif (b <= 5.8e-103)
		tmp = ((sqrt((a * (c * -3.0))) - b) / a) * 0.3333333333333333;
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4.6e-15], N[(N[(N[(N[(1.5 * N[(c * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5.8e-103], N[(N[(N[(N[Sqrt[N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / a), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.59999999999999981e-15

   1. Initial program 64.0%

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

      1. add-sqr-sqrt 63.9%

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

      2. pow2 63.9%

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

      3. pow1/2 63.9%

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

      4. sqrt-pow1 63.9%

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

      5. sub-neg 63.9%

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

      6. +-commutative 63.9%

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

      7. *-commutative 63.9%

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

      8. distribute-rgt-neg-in 63.9%

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

      9. fma-def 64.0%

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

      10. *-commutative 64.0%

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

      11. distribute-rgt-neg-in 64.0%

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

      12. metadata-eval 64.0%

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

      13. pow2 64.0%

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

      14. metadata-eval 64.0%

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

   3. Applied egg-rr 64.0%

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

   4. Taylor expanded in b around -inf 92.7%

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

      1. mul-1-neg 92.7%

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

      2. +-commutative 92.7%

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

      3. unsub-neg 92.7%

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

      4. associate-/l* 95.1%

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

      5. associate-/r/ 95.1%

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

      6. *-commutative 95.1%

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

   6. Simplified 95.1%

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

   if -4.59999999999999981e-15 < b < 5.7999999999999997e-103

   1. Initial program 79.0%

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

      1. add-sqr-sqrt 78.7%

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

      2. pow2 78.7%

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

      3. pow1/2 78.7%

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

      4. sqrt-pow1 78.7%

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

      5. sub-neg 78.7%

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

      6. +-commutative 78.7%

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

      7. *-commutative 78.7%

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

      8. distribute-rgt-neg-in 78.7%

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

      9. fma-def 78.7%

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

      10. *-commutative 78.7%

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

      11. distribute-rgt-neg-in 78.7%

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

      12. metadata-eval 78.7%

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

      13. pow2 78.7%

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

      14. metadata-eval 78.7%

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

   3. Applied egg-rr 78.7%

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

   4. Taylor expanded in c around inf 35.1%

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

   6. Simplified 69.4%

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

   if 5.7999999999999997e-103 < b

   1. Initial program 15.5%

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

   2. Taylor expanded in b around inf 89.7%

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

      1. *-commutative 89.7%

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

      2. associate-*l/ 89.7%

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

   4. Simplified 89.7%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.6 \cdot 10^{-15}:\\ \;\;\;\;\frac{\left(1.5 \cdot \left(c \cdot \frac{a}{b}\right) - b\right) - b}{a \cdot 3}\\ \mathbf{elif}\;b \leq 5.8 \cdot 10^{-103}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -3\right)} - b}{a} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 3: 80.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.6 \cdot 10^{-15}:\\ \;\;\;\;\frac{\left(1.5 \cdot \left(c \cdot \frac{a}{b}\right) - b\right) - b}{a \cdot 3}\\ \mathbf{elif}\;b \leq 3.1 \cdot 10^{-103}:\\ \;\;\;\;\frac{\sqrt{-3 \cdot \left(a \cdot c\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4.6e-15)
   (/ (- (- (* 1.5 (* c (/ a b))) b) b) (* a 3.0))
   (if (<= b 3.1e-103)
     (/ (- (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 <= -4.6e-15) {
		tmp = (((1.5 * (c * (a / b))) - b) - b) / (a * 3.0);
	} else if (b <= 3.1e-103) {
		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 <= (-4.6d-15)) then
        tmp = (((1.5d0 * (c * (a / b))) - b) - b) / (a * 3.0d0)
    else if (b <= 3.1d-103) 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 <= -4.6e-15) {
		tmp = (((1.5 * (c * (a / b))) - b) - b) / (a * 3.0);
	} else if (b <= 3.1e-103) {
		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 <= -4.6e-15:
		tmp = (((1.5 * (c * (a / b))) - b) - b) / (a * 3.0)
	elif b <= 3.1e-103:
		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 <= -4.6e-15)
		tmp = Float64(Float64(Float64(Float64(1.5 * Float64(c * Float64(a / b))) - b) - b) / Float64(a * 3.0));
	elseif (b <= 3.1e-103)
		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 <= -4.6e-15)
		tmp = (((1.5 * (c * (a / b))) - b) - b) / (a * 3.0);
	elseif (b <= 3.1e-103)
		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, -4.6e-15], N[(N[(N[(N[(1.5 * N[(c * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 3.1e-103], 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 < -4.59999999999999981e-15

   1. Initial program 64.0%

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

      1. add-sqr-sqrt 63.9%

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

      2. pow2 63.9%

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

      3. pow1/2 63.9%

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

      4. sqrt-pow1 63.9%

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

      5. sub-neg 63.9%

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

      6. +-commutative 63.9%

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

      7. *-commutative 63.9%

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

      8. distribute-rgt-neg-in 63.9%

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

      9. fma-def 64.0%

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

      10. *-commutative 64.0%

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

      11. distribute-rgt-neg-in 64.0%

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

      12. metadata-eval 64.0%

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

      13. pow2 64.0%

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

      14. metadata-eval 64.0%

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

   3. Applied egg-rr 64.0%

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

   4. Taylor expanded in b around -inf 92.7%

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

      1. mul-1-neg 92.7%

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

      2. +-commutative 92.7%

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

      3. unsub-neg 92.7%

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

      4. associate-/l* 95.1%

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

      5. associate-/r/ 95.1%

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

      6. *-commutative 95.1%

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

   6. Simplified 95.1%

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

   if -4.59999999999999981e-15 < b < 3.1000000000000001e-103

   1. Initial program 79.0%

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

      1. add-sqr-sqrt 78.7%

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

      2. pow2 78.7%

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

      3. pow1/2 78.7%

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

      4. sqrt-pow1 78.7%

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

      5. sub-neg 78.7%

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

      6. +-commutative 78.7%

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

      7. *-commutative 78.7%

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

      8. distribute-rgt-neg-in 78.7%

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

      9. fma-def 78.7%

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

      10. *-commutative 78.7%

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

      11. distribute-rgt-neg-in 78.7%

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

      12. metadata-eval 78.7%

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

      13. pow2 78.7%

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

      14. metadata-eval 78.7%

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

   3. Applied egg-rr 78.7%

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

   4. Taylor expanded in c around inf 35.1%

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

   6. Simplified 69.5%

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

   7. Taylor expanded in a around 0 69.5%

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

   if 3.1000000000000001e-103 < b

   1. Initial program 15.5%

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

   2. Taylor expanded in b around inf 89.7%

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

      1. *-commutative 89.7%

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

      2. associate-*l/ 89.7%

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

   4. Simplified 89.7%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.6 \cdot 10^{-15}:\\ \;\;\;\;\frac{\left(1.5 \cdot \left(c \cdot \frac{a}{b}\right) - b\right) - b}{a \cdot 3}\\ \mathbf{elif}\;b \leq 3.1 \cdot 10^{-103}:\\ \;\;\;\;\frac{\sqrt{-3 \cdot \left(a \cdot c\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 4: 81.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.6 \cdot 10^{-15}:\\ \;\;\;\;\frac{\left(1.5 \cdot \left(c \cdot \frac{a}{b}\right) - b\right) - b}{a \cdot 3}\\ \mathbf{elif}\;b \leq 5.2 \cdot 10^{-103}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4.6e-15)
   (/ (- (- (* 1.5 (* c (/ a b))) b) b) (* a 3.0))
   (if (<= b 5.2e-103)
     (/ (- (sqrt (* a (* c -3.0))) b) (* a 3.0))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.6e-15) {
		tmp = (((1.5 * (c * (a / b))) - b) - b) / (a * 3.0);
	} else if (b <= 5.2e-103) {
		tmp = (sqrt((a * (c * -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 <= (-4.6d-15)) then
        tmp = (((1.5d0 * (c * (a / b))) - b) - b) / (a * 3.0d0)
    else if (b <= 5.2d-103) then
        tmp = (sqrt((a * (c * (-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 <= -4.6e-15) {
		tmp = (((1.5 * (c * (a / b))) - b) - b) / (a * 3.0);
	} else if (b <= 5.2e-103) {
		tmp = (Math.sqrt((a * (c * -3.0))) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4.6e-15:
		tmp = (((1.5 * (c * (a / b))) - b) - b) / (a * 3.0)
	elif b <= 5.2e-103:
		tmp = (math.sqrt((a * (c * -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 <= -4.6e-15)
		tmp = Float64(Float64(Float64(Float64(1.5 * Float64(c * Float64(a / b))) - b) - b) / Float64(a * 3.0));
	elseif (b <= 5.2e-103)
		tmp = Float64(Float64(sqrt(Float64(a * Float64(c * -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 <= -4.6e-15)
		tmp = (((1.5 * (c * (a / b))) - b) - b) / (a * 3.0);
	elseif (b <= 5.2e-103)
		tmp = (sqrt((a * (c * -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, -4.6e-15], N[(N[(N[(N[(1.5 * N[(c * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5.2e-103], N[(N[(N[Sqrt[N[(a * N[(c * -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 < -4.59999999999999981e-15

   1. Initial program 64.0%

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

      1. add-sqr-sqrt 63.9%

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

      2. pow2 63.9%

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

      3. pow1/2 63.9%

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

      4. sqrt-pow1 63.9%

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

      5. sub-neg 63.9%

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

      6. +-commutative 63.9%

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

      7. *-commutative 63.9%

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

      8. distribute-rgt-neg-in 63.9%

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

      9. fma-def 64.0%

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

      10. *-commutative 64.0%

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

      11. distribute-rgt-neg-in 64.0%

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

      12. metadata-eval 64.0%

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

      13. pow2 64.0%

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

      14. metadata-eval 64.0%

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

   3. Applied egg-rr 64.0%

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

   4. Taylor expanded in b around -inf 92.7%

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

      1. mul-1-neg 92.7%

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

      2. +-commutative 92.7%

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

      3. unsub-neg 92.7%

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

      4. associate-/l* 95.1%

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

      5. associate-/r/ 95.1%

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

      6. *-commutative 95.1%

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

   6. Simplified 95.1%

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

   if -4.59999999999999981e-15 < b < 5.19999999999999993e-103

   1. Initial program 79.0%

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

      1. add-sqr-sqrt 78.7%

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

      2. pow2 78.7%

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

      3. pow1/2 78.7%

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

      4. sqrt-pow1 78.7%

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

      5. sub-neg 78.7%

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

      6. +-commutative 78.7%

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

      7. *-commutative 78.7%

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

      8. distribute-rgt-neg-in 78.7%

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

      9. fma-def 78.7%

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

      10. *-commutative 78.7%

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

      11. distribute-rgt-neg-in 78.7%

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

      12. metadata-eval 78.7%

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

      13. pow2 78.7%

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

      14. metadata-eval 78.7%

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

   3. Applied egg-rr 78.7%

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

   4. Taylor expanded in c around inf 35.1%

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

   6. Simplified 69.5%

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

   if 5.19999999999999993e-103 < b

   1. Initial program 15.5%

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

   2. Taylor expanded in b around inf 89.7%

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

      1. *-commutative 89.7%

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

      2. associate-*l/ 89.7%

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

   4. Simplified 89.7%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.6 \cdot 10^{-15}:\\ \;\;\;\;\frac{\left(1.5 \cdot \left(c \cdot \frac{a}{b}\right) - b\right) - b}{a \cdot 3}\\ \mathbf{elif}\;b \leq 5.2 \cdot 10^{-103}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 5: 81.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.6 \cdot 10^{-15}:\\ \;\;\;\;\frac{\left(1.5 \cdot \left(c \cdot \frac{a}{b}\right) - b\right) - b}{a \cdot 3}\\ \mathbf{elif}\;b \leq 6.5 \cdot 10^{-103}:\\ \;\;\;\;\frac{\frac{\sqrt{a \cdot \left(c \cdot -3\right)} - b}{a}}{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4.6e-15)
   (/ (- (- (* 1.5 (* c (/ a b))) b) b) (* a 3.0))
   (if (<= b 6.5e-103)
     (/ (/ (- (sqrt (* a (* c -3.0))) b) a) 3.0)
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.6e-15) {
		tmp = (((1.5 * (c * (a / b))) - b) - b) / (a * 3.0);
	} else if (b <= 6.5e-103) {
		tmp = ((sqrt((a * (c * -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 <= (-4.6d-15)) then
        tmp = (((1.5d0 * (c * (a / b))) - b) - b) / (a * 3.0d0)
    else if (b <= 6.5d-103) then
        tmp = ((sqrt((a * (c * (-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 <= -4.6e-15) {
		tmp = (((1.5 * (c * (a / b))) - b) - b) / (a * 3.0);
	} else if (b <= 6.5e-103) {
		tmp = ((Math.sqrt((a * (c * -3.0))) - b) / a) / 3.0;
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -4.6e-15:
		tmp = (((1.5 * (c * (a / b))) - b) - b) / (a * 3.0)
	elif b <= 6.5e-103:
		tmp = ((math.sqrt((a * (c * -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 <= -4.6e-15)
		tmp = Float64(Float64(Float64(Float64(1.5 * Float64(c * Float64(a / b))) - b) - b) / Float64(a * 3.0));
	elseif (b <= 6.5e-103)
		tmp = Float64(Float64(Float64(sqrt(Float64(a * Float64(c * -3.0))) - b) / 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 <= -4.6e-15)
		tmp = (((1.5 * (c * (a / b))) - b) - b) / (a * 3.0);
	elseif (b <= 6.5e-103)
		tmp = ((sqrt((a * (c * -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, -4.6e-15], N[(N[(N[(N[(1.5 * N[(c * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 6.5e-103], N[(N[(N[(N[Sqrt[N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / a), $MachinePrecision] / 3.0), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -4.59999999999999981e-15

   1. Initial program 64.0%

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

      1. add-sqr-sqrt 63.9%

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

      2. pow2 63.9%

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

      3. pow1/2 63.9%

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

      4. sqrt-pow1 63.9%

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

      5. sub-neg 63.9%

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

      6. +-commutative 63.9%

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

      7. *-commutative 63.9%

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

      8. distribute-rgt-neg-in 63.9%

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

      9. fma-def 64.0%

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

      10. *-commutative 64.0%

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

      11. distribute-rgt-neg-in 64.0%

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

      12. metadata-eval 64.0%

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

      13. pow2 64.0%

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

      14. metadata-eval 64.0%

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

   3. Applied egg-rr 64.0%

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

   4. Taylor expanded in b around -inf 92.7%

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

      1. mul-1-neg 92.7%

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

      2. +-commutative 92.7%

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

      3. unsub-neg 92.7%

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

      4. associate-/l* 95.1%

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

      5. associate-/r/ 95.1%

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

      6. *-commutative 95.1%

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

   6. Simplified 95.1%

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

   if -4.59999999999999981e-15 < b < 6.49999999999999966e-103

   1. Initial program 79.0%

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

      1. add-sqr-sqrt 78.7%

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

      2. pow2 78.7%

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

      3. pow1/2 78.7%

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

      4. sqrt-pow1 78.7%

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

      5. sub-neg 78.7%

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

      6. +-commutative 78.7%

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

      7. *-commutative 78.7%

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

      8. distribute-rgt-neg-in 78.7%

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

      9. fma-def 78.7%

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

      10. *-commutative 78.7%

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

      11. distribute-rgt-neg-in 78.7%

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

      12. metadata-eval 78.7%

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

      13. pow2 78.7%

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

      14. metadata-eval 78.7%

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

   3. Applied egg-rr 78.7%

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

   4. Taylor expanded in c around inf 35.1%

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

   6. Simplified 69.5%

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

      1. add-cube-cbrt 68.8%

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

      2. *-commutative 68.8%

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

      3. times-frac 68.7%

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

      4. pow2 68.7%

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

      5. *-commutative 68.7%

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

      6. associate-*r* 68.7%

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

      7. *-commutative 68.7%

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

   8. Applied egg-rr 68.7%

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

      1. associate-*l/ 68.7%

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

      2. associate-*r/ 68.7%

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

      3. unpow2 68.7%

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

      4. rem-3cbrt-lft 69.6%

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

      5. rem-square-sqrt 69.2%

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

      6. associate-*r/ 69.3%

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

      7. associate-*l/ 69.2%

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

      8. *-commutative 69.2%

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

      9. associate-*l/ 69.3%

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

   10. Simplified 69.6%

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

   if 6.49999999999999966e-103 < b

   1. Initial program 15.5%

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

   2. Taylor expanded in b around inf 89.7%

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

      1. *-commutative 89.7%

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

      2. associate-*l/ 89.7%

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

   4. Simplified 89.7%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.6 \cdot 10^{-15}:\\ \;\;\;\;\frac{\left(1.5 \cdot \left(c \cdot \frac{a}{b}\right) - b\right) - b}{a \cdot 3}\\ \mathbf{elif}\;b \leq 6.5 \cdot 10^{-103}:\\ \;\;\;\;\frac{\frac{\sqrt{a \cdot \left(c \cdot -3\right)} - b}{a}}{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 6: 80.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.12 \cdot 10^{-13}:\\ \;\;\;\;\frac{\left(1.5 \cdot \left(c \cdot \frac{a}{b}\right) - b\right) - b}{a \cdot 3}\\ \mathbf{elif}\;b \leq 6 \cdot 10^{-103}:\\ \;\;\;\;\frac{\frac{\sqrt{-3 \cdot \left(a \cdot c\right)} - b}{3}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.12e-13)
   (/ (- (- (* 1.5 (* c (/ a b))) b) b) (* a 3.0))
   (if (<= b 6e-103)
     (/ (/ (- (sqrt (* -3.0 (* a c))) b) 3.0) a)
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.12e-13) {
		tmp = (((1.5 * (c * (a / b))) - b) - b) / (a * 3.0);
	} else if (b <= 6e-103) {
		tmp = ((sqrt((-3.0 * (a * c))) - b) / 3.0) / 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 <= (-1.12d-13)) then
        tmp = (((1.5d0 * (c * (a / b))) - b) - b) / (a * 3.0d0)
    else if (b <= 6d-103) then
        tmp = ((sqrt(((-3.0d0) * (a * c))) - b) / 3.0d0) / 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 <= -1.12e-13) {
		tmp = (((1.5 * (c * (a / b))) - b) - b) / (a * 3.0);
	} else if (b <= 6e-103) {
		tmp = ((Math.sqrt((-3.0 * (a * c))) - b) / 3.0) / a;
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.12e-13:
		tmp = (((1.5 * (c * (a / b))) - b) - b) / (a * 3.0)
	elif b <= 6e-103:
		tmp = ((math.sqrt((-3.0 * (a * c))) - b) / 3.0) / a
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.12e-13)
		tmp = Float64(Float64(Float64(Float64(1.5 * Float64(c * Float64(a / b))) - b) - b) / Float64(a * 3.0));
	elseif (b <= 6e-103)
		tmp = Float64(Float64(Float64(sqrt(Float64(-3.0 * Float64(a * c))) - b) / 3.0) / 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 <= -1.12e-13)
		tmp = (((1.5 * (c * (a / b))) - b) - b) / (a * 3.0);
	elseif (b <= 6e-103)
		tmp = ((sqrt((-3.0 * (a * c))) - b) / 3.0) / a;
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.12e-13], N[(N[(N[(N[(1.5 * N[(c * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 6e-103], N[(N[(N[(N[Sqrt[N[(-3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / 3.0), $MachinePrecision] / a), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.12e-13

   1. Initial program 64.0%

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

      1. add-sqr-sqrt 63.9%

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

      2. pow2 63.9%

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

      3. pow1/2 63.9%

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

      4. sqrt-pow1 63.9%

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

      5. sub-neg 63.9%

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

      6. +-commutative 63.9%

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

      7. *-commutative 63.9%

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

      8. distribute-rgt-neg-in 63.9%

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

      9. fma-def 64.0%

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

      10. *-commutative 64.0%

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

      11. distribute-rgt-neg-in 64.0%

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

      12. metadata-eval 64.0%

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

      13. pow2 64.0%

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

      14. metadata-eval 64.0%

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

   3. Applied egg-rr 64.0%

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

   4. Taylor expanded in b around -inf 92.7%

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

      1. mul-1-neg 92.7%

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

      2. +-commutative 92.7%

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

      3. unsub-neg 92.7%

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

      4. associate-/l* 95.1%

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

      5. associate-/r/ 95.1%

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

      6. *-commutative 95.1%

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

   6. Simplified 95.1%

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

   if -1.12e-13 < b < 6e-103

   1. Initial program 79.0%

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

      1. add-sqr-sqrt 78.7%

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

      2. pow2 78.7%

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

      3. pow1/2 78.7%

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

      4. sqrt-pow1 78.7%

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

      5. sub-neg 78.7%

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

      6. +-commutative 78.7%

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

      7. *-commutative 78.7%

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

      8. distribute-rgt-neg-in 78.7%

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

      9. fma-def 78.7%

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

      10. *-commutative 78.7%

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

      11. distribute-rgt-neg-in 78.7%

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

      12. metadata-eval 78.7%

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

      13. pow2 78.7%

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

      14. metadata-eval 78.7%

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

   3. Applied egg-rr 78.7%

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

   4. Taylor expanded in c around inf 35.1%

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

   6. Simplified 69.5%

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

   7. Taylor expanded in a around 0 69.5%

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

      1. add-cube-cbrt 68.6%

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

      2. *-commutative 68.6%

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

      3. times-frac 68.6%

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

      4. pow2 68.6%

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

      5. associate-*r* 68.8%

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

      6. *-commutative 68.8%

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

      7. *-commutative 68.8%

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

   9. Applied egg-rr 68.7%

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

       1. associate-*l/ 68.7%

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

       2. associate-*r/ 68.7%

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

       3. unpow2 68.7%

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

       4. rem-3cbrt-lft 69.6%

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

       5. associate-*r* 69.6%

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

   11. Simplified 69.6%

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

   if 6e-103 < b

   1. Initial program 15.5%

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

   2. Taylor expanded in b around inf 89.7%

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

      1. *-commutative 89.7%

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

      2. associate-*l/ 89.7%

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

   4. Simplified 89.7%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.12 \cdot 10^{-13}:\\ \;\;\;\;\frac{\left(1.5 \cdot \left(c \cdot \frac{a}{b}\right) - b\right) - b}{a \cdot 3}\\ \mathbf{elif}\;b \leq 6 \cdot 10^{-103}:\\ \;\;\;\;\frac{\frac{\sqrt{-3 \cdot \left(a \cdot c\right)} - b}{3}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 7: 81.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.6 \cdot 10^{-15}:\\ \;\;\;\;\frac{\left(1.5 \cdot \left(c \cdot \frac{a}{b}\right) - b\right) - b}{a \cdot 3}\\ \mathbf{elif}\;b \leq 7 \cdot 10^{-103}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4.6e-15)
   (/ (- (- (* 1.5 (* c (/ a b))) b) b) (* a 3.0))
   (if (<= b 7e-103)
     (/ (- (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 <= -4.6e-15) {
		tmp = (((1.5 * (c * (a / b))) - b) - b) / (a * 3.0);
	} else if (b <= 7e-103) {
		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 <= (-4.6d-15)) then
        tmp = (((1.5d0 * (c * (a / b))) - b) - b) / (a * 3.0d0)
    else if (b <= 7d-103) 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 <= -4.6e-15) {
		tmp = (((1.5 * (c * (a / b))) - b) - b) / (a * 3.0);
	} else if (b <= 7e-103) {
		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 <= -4.6e-15:
		tmp = (((1.5 * (c * (a / b))) - b) - b) / (a * 3.0)
	elif b <= 7e-103:
		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 <= -4.6e-15)
		tmp = Float64(Float64(Float64(Float64(1.5 * Float64(c * Float64(a / b))) - b) - b) / Float64(a * 3.0));
	elseif (b <= 7e-103)
		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 <= -4.6e-15)
		tmp = (((1.5 * (c * (a / b))) - b) - b) / (a * 3.0);
	elseif (b <= 7e-103)
		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, -4.6e-15], N[(N[(N[(N[(1.5 * N[(c * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 7e-103], 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 < -4.59999999999999981e-15

   1. Initial program 64.0%

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

      1. add-sqr-sqrt 63.9%

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

      2. pow2 63.9%

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

      3. pow1/2 63.9%

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

      4. sqrt-pow1 63.9%

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

      5. sub-neg 63.9%

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

      6. +-commutative 63.9%

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

      7. *-commutative 63.9%

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

      8. distribute-rgt-neg-in 63.9%

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

      9. fma-def 64.0%

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

      10. *-commutative 64.0%

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

      11. distribute-rgt-neg-in 64.0%

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

      12. metadata-eval 64.0%

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

      13. pow2 64.0%

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

      14. metadata-eval 64.0%

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

   3. Applied egg-rr 64.0%

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

   4. Taylor expanded in b around -inf 92.7%

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

      1. mul-1-neg 92.7%

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

      2. +-commutative 92.7%

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

      3. unsub-neg 92.7%

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

      4. associate-/l* 95.1%

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

      5. associate-/r/ 95.1%

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

      6. *-commutative 95.1%

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

   6. Simplified 95.1%

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

   if -4.59999999999999981e-15 < b < 7.00000000000000032e-103

   1. Initial program 79.0%

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

   2. Taylor expanded in b around 0 69.5%

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

      1. associate-*r* 69.7%

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

      2. *-commutative 69.7%

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

      3. *-commutative 69.7%

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

   4. Simplified 69.7%

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

   if 7.00000000000000032e-103 < b

   1. Initial program 15.5%

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

   2. Taylor expanded in b around inf 89.7%

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

      1. *-commutative 89.7%

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

      2. associate-*l/ 89.7%

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

   4. Simplified 89.7%

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

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.6 \cdot 10^{-15}:\\ \;\;\;\;\frac{\left(1.5 \cdot \left(c \cdot \frac{a}{b}\right) - b\right) - b}{a \cdot 3}\\ \mathbf{elif}\;b \leq 7 \cdot 10^{-103}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 8: 67.2% accurate, 8.9× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-2d-310)) then
        tmp = ((-0.6666666666666666d0) * (b / a)) + (0.5d0 * (c / b))
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

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

Python

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

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2e-310)
		tmp = Float64(Float64(-0.6666666666666666 * Float64(b / a)) + Float64(0.5 * Float64(c / b)));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

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

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2e-310], N[(N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.999999999999994e-310

   1. Initial program 70.0%

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

   2. Taylor expanded in b around -inf 73.0%

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

   if -1.999999999999994e-310 < b

   1. Initial program 27.6%

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

   2. Taylor expanded in b around inf 73.4%

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

      1. *-commutative 73.4%

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

      2. associate-*l/ 73.4%

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

   4. Simplified 73.4%

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

4. Final simplification 73.2%

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

Alternative 9: 67.1% accurate, 16.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.9 \cdot 10^{-279}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.9e-279) (/ b (* a -1.5)) (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.9e-279) {
		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 <= 1.9d-279) 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 <= 1.9e-279) {
		tmp = b / (a * -1.5);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 1.9e-279:
		tmp = b / (a * -1.5)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.9e-279)
		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 <= 1.9e-279)
		tmp = b / (a * -1.5);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.9e-279], 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 < 1.90000000000000016e-279

   1. Initial program 69.9%

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

      1. add-sqr-sqrt 69.8%

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

      2. pow2 69.8%

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

      3. pow1/2 69.8%

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

      4. sqrt-pow1 69.8%

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

      5. sub-neg 69.8%

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

      6. +-commutative 69.8%

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

      7. *-commutative 69.8%

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

      8. distribute-rgt-neg-in 69.8%

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

      9. fma-def 69.8%

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

      10. *-commutative 69.8%

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

      11. distribute-rgt-neg-in 69.8%

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

      12. metadata-eval 69.8%

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

      13. pow2 69.8%

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

      14. metadata-eval 69.8%

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

   3. Applied egg-rr 69.8%

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

   4. Taylor expanded in b around -inf 71.1%

      \[\leadsto \color{blue}{2 \cdot \left(-0.3333333333333333 \cdot \frac{b}{a}\right)} \]
   5. Step-by-step derivation

      1. associate-*r* 71.1%

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

      2. metadata-eval 71.1%

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

      3. *-commutative 71.1%

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

      4. associate-*l/ 71.0%

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

      5. associate-/l* 71.0%

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

   6. Simplified 71.0%

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

   7. Taylor expanded in a around 0 71.1%

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

      1. *-commutative 71.1%

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

   9. Simplified 71.1%

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

   if 1.90000000000000016e-279 < b

   1. Initial program 26.7%

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

   2. Taylor expanded in b around inf 75.1%

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

      1. *-commutative 75.1%

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

      2. associate-*l/ 75.1%

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

   4. Simplified 75.1%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.9 \cdot 10^{-279}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]

Alternative 10: 34.5% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 48.5%

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

   1. frac-2neg 48.5%

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

   2. div-inv 48.4%

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

3. Applied egg-rr 43.9%

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

   1. associate-/r* 43.9%

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

5. Simplified 43.9%

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

   1. associate-*r/ 43.9%

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

   2. clear-num 43.8%

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

   3. un-div-inv 43.9%

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

7. Applied egg-rr 43.9%

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

8. Taylor expanded in b around -inf 37.1%

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

   1. *-commutative 37.1%

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

   2. associate-*l/ 37.0%

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

   3. associate-*r/ 37.1%

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

10. Simplified 37.1%

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

11. Final simplification 37.1%

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

Alternative 11: 34.5% accurate, 23.2× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return -0.6666666666666666 * (b / 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 = (-0.6666666666666666d0) * (b / a)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 48.5%

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

2. Taylor expanded in b around -inf 37.1%

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

   1. *-commutative 37.1%

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

4. Simplified 37.1%

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

5. Final simplification 37.1%

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

Alternative 12: 34.6% accurate, 23.2× speedup

Math

\[\begin{array}{l} \\ \frac{b}{a \cdot -1.5} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (/ b (* a -1.5)))

C

double code(double a, double b, double c) {
	return b / (a * -1.5);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = b / (a * (-1.5d0))
end function

Java

public static double code(double a, double b, double c) {
	return b / (a * -1.5);
}

Python

def code(a, b, c):
	return b / (a * -1.5)

Julia

function code(a, b, c)
	return Float64(b / Float64(a * -1.5))
end

MATLAB

function tmp = code(a, b, c)
	tmp = b / (a * -1.5);
end

Wolfram

code[a_, b_, c_] := N[(b / N[(a * -1.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 48.5%

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

   1. add-sqr-sqrt 46.7%

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

   2. pow2 46.7%

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

   3. pow1/2 46.7%

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

   4. sqrt-pow1 46.7%

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

   5. sub-neg 46.7%

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

   6. +-commutative 46.7%

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

   7. *-commutative 46.7%

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

   8. distribute-rgt-neg-in 46.7%

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

   9. fma-def 46.8%

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

   10. *-commutative 46.8%

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

   11. distribute-rgt-neg-in 46.8%

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

   12. metadata-eval 46.8%

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

   13. pow2 46.8%

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

   14. metadata-eval 46.8%

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

3. Applied egg-rr 46.8%

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

4. Taylor expanded in b around -inf 37.1%

   \[\leadsto \color{blue}{2 \cdot \left(-0.3333333333333333 \cdot \frac{b}{a}\right)} \]
5. Step-by-step derivation

   1. associate-*r* 37.1%

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

   2. metadata-eval 37.1%

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

   3. *-commutative 37.1%

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

   4. associate-*l/ 37.0%

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

   5. associate-/l* 37.1%

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

6. Simplified 37.1%

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

7. Taylor expanded in a around 0 37.1%

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

   1. *-commutative 37.1%

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

9. Simplified 37.1%

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

10. Final simplification 37.1%

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

Reproduce

herbie shell --seed 2023339 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))