Cubic critical

Percentage Accurate: 52.3% → 86.9%

Time: 16.0s

Alternatives: 11

Speedup: 16.4×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 52.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 86.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -4.5 \cdot 10^{+94}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 8 \cdot 10^{-149}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{elif}\;b \leq 105000:\\ \;\;\;\;\frac{a \cdot \left(c \cdot -3\right)}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}} \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -4.5e+94)
   (+ (* -0.6666666666666666 (/ b a)) (* 0.5 (/ c b)))
   (if (<= b 8e-149)
     (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0))
     (if (<= b 105000.0)
       (*
        (/ (* a (* c -3.0)) (+ b (sqrt (fma a (* c -3.0) (pow b 2.0)))))
        (/ 0.3333333333333333 a))
       (/ 1.0 (+ (* -2.0 (/ b c)) (* 1.5 (/ a b))))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -4.5e+94) {
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	} else if (b <= 8e-149) {
		tmp = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	} else if (b <= 105000.0) {
		tmp = ((a * (c * -3.0)) / (b + sqrt(fma(a, (c * -3.0), pow(b, 2.0))))) * (0.3333333333333333 / a);
	} else {
		tmp = 1.0 / ((-2.0 * (b / c)) + (1.5 * (a / b)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -4.5e+94)
		tmp = Float64(Float64(-0.6666666666666666 * Float64(b / a)) + Float64(0.5 * Float64(c / b)));
	elseif (b <= 8e-149)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0));
	elseif (b <= 105000.0)
		tmp = Float64(Float64(Float64(a * Float64(c * -3.0)) / Float64(b + sqrt(fma(a, Float64(c * -3.0), (b ^ 2.0))))) * Float64(0.3333333333333333 / a));
	else
		tmp = Float64(1.0 / Float64(Float64(-2.0 * Float64(b / c)) + Float64(1.5 * Float64(a / b))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -4.5e+94], N[(N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 8e-149], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 105000.0], N[(N[(N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision] / N[(b + N[Sqrt[N[(a * N[(c * -3.0), $MachinePrecision] + N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.3333333333333333 / a), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(-2.0 * N[(b / c), $MachinePrecision]), $MachinePrecision] + N[(1.5 * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if b < -4.49999999999999972e94

   1. Initial program 59.3%

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

   2. Taylor expanded in b around -inf 91.5%

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

   if -4.49999999999999972e94 < b < 7.99999999999999983e-149

   1. Initial program 91.3%

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

   if 7.99999999999999983e-149 < b < 105000

   1. Initial program 57.1%

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

   3. Applied egg-rr 57.1%

      \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)} \cdot \frac{0.3333333333333333}{a} + \left(-b \cdot \frac{0.3333333333333333}{a}\right)} \]
   4. Step-by-step derivation

      1. sub-neg 57.1%

         \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)} \cdot \frac{0.3333333333333333}{a} - b \cdot \frac{0.3333333333333333}{a}} \]

      2. distribute-rgt-out-- 57.1%

         \[\leadsto \color{blue}{\frac{0.3333333333333333}{a} \cdot \left(\sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)} - b\right)} \]

   5. Simplified 57.1%

      \[\leadsto \color{blue}{\frac{0.3333333333333333}{a} \cdot \left(\sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)} - b\right)} \]
   6. Step-by-step derivation

      1. *-commutative 57.1%

         \[\leadsto \color{blue}{\left(\sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)} - b\right) \cdot \frac{0.3333333333333333}{a}} \]

      2. flip-- 56.8%

         \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)} \cdot \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)} - b \cdot b}{\sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)} + b}} \cdot \frac{0.3333333333333333}{a} \]

      3. frac-times 46.4%

         \[\leadsto \color{blue}{\frac{\left(\sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)} \cdot \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)} - b \cdot b\right) \cdot 0.3333333333333333}{\left(\sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)} + b\right) \cdot a}} \]

      4. add-sqr-sqrt 46.3%

         \[\leadsto \frac{\left(\color{blue}{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)} - b \cdot b\right) \cdot 0.3333333333333333}{\left(\sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)} + b\right) \cdot a} \]

      5. unpow2 46.3%

         \[\leadsto \frac{\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right) - \color{blue}{{b}^{2}}\right) \cdot 0.3333333333333333}{\left(\sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)} + b\right) \cdot a} \]

      6. +-commutative 46.3%

         \[\leadsto \frac{\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right) - {b}^{2}\right) \cdot 0.3333333333333333}{\color{blue}{\left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}\right)} \cdot a} \]

   7. Applied egg-rr 46.3%

      \[\leadsto \color{blue}{\frac{\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right) - {b}^{2}\right) \cdot 0.3333333333333333}{\left(b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}\right) \cdot a}} \]
   8. Step-by-step derivation

      1. times-frac 56.7%

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right) - {b}^{2}}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}} \cdot \frac{0.3333333333333333}{a}} \]

      2. unpow2 56.7%

         \[\leadsto \frac{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right) - \color{blue}{b \cdot b}}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}} \cdot \frac{0.3333333333333333}{a} \]

      3. sqr-neg 56.7%

         \[\leadsto \frac{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right) - \color{blue}{\left(-b\right) \cdot \left(-b\right)}}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}} \cdot \frac{0.3333333333333333}{a} \]

      4. cancel-sign-sub 56.7%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right) + b \cdot \left(-b\right)}}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}} \cdot \frac{0.3333333333333333}{a} \]

      5. fma-udef 56.7%

         \[\leadsto \frac{\color{blue}{\left(a \cdot \left(c \cdot -3\right) + {b}^{2}\right)} + b \cdot \left(-b\right)}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}} \cdot \frac{0.3333333333333333}{a} \]

      6. associate-+l+ 75.6%

         \[\leadsto \frac{\color{blue}{a \cdot \left(c \cdot -3\right) + \left({b}^{2} + b \cdot \left(-b\right)\right)}}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}} \cdot \frac{0.3333333333333333}{a} \]

      7. unpow2 75.6%

         \[\leadsto \frac{a \cdot \left(c \cdot -3\right) + \left(\color{blue}{b \cdot b} + b \cdot \left(-b\right)\right)}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}} \cdot \frac{0.3333333333333333}{a} \]

      8. distribute-lft-in 75.6%

         \[\leadsto \frac{a \cdot \left(c \cdot -3\right) + \color{blue}{b \cdot \left(b + \left(-b\right)\right)}}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}} \cdot \frac{0.3333333333333333}{a} \]

      9. neg-mul-1 75.6%

         \[\leadsto \frac{a \cdot \left(c \cdot -3\right) + b \cdot \left(b + \color{blue}{-1 \cdot b}\right)}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}} \cdot \frac{0.3333333333333333}{a} \]

      10. *-commutative 75.6%

          \[\leadsto \frac{a \cdot \left(c \cdot -3\right) + \color{blue}{\left(b + -1 \cdot b\right) \cdot b}}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}} \cdot \frac{0.3333333333333333}{a} \]

      11. distribute-rgt1-in 75.6%

          \[\leadsto \frac{a \cdot \left(c \cdot -3\right) + \color{blue}{\left(\left(-1 + 1\right) \cdot b\right)} \cdot b}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}} \cdot \frac{0.3333333333333333}{a} \]

      12. metadata-eval 75.6%

          \[\leadsto \frac{a \cdot \left(c \cdot -3\right) + \left(\color{blue}{0} \cdot b\right) \cdot b}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}} \cdot \frac{0.3333333333333333}{a} \]

      13. mul0-lft 75.6%

          \[\leadsto \frac{a \cdot \left(c \cdot -3\right) + \color{blue}{0} \cdot b}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}} \cdot \frac{0.3333333333333333}{a} \]

      14. mul0-lft 75.6%

          \[\leadsto \frac{a \cdot \left(c \cdot -3\right) + \color{blue}{0}}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}} \cdot \frac{0.3333333333333333}{a} \]

   9. Simplified 75.6%

      \[\leadsto \color{blue}{\frac{a \cdot \left(c \cdot -3\right) + 0}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}} \cdot \frac{0.3333333333333333}{a}} \]

   if 105000 < b

   1. Initial program 13.8%

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

      1. add-sqr-sqrt 10.2%

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

      2. pow2 10.2%

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

      3. pow1/2 10.2%

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

      4. sqrt-pow1 10.3%

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

      5. sub-neg 10.3%

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

      6. +-commutative 10.3%

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

      7. distribute-lft-neg-in 10.3%

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

      8. *-commutative 10.3%

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

      9. distribute-rgt-neg-in 10.3%

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

      10. metadata-eval 10.3%

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

      11. associate-*r* 10.3%

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

      12. *-commutative 10.3%

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

      13. fma-udef 10.3%

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

      14. pow2 10.3%

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

      15. metadata-eval 10.3%

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

   3. Applied egg-rr 10.3%

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

      1. clear-num 10.3%

         \[\leadsto \color{blue}{\frac{1}{\frac{3 \cdot a}{\left(-b\right) + {\left({\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{0.25}\right)}^{2}}}} \]

      2. inv-pow 10.3%

         \[\leadsto \color{blue}{{\left(\frac{3 \cdot a}{\left(-b\right) + {\left({\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{0.25}\right)}^{2}}\right)}^{-1}} \]

      3. *-commutative 10.3%

         \[\leadsto {\left(\frac{\color{blue}{a \cdot 3}}{\left(-b\right) + {\left({\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{0.25}\right)}^{2}}\right)}^{-1} \]

      4. neg-mul-1 10.3%

         \[\leadsto {\left(\frac{a \cdot 3}{\color{blue}{-1 \cdot b} + {\left({\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{0.25}\right)}^{2}}\right)}^{-1} \]

      5. pow-pow 13.8%

         \[\leadsto {\left(\frac{a \cdot 3}{-1 \cdot b + \color{blue}{{\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{\left(0.25 \cdot 2\right)}}}\right)}^{-1} \]

      6. metadata-eval 13.8%

         \[\leadsto {\left(\frac{a \cdot 3}{-1 \cdot b + {\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{\color{blue}{0.5}}}\right)}^{-1} \]

      7. pow1/2 13.8%

         \[\leadsto {\left(\frac{a \cdot 3}{-1 \cdot b + \color{blue}{\sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}}}\right)}^{-1} \]

      8. fma-def 13.8%

         \[\leadsto {\left(\frac{a \cdot 3}{\color{blue}{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}\right)}}\right)}^{-1} \]

   5. Applied egg-rr 13.8%

      \[\leadsto \color{blue}{{\left(\frac{a \cdot 3}{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}\right)}\right)}^{-1}} \]
   6. Step-by-step derivation

      1. unpow-1 13.8%

         \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 3}{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}\right)}}} \]

   7. Simplified 13.8%

      \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 3}{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}\right)}}} \]

   8. Taylor expanded in b around inf 93.5%

      \[\leadsto \frac{1}{\color{blue}{-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.5 \cdot 10^{+94}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 8 \cdot 10^{-149}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{elif}\;b \leq 105000:\\ \;\;\;\;\frac{a \cdot \left(c \cdot -3\right)}{b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}} \cdot \frac{0.3333333333333333}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}}\\ \end{array} \]

Alternative 2: 78.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{if}\;b \leq -4.5 \cdot 10^{-14}:\\ \;\;\;\;\frac{1.5 \cdot \left(c \cdot \frac{a}{b}\right) + b \cdot -2}{a \cdot 3}\\ \mathbf{elif}\;b \leq -8 \cdot 10^{-64}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq -4.5 \cdot 10^{-133}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{elif}\;b \leq 3 \cdot 10^{-12}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (* c (* a -3.0))) b) (* a 3.0))))
   (if (<= b -4.5e-14)
     (/ (+ (* 1.5 (* c (/ a b))) (* b -2.0)) (* a 3.0))
     (if (<= b -8e-64)
       t_0
       (if (<= b -4.5e-133)
         (* b (/ -0.6666666666666666 a))
         (if (<= b 3e-12)
           t_0
           (/ 1.0 (+ (* -2.0 (/ b c)) (* 1.5 (/ a b))))))))))

C

double code(double a, double b, double c) {
	double t_0 = (sqrt((c * (a * -3.0))) - b) / (a * 3.0);
	double tmp;
	if (b <= -4.5e-14) {
		tmp = ((1.5 * (c * (a / b))) + (b * -2.0)) / (a * 3.0);
	} else if (b <= -8e-64) {
		tmp = t_0;
	} else if (b <= -4.5e-133) {
		tmp = b * (-0.6666666666666666 / a);
	} else if (b <= 3e-12) {
		tmp = t_0;
	} else {
		tmp = 1.0 / ((-2.0 * (b / c)) + (1.5 * (a / 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) :: t_0
    real(8) :: tmp
    t_0 = (sqrt((c * (a * (-3.0d0)))) - b) / (a * 3.0d0)
    if (b <= (-4.5d-14)) then
        tmp = ((1.5d0 * (c * (a / b))) + (b * (-2.0d0))) / (a * 3.0d0)
    else if (b <= (-8d-64)) then
        tmp = t_0
    else if (b <= (-4.5d-133)) then
        tmp = b * ((-0.6666666666666666d0) / a)
    else if (b <= 3d-12) then
        tmp = t_0
    else
        tmp = 1.0d0 / (((-2.0d0) * (b / c)) + (1.5d0 * (a / b)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (Math.sqrt((c * (a * -3.0))) - b) / (a * 3.0);
	double tmp;
	if (b <= -4.5e-14) {
		tmp = ((1.5 * (c * (a / b))) + (b * -2.0)) / (a * 3.0);
	} else if (b <= -8e-64) {
		tmp = t_0;
	} else if (b <= -4.5e-133) {
		tmp = b * (-0.6666666666666666 / a);
	} else if (b <= 3e-12) {
		tmp = t_0;
	} else {
		tmp = 1.0 / ((-2.0 * (b / c)) + (1.5 * (a / b)));
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (math.sqrt((c * (a * -3.0))) - b) / (a * 3.0)
	tmp = 0
	if b <= -4.5e-14:
		tmp = ((1.5 * (c * (a / b))) + (b * -2.0)) / (a * 3.0)
	elif b <= -8e-64:
		tmp = t_0
	elif b <= -4.5e-133:
		tmp = b * (-0.6666666666666666 / a)
	elif b <= 3e-12:
		tmp = t_0
	else:
		tmp = 1.0 / ((-2.0 * (b / c)) + (1.5 * (a / b)))
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(sqrt(Float64(c * Float64(a * -3.0))) - b) / Float64(a * 3.0))
	tmp = 0.0
	if (b <= -4.5e-14)
		tmp = Float64(Float64(Float64(1.5 * Float64(c * Float64(a / b))) + Float64(b * -2.0)) / Float64(a * 3.0));
	elseif (b <= -8e-64)
		tmp = t_0;
	elseif (b <= -4.5e-133)
		tmp = Float64(b * Float64(-0.6666666666666666 / a));
	elseif (b <= 3e-12)
		tmp = t_0;
	else
		tmp = Float64(1.0 / Float64(Float64(-2.0 * Float64(b / c)) + Float64(1.5 * Float64(a / b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (sqrt((c * (a * -3.0))) - b) / (a * 3.0);
	tmp = 0.0;
	if (b <= -4.5e-14)
		tmp = ((1.5 * (c * (a / b))) + (b * -2.0)) / (a * 3.0);
	elseif (b <= -8e-64)
		tmp = t_0;
	elseif (b <= -4.5e-133)
		tmp = b * (-0.6666666666666666 / a);
	elseif (b <= 3e-12)
		tmp = t_0;
	else
		tmp = 1.0 / ((-2.0 * (b / c)) + (1.5 * (a / b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[Sqrt[N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -4.5e-14], N[(N[(N[(1.5 * N[(c * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(b * -2.0), $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, -8e-64], t$95$0, If[LessEqual[b, -4.5e-133], N[(b * N[(-0.6666666666666666 / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 3e-12], t$95$0, N[(1.0 / N[(N[(-2.0 * N[(b / c), $MachinePrecision]), $MachinePrecision] + N[(1.5 * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -4.4999999999999998e-14

   1. Initial program 72.3%

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

      1. add-sqr-sqrt 72.1%

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

      2. pow2 72.1%

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

      3. pow1/2 72.1%

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

      4. sqrt-pow1 72.1%

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

      5. sub-neg 72.1%

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

      6. +-commutative 72.1%

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

      7. distribute-lft-neg-in 72.1%

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

      8. *-commutative 72.1%

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

      9. distribute-rgt-neg-in 72.1%

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

      10. metadata-eval 72.1%

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

      11. associate-*r* 72.1%

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

      12. *-commutative 72.1%

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

      13. fma-udef 72.2%

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

      14. pow2 72.2%

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

      15. metadata-eval 72.2%

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

   3. Applied egg-rr 72.2%

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

   4. Taylor expanded in b around -inf 82.5%

      \[\leadsto \frac{\color{blue}{2 \cdot \left(-1 \cdot b\right) + 1.5 \cdot \frac{a \cdot c}{b}}}{3 \cdot a} \]
   5. Step-by-step derivation

      1. +-commutative 82.5%

         \[\leadsto \frac{\color{blue}{1.5 \cdot \frac{a \cdot c}{b} + 2 \cdot \left(-1 \cdot b\right)}}{3 \cdot a} \]

      2. associate-/l* 86.1%

         \[\leadsto \frac{1.5 \cdot \color{blue}{\frac{a}{\frac{b}{c}}} + 2 \cdot \left(-1 \cdot b\right)}{3 \cdot a} \]

      3. associate-/r/ 86.1%

         \[\leadsto \frac{1.5 \cdot \color{blue}{\left(\frac{a}{b} \cdot c\right)} + 2 \cdot \left(-1 \cdot b\right)}{3 \cdot a} \]

      4. associate-*r* 86.1%

         \[\leadsto \frac{1.5 \cdot \left(\frac{a}{b} \cdot c\right) + \color{blue}{\left(2 \cdot -1\right) \cdot b}}{3 \cdot a} \]

      5. metadata-eval 86.1%

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

      6. *-commutative 86.1%

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

   6. Simplified 86.1%

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

   if -4.4999999999999998e-14 < b < -7.99999999999999972e-64 or -4.50000000000000009e-133 < b < 3.0000000000000001e-12

   1. Initial program 81.7%

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

      1. prod-diff 81.4%

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

      2. *-commutative 81.4%

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

      3. fma-def 81.4%

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

      4. associate-+l+ 81.4%

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

      5. pow2 81.4%

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

      6. *-commutative 81.4%

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

      7. distribute-rgt-neg-in 81.4%

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

      8. *-commutative 81.4%

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

      9. distribute-rgt-neg-in 81.4%

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

      10. metadata-eval 81.4%

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

      11. *-commutative 81.4%

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

      12. fma-udef 81.4%

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

      13. distribute-lft-neg-in 81.4%

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

      14. distribute-rgt-neg-in 81.4%

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

      15. *-commutative 81.4%

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

      16. distribute-rgt-neg-in 81.4%

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

      17. metadata-eval 81.4%

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

      18. associate-*l* 81.4%

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

   3. Applied egg-rr 81.4%

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

   4. Taylor expanded in b around 0 78.6%

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

      1. neg-mul-1 78.6%

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

      2. unsub-neg 78.6%

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

      3. distribute-rgt-out 78.9%

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

      4. metadata-eval 78.9%

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

      5. *-commutative 78.9%

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

      6. associate-*r* 79.0%

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

   6. Simplified 79.0%

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

   if -7.99999999999999972e-64 < b < -4.50000000000000009e-133

   1. Initial program 89.8%

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

   3. Applied egg-rr 89.6%

      \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)} \cdot \frac{0.3333333333333333}{a} + \left(-b \cdot \frac{0.3333333333333333}{a}\right)} \]
   4. Step-by-step derivation

      1. sub-neg 89.6%

         \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)} \cdot \frac{0.3333333333333333}{a} - b \cdot \frac{0.3333333333333333}{a}} \]

      2. distribute-rgt-out-- 89.6%

         \[\leadsto \color{blue}{\frac{0.3333333333333333}{a} \cdot \left(\sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)} - b\right)} \]

   5. Simplified 89.6%

      \[\leadsto \color{blue}{\frac{0.3333333333333333}{a} \cdot \left(\sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)} - b\right)} \]
   6. Step-by-step derivation

      1. associate-*l/ 89.3%

         \[\leadsto \color{blue}{\frac{0.3333333333333333 \cdot \left(\sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)} - b\right)}{a}} \]

      2. *-commutative 89.3%

         \[\leadsto \frac{\color{blue}{\left(\sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)} - b\right) \cdot 0.3333333333333333}}{a} \]

      3. sub-neg 89.3%

         \[\leadsto \frac{\color{blue}{\left(\sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)} + \left(-b\right)\right)} \cdot 0.3333333333333333}{a} \]

      4. +-commutative 89.3%

         \[\leadsto \frac{\color{blue}{\left(\left(-b\right) + \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}\right)} \cdot 0.3333333333333333}{a} \]

      5. neg-mul-1 89.3%

         \[\leadsto \frac{\left(\color{blue}{-1 \cdot b} + \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}\right) \cdot 0.3333333333333333}{a} \]

      6. fma-udef 89.3%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}\right)} \cdot 0.3333333333333333}{a} \]

      7. metadata-eval 89.3%

         \[\leadsto \frac{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}\right) \cdot \color{blue}{\frac{1}{3}}}{a} \]

      8. div-inv 89.5%

         \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}\right)}{3}}}{a} \]

      9. associate-/r* 89.8%

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}\right)}{3 \cdot a}} \]

      10. *-commutative 89.8%

          \[\leadsto \frac{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}\right)}{\color{blue}{a \cdot 3}} \]

      11. associate-/r* 89.5%

          \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}\right)}{a}}{3}} \]

      12. fma-udef 89.5%

          \[\leadsto \frac{\frac{\color{blue}{-1 \cdot b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}}}{a}}{3} \]

      13. neg-mul-1 89.5%

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

      14. +-commutative 89.5%

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

      15. sub-neg 89.5%

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

   7. Applied egg-rr 89.5%

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

   8. Taylor expanded in b around -inf 71.7%

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

      1. *-commutative 71.7%

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

      2. associate-*l/ 71.8%

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

      3. associate-*r/ 72.1%

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

   10. Simplified 72.1%

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

   if 3.0000000000000001e-12 < b

   1. Initial program 14.6%

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

      1. add-sqr-sqrt 11.2%

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

      2. pow2 11.2%

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

      3. pow1/2 11.2%

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

      4. sqrt-pow1 11.2%

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

      5. sub-neg 11.2%

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

      6. +-commutative 11.2%

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

      7. distribute-lft-neg-in 11.2%

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

      8. *-commutative 11.2%

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

      9. distribute-rgt-neg-in 11.2%

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

      10. metadata-eval 11.2%

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

      11. associate-*r* 11.2%

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

      12. *-commutative 11.2%

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

      13. fma-udef 11.3%

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

      14. pow2 11.3%

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

      15. metadata-eval 11.3%

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

   3. Applied egg-rr 11.3%

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

      1. clear-num 11.2%

         \[\leadsto \color{blue}{\frac{1}{\frac{3 \cdot a}{\left(-b\right) + {\left({\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{0.25}\right)}^{2}}}} \]

      2. inv-pow 11.2%

         \[\leadsto \color{blue}{{\left(\frac{3 \cdot a}{\left(-b\right) + {\left({\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{0.25}\right)}^{2}}\right)}^{-1}} \]

      3. *-commutative 11.2%

         \[\leadsto {\left(\frac{\color{blue}{a \cdot 3}}{\left(-b\right) + {\left({\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{0.25}\right)}^{2}}\right)}^{-1} \]

      4. neg-mul-1 11.2%

         \[\leadsto {\left(\frac{a \cdot 3}{\color{blue}{-1 \cdot b} + {\left({\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{0.25}\right)}^{2}}\right)}^{-1} \]

      5. pow-pow 14.7%

         \[\leadsto {\left(\frac{a \cdot 3}{-1 \cdot b + \color{blue}{{\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{\left(0.25 \cdot 2\right)}}}\right)}^{-1} \]

      6. metadata-eval 14.7%

         \[\leadsto {\left(\frac{a \cdot 3}{-1 \cdot b + {\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{\color{blue}{0.5}}}\right)}^{-1} \]

      7. pow1/2 14.7%

         \[\leadsto {\left(\frac{a \cdot 3}{-1 \cdot b + \color{blue}{\sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}}}\right)}^{-1} \]

      8. fma-def 14.7%

         \[\leadsto {\left(\frac{a \cdot 3}{\color{blue}{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}\right)}}\right)}^{-1} \]

   5. Applied egg-rr 14.7%

      \[\leadsto \color{blue}{{\left(\frac{a \cdot 3}{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}\right)}\right)}^{-1}} \]
   6. Step-by-step derivation

      1. unpow-1 14.7%

         \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 3}{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}\right)}}} \]

   7. Simplified 14.7%

      \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 3}{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}\right)}}} \]

   8. Taylor expanded in b around inf 91.0%

      \[\leadsto \frac{1}{\color{blue}{-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -4.5 \cdot 10^{-14}:\\ \;\;\;\;\frac{1.5 \cdot \left(c \cdot \frac{a}{b}\right) + b \cdot -2}{a \cdot 3}\\ \mathbf{elif}\;b \leq -8 \cdot 10^{-64}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{elif}\;b \leq -4.5 \cdot 10^{-133}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{elif}\;b \leq 3 \cdot 10^{-12}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}}\\ \end{array} \]

Alternative 3: 84.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.3 \cdot 10^{+97}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 3.2 \cdot 10^{-12}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.3e+97)
   (+ (* -0.6666666666666666 (/ b a)) (* 0.5 (/ c b)))
   (if (<= b 3.2e-12)
     (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0))
     (/ 1.0 (+ (* -2.0 (/ b c)) (* 1.5 (/ a b)))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.3e+97) {
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	} else if (b <= 3.2e-12) {
		tmp = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	} else {
		tmp = 1.0 / ((-2.0 * (b / c)) + (1.5 * (a / 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 <= (-1.3d+97)) then
        tmp = ((-0.6666666666666666d0) * (b / a)) + (0.5d0 * (c / b))
    else if (b <= 3.2d-12) then
        tmp = (sqrt(((b * b) - (c * (a * 3.0d0)))) - b) / (a * 3.0d0)
    else
        tmp = 1.0d0 / (((-2.0d0) * (b / c)) + (1.5d0 * (a / b)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.3e+97) {
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	} else if (b <= 3.2e-12) {
		tmp = (Math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	} else {
		tmp = 1.0 / ((-2.0 * (b / c)) + (1.5 * (a / b)));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.3e+97:
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b))
	elif b <= 3.2e-12:
		tmp = (math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)
	else:
		tmp = 1.0 / ((-2.0 * (b / c)) + (1.5 * (a / b)))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.3e+97)
		tmp = Float64(Float64(-0.6666666666666666 * Float64(b / a)) + Float64(0.5 * Float64(c / b)));
	elseif (b <= 3.2e-12)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(1.0 / Float64(Float64(-2.0 * Float64(b / c)) + Float64(1.5 * Float64(a / b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.3e+97)
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	elseif (b <= 3.2e-12)
		tmp = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	else
		tmp = 1.0 / ((-2.0 * (b / c)) + (1.5 * (a / b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.3e+97], N[(N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 3.2e-12], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(-2.0 * N[(b / c), $MachinePrecision]), $MachinePrecision] + N[(1.5 * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.3e97

   1. Initial program 59.3%

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

   2. Taylor expanded in b around -inf 91.5%

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

   if -1.3e97 < b < 3.2000000000000001e-12

   1. Initial program 86.0%

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

   if 3.2000000000000001e-12 < b

   1. Initial program 14.6%

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

      1. add-sqr-sqrt 11.2%

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

      2. pow2 11.2%

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

      3. pow1/2 11.2%

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

      4. sqrt-pow1 11.2%

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

      5. sub-neg 11.2%

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

      6. +-commutative 11.2%

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

      7. distribute-lft-neg-in 11.2%

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

      8. *-commutative 11.2%

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

      9. distribute-rgt-neg-in 11.2%

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

      10. metadata-eval 11.2%

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

      11. associate-*r* 11.2%

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

      12. *-commutative 11.2%

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

      13. fma-udef 11.3%

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

      14. pow2 11.3%

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

      15. metadata-eval 11.3%

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

   3. Applied egg-rr 11.3%

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

      1. clear-num 11.2%

         \[\leadsto \color{blue}{\frac{1}{\frac{3 \cdot a}{\left(-b\right) + {\left({\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{0.25}\right)}^{2}}}} \]

      2. inv-pow 11.2%

         \[\leadsto \color{blue}{{\left(\frac{3 \cdot a}{\left(-b\right) + {\left({\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{0.25}\right)}^{2}}\right)}^{-1}} \]

      3. *-commutative 11.2%

         \[\leadsto {\left(\frac{\color{blue}{a \cdot 3}}{\left(-b\right) + {\left({\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{0.25}\right)}^{2}}\right)}^{-1} \]

      4. neg-mul-1 11.2%

         \[\leadsto {\left(\frac{a \cdot 3}{\color{blue}{-1 \cdot b} + {\left({\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{0.25}\right)}^{2}}\right)}^{-1} \]

      5. pow-pow 14.7%

         \[\leadsto {\left(\frac{a \cdot 3}{-1 \cdot b + \color{blue}{{\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{\left(0.25 \cdot 2\right)}}}\right)}^{-1} \]

      6. metadata-eval 14.7%

         \[\leadsto {\left(\frac{a \cdot 3}{-1 \cdot b + {\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{\color{blue}{0.5}}}\right)}^{-1} \]

      7. pow1/2 14.7%

         \[\leadsto {\left(\frac{a \cdot 3}{-1 \cdot b + \color{blue}{\sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}}}\right)}^{-1} \]

      8. fma-def 14.7%

         \[\leadsto {\left(\frac{a \cdot 3}{\color{blue}{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}\right)}}\right)}^{-1} \]

   5. Applied egg-rr 14.7%

      \[\leadsto \color{blue}{{\left(\frac{a \cdot 3}{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}\right)}\right)}^{-1}} \]
   6. Step-by-step derivation

      1. unpow-1 14.7%

         \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 3}{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}\right)}}} \]

   7. Simplified 14.7%

      \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 3}{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}\right)}}} \]

   8. Taylor expanded in b around inf 91.0%

      \[\leadsto \frac{1}{\color{blue}{-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.3 \cdot 10^{+97}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 3.2 \cdot 10^{-12}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}}\\ \end{array} \]

Alternative 4: 78.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.3333333333333333 \cdot \frac{\sqrt{c \cdot \left(a \cdot -3\right)}}{a}\\ \mathbf{if}\;b \leq -1.4 \cdot 10^{-17}:\\ \;\;\;\;\frac{1.5 \cdot \left(c \cdot \frac{a}{b}\right) + b \cdot -2}{a \cdot 3}\\ \mathbf{elif}\;b \leq -5.5 \cdot 10^{-63}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq -4.5 \cdot 10^{-133}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{elif}\;b \leq 3 \cdot 10^{-12}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* 0.3333333333333333 (/ (sqrt (* c (* a -3.0))) a))))
   (if (<= b -1.4e-17)
     (/ (+ (* 1.5 (* c (/ a b))) (* b -2.0)) (* a 3.0))
     (if (<= b -5.5e-63)
       t_0
       (if (<= b -4.5e-133)
         (* b (/ -0.6666666666666666 a))
         (if (<= b 3e-12)
           t_0
           (/ 1.0 (+ (* -2.0 (/ b c)) (* 1.5 (/ a b))))))))))

C

double code(double a, double b, double c) {
	double t_0 = 0.3333333333333333 * (sqrt((c * (a * -3.0))) / a);
	double tmp;
	if (b <= -1.4e-17) {
		tmp = ((1.5 * (c * (a / b))) + (b * -2.0)) / (a * 3.0);
	} else if (b <= -5.5e-63) {
		tmp = t_0;
	} else if (b <= -4.5e-133) {
		tmp = b * (-0.6666666666666666 / a);
	} else if (b <= 3e-12) {
		tmp = t_0;
	} else {
		tmp = 1.0 / ((-2.0 * (b / c)) + (1.5 * (a / 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) :: t_0
    real(8) :: tmp
    t_0 = 0.3333333333333333d0 * (sqrt((c * (a * (-3.0d0)))) / a)
    if (b <= (-1.4d-17)) then
        tmp = ((1.5d0 * (c * (a / b))) + (b * (-2.0d0))) / (a * 3.0d0)
    else if (b <= (-5.5d-63)) then
        tmp = t_0
    else if (b <= (-4.5d-133)) then
        tmp = b * ((-0.6666666666666666d0) / a)
    else if (b <= 3d-12) then
        tmp = t_0
    else
        tmp = 1.0d0 / (((-2.0d0) * (b / c)) + (1.5d0 * (a / b)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = 0.3333333333333333 * (Math.sqrt((c * (a * -3.0))) / a);
	double tmp;
	if (b <= -1.4e-17) {
		tmp = ((1.5 * (c * (a / b))) + (b * -2.0)) / (a * 3.0);
	} else if (b <= -5.5e-63) {
		tmp = t_0;
	} else if (b <= -4.5e-133) {
		tmp = b * (-0.6666666666666666 / a);
	} else if (b <= 3e-12) {
		tmp = t_0;
	} else {
		tmp = 1.0 / ((-2.0 * (b / c)) + (1.5 * (a / b)));
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = 0.3333333333333333 * (math.sqrt((c * (a * -3.0))) / a)
	tmp = 0
	if b <= -1.4e-17:
		tmp = ((1.5 * (c * (a / b))) + (b * -2.0)) / (a * 3.0)
	elif b <= -5.5e-63:
		tmp = t_0
	elif b <= -4.5e-133:
		tmp = b * (-0.6666666666666666 / a)
	elif b <= 3e-12:
		tmp = t_0
	else:
		tmp = 1.0 / ((-2.0 * (b / c)) + (1.5 * (a / b)))
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(0.3333333333333333 * Float64(sqrt(Float64(c * Float64(a * -3.0))) / a))
	tmp = 0.0
	if (b <= -1.4e-17)
		tmp = Float64(Float64(Float64(1.5 * Float64(c * Float64(a / b))) + Float64(b * -2.0)) / Float64(a * 3.0));
	elseif (b <= -5.5e-63)
		tmp = t_0;
	elseif (b <= -4.5e-133)
		tmp = Float64(b * Float64(-0.6666666666666666 / a));
	elseif (b <= 3e-12)
		tmp = t_0;
	else
		tmp = Float64(1.0 / Float64(Float64(-2.0 * Float64(b / c)) + Float64(1.5 * Float64(a / b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = 0.3333333333333333 * (sqrt((c * (a * -3.0))) / a);
	tmp = 0.0;
	if (b <= -1.4e-17)
		tmp = ((1.5 * (c * (a / b))) + (b * -2.0)) / (a * 3.0);
	elseif (b <= -5.5e-63)
		tmp = t_0;
	elseif (b <= -4.5e-133)
		tmp = b * (-0.6666666666666666 / a);
	elseif (b <= 3e-12)
		tmp = t_0;
	else
		tmp = 1.0 / ((-2.0 * (b / c)) + (1.5 * (a / b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(0.3333333333333333 * N[(N[Sqrt[N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -1.4e-17], N[(N[(N[(1.5 * N[(c * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(b * -2.0), $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, -5.5e-63], t$95$0, If[LessEqual[b, -4.5e-133], N[(b * N[(-0.6666666666666666 / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 3e-12], t$95$0, N[(1.0 / N[(N[(-2.0 * N[(b / c), $MachinePrecision]), $MachinePrecision] + N[(1.5 * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -1.3999999999999999e-17

   1. Initial program 72.3%

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

      1. add-sqr-sqrt 72.1%

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

      2. pow2 72.1%

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

      3. pow1/2 72.1%

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

      4. sqrt-pow1 72.1%

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

      5. sub-neg 72.1%

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

      6. +-commutative 72.1%

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

      7. distribute-lft-neg-in 72.1%

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

      8. *-commutative 72.1%

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

      9. distribute-rgt-neg-in 72.1%

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

      10. metadata-eval 72.1%

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

      11. associate-*r* 72.1%

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

      12. *-commutative 72.1%

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

      13. fma-udef 72.2%

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

      14. pow2 72.2%

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

      15. metadata-eval 72.2%

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

   3. Applied egg-rr 72.2%

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

   4. Taylor expanded in b around -inf 82.5%

      \[\leadsto \frac{\color{blue}{2 \cdot \left(-1 \cdot b\right) + 1.5 \cdot \frac{a \cdot c}{b}}}{3 \cdot a} \]
   5. Step-by-step derivation

      1. +-commutative 82.5%

         \[\leadsto \frac{\color{blue}{1.5 \cdot \frac{a \cdot c}{b} + 2 \cdot \left(-1 \cdot b\right)}}{3 \cdot a} \]

      2. associate-/l* 86.1%

         \[\leadsto \frac{1.5 \cdot \color{blue}{\frac{a}{\frac{b}{c}}} + 2 \cdot \left(-1 \cdot b\right)}{3 \cdot a} \]

      3. associate-/r/ 86.1%

         \[\leadsto \frac{1.5 \cdot \color{blue}{\left(\frac{a}{b} \cdot c\right)} + 2 \cdot \left(-1 \cdot b\right)}{3 \cdot a} \]

      4. associate-*r* 86.1%

         \[\leadsto \frac{1.5 \cdot \left(\frac{a}{b} \cdot c\right) + \color{blue}{\left(2 \cdot -1\right) \cdot b}}{3 \cdot a} \]

      5. metadata-eval 86.1%

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

      6. *-commutative 86.1%

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

   6. Simplified 86.1%

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

   if -1.3999999999999999e-17 < b < -5.50000000000000043e-63 or -4.50000000000000009e-133 < b < 3.0000000000000001e-12

   1. Initial program 81.7%

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

      1. prod-diff 81.4%

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

      2. *-commutative 81.4%

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

      3. fma-def 81.4%

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

      4. associate-+l+ 81.4%

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

      5. pow2 81.4%

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

      6. *-commutative 81.4%

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

      7. distribute-rgt-neg-in 81.4%

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

      8. *-commutative 81.4%

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

      9. distribute-rgt-neg-in 81.4%

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

      10. metadata-eval 81.4%

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

      11. *-commutative 81.4%

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

      12. fma-udef 81.4%

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

      13. distribute-lft-neg-in 81.4%

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

      14. distribute-rgt-neg-in 81.4%

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

      15. *-commutative 81.4%

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

      16. distribute-rgt-neg-in 81.4%

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

      17. metadata-eval 81.4%

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

      18. associate-*l* 81.4%

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

   3. Applied egg-rr 81.4%

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

   4. Taylor expanded in b around 0 77.6%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \left(\frac{1}{a} \cdot \sqrt{-6 \cdot \left(a \cdot c\right) + 3 \cdot \left(a \cdot c\right)}\right)} \]
   5. Step-by-step derivation

      1. associate-*l/ 77.8%

         \[\leadsto 0.3333333333333333 \cdot \color{blue}{\frac{1 \cdot \sqrt{-6 \cdot \left(a \cdot c\right) + 3 \cdot \left(a \cdot c\right)}}{a}} \]

      2. *-lft-identity 77.8%

         \[\leadsto 0.3333333333333333 \cdot \frac{\color{blue}{\sqrt{-6 \cdot \left(a \cdot c\right) + 3 \cdot \left(a \cdot c\right)}}}{a} \]

      3. distribute-rgt-out 78.0%

         \[\leadsto 0.3333333333333333 \cdot \frac{\sqrt{\color{blue}{\left(a \cdot c\right) \cdot \left(-6 + 3\right)}}}{a} \]

      4. metadata-eval 78.0%

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

      5. *-commutative 78.0%

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

      6. associate-*r* 78.0%

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

   6. Simplified 78.0%

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

   if -5.50000000000000043e-63 < b < -4.50000000000000009e-133

   1. Initial program 89.8%

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

   3. Applied egg-rr 89.6%

      \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)} \cdot \frac{0.3333333333333333}{a} + \left(-b \cdot \frac{0.3333333333333333}{a}\right)} \]
   4. Step-by-step derivation

      1. sub-neg 89.6%

         \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)} \cdot \frac{0.3333333333333333}{a} - b \cdot \frac{0.3333333333333333}{a}} \]

      2. distribute-rgt-out-- 89.6%

         \[\leadsto \color{blue}{\frac{0.3333333333333333}{a} \cdot \left(\sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)} - b\right)} \]

   5. Simplified 89.6%

      \[\leadsto \color{blue}{\frac{0.3333333333333333}{a} \cdot \left(\sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)} - b\right)} \]
   6. Step-by-step derivation

      1. associate-*l/ 89.3%

         \[\leadsto \color{blue}{\frac{0.3333333333333333 \cdot \left(\sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)} - b\right)}{a}} \]

      2. *-commutative 89.3%

         \[\leadsto \frac{\color{blue}{\left(\sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)} - b\right) \cdot 0.3333333333333333}}{a} \]

      3. sub-neg 89.3%

         \[\leadsto \frac{\color{blue}{\left(\sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)} + \left(-b\right)\right)} \cdot 0.3333333333333333}{a} \]

      4. +-commutative 89.3%

         \[\leadsto \frac{\color{blue}{\left(\left(-b\right) + \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}\right)} \cdot 0.3333333333333333}{a} \]

      5. neg-mul-1 89.3%

         \[\leadsto \frac{\left(\color{blue}{-1 \cdot b} + \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}\right) \cdot 0.3333333333333333}{a} \]

      6. fma-udef 89.3%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}\right)} \cdot 0.3333333333333333}{a} \]

      7. metadata-eval 89.3%

         \[\leadsto \frac{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}\right) \cdot \color{blue}{\frac{1}{3}}}{a} \]

      8. div-inv 89.5%

         \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}\right)}{3}}}{a} \]

      9. associate-/r* 89.8%

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}\right)}{3 \cdot a}} \]

      10. *-commutative 89.8%

          \[\leadsto \frac{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}\right)}{\color{blue}{a \cdot 3}} \]

      11. associate-/r* 89.5%

          \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}\right)}{a}}{3}} \]

      12. fma-udef 89.5%

          \[\leadsto \frac{\frac{\color{blue}{-1 \cdot b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}}}{a}}{3} \]

      13. neg-mul-1 89.5%

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

      14. +-commutative 89.5%

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

      15. sub-neg 89.5%

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

   7. Applied egg-rr 89.5%

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

   8. Taylor expanded in b around -inf 71.7%

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

      1. *-commutative 71.7%

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

      2. associate-*l/ 71.8%

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

      3. associate-*r/ 72.1%

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

   10. Simplified 72.1%

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

   if 3.0000000000000001e-12 < b

   1. Initial program 14.6%

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

      1. add-sqr-sqrt 11.2%

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

      2. pow2 11.2%

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

      3. pow1/2 11.2%

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

      4. sqrt-pow1 11.2%

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

      5. sub-neg 11.2%

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

      6. +-commutative 11.2%

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

      7. distribute-lft-neg-in 11.2%

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

      8. *-commutative 11.2%

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

      9. distribute-rgt-neg-in 11.2%

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

      10. metadata-eval 11.2%

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

      11. associate-*r* 11.2%

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

      12. *-commutative 11.2%

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

      13. fma-udef 11.3%

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

      14. pow2 11.3%

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

      15. metadata-eval 11.3%

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

   3. Applied egg-rr 11.3%

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

      1. clear-num 11.2%

         \[\leadsto \color{blue}{\frac{1}{\frac{3 \cdot a}{\left(-b\right) + {\left({\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{0.25}\right)}^{2}}}} \]

      2. inv-pow 11.2%

         \[\leadsto \color{blue}{{\left(\frac{3 \cdot a}{\left(-b\right) + {\left({\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{0.25}\right)}^{2}}\right)}^{-1}} \]

      3. *-commutative 11.2%

         \[\leadsto {\left(\frac{\color{blue}{a \cdot 3}}{\left(-b\right) + {\left({\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{0.25}\right)}^{2}}\right)}^{-1} \]

      4. neg-mul-1 11.2%

         \[\leadsto {\left(\frac{a \cdot 3}{\color{blue}{-1 \cdot b} + {\left({\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{0.25}\right)}^{2}}\right)}^{-1} \]

      5. pow-pow 14.7%

         \[\leadsto {\left(\frac{a \cdot 3}{-1 \cdot b + \color{blue}{{\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{\left(0.25 \cdot 2\right)}}}\right)}^{-1} \]

      6. metadata-eval 14.7%

         \[\leadsto {\left(\frac{a \cdot 3}{-1 \cdot b + {\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{\color{blue}{0.5}}}\right)}^{-1} \]

      7. pow1/2 14.7%

         \[\leadsto {\left(\frac{a \cdot 3}{-1 \cdot b + \color{blue}{\sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}}}\right)}^{-1} \]

      8. fma-def 14.7%

         \[\leadsto {\left(\frac{a \cdot 3}{\color{blue}{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}\right)}}\right)}^{-1} \]

   5. Applied egg-rr 14.7%

      \[\leadsto \color{blue}{{\left(\frac{a \cdot 3}{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}\right)}\right)}^{-1}} \]
   6. Step-by-step derivation

      1. unpow-1 14.7%

         \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 3}{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}\right)}}} \]

   7. Simplified 14.7%

      \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 3}{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}\right)}}} \]

   8. Taylor expanded in b around inf 91.0%

      \[\leadsto \frac{1}{\color{blue}{-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.4 \cdot 10^{-17}:\\ \;\;\;\;\frac{1.5 \cdot \left(c \cdot \frac{a}{b}\right) + b \cdot -2}{a \cdot 3}\\ \mathbf{elif}\;b \leq -5.5 \cdot 10^{-63}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\sqrt{c \cdot \left(a \cdot -3\right)}}{a}\\ \mathbf{elif}\;b \leq -4.5 \cdot 10^{-133}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{elif}\;b \leq 3 \cdot 10^{-12}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\sqrt{c \cdot \left(a \cdot -3\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}}\\ \end{array} \]

Alternative 5: 78.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{c \cdot \left(a \cdot -3\right)}\\ \mathbf{if}\;b \leq -1.9 \cdot 10^{-17}:\\ \;\;\;\;\frac{1.5 \cdot \left(c \cdot \frac{a}{b}\right) + b \cdot -2}{a \cdot 3}\\ \mathbf{elif}\;b \leq -7.8 \cdot 10^{-64}:\\ \;\;\;\;t_0 \cdot \left(0.3333333333333333 \cdot \frac{1}{a}\right)\\ \mathbf{elif}\;b \leq -4.5 \cdot 10^{-133}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{elif}\;b \leq 5.5 \cdot 10^{-11}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{t_0}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (sqrt (* c (* a -3.0)))))
   (if (<= b -1.9e-17)
     (/ (+ (* 1.5 (* c (/ a b))) (* b -2.0)) (* a 3.0))
     (if (<= b -7.8e-64)
       (* t_0 (* 0.3333333333333333 (/ 1.0 a)))
       (if (<= b -4.5e-133)
         (* b (/ -0.6666666666666666 a))
         (if (<= b 5.5e-11)
           (* 0.3333333333333333 (/ t_0 a))
           (/ 1.0 (+ (* -2.0 (/ b c)) (* 1.5 (/ a b))))))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt((c * (a * -3.0)));
	double tmp;
	if (b <= -1.9e-17) {
		tmp = ((1.5 * (c * (a / b))) + (b * -2.0)) / (a * 3.0);
	} else if (b <= -7.8e-64) {
		tmp = t_0 * (0.3333333333333333 * (1.0 / a));
	} else if (b <= -4.5e-133) {
		tmp = b * (-0.6666666666666666 / a);
	} else if (b <= 5.5e-11) {
		tmp = 0.3333333333333333 * (t_0 / a);
	} else {
		tmp = 1.0 / ((-2.0 * (b / c)) + (1.5 * (a / 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) :: t_0
    real(8) :: tmp
    t_0 = sqrt((c * (a * (-3.0d0))))
    if (b <= (-1.9d-17)) then
        tmp = ((1.5d0 * (c * (a / b))) + (b * (-2.0d0))) / (a * 3.0d0)
    else if (b <= (-7.8d-64)) then
        tmp = t_0 * (0.3333333333333333d0 * (1.0d0 / a))
    else if (b <= (-4.5d-133)) then
        tmp = b * ((-0.6666666666666666d0) / a)
    else if (b <= 5.5d-11) then
        tmp = 0.3333333333333333d0 * (t_0 / a)
    else
        tmp = 1.0d0 / (((-2.0d0) * (b / c)) + (1.5d0 * (a / b)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = Math.sqrt((c * (a * -3.0)));
	double tmp;
	if (b <= -1.9e-17) {
		tmp = ((1.5 * (c * (a / b))) + (b * -2.0)) / (a * 3.0);
	} else if (b <= -7.8e-64) {
		tmp = t_0 * (0.3333333333333333 * (1.0 / a));
	} else if (b <= -4.5e-133) {
		tmp = b * (-0.6666666666666666 / a);
	} else if (b <= 5.5e-11) {
		tmp = 0.3333333333333333 * (t_0 / a);
	} else {
		tmp = 1.0 / ((-2.0 * (b / c)) + (1.5 * (a / b)));
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = math.sqrt((c * (a * -3.0)))
	tmp = 0
	if b <= -1.9e-17:
		tmp = ((1.5 * (c * (a / b))) + (b * -2.0)) / (a * 3.0)
	elif b <= -7.8e-64:
		tmp = t_0 * (0.3333333333333333 * (1.0 / a))
	elif b <= -4.5e-133:
		tmp = b * (-0.6666666666666666 / a)
	elif b <= 5.5e-11:
		tmp = 0.3333333333333333 * (t_0 / a)
	else:
		tmp = 1.0 / ((-2.0 * (b / c)) + (1.5 * (a / b)))
	return tmp

Julia

function code(a, b, c)
	t_0 = sqrt(Float64(c * Float64(a * -3.0)))
	tmp = 0.0
	if (b <= -1.9e-17)
		tmp = Float64(Float64(Float64(1.5 * Float64(c * Float64(a / b))) + Float64(b * -2.0)) / Float64(a * 3.0));
	elseif (b <= -7.8e-64)
		tmp = Float64(t_0 * Float64(0.3333333333333333 * Float64(1.0 / a)));
	elseif (b <= -4.5e-133)
		tmp = Float64(b * Float64(-0.6666666666666666 / a));
	elseif (b <= 5.5e-11)
		tmp = Float64(0.3333333333333333 * Float64(t_0 / a));
	else
		tmp = Float64(1.0 / Float64(Float64(-2.0 * Float64(b / c)) + Float64(1.5 * Float64(a / b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = sqrt((c * (a * -3.0)));
	tmp = 0.0;
	if (b <= -1.9e-17)
		tmp = ((1.5 * (c * (a / b))) + (b * -2.0)) / (a * 3.0);
	elseif (b <= -7.8e-64)
		tmp = t_0 * (0.3333333333333333 * (1.0 / a));
	elseif (b <= -4.5e-133)
		tmp = b * (-0.6666666666666666 / a);
	elseif (b <= 5.5e-11)
		tmp = 0.3333333333333333 * (t_0 / a);
	else
		tmp = 1.0 / ((-2.0 * (b / c)) + (1.5 * (a / b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Sqrt[N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[b, -1.9e-17], N[(N[(N[(1.5 * N[(c * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(b * -2.0), $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, -7.8e-64], N[(t$95$0 * N[(0.3333333333333333 * N[(1.0 / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, -4.5e-133], N[(b * N[(-0.6666666666666666 / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5.5e-11], N[(0.3333333333333333 * N[(t$95$0 / a), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(-2.0 * N[(b / c), $MachinePrecision]), $MachinePrecision] + N[(1.5 * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if b < -1.9000000000000001e-17

   1. Initial program 72.3%

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

      1. add-sqr-sqrt 72.1%

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

      2. pow2 72.1%

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

      3. pow1/2 72.1%

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

      4. sqrt-pow1 72.1%

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

      5. sub-neg 72.1%

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

      6. +-commutative 72.1%

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

      7. distribute-lft-neg-in 72.1%

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

      8. *-commutative 72.1%

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

      9. distribute-rgt-neg-in 72.1%

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

      10. metadata-eval 72.1%

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

      11. associate-*r* 72.1%

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

      12. *-commutative 72.1%

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

      13. fma-udef 72.2%

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

      14. pow2 72.2%

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

      15. metadata-eval 72.2%

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

   3. Applied egg-rr 72.2%

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

   4. Taylor expanded in b around -inf 82.5%

      \[\leadsto \frac{\color{blue}{2 \cdot \left(-1 \cdot b\right) + 1.5 \cdot \frac{a \cdot c}{b}}}{3 \cdot a} \]
   5. Step-by-step derivation

      1. +-commutative 82.5%

         \[\leadsto \frac{\color{blue}{1.5 \cdot \frac{a \cdot c}{b} + 2 \cdot \left(-1 \cdot b\right)}}{3 \cdot a} \]

      2. associate-/l* 86.1%

         \[\leadsto \frac{1.5 \cdot \color{blue}{\frac{a}{\frac{b}{c}}} + 2 \cdot \left(-1 \cdot b\right)}{3 \cdot a} \]

      3. associate-/r/ 86.1%

         \[\leadsto \frac{1.5 \cdot \color{blue}{\left(\frac{a}{b} \cdot c\right)} + 2 \cdot \left(-1 \cdot b\right)}{3 \cdot a} \]

      4. associate-*r* 86.1%

         \[\leadsto \frac{1.5 \cdot \left(\frac{a}{b} \cdot c\right) + \color{blue}{\left(2 \cdot -1\right) \cdot b}}{3 \cdot a} \]

      5. metadata-eval 86.1%

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

      6. *-commutative 86.1%

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

   6. Simplified 86.1%

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

   if -1.9000000000000001e-17 < b < -7.7999999999999994e-64

   1. Initial program 99.4%

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

      1. prod-diff 99.4%

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

      2. *-commutative 99.4%

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

      3. fma-def 99.4%

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

      4. associate-+l+ 99.4%

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

      5. pow2 99.4%

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

      6. *-commutative 99.4%

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

      7. distribute-rgt-neg-in 99.4%

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

      8. *-commutative 99.4%

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

      9. distribute-rgt-neg-in 99.4%

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

      10. metadata-eval 99.4%

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

      11. *-commutative 99.4%

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

      12. fma-udef 99.4%

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

      13. distribute-lft-neg-in 99.4%

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

      14. distribute-rgt-neg-in 99.4%

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

      15. *-commutative 99.4%

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

      16. distribute-rgt-neg-in 99.4%

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

      17. metadata-eval 99.4%

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

      18. associate-*l* 99.4%

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

   3. Applied egg-rr 99.4%

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

   4. Taylor expanded in b around 0 98.8%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \left(\frac{1}{a} \cdot \sqrt{-6 \cdot \left(a \cdot c\right) + 3 \cdot \left(a \cdot c\right)}\right)} \]
   5. Step-by-step derivation

      1. associate-*r* 99.4%

         \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot \frac{1}{a}\right) \cdot \sqrt{-6 \cdot \left(a \cdot c\right) + 3 \cdot \left(a \cdot c\right)}} \]

      2. distribute-rgt-out 99.4%

         \[\leadsto \left(0.3333333333333333 \cdot \frac{1}{a}\right) \cdot \sqrt{\color{blue}{\left(a \cdot c\right) \cdot \left(-6 + 3\right)}} \]

      3. metadata-eval 99.4%

         \[\leadsto \left(0.3333333333333333 \cdot \frac{1}{a}\right) \cdot \sqrt{\left(a \cdot c\right) \cdot \color{blue}{-3}} \]

      4. *-commutative 99.4%

         \[\leadsto \left(0.3333333333333333 \cdot \frac{1}{a}\right) \cdot \sqrt{\color{blue}{\left(c \cdot a\right)} \cdot -3} \]

      5. associate-*r* 99.1%

         \[\leadsto \left(0.3333333333333333 \cdot \frac{1}{a}\right) \cdot \sqrt{\color{blue}{c \cdot \left(a \cdot -3\right)}} \]

   6. Simplified 99.1%

      \[\leadsto \color{blue}{\left(0.3333333333333333 \cdot \frac{1}{a}\right) \cdot \sqrt{c \cdot \left(a \cdot -3\right)}} \]

   if -7.7999999999999994e-64 < b < -4.50000000000000009e-133

   1. Initial program 89.8%

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

   3. Applied egg-rr 89.6%

      \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)} \cdot \frac{0.3333333333333333}{a} + \left(-b \cdot \frac{0.3333333333333333}{a}\right)} \]
   4. Step-by-step derivation

      1. sub-neg 89.6%

         \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)} \cdot \frac{0.3333333333333333}{a} - b \cdot \frac{0.3333333333333333}{a}} \]

      2. distribute-rgt-out-- 89.6%

         \[\leadsto \color{blue}{\frac{0.3333333333333333}{a} \cdot \left(\sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)} - b\right)} \]

   5. Simplified 89.6%

      \[\leadsto \color{blue}{\frac{0.3333333333333333}{a} \cdot \left(\sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)} - b\right)} \]
   6. Step-by-step derivation

      1. associate-*l/ 89.3%

         \[\leadsto \color{blue}{\frac{0.3333333333333333 \cdot \left(\sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)} - b\right)}{a}} \]

      2. *-commutative 89.3%

         \[\leadsto \frac{\color{blue}{\left(\sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)} - b\right) \cdot 0.3333333333333333}}{a} \]

      3. sub-neg 89.3%

         \[\leadsto \frac{\color{blue}{\left(\sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)} + \left(-b\right)\right)} \cdot 0.3333333333333333}{a} \]

      4. +-commutative 89.3%

         \[\leadsto \frac{\color{blue}{\left(\left(-b\right) + \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}\right)} \cdot 0.3333333333333333}{a} \]

      5. neg-mul-1 89.3%

         \[\leadsto \frac{\left(\color{blue}{-1 \cdot b} + \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}\right) \cdot 0.3333333333333333}{a} \]

      6. fma-udef 89.3%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}\right)} \cdot 0.3333333333333333}{a} \]

      7. metadata-eval 89.3%

         \[\leadsto \frac{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}\right) \cdot \color{blue}{\frac{1}{3}}}{a} \]

      8. div-inv 89.5%

         \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}\right)}{3}}}{a} \]

      9. associate-/r* 89.8%

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}\right)}{3 \cdot a}} \]

      10. *-commutative 89.8%

          \[\leadsto \frac{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}\right)}{\color{blue}{a \cdot 3}} \]

      11. associate-/r* 89.5%

          \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}\right)}{a}}{3}} \]

      12. fma-udef 89.5%

          \[\leadsto \frac{\frac{\color{blue}{-1 \cdot b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}}}{a}}{3} \]

      13. neg-mul-1 89.5%

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

      14. +-commutative 89.5%

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

      15. sub-neg 89.5%

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

   7. Applied egg-rr 89.5%

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

   8. Taylor expanded in b around -inf 71.7%

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

      1. *-commutative 71.7%

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

      2. associate-*l/ 71.8%

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

      3. associate-*r/ 72.1%

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

   10. Simplified 72.1%

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

   if -4.50000000000000009e-133 < b < 5.49999999999999975e-11

   1. Initial program 80.5%

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

      1. prod-diff 80.2%

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

      2. *-commutative 80.2%

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

      3. fma-def 80.2%

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

      4. associate-+l+ 80.2%

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

      5. pow2 80.2%

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

      6. *-commutative 80.2%

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

      7. distribute-rgt-neg-in 80.2%

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

      8. *-commutative 80.2%

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

      9. distribute-rgt-neg-in 80.2%

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

      10. metadata-eval 80.2%

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

      11. *-commutative 80.2%

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

      12. fma-udef 80.2%

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

      13. distribute-lft-neg-in 80.2%

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

      14. distribute-rgt-neg-in 80.2%

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

      15. *-commutative 80.2%

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

      16. distribute-rgt-neg-in 80.2%

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

      17. metadata-eval 80.2%

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

      18. associate-*l* 80.2%

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

   3. Applied egg-rr 80.2%

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

   4. Taylor expanded in b around 0 76.2%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \left(\frac{1}{a} \cdot \sqrt{-6 \cdot \left(a \cdot c\right) + 3 \cdot \left(a \cdot c\right)}\right)} \]
   5. Step-by-step derivation

      1. associate-*l/ 76.3%

         \[\leadsto 0.3333333333333333 \cdot \color{blue}{\frac{1 \cdot \sqrt{-6 \cdot \left(a \cdot c\right) + 3 \cdot \left(a \cdot c\right)}}{a}} \]

      2. *-lft-identity 76.3%

         \[\leadsto 0.3333333333333333 \cdot \frac{\color{blue}{\sqrt{-6 \cdot \left(a \cdot c\right) + 3 \cdot \left(a \cdot c\right)}}}{a} \]

      3. distribute-rgt-out 76.6%

         \[\leadsto 0.3333333333333333 \cdot \frac{\sqrt{\color{blue}{\left(a \cdot c\right) \cdot \left(-6 + 3\right)}}}{a} \]

      4. metadata-eval 76.6%

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

      5. *-commutative 76.6%

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

      6. associate-*r* 76.6%

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

   6. Simplified 76.6%

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

   if 5.49999999999999975e-11 < b

   1. Initial program 14.6%

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

      1. add-sqr-sqrt 11.2%

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

      2. pow2 11.2%

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

      3. pow1/2 11.2%

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

      4. sqrt-pow1 11.2%

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

      5. sub-neg 11.2%

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

      6. +-commutative 11.2%

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

      7. distribute-lft-neg-in 11.2%

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

      8. *-commutative 11.2%

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

      9. distribute-rgt-neg-in 11.2%

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

      10. metadata-eval 11.2%

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

      11. associate-*r* 11.2%

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

      12. *-commutative 11.2%

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

      13. fma-udef 11.3%

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

      14. pow2 11.3%

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

      15. metadata-eval 11.3%

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

   3. Applied egg-rr 11.3%

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

      1. clear-num 11.2%

         \[\leadsto \color{blue}{\frac{1}{\frac{3 \cdot a}{\left(-b\right) + {\left({\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{0.25}\right)}^{2}}}} \]

      2. inv-pow 11.2%

         \[\leadsto \color{blue}{{\left(\frac{3 \cdot a}{\left(-b\right) + {\left({\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{0.25}\right)}^{2}}\right)}^{-1}} \]

      3. *-commutative 11.2%

         \[\leadsto {\left(\frac{\color{blue}{a \cdot 3}}{\left(-b\right) + {\left({\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{0.25}\right)}^{2}}\right)}^{-1} \]

      4. neg-mul-1 11.2%

         \[\leadsto {\left(\frac{a \cdot 3}{\color{blue}{-1 \cdot b} + {\left({\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{0.25}\right)}^{2}}\right)}^{-1} \]

      5. pow-pow 14.7%

         \[\leadsto {\left(\frac{a \cdot 3}{-1 \cdot b + \color{blue}{{\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{\left(0.25 \cdot 2\right)}}}\right)}^{-1} \]

      6. metadata-eval 14.7%

         \[\leadsto {\left(\frac{a \cdot 3}{-1 \cdot b + {\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{\color{blue}{0.5}}}\right)}^{-1} \]

      7. pow1/2 14.7%

         \[\leadsto {\left(\frac{a \cdot 3}{-1 \cdot b + \color{blue}{\sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}}}\right)}^{-1} \]

      8. fma-def 14.7%

         \[\leadsto {\left(\frac{a \cdot 3}{\color{blue}{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}\right)}}\right)}^{-1} \]

   5. Applied egg-rr 14.7%

      \[\leadsto \color{blue}{{\left(\frac{a \cdot 3}{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}\right)}\right)}^{-1}} \]
   6. Step-by-step derivation

      1. unpow-1 14.7%

         \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 3}{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}\right)}}} \]

   7. Simplified 14.7%

      \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 3}{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}\right)}}} \]

   8. Taylor expanded in b around inf 91.0%

      \[\leadsto \frac{1}{\color{blue}{-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.9 \cdot 10^{-17}:\\ \;\;\;\;\frac{1.5 \cdot \left(c \cdot \frac{a}{b}\right) + b \cdot -2}{a \cdot 3}\\ \mathbf{elif}\;b \leq -7.8 \cdot 10^{-64}:\\ \;\;\;\;\sqrt{c \cdot \left(a \cdot -3\right)} \cdot \left(0.3333333333333333 \cdot \frac{1}{a}\right)\\ \mathbf{elif}\;b \leq -4.5 \cdot 10^{-133}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{elif}\;b \leq 5.5 \cdot 10^{-11}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\sqrt{c \cdot \left(a \cdot -3\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}}\\ \end{array} \]

Alternative 6: 67.2% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.05 \cdot 10^{-268}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.05e-268)
   (+ (* -0.6666666666666666 (/ b a)) (* 0.5 (/ c b)))
   (/ 1.0 (+ (* -2.0 (/ b c)) (* 1.5 (/ a b))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.05e-268) {
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	} else {
		tmp = 1.0 / ((-2.0 * (b / c)) + (1.5 * (a / b)));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-2.05d-268)) then
        tmp = ((-0.6666666666666666d0) * (b / a)) + (0.5d0 * (c / b))
    else
        tmp = 1.0d0 / (((-2.0d0) * (b / c)) + (1.5d0 * (a / b)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.05e-268) {
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	} else {
		tmp = 1.0 / ((-2.0 * (b / c)) + (1.5 * (a / b)));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.05e-268:
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b))
	else:
		tmp = 1.0 / ((-2.0 * (b / c)) + (1.5 * (a / b)))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.05e-268)
		tmp = Float64(Float64(-0.6666666666666666 * Float64(b / a)) + Float64(0.5 * Float64(c / b)));
	else
		tmp = Float64(1.0 / Float64(Float64(-2.0 * Float64(b / c)) + Float64(1.5 * Float64(a / b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.05e-268)
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	else
		tmp = 1.0 / ((-2.0 * (b / c)) + (1.5 * (a / b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.05e-268], N[(N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(-2.0 * N[(b / c), $MachinePrecision]), $MachinePrecision] + N[(1.5 * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -2.0499999999999999e-268

   1. Initial program 77.1%

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

   2. Taylor expanded in b around -inf 70.0%

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

   if -2.0499999999999999e-268 < b

   1. Initial program 42.3%

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

      1. add-sqr-sqrt 40.2%

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

      2. pow2 40.2%

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

      3. pow1/2 40.2%

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

      4. sqrt-pow1 40.3%

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

      5. sub-neg 40.3%

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

      6. +-commutative 40.3%

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

      7. distribute-lft-neg-in 40.3%

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

      8. *-commutative 40.3%

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

      9. distribute-rgt-neg-in 40.3%

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

      10. metadata-eval 40.3%

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

      11. associate-*r* 40.3%

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

      12. *-commutative 40.3%

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

      13. fma-udef 40.3%

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

      14. pow2 40.3%

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

      15. metadata-eval 40.3%

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

   3. Applied egg-rr 40.3%

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

      1. clear-num 40.3%

         \[\leadsto \color{blue}{\frac{1}{\frac{3 \cdot a}{\left(-b\right) + {\left({\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{0.25}\right)}^{2}}}} \]

      2. inv-pow 40.3%

         \[\leadsto \color{blue}{{\left(\frac{3 \cdot a}{\left(-b\right) + {\left({\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{0.25}\right)}^{2}}\right)}^{-1}} \]

      3. *-commutative 40.3%

         \[\leadsto {\left(\frac{\color{blue}{a \cdot 3}}{\left(-b\right) + {\left({\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{0.25}\right)}^{2}}\right)}^{-1} \]

      4. neg-mul-1 40.3%

         \[\leadsto {\left(\frac{a \cdot 3}{\color{blue}{-1 \cdot b} + {\left({\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{0.25}\right)}^{2}}\right)}^{-1} \]

      5. pow-pow 42.2%

         \[\leadsto {\left(\frac{a \cdot 3}{-1 \cdot b + \color{blue}{{\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{\left(0.25 \cdot 2\right)}}}\right)}^{-1} \]

      6. metadata-eval 42.2%

         \[\leadsto {\left(\frac{a \cdot 3}{-1 \cdot b + {\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{\color{blue}{0.5}}}\right)}^{-1} \]

      7. pow1/2 42.2%

         \[\leadsto {\left(\frac{a \cdot 3}{-1 \cdot b + \color{blue}{\sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}}}\right)}^{-1} \]

      8. fma-def 42.2%

         \[\leadsto {\left(\frac{a \cdot 3}{\color{blue}{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}\right)}}\right)}^{-1} \]

   5. Applied egg-rr 42.2%

      \[\leadsto \color{blue}{{\left(\frac{a \cdot 3}{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}\right)}\right)}^{-1}} \]
   6. Step-by-step derivation

      1. unpow-1 42.2%

         \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 3}{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}\right)}}} \]

   7. Simplified 42.2%

      \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 3}{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}\right)}}} \]

   8. Taylor expanded in b around inf 58.6%

      \[\leadsto \frac{1}{\color{blue}{-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.05 \cdot 10^{-268}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}}\\ \end{array} \]

Alternative 7: 23.8% accurate, 16.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.4 \cdot 10^{-291}:\\ \;\;\;\;\frac{b}{a} \cdot -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.4e-291) (* (/ b a) -0.3333333333333333) (/ 0.0 a)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.4e-291) {
		tmp = (b / a) * -0.3333333333333333;
	} else {
		tmp = 0.0 / a;
	}
	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 <= (-1.4d-291)) then
        tmp = (b / a) * (-0.3333333333333333d0)
    else
        tmp = 0.0d0 / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.4e-291) {
		tmp = (b / a) * -0.3333333333333333;
	} else {
		tmp = 0.0 / a;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.4e-291:
		tmp = (b / a) * -0.3333333333333333
	else:
		tmp = 0.0 / a
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.4e-291)
		tmp = Float64(Float64(b / a) * -0.3333333333333333);
	else
		tmp = Float64(0.0 / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.4e-291)
		tmp = (b / a) * -0.3333333333333333;
	else
		tmp = 0.0 / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.4e-291], N[(N[(b / a), $MachinePrecision] * -0.3333333333333333), $MachinePrecision], N[(0.0 / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -1.4e-291

   1. Initial program 77.6%

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

      1. add-sqr-sqrt 77.3%

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

      2. pow2 77.3%

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

      3. pow1/2 77.3%

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

      4. sqrt-pow1 77.3%

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

      5. sub-neg 77.3%

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

      6. +-commutative 77.3%

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

      7. distribute-lft-neg-in 77.3%

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

      8. *-commutative 77.3%

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

      9. distribute-rgt-neg-in 77.3%

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

      10. metadata-eval 77.3%

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

      11. associate-*r* 77.3%

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

      12. *-commutative 77.3%

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

      13. fma-udef 77.4%

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

      14. pow2 77.4%

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

      15. metadata-eval 77.4%

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

   3. Applied egg-rr 77.4%

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

   4. Taylor expanded in b around inf 30.7%

      \[\leadsto \color{blue}{-0.3333333333333333 \cdot \frac{b}{a}} \]

   if -1.4e-291 < b

   1. Initial program 40.8%

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

      1. add-sqr-sqrt 38.7%

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

      2. pow2 38.7%

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

      3. pow1/2 38.7%

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

      4. sqrt-pow1 38.8%

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

      5. sub-neg 38.8%

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

      6. +-commutative 38.8%

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

      7. distribute-lft-neg-in 38.8%

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

      8. *-commutative 38.8%

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

      9. distribute-rgt-neg-in 38.8%

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

      10. metadata-eval 38.8%

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

      11. associate-*r* 38.7%

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

      12. *-commutative 38.7%

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

      13. fma-udef 38.8%

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

      14. pow2 38.8%

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

      15. metadata-eval 38.8%

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

   3. Applied egg-rr 38.8%

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

      1. clear-num 38.7%

         \[\leadsto \color{blue}{\frac{1}{\frac{3 \cdot a}{\left(-b\right) + {\left({\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{0.25}\right)}^{2}}}} \]

      2. inv-pow 38.7%

         \[\leadsto \color{blue}{{\left(\frac{3 \cdot a}{\left(-b\right) + {\left({\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{0.25}\right)}^{2}}\right)}^{-1}} \]

      3. *-commutative 38.7%

         \[\leadsto {\left(\frac{\color{blue}{a \cdot 3}}{\left(-b\right) + {\left({\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{0.25}\right)}^{2}}\right)}^{-1} \]

      4. neg-mul-1 38.7%

         \[\leadsto {\left(\frac{a \cdot 3}{\color{blue}{-1 \cdot b} + {\left({\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{0.25}\right)}^{2}}\right)}^{-1} \]

      5. pow-pow 40.7%

         \[\leadsto {\left(\frac{a \cdot 3}{-1 \cdot b + \color{blue}{{\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{\left(0.25 \cdot 2\right)}}}\right)}^{-1} \]

      6. metadata-eval 40.7%

         \[\leadsto {\left(\frac{a \cdot 3}{-1 \cdot b + {\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{\color{blue}{0.5}}}\right)}^{-1} \]

      7. pow1/2 40.7%

         \[\leadsto {\left(\frac{a \cdot 3}{-1 \cdot b + \color{blue}{\sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}}}\right)}^{-1} \]

      8. fma-def 40.7%

         \[\leadsto {\left(\frac{a \cdot 3}{\color{blue}{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}\right)}}\right)}^{-1} \]

   5. Applied egg-rr 40.7%

      \[\leadsto \color{blue}{{\left(\frac{a \cdot 3}{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}\right)}\right)}^{-1}} \]
   6. Step-by-step derivation

      1. unpow-1 40.7%

         \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 3}{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}\right)}}} \]

   7. Simplified 40.7%

      \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 3}{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}\right)}}} \]

   8. Taylor expanded in a around 0 16.9%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{b + -1 \cdot b}{a}} \]
   9. Step-by-step derivation

      1. associate-*r/ 16.9%

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

      2. distribute-rgt1-in 16.9%

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

      3. metadata-eval 16.9%

         \[\leadsto \frac{0.3333333333333333 \cdot \left(\color{blue}{0} \cdot b\right)}{a} \]

      4. mul0-lft 16.9%

         \[\leadsto \frac{0.3333333333333333 \cdot \color{blue}{0}}{a} \]

      5. metadata-eval 16.9%

         \[\leadsto \frac{\color{blue}{0}}{a} \]

   10. Simplified 16.9%

       \[\leadsto \color{blue}{\frac{0}{a}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 24.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.4 \cdot 10^{-291}:\\ \;\;\;\;\frac{b}{a} \cdot -0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{a}\\ \end{array} \]

Alternative 8: 43.4% accurate, 16.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e-310) (* b (/ -0.6666666666666666 a)) (/ 0.0 a)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = b * (-0.6666666666666666 / a);
	} else {
		tmp = 0.0 / a;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-5d-310)) then
        tmp = b * ((-0.6666666666666666d0) / a)
    else
        tmp = 0.0d0 / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = b * (-0.6666666666666666 / a);
	} else {
		tmp = 0.0 / a;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e-310:
		tmp = b * (-0.6666666666666666 / a)
	else:
		tmp = 0.0 / a
	return tmp

Julia

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

MATLAB

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

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-310], N[(b * N[(-0.6666666666666666 / a), $MachinePrecision]), $MachinePrecision], N[(0.0 / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 77.7%

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

   3. Applied egg-rr 77.6%

      \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)} \cdot \frac{0.3333333333333333}{a} + \left(-b \cdot \frac{0.3333333333333333}{a}\right)} \]
   4. Step-by-step derivation

      1. sub-neg 77.6%

         \[\leadsto \color{blue}{\sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)} \cdot \frac{0.3333333333333333}{a} - b \cdot \frac{0.3333333333333333}{a}} \]

      2. distribute-rgt-out-- 77.6%

         \[\leadsto \color{blue}{\frac{0.3333333333333333}{a} \cdot \left(\sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)} - b\right)} \]

   5. Simplified 77.6%

      \[\leadsto \color{blue}{\frac{0.3333333333333333}{a} \cdot \left(\sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)} - b\right)} \]
   6. Step-by-step derivation

      1. associate-*l/ 77.6%

         \[\leadsto \color{blue}{\frac{0.3333333333333333 \cdot \left(\sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)} - b\right)}{a}} \]

      2. *-commutative 77.6%

         \[\leadsto \frac{\color{blue}{\left(\sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)} - b\right) \cdot 0.3333333333333333}}{a} \]

      3. sub-neg 77.6%

         \[\leadsto \frac{\color{blue}{\left(\sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)} + \left(-b\right)\right)} \cdot 0.3333333333333333}{a} \]

      4. +-commutative 77.6%

         \[\leadsto \frac{\color{blue}{\left(\left(-b\right) + \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}\right)} \cdot 0.3333333333333333}{a} \]

      5. neg-mul-1 77.6%

         \[\leadsto \frac{\left(\color{blue}{-1 \cdot b} + \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}\right) \cdot 0.3333333333333333}{a} \]

      6. fma-udef 77.6%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}\right)} \cdot 0.3333333333333333}{a} \]

      7. metadata-eval 77.6%

         \[\leadsto \frac{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}\right) \cdot \color{blue}{\frac{1}{3}}}{a} \]

      8. div-inv 77.7%

         \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}\right)}{3}}}{a} \]

      9. associate-/r* 77.8%

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}\right)}{3 \cdot a}} \]

      10. *-commutative 77.8%

          \[\leadsto \frac{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}\right)}{\color{blue}{a \cdot 3}} \]

      11. associate-/r* 77.7%

          \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}\right)}{a}}{3}} \]

      12. fma-udef 77.7%

          \[\leadsto \frac{\frac{\color{blue}{-1 \cdot b + \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}}}{a}}{3} \]

      13. neg-mul-1 77.7%

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

      14. +-commutative 77.7%

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

      15. sub-neg 77.7%

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

   7. Applied egg-rr 77.7%

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

   8. Taylor expanded in b around -inf 68.1%

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

      1. *-commutative 68.1%

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

      2. associate-*l/ 68.0%

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

      3. associate-*r/ 68.1%

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

   10. Simplified 68.1%

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

   if -4.999999999999985e-310 < b

   1. Initial program 40.3%

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

      1. add-sqr-sqrt 38.2%

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

      2. pow2 38.2%

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

      3. pow1/2 38.2%

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

      4. sqrt-pow1 38.3%

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

      5. sub-neg 38.3%

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

      6. +-commutative 38.3%

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

      7. distribute-lft-neg-in 38.3%

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

      8. *-commutative 38.3%

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

      9. distribute-rgt-neg-in 38.3%

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

      10. metadata-eval 38.3%

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

      11. associate-*r* 38.2%

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

      12. *-commutative 38.2%

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

      13. fma-udef 38.2%

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

      14. pow2 38.2%

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

      15. metadata-eval 38.2%

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

   3. Applied egg-rr 38.2%

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

      1. clear-num 38.2%

         \[\leadsto \color{blue}{\frac{1}{\frac{3 \cdot a}{\left(-b\right) + {\left({\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{0.25}\right)}^{2}}}} \]

      2. inv-pow 38.2%

         \[\leadsto \color{blue}{{\left(\frac{3 \cdot a}{\left(-b\right) + {\left({\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{0.25}\right)}^{2}}\right)}^{-1}} \]

      3. *-commutative 38.2%

         \[\leadsto {\left(\frac{\color{blue}{a \cdot 3}}{\left(-b\right) + {\left({\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{0.25}\right)}^{2}}\right)}^{-1} \]

      4. neg-mul-1 38.2%

         \[\leadsto {\left(\frac{a \cdot 3}{\color{blue}{-1 \cdot b} + {\left({\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{0.25}\right)}^{2}}\right)}^{-1} \]

      5. pow-pow 40.2%

         \[\leadsto {\left(\frac{a \cdot 3}{-1 \cdot b + \color{blue}{{\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{\left(0.25 \cdot 2\right)}}}\right)}^{-1} \]

      6. metadata-eval 40.2%

         \[\leadsto {\left(\frac{a \cdot 3}{-1 \cdot b + {\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{\color{blue}{0.5}}}\right)}^{-1} \]

      7. pow1/2 40.2%

         \[\leadsto {\left(\frac{a \cdot 3}{-1 \cdot b + \color{blue}{\sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}}}\right)}^{-1} \]

      8. fma-def 40.2%

         \[\leadsto {\left(\frac{a \cdot 3}{\color{blue}{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}\right)}}\right)}^{-1} \]

   5. Applied egg-rr 40.2%

      \[\leadsto \color{blue}{{\left(\frac{a \cdot 3}{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}\right)}\right)}^{-1}} \]
   6. Step-by-step derivation

      1. unpow-1 40.2%

         \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 3}{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}\right)}}} \]

   7. Simplified 40.2%

      \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 3}{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}\right)}}} \]

   8. Taylor expanded in a around 0 17.0%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{b + -1 \cdot b}{a}} \]
   9. Step-by-step derivation

      1. associate-*r/ 17.0%

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

      2. distribute-rgt1-in 17.0%

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

      3. metadata-eval 17.0%

         \[\leadsto \frac{0.3333333333333333 \cdot \left(\color{blue}{0} \cdot b\right)}{a} \]

      4. mul0-lft 17.0%

         \[\leadsto \frac{0.3333333333333333 \cdot \color{blue}{0}}{a} \]

      5. metadata-eval 17.0%

         \[\leadsto \frac{\color{blue}{0}}{a} \]

   10. Simplified 17.0%

       \[\leadsto \color{blue}{\frac{0}{a}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 45.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{a}\\ \end{array} \]

Alternative 9: 43.5% accurate, 16.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e-310) (/ b (* a -1.5)) (/ 0.0 a)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = b / (a * -1.5);
	} else {
		tmp = 0.0 / a;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-5d-310)) then
        tmp = b / (a * (-1.5d0))
    else
        tmp = 0.0d0 / a
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = b / (a * -1.5);
	} else {
		tmp = 0.0 / a;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e-310:
		tmp = b / (a * -1.5)
	else:
		tmp = 0.0 / a
	return tmp

Julia

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

MATLAB

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

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-310], N[(b / N[(a * -1.5), $MachinePrecision]), $MachinePrecision], N[(0.0 / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 77.7%

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

      1. frac-2neg 77.7%

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

      2. div-inv 77.6%

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

   3. Applied egg-rr 77.7%

      \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}\right) \cdot \frac{1}{a \cdot -3}} \]

   4. Taylor expanded in b around -inf 68.1%

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

      1. *-commutative 68.1%

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

   6. Simplified 68.1%

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

      1. un-div-inv 68.2%

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

      2. times-frac 68.1%

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

      3. metadata-eval 68.1%

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

      4. associate-/r/ 68.1%

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

      5. div-inv 68.2%

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

      6. metadata-eval 68.2%

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

   8. Applied egg-rr 68.2%

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

   if -4.999999999999985e-310 < b

   1. Initial program 40.3%

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

      1. add-sqr-sqrt 38.2%

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

      2. pow2 38.2%

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

      3. pow1/2 38.2%

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

      4. sqrt-pow1 38.3%

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

      5. sub-neg 38.3%

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

      6. +-commutative 38.3%

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

      7. distribute-lft-neg-in 38.3%

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

      8. *-commutative 38.3%

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

      9. distribute-rgt-neg-in 38.3%

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

      10. metadata-eval 38.3%

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

      11. associate-*r* 38.2%

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

      12. *-commutative 38.2%

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

      13. fma-udef 38.2%

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

      14. pow2 38.2%

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

      15. metadata-eval 38.2%

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

   3. Applied egg-rr 38.2%

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

      1. clear-num 38.2%

         \[\leadsto \color{blue}{\frac{1}{\frac{3 \cdot a}{\left(-b\right) + {\left({\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{0.25}\right)}^{2}}}} \]

      2. inv-pow 38.2%

         \[\leadsto \color{blue}{{\left(\frac{3 \cdot a}{\left(-b\right) + {\left({\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{0.25}\right)}^{2}}\right)}^{-1}} \]

      3. *-commutative 38.2%

         \[\leadsto {\left(\frac{\color{blue}{a \cdot 3}}{\left(-b\right) + {\left({\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{0.25}\right)}^{2}}\right)}^{-1} \]

      4. neg-mul-1 38.2%

         \[\leadsto {\left(\frac{a \cdot 3}{\color{blue}{-1 \cdot b} + {\left({\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{0.25}\right)}^{2}}\right)}^{-1} \]

      5. pow-pow 40.2%

         \[\leadsto {\left(\frac{a \cdot 3}{-1 \cdot b + \color{blue}{{\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{\left(0.25 \cdot 2\right)}}}\right)}^{-1} \]

      6. metadata-eval 40.2%

         \[\leadsto {\left(\frac{a \cdot 3}{-1 \cdot b + {\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{\color{blue}{0.5}}}\right)}^{-1} \]

      7. pow1/2 40.2%

         \[\leadsto {\left(\frac{a \cdot 3}{-1 \cdot b + \color{blue}{\sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}}}\right)}^{-1} \]

      8. fma-def 40.2%

         \[\leadsto {\left(\frac{a \cdot 3}{\color{blue}{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}\right)}}\right)}^{-1} \]

   5. Applied egg-rr 40.2%

      \[\leadsto \color{blue}{{\left(\frac{a \cdot 3}{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}\right)}\right)}^{-1}} \]
   6. Step-by-step derivation

      1. unpow-1 40.2%

         \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 3}{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}\right)}}} \]

   7. Simplified 40.2%

      \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 3}{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}\right)}}} \]

   8. Taylor expanded in a around 0 17.0%

      \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{b + -1 \cdot b}{a}} \]
   9. Step-by-step derivation

      1. associate-*r/ 17.0%

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

      2. distribute-rgt1-in 17.0%

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

      3. metadata-eval 17.0%

         \[\leadsto \frac{0.3333333333333333 \cdot \left(\color{blue}{0} \cdot b\right)}{a} \]

      4. mul0-lft 17.0%

         \[\leadsto \frac{0.3333333333333333 \cdot \color{blue}{0}}{a} \]

      5. metadata-eval 17.0%

         \[\leadsto \frac{\color{blue}{0}}{a} \]

   10. Simplified 17.0%

       \[\leadsto \color{blue}{\frac{0}{a}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 45.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{b}{a \cdot -1.5}\\ \mathbf{else}:\\ \;\;\;\;\frac{0}{a}\\ \end{array} \]

Alternative 10: 67.3% accurate, 16.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 77.7%

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

      1. frac-2neg 77.7%

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

      2. div-inv 77.6%

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

   3. Applied egg-rr 77.7%

      \[\leadsto \color{blue}{\left(b - \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}\right) \cdot \frac{1}{a \cdot -3}} \]

   4. Taylor expanded in b around -inf 68.1%

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

      1. *-commutative 68.1%

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

   6. Simplified 68.1%

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

      1. un-div-inv 68.2%

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

      2. times-frac 68.1%

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

      3. metadata-eval 68.1%

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

      4. associate-/r/ 68.1%

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

      5. div-inv 68.2%

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

      6. metadata-eval 68.2%

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

   8. Applied egg-rr 68.2%

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

   if -4.999999999999985e-310 < b

   1. Initial program 40.3%

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

   2. Taylor expanded in b around inf 60.4%

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

      1. *-commutative 60.4%

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

      2. associate-*l/ 60.4%

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

   4. Simplified 60.4%

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

4. Final simplification 64.7%

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

Alternative 11: 11.2% accurate, 38.7× speedup

Math

\[\begin{array}{l} \\ \frac{0}{a} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (/ 0.0 a))

C

double code(double a, double b, double c) {
	return 0.0 / 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 = 0.0d0 / a
end function

Java

public static double code(double a, double b, double c) {
	return 0.0 / a;
}

Python

def code(a, b, c):
	return 0.0 / a

Julia

function code(a, b, c)
	return Float64(0.0 / a)
end

MATLAB

function tmp = code(a, b, c)
	tmp = 0.0 / a;
end

Wolfram

code[a_, b_, c_] := N[(0.0 / a), $MachinePrecision]

Derivation

1. Initial program 61.0%

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

   1. add-sqr-sqrt 60.0%

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

   2. pow2 60.0%

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

   3. pow1/2 60.0%

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

   4. sqrt-pow1 60.0%

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

   5. sub-neg 60.0%

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

   6. +-commutative 60.0%

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

   7. distribute-lft-neg-in 60.0%

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

   8. *-commutative 60.0%

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

   9. distribute-rgt-neg-in 60.0%

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

   10. metadata-eval 60.0%

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

   11. associate-*r* 60.0%

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

   12. *-commutative 60.0%

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

   13. fma-udef 60.0%

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

   14. pow2 60.0%

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

   15. metadata-eval 60.0%

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

3. Applied egg-rr 60.0%

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

   1. clear-num 60.0%

      \[\leadsto \color{blue}{\frac{1}{\frac{3 \cdot a}{\left(-b\right) + {\left({\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{0.25}\right)}^{2}}}} \]

   2. inv-pow 60.0%

      \[\leadsto \color{blue}{{\left(\frac{3 \cdot a}{\left(-b\right) + {\left({\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{0.25}\right)}^{2}}\right)}^{-1}} \]

   3. *-commutative 60.0%

      \[\leadsto {\left(\frac{\color{blue}{a \cdot 3}}{\left(-b\right) + {\left({\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{0.25}\right)}^{2}}\right)}^{-1} \]

   4. neg-mul-1 60.0%

      \[\leadsto {\left(\frac{a \cdot 3}{\color{blue}{-1 \cdot b} + {\left({\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{0.25}\right)}^{2}}\right)}^{-1} \]

   5. pow-pow 61.0%

      \[\leadsto {\left(\frac{a \cdot 3}{-1 \cdot b + \color{blue}{{\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{\left(0.25 \cdot 2\right)}}}\right)}^{-1} \]

   6. metadata-eval 61.0%

      \[\leadsto {\left(\frac{a \cdot 3}{-1 \cdot b + {\left(\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)\right)}^{\color{blue}{0.5}}}\right)}^{-1} \]

   7. pow1/2 61.0%

      \[\leadsto {\left(\frac{a \cdot 3}{-1 \cdot b + \color{blue}{\sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}}}\right)}^{-1} \]

   8. fma-def 61.0%

      \[\leadsto {\left(\frac{a \cdot 3}{\color{blue}{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}\right)}}\right)}^{-1} \]

5. Applied egg-rr 61.0%

   \[\leadsto \color{blue}{{\left(\frac{a \cdot 3}{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}\right)}\right)}^{-1}} \]
6. Step-by-step derivation

   1. unpow-1 61.0%

      \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 3}{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}\right)}}} \]

7. Simplified 61.0%

   \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 3}{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(a, c \cdot -3, {b}^{2}\right)}\right)}}} \]

8. Taylor expanded in a around 0 9.1%

   \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{b + -1 \cdot b}{a}} \]
9. Step-by-step derivation

   1. associate-*r/ 9.1%

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

   2. distribute-rgt1-in 9.1%

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

   3. metadata-eval 9.1%

      \[\leadsto \frac{0.3333333333333333 \cdot \left(\color{blue}{0} \cdot b\right)}{a} \]

   4. mul0-lft 9.1%

      \[\leadsto \frac{0.3333333333333333 \cdot \color{blue}{0}}{a} \]

   5. metadata-eval 9.1%

      \[\leadsto \frac{\color{blue}{0}}{a} \]

10. Simplified 9.1%

    \[\leadsto \color{blue}{\frac{0}{a}} \]

11. Final simplification 9.1%

    \[\leadsto \frac{0}{a} \]

Reproduce

herbie shell --seed 2023305 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))