Cubic critical

Percentage Accurate: 52.8% → 86.0%

Time: 18.7s

Alternatives: 14

Speedup: 11.6×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 52.8% 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: 86.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.8 \cdot 10^{+139}:\\ \;\;\;\;\frac{\frac{b}{-1.5}}{a}\\ \mathbf{elif}\;b \leq 1.1 \cdot 10^{-50}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 3\right) \cdot c} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(-0.5, \frac{a}{b}, 0.6666666666666666 \cdot \frac{b}{c}\right)\right)}^{-1} \cdot -0.3333333333333333\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.8e+139)
   (/ (/ b -1.5) a)
   (if (<= b 1.1e-50)
     (/ (- (sqrt (- (* b b) (* (* a 3.0) c))) b) (* a 3.0))
     (*
      (pow (fma -0.5 (/ a b) (* 0.6666666666666666 (/ b c))) -1.0)
      -0.3333333333333333))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.8e+139) {
		tmp = (b / -1.5) / a;
	} else if (b <= 1.1e-50) {
		tmp = (sqrt(((b * b) - ((a * 3.0) * c))) - b) / (a * 3.0);
	} else {
		tmp = pow(fma(-0.5, (a / b), (0.6666666666666666 * (b / c))), -1.0) * -0.3333333333333333;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.8e+139)
		tmp = Float64(Float64(b / -1.5) / a);
	elseif (b <= 1.1e-50)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(a * 3.0) * c))) - b) / Float64(a * 3.0));
	else
		tmp = Float64((fma(-0.5, Float64(a / b), Float64(0.6666666666666666 * Float64(b / c))) ^ -1.0) * -0.3333333333333333);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.8e+139], N[(N[(b / -1.5), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 1.1e-50], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(a * 3.0), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(-0.5 * N[(a / b), $MachinePrecision] + N[(0.6666666666666666 * N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], -1.0], $MachinePrecision] * -0.3333333333333333), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.7999999999999998e139

   1. Initial program 41.9%

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

   3. Taylor expanded in b around -inf 94.5%

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

      1. *-commutative 94.5%

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

   5. Simplified 94.5%

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

      1. add-cbrt-cube 60.9%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\left(\frac{b}{a} \cdot -0.6666666666666666\right) \cdot \left(\frac{b}{a} \cdot -0.6666666666666666\right)\right) \cdot \left(\frac{b}{a} \cdot -0.6666666666666666\right)}} \]

      2. pow1/3 34.3%

         \[\leadsto \color{blue}{{\left(\left(\left(\frac{b}{a} \cdot -0.6666666666666666\right) \cdot \left(\frac{b}{a} \cdot -0.6666666666666666\right)\right) \cdot \left(\frac{b}{a} \cdot -0.6666666666666666\right)\right)}^{0.3333333333333333}} \]

      3. pow3 34.3%

         \[\leadsto {\color{blue}{\left({\left(\frac{b}{a} \cdot -0.6666666666666666\right)}^{3}\right)}}^{0.3333333333333333} \]

   7. Applied egg-rr 34.3%

      \[\leadsto \color{blue}{{\left({\left(\frac{b}{a} \cdot -0.6666666666666666\right)}^{3}\right)}^{0.3333333333333333}} \]
   8. Step-by-step derivation

      1. unpow1/3 60.9%

         \[\leadsto \color{blue}{\sqrt[3]{{\left(\frac{b}{a} \cdot -0.6666666666666666\right)}^{3}}} \]

      2. rem-cbrt-cube 94.5%

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

      3. metadata-eval 94.5%

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

      4. times-frac 94.7%

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

      5. associate-/l/ 94.7%

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

      6. associate-/l* 94.7%

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

      7. metadata-eval 94.7%

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

   9. Applied egg-rr 94.7%

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

   if -2.7999999999999998e139 < b < 1.0999999999999999e-50

   1. Initial program 80.7%

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

   if 1.0999999999999999e-50 < b

   1. Initial program 16.1%

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

      1. fma-neg 16.1%

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

      2. associate-*l* 16.1%

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

      3. distribute-lft-neg-in 16.1%

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

      4. metadata-eval 16.1%

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

      5. *-commutative 16.1%

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

      6. associate-*r* 16.1%

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

      7. add-cube-cbrt 12.4%

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

      8. pow3 12.4%

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

      9. *-commutative 12.4%

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

      10. associate-*r* 12.4%

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

      11. metadata-eval 12.4%

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

      12. distribute-rgt-neg-in 12.4%

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

      13. *-commutative 12.4%

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

      14. *-commutative 12.4%

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

      15. *-commutative 12.4%

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

      16. distribute-rgt-neg-in 12.4%

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

      17. metadata-eval 12.4%

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

   4. Applied egg-rr 12.4%

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

      1. frac-2neg 12.4%

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

      2. div-inv 12.4%

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

   6. Applied egg-rr 16.6%

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

      1. associate-*r/ 16.6%

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

      2. times-frac 16.6%

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

      3. associate-*r* 16.6%

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

      4. *-commutative 16.6%

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

      5. rem-square-sqrt 0.0%

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

      6. unpow2 0.0%

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

      7. associate-*r* 0.0%

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

      8. unpow2 0.0%

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

      9. rem-square-sqrt 16.6%

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

      10. metadata-eval 16.6%

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

   8. Simplified 16.6%

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

      1. clear-num 16.6%

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

      2. inv-pow 16.6%

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

   10. Applied egg-rr 16.6%

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

   11. Taylor expanded in b around inf 0.0%

       \[\leadsto {\color{blue}{\left(-2 \cdot \frac{b}{c \cdot {\left(\sqrt{-3}\right)}^{2}} + -0.5 \cdot \frac{a}{b}\right)}}^{-1} \cdot -0.3333333333333333 \]
   12. Step-by-step derivation

       1. +-commutative 0.0%

          \[\leadsto {\color{blue}{\left(-0.5 \cdot \frac{a}{b} + -2 \cdot \frac{b}{c \cdot {\left(\sqrt{-3}\right)}^{2}}\right)}}^{-1} \cdot -0.3333333333333333 \]

       2. fma-def 0.0%

          \[\leadsto {\color{blue}{\left(\mathsf{fma}\left(-0.5, \frac{a}{b}, -2 \cdot \frac{b}{c \cdot {\left(\sqrt{-3}\right)}^{2}}\right)\right)}}^{-1} \cdot -0.3333333333333333 \]

       3. associate-*r/ 0.0%

          \[\leadsto {\left(\mathsf{fma}\left(-0.5, \frac{a}{b}, \color{blue}{\frac{-2 \cdot b}{c \cdot {\left(\sqrt{-3}\right)}^{2}}}\right)\right)}^{-1} \cdot -0.3333333333333333 \]

       4. *-commutative 0.0%

          \[\leadsto {\left(\mathsf{fma}\left(-0.5, \frac{a}{b}, \frac{-2 \cdot b}{\color{blue}{{\left(\sqrt{-3}\right)}^{2} \cdot c}}\right)\right)}^{-1} \cdot -0.3333333333333333 \]

       5. unpow2 0.0%

          \[\leadsto {\left(\mathsf{fma}\left(-0.5, \frac{a}{b}, \frac{-2 \cdot b}{\color{blue}{\left(\sqrt{-3} \cdot \sqrt{-3}\right)} \cdot c}\right)\right)}^{-1} \cdot -0.3333333333333333 \]

       6. rem-square-sqrt 82.7%

          \[\leadsto {\left(\mathsf{fma}\left(-0.5, \frac{a}{b}, \frac{-2 \cdot b}{\color{blue}{-3} \cdot c}\right)\right)}^{-1} \cdot -0.3333333333333333 \]

       7. times-frac 82.8%

          \[\leadsto {\left(\mathsf{fma}\left(-0.5, \frac{a}{b}, \color{blue}{\frac{-2}{-3} \cdot \frac{b}{c}}\right)\right)}^{-1} \cdot -0.3333333333333333 \]

       8. metadata-eval 82.8%

          \[\leadsto {\left(\mathsf{fma}\left(-0.5, \frac{a}{b}, \color{blue}{0.6666666666666666} \cdot \frac{b}{c}\right)\right)}^{-1} \cdot -0.3333333333333333 \]

   13. Simplified 82.8%

       \[\leadsto {\color{blue}{\left(\mathsf{fma}\left(-0.5, \frac{a}{b}, 0.6666666666666666 \cdot \frac{b}{c}\right)\right)}}^{-1} \cdot -0.3333333333333333 \]
3. Recombined 3 regimes into one program.

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.8 \cdot 10^{+139}:\\ \;\;\;\;\frac{\frac{b}{-1.5}}{a}\\ \mathbf{elif}\;b \leq 1.1 \cdot 10^{-50}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 3\right) \cdot c} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;{\left(\mathsf{fma}\left(-0.5, \frac{a}{b}, 0.6666666666666666 \cdot \frac{b}{c}\right)\right)}^{-1} \cdot -0.3333333333333333\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 77.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.3333333333333333 \cdot \frac{\sqrt{a \cdot \left(c \cdot -3\right)} - b}{a}\\ \mathbf{if}\;b \leq -1.18 \cdot 10^{+80}:\\ \;\;\;\;\frac{\left(\frac{a \cdot 1.5}{\frac{b}{c}} - b\right) - b}{a \cdot 3}\\ \mathbf{elif}\;b \leq -7.2 \cdot 10^{+14}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;b \leq -1.6 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(b, \frac{-0.6666666666666666}{a}, \frac{0.5}{\frac{b}{c}}\right)\\ \mathbf{elif}\;b \leq 1.8 \cdot 10^{-52}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* 0.3333333333333333 (/ (- (sqrt (* a (* c -3.0))) b) a))))
   (if (<= b -1.18e+80)
     (/ (- (- (/ (* a 1.5) (/ b c)) b) b) (* a 3.0))
     (if (<= b -7.2e+14)
       t_0
       (if (<= b -1.6e-8)
         (fma b (/ -0.6666666666666666 a) (/ 0.5 (/ b c)))
         (if (<= b 1.8e-52) t_0 (/ (* c -0.5) b)))))))

C

double code(double a, double b, double c) {
	double t_0 = 0.3333333333333333 * ((sqrt((a * (c * -3.0))) - b) / a);
	double tmp;
	if (b <= -1.18e+80) {
		tmp = ((((a * 1.5) / (b / c)) - b) - b) / (a * 3.0);
	} else if (b <= -7.2e+14) {
		tmp = t_0;
	} else if (b <= -1.6e-8) {
		tmp = fma(b, (-0.6666666666666666 / a), (0.5 / (b / c)));
	} else if (b <= 1.8e-52) {
		tmp = t_0;
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(0.3333333333333333 * Float64(Float64(sqrt(Float64(a * Float64(c * -3.0))) - b) / a))
	tmp = 0.0
	if (b <= -1.18e+80)
		tmp = Float64(Float64(Float64(Float64(Float64(a * 1.5) / Float64(b / c)) - b) - b) / Float64(a * 3.0));
	elseif (b <= -7.2e+14)
		tmp = t_0;
	elseif (b <= -1.6e-8)
		tmp = fma(b, Float64(-0.6666666666666666 / a), Float64(0.5 / Float64(b / c)));
	elseif (b <= 1.8e-52)
		tmp = t_0;
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(0.3333333333333333 * N[(N[(N[Sqrt[N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -1.18e+80], N[(N[(N[(N[(N[(a * 1.5), $MachinePrecision] / N[(b / c), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, -7.2e+14], t$95$0, If[LessEqual[b, -1.6e-8], N[(b * N[(-0.6666666666666666 / a), $MachinePrecision] + N[(0.5 / N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.8e-52], t$95$0, N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -1.18e80

   1. Initial program 58.1%

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

   3. Taylor expanded in b around -inf 91.8%

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

      1. +-commutative 91.8%

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

      2. mul-1-neg 91.8%

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

      3. unsub-neg 91.8%

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

      4. associate-/l* 95.9%

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

      5. associate-*r/ 95.9%

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

   5. Simplified 95.9%

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

   if -1.18e80 < b < -7.2e14 or -1.6000000000000001e-8 < b < 1.79999999999999994e-52

   1. Initial program 78.0%

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

   3. Taylor expanded in b around 0 69.9%

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

      1. associate-*r* 70.0%

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

      2. *-commutative 70.0%

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

      3. *-commutative 70.0%

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

   5. Simplified 70.0%

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

      1. +-commutative 70.0%

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

      2. unsub-neg 70.0%

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

   7. Applied egg-rr 70.0%

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

      1. associate-*r* 69.9%

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

      2. *-commutative 69.9%

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

      3. rem-square-sqrt 0.0%

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

      4. unpow2 0.0%

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

      5. associate-*r* 0.0%

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

      6. unpow2 0.0%

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

      7. rem-square-sqrt 69.9%

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

   9. Simplified 69.9%

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

       1. *-un-lft-identity 69.9%

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

       2. metadata-eval 69.9%

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

       3. times-frac 70.0%

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

       4. metadata-eval 70.0%

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

       5. metadata-eval 70.0%

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

   11. Applied egg-rr 70.0%

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

   if -7.2e14 < b < -1.6000000000000001e-8

   1. Initial program 83.5%

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

   3. Taylor expanded in b around -inf 83.5%

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

      1. +-commutative 83.5%

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

      2. mul-1-neg 83.5%

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

      3. unsub-neg 83.5%

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

      4. associate-/l* 83.5%

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

      5. associate-*r/ 83.5%

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

   5. Simplified 83.5%

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

   6. Taylor expanded in b around 0 83.5%

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

      1. associate-*r/ 83.8%

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

      2. *-commutative 83.8%

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

      3. associate-*r/ 83.8%

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

      4. fma-def 83.8%

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

      5. associate-*r/ 83.8%

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

      6. associate-/l* 83.8%

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

   8. Simplified 83.8%

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

   if 1.79999999999999994e-52 < b

   1. Initial program 16.1%

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

   3. Taylor expanded in b around inf 82.6%

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

      1. associate-*r/ 82.6%

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

   5. Simplified 82.6%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.18 \cdot 10^{+80}:\\ \;\;\;\;\frac{\left(\frac{a \cdot 1.5}{\frac{b}{c}} - b\right) - b}{a \cdot 3}\\ \mathbf{elif}\;b \leq -7.2 \cdot 10^{+14}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\sqrt{a \cdot \left(c \cdot -3\right)} - b}{a}\\ \mathbf{elif}\;b \leq -1.6 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(b, \frac{-0.6666666666666666}{a}, \frac{0.5}{\frac{b}{c}}\right)\\ \mathbf{elif}\;b \leq 1.8 \cdot 10^{-52}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\sqrt{a \cdot \left(c \cdot -3\right)} - b}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 77.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{a \cdot \left(c \cdot -3\right)} - b\\ \mathbf{if}\;b \leq -1.18 \cdot 10^{+80}:\\ \;\;\;\;\frac{\left(\frac{a \cdot 1.5}{\frac{b}{c}} - b\right) - b}{a \cdot 3}\\ \mathbf{elif}\;b \leq -7.2 \cdot 10^{+14}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{t_0}{a}\\ \mathbf{elif}\;b \leq -3.8 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(b, \frac{-0.6666666666666666}{a}, \frac{0.5}{\frac{b}{c}}\right)\\ \mathbf{elif}\;b \leq 3.6 \cdot 10^{-52}:\\ \;\;\;\;\frac{t_0}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (- (sqrt (* a (* c -3.0))) b)))
   (if (<= b -1.18e+80)
     (/ (- (- (/ (* a 1.5) (/ b c)) b) b) (* a 3.0))
     (if (<= b -7.2e+14)
       (* 0.3333333333333333 (/ t_0 a))
       (if (<= b -3.8e-8)
         (fma b (/ -0.6666666666666666 a) (/ 0.5 (/ b c)))
         (if (<= b 3.6e-52) (/ t_0 (* a 3.0)) (/ (* c -0.5) b)))))))

C

double code(double a, double b, double c) {
	double t_0 = sqrt((a * (c * -3.0))) - b;
	double tmp;
	if (b <= -1.18e+80) {
		tmp = ((((a * 1.5) / (b / c)) - b) - b) / (a * 3.0);
	} else if (b <= -7.2e+14) {
		tmp = 0.3333333333333333 * (t_0 / a);
	} else if (b <= -3.8e-8) {
		tmp = fma(b, (-0.6666666666666666 / a), (0.5 / (b / c)));
	} else if (b <= 3.6e-52) {
		tmp = t_0 / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(sqrt(Float64(a * Float64(c * -3.0))) - b)
	tmp = 0.0
	if (b <= -1.18e+80)
		tmp = Float64(Float64(Float64(Float64(Float64(a * 1.5) / Float64(b / c)) - b) - b) / Float64(a * 3.0));
	elseif (b <= -7.2e+14)
		tmp = Float64(0.3333333333333333 * Float64(t_0 / a));
	elseif (b <= -3.8e-8)
		tmp = fma(b, Float64(-0.6666666666666666 / a), Float64(0.5 / Float64(b / c)));
	elseif (b <= 3.6e-52)
		tmp = Float64(t_0 / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[Sqrt[N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]}, If[LessEqual[b, -1.18e+80], N[(N[(N[(N[(N[(a * 1.5), $MachinePrecision] / N[(b / c), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, -7.2e+14], N[(0.3333333333333333 * N[(t$95$0 / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, -3.8e-8], N[(b * N[(-0.6666666666666666 / a), $MachinePrecision] + N[(0.5 / N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 3.6e-52], N[(t$95$0 / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]]]]

Derivation

1. Split input into 5 regimes

2. if b < -1.18e80

   1. Initial program 58.1%

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

   3. Taylor expanded in b around -inf 91.8%

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

      1. +-commutative 91.8%

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

      2. mul-1-neg 91.8%

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

      3. unsub-neg 91.8%

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

      4. associate-/l* 95.9%

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

      5. associate-*r/ 95.9%

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

   5. Simplified 95.9%

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

   if -1.18e80 < b < -7.2e14

   1. Initial program 99.2%

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

   3. Taylor expanded in b around 0 80.6%

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

      1. associate-*r* 80.7%

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

      2. *-commutative 80.7%

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

      3. *-commutative 80.7%

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

   5. Simplified 80.7%

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

      1. +-commutative 80.7%

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

      2. unsub-neg 80.7%

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

   7. Applied egg-rr 80.7%

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

      1. associate-*r* 80.6%

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

      2. *-commutative 80.6%

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

      3. rem-square-sqrt 0.0%

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

      4. unpow2 0.0%

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

      5. associate-*r* 0.0%

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

      6. unpow2 0.0%

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

      7. rem-square-sqrt 80.6%

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

   9. Simplified 80.6%

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

       1. *-un-lft-identity 80.6%

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

       2. metadata-eval 80.6%

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

       3. times-frac 80.9%

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

       4. metadata-eval 80.9%

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

       5. metadata-eval 80.9%

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

   11. Applied egg-rr 80.9%

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

   if -7.2e14 < b < -3.80000000000000028e-8

   1. Initial program 83.5%

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

   3. Taylor expanded in b around -inf 83.5%

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

      1. +-commutative 83.5%

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

      2. mul-1-neg 83.5%

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

      3. unsub-neg 83.5%

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

      4. associate-/l* 83.5%

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

      5. associate-*r/ 83.5%

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

   5. Simplified 83.5%

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

   6. Taylor expanded in b around 0 83.5%

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

      1. associate-*r/ 83.8%

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

      2. *-commutative 83.8%

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

      3. associate-*r/ 83.8%

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

      4. fma-def 83.8%

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

      5. associate-*r/ 83.8%

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

      6. associate-/l* 83.8%

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

   8. Simplified 83.8%

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

   if -3.80000000000000028e-8 < b < 3.59999999999999988e-52

   1. Initial program 74.7%

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

   3. Taylor expanded in b around 0 68.3%

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

      1. associate-*r* 68.4%

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

      2. *-commutative 68.4%

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

      3. *-commutative 68.4%

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

   5. Simplified 68.4%

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

      1. +-commutative 68.4%

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

      2. unsub-neg 68.4%

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

   7. Applied egg-rr 68.4%

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

      1. associate-*r* 68.3%

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

      2. *-commutative 68.3%

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

      3. rem-square-sqrt 0.0%

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

      4. unpow2 0.0%

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

      5. associate-*r* 0.0%

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

      6. unpow2 0.0%

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

      7. rem-square-sqrt 68.3%

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

   9. Simplified 68.3%

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

   if 3.59999999999999988e-52 < b

   1. Initial program 16.1%

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

   3. Taylor expanded in b around inf 82.6%

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

      1. associate-*r/ 82.6%

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

   5. Simplified 82.6%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.18 \cdot 10^{+80}:\\ \;\;\;\;\frac{\left(\frac{a \cdot 1.5}{\frac{b}{c}} - b\right) - b}{a \cdot 3}\\ \mathbf{elif}\;b \leq -7.2 \cdot 10^{+14}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\sqrt{a \cdot \left(c \cdot -3\right)} - b}{a}\\ \mathbf{elif}\;b \leq -3.8 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(b, \frac{-0.6666666666666666}{a}, \frac{0.5}{\frac{b}{c}}\right)\\ \mathbf{elif}\;b \leq 3.6 \cdot 10^{-52}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 77.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.18 \cdot 10^{+80}:\\ \;\;\;\;\frac{\left(\frac{a \cdot 1.5}{\frac{b}{c}} - b\right) - b}{a \cdot 3}\\ \mathbf{elif}\;b \leq -7.2 \cdot 10^{+14}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\sqrt{a \cdot \left(c \cdot -3\right)} - b}{a}\\ \mathbf{elif}\;b \leq -2.6 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(b, \frac{-0.6666666666666666}{a}, \frac{0.5}{\frac{b}{c}}\right)\\ \mathbf{elif}\;b \leq 1.45 \cdot 10^{-54}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.18e+80)
   (/ (- (- (/ (* a 1.5) (/ b c)) b) b) (* a 3.0))
   (if (<= b -7.2e+14)
     (* 0.3333333333333333 (/ (- (sqrt (* a (* c -3.0))) b) a))
     (if (<= b -2.6e-8)
       (fma b (/ -0.6666666666666666 a) (/ 0.5 (/ b c)))
       (if (<= b 1.45e-54)
         (/ (- (sqrt (* c (* a -3.0))) b) (* a 3.0))
         (/ (* c -0.5) b))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.18e+80) {
		tmp = ((((a * 1.5) / (b / c)) - b) - b) / (a * 3.0);
	} else if (b <= -7.2e+14) {
		tmp = 0.3333333333333333 * ((sqrt((a * (c * -3.0))) - b) / a);
	} else if (b <= -2.6e-8) {
		tmp = fma(b, (-0.6666666666666666 / a), (0.5 / (b / c)));
	} else if (b <= 1.45e-54) {
		tmp = (sqrt((c * (a * -3.0))) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.18e+80)
		tmp = Float64(Float64(Float64(Float64(Float64(a * 1.5) / Float64(b / c)) - b) - b) / Float64(a * 3.0));
	elseif (b <= -7.2e+14)
		tmp = Float64(0.3333333333333333 * Float64(Float64(sqrt(Float64(a * Float64(c * -3.0))) - b) / a));
	elseif (b <= -2.6e-8)
		tmp = fma(b, Float64(-0.6666666666666666 / a), Float64(0.5 / Float64(b / c)));
	elseif (b <= 1.45e-54)
		tmp = Float64(Float64(sqrt(Float64(c * Float64(a * -3.0))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.18e+80], N[(N[(N[(N[(N[(a * 1.5), $MachinePrecision] / N[(b / c), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, -7.2e+14], N[(0.3333333333333333 * N[(N[(N[Sqrt[N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, -2.6e-8], N[(b * N[(-0.6666666666666666 / a), $MachinePrecision] + N[(0.5 / N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1.45e-54], N[(N[(N[Sqrt[N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if b < -1.18e80

   1. Initial program 58.1%

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

   3. Taylor expanded in b around -inf 91.8%

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

      1. +-commutative 91.8%

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

      2. mul-1-neg 91.8%

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

      3. unsub-neg 91.8%

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

      4. associate-/l* 95.9%

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

      5. associate-*r/ 95.9%

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

   5. Simplified 95.9%

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

   if -1.18e80 < b < -7.2e14

   1. Initial program 99.2%

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

   3. Taylor expanded in b around 0 80.6%

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

      1. associate-*r* 80.7%

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

      2. *-commutative 80.7%

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

      3. *-commutative 80.7%

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

   5. Simplified 80.7%

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

      1. +-commutative 80.7%

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

      2. unsub-neg 80.7%

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

   7. Applied egg-rr 80.7%

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

      1. associate-*r* 80.6%

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

      2. *-commutative 80.6%

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

      3. rem-square-sqrt 0.0%

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

      4. unpow2 0.0%

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

      5. associate-*r* 0.0%

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

      6. unpow2 0.0%

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

      7. rem-square-sqrt 80.6%

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

   9. Simplified 80.6%

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

       1. *-un-lft-identity 80.6%

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

       2. metadata-eval 80.6%

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

       3. times-frac 80.9%

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

       4. metadata-eval 80.9%

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

       5. metadata-eval 80.9%

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

   11. Applied egg-rr 80.9%

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

   if -7.2e14 < b < -2.6000000000000001e-8

   1. Initial program 83.5%

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

   3. Taylor expanded in b around -inf 83.5%

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

      1. +-commutative 83.5%

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

      2. mul-1-neg 83.5%

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

      3. unsub-neg 83.5%

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

      4. associate-/l* 83.5%

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

      5. associate-*r/ 83.5%

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

   5. Simplified 83.5%

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

   6. Taylor expanded in b around 0 83.5%

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

      1. associate-*r/ 83.8%

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

      2. *-commutative 83.8%

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

      3. associate-*r/ 83.8%

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

      4. fma-def 83.8%

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

      5. associate-*r/ 83.8%

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

      6. associate-/l* 83.8%

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

   8. Simplified 83.8%

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

   if -2.6000000000000001e-8 < b < 1.45000000000000007e-54

   1. Initial program 74.7%

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

   3. Taylor expanded in b around 0 68.3%

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

      1. associate-*r* 68.4%

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

      2. *-commutative 68.4%

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

      3. *-commutative 68.4%

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

   5. Simplified 68.4%

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

      1. +-commutative 68.4%

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

      2. unsub-neg 68.4%

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

   7. Applied egg-rr 68.4%

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

   if 1.45000000000000007e-54 < b

   1. Initial program 16.1%

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

   3. Taylor expanded in b around inf 82.6%

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

      1. associate-*r/ 82.6%

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

   5. Simplified 82.6%

      \[\leadsto \color{blue}{\frac{-0.5 \cdot c}{b}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.18 \cdot 10^{+80}:\\ \;\;\;\;\frac{\left(\frac{a \cdot 1.5}{\frac{b}{c}} - b\right) - b}{a \cdot 3}\\ \mathbf{elif}\;b \leq -7.2 \cdot 10^{+14}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\sqrt{a \cdot \left(c \cdot -3\right)} - b}{a}\\ \mathbf{elif}\;b \leq -2.6 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(b, \frac{-0.6666666666666666}{a}, \frac{0.5}{\frac{b}{c}}\right)\\ \mathbf{elif}\;b \leq 1.45 \cdot 10^{-54}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.1 \cdot 10^{+140}:\\ \;\;\;\;\frac{\frac{b}{-1.5}}{a}\\ \mathbf{elif}\;b \leq 4.2 \cdot 10^{-49}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 3\right) \cdot c} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.1e+140)
   (/ (/ b -1.5) a)
   (if (<= b 4.2e-49)
     (/ (- (sqrt (- (* b b) (* (* a 3.0) c))) b) (* a 3.0))
     (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.1e+140) {
		tmp = (b / -1.5) / a;
	} else if (b <= 4.2e-49) {
		tmp = (sqrt(((b * b) - ((a * 3.0) * c))) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-2.1d+140)) then
        tmp = (b / (-1.5d0)) / a
    else if (b <= 4.2d-49) then
        tmp = (sqrt(((b * b) - ((a * 3.0d0) * c))) - b) / (a * 3.0d0)
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.1e+140) {
		tmp = (b / -1.5) / a;
	} else if (b <= 4.2e-49) {
		tmp = (Math.sqrt(((b * b) - ((a * 3.0) * c))) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.1e+140:
		tmp = (b / -1.5) / a
	elif b <= 4.2e-49:
		tmp = (math.sqrt(((b * b) - ((a * 3.0) * c))) - b) / (a * 3.0)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.1e+140)
		tmp = Float64(Float64(b / -1.5) / a);
	elseif (b <= 4.2e-49)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(Float64(a * 3.0) * c))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.1e+140)
		tmp = (b / -1.5) / a;
	elseif (b <= 4.2e-49)
		tmp = (sqrt(((b * b) - ((a * 3.0) * c))) - b) / (a * 3.0);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.1e+140], N[(N[(b / -1.5), $MachinePrecision] / a), $MachinePrecision], If[LessEqual[b, 4.2e-49], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(a * 3.0), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -2.1000000000000002e140

   1. Initial program 41.9%

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

   3. Taylor expanded in b around -inf 94.5%

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

      1. *-commutative 94.5%

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

   5. Simplified 94.5%

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

      1. add-cbrt-cube 60.9%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\left(\frac{b}{a} \cdot -0.6666666666666666\right) \cdot \left(\frac{b}{a} \cdot -0.6666666666666666\right)\right) \cdot \left(\frac{b}{a} \cdot -0.6666666666666666\right)}} \]

      2. pow1/3 34.3%

         \[\leadsto \color{blue}{{\left(\left(\left(\frac{b}{a} \cdot -0.6666666666666666\right) \cdot \left(\frac{b}{a} \cdot -0.6666666666666666\right)\right) \cdot \left(\frac{b}{a} \cdot -0.6666666666666666\right)\right)}^{0.3333333333333333}} \]

      3. pow3 34.3%

         \[\leadsto {\color{blue}{\left({\left(\frac{b}{a} \cdot -0.6666666666666666\right)}^{3}\right)}}^{0.3333333333333333} \]

   7. Applied egg-rr 34.3%

      \[\leadsto \color{blue}{{\left({\left(\frac{b}{a} \cdot -0.6666666666666666\right)}^{3}\right)}^{0.3333333333333333}} \]
   8. Step-by-step derivation

      1. unpow1/3 60.9%

         \[\leadsto \color{blue}{\sqrt[3]{{\left(\frac{b}{a} \cdot -0.6666666666666666\right)}^{3}}} \]

      2. rem-cbrt-cube 94.5%

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

      3. metadata-eval 94.5%

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

      4. times-frac 94.7%

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

      5. associate-/l/ 94.7%

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

      6. associate-/l* 94.7%

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

      7. metadata-eval 94.7%

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

   9. Applied egg-rr 94.7%

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

   if -2.1000000000000002e140 < b < 4.1999999999999998e-49

   1. Initial program 80.7%

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

   if 4.1999999999999998e-49 < b

   1. Initial program 16.1%

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

   3. Taylor expanded in b around inf 82.6%

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

      1. associate-*r/ 82.6%

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

   5. Simplified 82.6%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.1 \cdot 10^{+140}:\\ \;\;\;\;\frac{\frac{b}{-1.5}}{a}\\ \mathbf{elif}\;b \leq 4.2 \cdot 10^{-49}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - \left(a \cdot 3\right) \cdot c} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 68.4% accurate, 5.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 74.3%

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

   3. Taylor expanded in b around -inf 60.3%

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

      1. +-commutative 60.3%

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

      2. mul-1-neg 60.3%

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

      3. unsub-neg 60.3%

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

      4. associate-/l* 62.1%

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

      5. associate-*r/ 62.1%

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

   5. Simplified 62.1%

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

   if -4.999999999999985e-310 < b

   1. Initial program 33.5%

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

   3. Taylor expanded in b around inf 63.2%

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

      1. associate-*r/ 63.2%

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

   5. Simplified 63.2%

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

4. Final simplification 62.7%

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

Alternative 7: 68.5% accurate, 7.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 74.3%

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

   3. Taylor expanded in b around -inf 62.0%

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

   if -4.999999999999985e-310 < b

   1. Initial program 33.5%

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

   3. Taylor expanded in b around inf 63.2%

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

      1. associate-*r/ 63.2%

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

   5. Simplified 63.2%

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

4. Final simplification 62.7%

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

Alternative 8: 44.3% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.5 \cdot 10^{+34}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 3.5e+34) (* b (/ -0.6666666666666666 a)) (* 0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 3.5e+34) {
		tmp = b * (-0.6666666666666666 / a);
	} else {
		tmp = 0.5 * (c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 3.5d+34) then
        tmp = b * ((-0.6666666666666666d0) / a)
    else
        tmp = 0.5d0 * (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 3.5e+34) {
		tmp = b * (-0.6666666666666666 / a);
	} else {
		tmp = 0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 3.5e+34:
		tmp = b * (-0.6666666666666666 / a)
	else:
		tmp = 0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 3.5e+34)
		tmp = Float64(b * Float64(-0.6666666666666666 / a));
	else
		tmp = Float64(0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 3.5e+34)
		tmp = b * (-0.6666666666666666 / a);
	else
		tmp = 0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 3.5e+34], N[(b * N[(-0.6666666666666666 / a), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.49999999999999998e34

   1. Initial program 66.1%

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

   3. Taylor expanded in b around -inf 38.5%

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

      1. *-commutative 38.5%

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

   5. Simplified 38.5%

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

      1. *-commutative 38.5%

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

      2. clear-num 38.5%

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

      3. un-div-inv 38.6%

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

   7. Applied egg-rr 38.6%

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

      1. associate-/r/ 38.5%

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

   9. Simplified 38.5%

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

   if 3.49999999999999998e34 < 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. Add Preprocessing

   3. Taylor expanded in b around -inf 2.2%

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

      1. +-commutative 2.2%

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

      2. mul-1-neg 2.2%

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

      3. unsub-neg 2.2%

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

      4. associate-/l* 2.5%

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

      5. associate-*r/ 2.5%

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

   5. Simplified 2.5%

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

   6. Taylor expanded in b around 0 27.1%

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

4. Final simplification 35.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.5 \cdot 10^{+34}:\\ \;\;\;\;b \cdot \frac{-0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 44.3% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 6.8 \cdot 10^{+35}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 6.8e+35) (* -0.6666666666666666 (/ b a)) (* 0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 6.8e+35) {
		tmp = -0.6666666666666666 * (b / a);
	} else {
		tmp = 0.5 * (c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 6.8d+35) then
        tmp = (-0.6666666666666666d0) * (b / a)
    else
        tmp = 0.5d0 * (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 6.8e+35) {
		tmp = -0.6666666666666666 * (b / a);
	} else {
		tmp = 0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 6.8e+35:
		tmp = -0.6666666666666666 * (b / a)
	else:
		tmp = 0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 6.8e+35)
		tmp = Float64(-0.6666666666666666 * Float64(b / a));
	else
		tmp = Float64(0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 6.8e+35)
		tmp = -0.6666666666666666 * (b / a);
	else
		tmp = 0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 6.8e+35], N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 6.8000000000000002e35

   1. Initial program 66.1%

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

   3. Taylor expanded in b around -inf 38.5%

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

      1. *-commutative 38.5%

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

   5. Simplified 38.5%

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

   if 6.8000000000000002e35 < 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. Add Preprocessing

   3. Taylor expanded in b around -inf 2.2%

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

      1. +-commutative 2.2%

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

      2. mul-1-neg 2.2%

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

      3. unsub-neg 2.2%

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

      4. associate-/l* 2.5%

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

      5. associate-*r/ 2.5%

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

   5. Simplified 2.5%

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

   6. Taylor expanded in b around 0 27.1%

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

4. Final simplification 35.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 6.8 \cdot 10^{+35}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 44.3% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 3.4 \cdot 10^{+34}:\\ \;\;\;\;\frac{-0.6666666666666666}{\frac{a}{b}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 3.4e+34) (/ -0.6666666666666666 (/ a b)) (* 0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 3.4e+34) {
		tmp = -0.6666666666666666 / (a / b);
	} else {
		tmp = 0.5 * (c / b);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 3.4d+34) then
        tmp = (-0.6666666666666666d0) / (a / b)
    else
        tmp = 0.5d0 * (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 3.4e+34) {
		tmp = -0.6666666666666666 / (a / b);
	} else {
		tmp = 0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 3.4e+34:
		tmp = -0.6666666666666666 / (a / b)
	else:
		tmp = 0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 3.4e+34)
		tmp = Float64(-0.6666666666666666 / Float64(a / b));
	else
		tmp = Float64(0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 3.4e+34)
		tmp = -0.6666666666666666 / (a / b);
	else
		tmp = 0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 3.4e+34], N[(-0.6666666666666666 / N[(a / b), $MachinePrecision]), $MachinePrecision], N[(0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 3.3999999999999999e34

   1. Initial program 66.1%

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

   3. Taylor expanded in b around -inf 38.5%

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

      1. *-commutative 38.5%

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

   5. Simplified 38.5%

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

      1. *-commutative 38.5%

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

      2. clear-num 38.5%

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

      3. un-div-inv 38.6%

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

   7. Applied egg-rr 38.6%

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

   if 3.3999999999999999e34 < 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. Add Preprocessing

   3. Taylor expanded in b around -inf 2.2%

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

      1. +-commutative 2.2%

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

      2. mul-1-neg 2.2%

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

      3. unsub-neg 2.2%

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

      4. associate-/l* 2.5%

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

      5. associate-*r/ 2.5%

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

   5. Simplified 2.5%

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

   6. Taylor expanded in b around 0 27.1%

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

4. Final simplification 35.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 3.4 \cdot 10^{+34}:\\ \;\;\;\;\frac{-0.6666666666666666}{\frac{a}{b}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{c}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 67.9% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.35 \cdot 10^{-290}:\\ \;\;\;\;\frac{-0.6666666666666666}{\frac{a}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{\frac{b}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.35e-290) (/ -0.6666666666666666 (/ a b)) (/ -0.5 (/ b c))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.35e-290) {
		tmp = -0.6666666666666666 / (a / b);
	} else {
		tmp = -0.5 / (b / c);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 1.35d-290) then
        tmp = (-0.6666666666666666d0) / (a / b)
    else
        tmp = (-0.5d0) / (b / c)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.35e-290) {
		tmp = -0.6666666666666666 / (a / b);
	} else {
		tmp = -0.5 / (b / c);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 1.35e-290:
		tmp = -0.6666666666666666 / (a / b)
	else:
		tmp = -0.5 / (b / c)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.35e-290)
		tmp = Float64(-0.6666666666666666 / Float64(a / b));
	else
		tmp = Float64(-0.5 / Float64(b / c));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 1.35e-290)
		tmp = -0.6666666666666666 / (a / b);
	else
		tmp = -0.5 / (b / c);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.35e-290], N[(-0.6666666666666666 / N[(a / b), $MachinePrecision]), $MachinePrecision], N[(-0.5 / N[(b / c), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.34999999999999999e-290

   1. Initial program 74.1%

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

   3. Taylor expanded in b around -inf 60.2%

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

      1. *-commutative 60.2%

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

   5. Simplified 60.2%

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

      1. *-commutative 60.2%

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

      2. clear-num 60.2%

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

      3. un-div-inv 60.3%

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

   7. Applied egg-rr 60.3%

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

   if 1.34999999999999999e-290 < b

   1. Initial program 32.8%

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

   3. Taylor expanded in b around inf 64.5%

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

      1. associate-*r/ 64.5%

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

      2. associate-/l* 64.0%

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

   5. Simplified 64.0%

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

4. Final simplification 62.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.35 \cdot 10^{-290}:\\ \;\;\;\;\frac{-0.6666666666666666}{\frac{a}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5}{\frac{b}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 68.3% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.35 \cdot 10^{-290}:\\ \;\;\;\;\frac{-0.6666666666666666}{\frac{a}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.35e-290) (/ -0.6666666666666666 (/ a b)) (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.35e-290) {
		tmp = -0.6666666666666666 / (a / b);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 1.35d-290) then
        tmp = (-0.6666666666666666d0) / (a / b)
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.35e-290) {
		tmp = -0.6666666666666666 / (a / b);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 1.35e-290:
		tmp = -0.6666666666666666 / (a / b)
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.35e-290)
		tmp = Float64(-0.6666666666666666 / Float64(a / b));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 1.35e-290)
		tmp = -0.6666666666666666 / (a / b);
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.35e-290], N[(-0.6666666666666666 / N[(a / b), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.34999999999999999e-290

   1. Initial program 74.1%

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

   3. Taylor expanded in b around -inf 60.2%

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

      1. *-commutative 60.2%

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

   5. Simplified 60.2%

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

      1. *-commutative 60.2%

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

      2. clear-num 60.2%

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

      3. un-div-inv 60.3%

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

   7. Applied egg-rr 60.3%

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

   if 1.34999999999999999e-290 < b

   1. Initial program 32.8%

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

   3. Taylor expanded in b around inf 64.5%

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

      1. associate-*r/ 64.5%

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

   5. Simplified 64.5%

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

4. Final simplification 62.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.35 \cdot 10^{-290}:\\ \;\;\;\;\frac{-0.6666666666666666}{\frac{a}{b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 68.4% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 1.35 \cdot 10^{-290}:\\ \;\;\;\;\frac{\frac{b}{-1.5}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 1.35e-290) (/ (/ b -1.5) a) (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.35e-290) {
		tmp = (b / -1.5) / a;
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 1.35d-290) then
        tmp = (b / (-1.5d0)) / a
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 1.35e-290) {
		tmp = (b / -1.5) / a;
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 1.35e-290:
		tmp = (b / -1.5) / a
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 1.35e-290)
		tmp = Float64(Float64(b / -1.5) / a);
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 1.35e-290)
		tmp = (b / -1.5) / a;
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 1.35e-290], N[(N[(b / -1.5), $MachinePrecision] / a), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 1.34999999999999999e-290

   1. Initial program 74.1%

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

   3. Taylor expanded in b around -inf 60.2%

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

      1. *-commutative 60.2%

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

   5. Simplified 60.2%

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

      1. add-cbrt-cube 33.8%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\left(\frac{b}{a} \cdot -0.6666666666666666\right) \cdot \left(\frac{b}{a} \cdot -0.6666666666666666\right)\right) \cdot \left(\frac{b}{a} \cdot -0.6666666666666666\right)}} \]

      2. pow1/3 17.9%

         \[\leadsto \color{blue}{{\left(\left(\left(\frac{b}{a} \cdot -0.6666666666666666\right) \cdot \left(\frac{b}{a} \cdot -0.6666666666666666\right)\right) \cdot \left(\frac{b}{a} \cdot -0.6666666666666666\right)\right)}^{0.3333333333333333}} \]

      3. pow3 17.9%

         \[\leadsto {\color{blue}{\left({\left(\frac{b}{a} \cdot -0.6666666666666666\right)}^{3}\right)}}^{0.3333333333333333} \]

   7. Applied egg-rr 17.9%

      \[\leadsto \color{blue}{{\left({\left(\frac{b}{a} \cdot -0.6666666666666666\right)}^{3}\right)}^{0.3333333333333333}} \]
   8. Step-by-step derivation

      1. unpow1/3 33.8%

         \[\leadsto \color{blue}{\sqrt[3]{{\left(\frac{b}{a} \cdot -0.6666666666666666\right)}^{3}}} \]

      2. rem-cbrt-cube 60.2%

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

      3. metadata-eval 60.2%

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

      4. times-frac 60.3%

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

      5. associate-/l/ 60.3%

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

      6. associate-/l* 60.3%

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

      7. metadata-eval 60.3%

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

   9. Applied egg-rr 60.3%

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

   if 1.34999999999999999e-290 < b

   1. Initial program 32.8%

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

   3. Taylor expanded in b around inf 64.5%

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

      1. associate-*r/ 64.5%

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

   5. Simplified 64.5%

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

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 1.35 \cdot 10^{-290}:\\ \;\;\;\;\frac{\frac{b}{-1.5}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 10.9% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.0%

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

3. Taylor expanded in b around -inf 27.1%

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

   1. +-commutative 27.1%

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

   2. mul-1-neg 27.1%

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

   3. unsub-neg 27.1%

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

   4. associate-/l* 28.0%

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

   5. associate-*r/ 28.0%

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

5. Simplified 28.0%

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

6. Taylor expanded in b around 0 10.0%

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

7. Final simplification 10.0%

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

Reproduce

herbie shell --seed 2024019 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))