Cubic critical

Percentage Accurate: 63.5% → 87.4%

Time: 21.6s

Alternatives: 16

Speedup: 11.6×

• could not determine a ground truth

  1. a = 9.556233487546083e-176
  2. b = -9.718752567681072e-233
  3. c = -3.3009486355403756e-298

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: 63.5% 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: 87.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;c \leq 2.65 \cdot 10^{+234}:\\ \;\;\;\;\left(b - \mathsf{hypot}\left(b, \sqrt{\left|c \cdot \left(a \cdot -3\right)\right|}\right)\right) \cdot \frac{\frac{1}{a}}{-3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{b - \mathsf{hypot}\left(b, \sqrt{a \cdot -3} \cdot \sqrt{c}\right)}{a}}{-3}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= c 2.65e+234)
   (* (- b (hypot b (sqrt (fabs (* c (* a -3.0)))))) (/ (/ 1.0 a) -3.0))
   (/ (/ (- b (hypot b (* (sqrt (* a -3.0)) (sqrt c)))) a) -3.0)))

C

double code(double a, double b, double c) {
	double tmp;
	if (c <= 2.65e+234) {
		tmp = (b - hypot(b, sqrt(fabs((c * (a * -3.0)))))) * ((1.0 / a) / -3.0);
	} else {
		tmp = ((b - hypot(b, (sqrt((a * -3.0)) * sqrt(c)))) / a) / -3.0;
	}
	return tmp;
}

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (c <= 2.65e+234) {
		tmp = (b - Math.hypot(b, Math.sqrt(Math.abs((c * (a * -3.0)))))) * ((1.0 / a) / -3.0);
	} else {
		tmp = ((b - Math.hypot(b, (Math.sqrt((a * -3.0)) * Math.sqrt(c)))) / a) / -3.0;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if c <= 2.65e+234:
		tmp = (b - math.hypot(b, math.sqrt(math.fabs((c * (a * -3.0)))))) * ((1.0 / a) / -3.0)
	else:
		tmp = ((b - math.hypot(b, (math.sqrt((a * -3.0)) * math.sqrt(c)))) / a) / -3.0
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (c <= 2.65e+234)
		tmp = Float64(Float64(b - hypot(b, sqrt(abs(Float64(c * Float64(a * -3.0)))))) * Float64(Float64(1.0 / a) / -3.0));
	else
		tmp = Float64(Float64(Float64(b - hypot(b, Float64(sqrt(Float64(a * -3.0)) * sqrt(c)))) / a) / -3.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (c <= 2.65e+234)
		tmp = (b - hypot(b, sqrt(abs((c * (a * -3.0)))))) * ((1.0 / a) / -3.0);
	else
		tmp = ((b - hypot(b, (sqrt((a * -3.0)) * sqrt(c)))) / a) / -3.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[c, 2.65e+234], N[(N[(b - N[Sqrt[b ^ 2 + N[Sqrt[N[Abs[N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[(N[(1.0 / a), $MachinePrecision] / -3.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b - N[Sqrt[b ^ 2 + N[(N[Sqrt[N[(a * -3.0), $MachinePrecision]], $MachinePrecision] * N[Sqrt[c], $MachinePrecision]), $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] / -3.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if c < 2.64999999999999999e234

   1. Initial program 67.9%

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

      1. sqr-neg 67.9%

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

      2. sqr-neg 67.9%

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

      3. associate-*l* 67.8%

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

   3. Simplified 67.8%

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

   6. Applied egg-rr 64.6%

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

      1. associate-/r* 64.6%

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

   8. Simplified 64.6%

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

      1. associate-*r* 64.6%

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

      2. *-commutative 64.6%

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

      3. add-sqr-sqrt 64.6%

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

      4. pow1/2 64.6%

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

      5. pow1/2 64.6%

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

      6. pow-prod-down 77.9%

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

      7. pow2 77.9%

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

      8. *-commutative 77.9%

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

      9. associate-*r* 77.9%

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

   10. Applied egg-rr 77.9%

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

       1. unpow1/2 77.9%

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

       2. unpow2 77.9%

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

       3. rem-sqrt-square 91.6%

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

       4. associate-*r* 91.6%

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

       5. *-commutative 91.6%

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

       6. associate-*l* 91.5%

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

   12. Simplified 91.5%

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

   if 2.64999999999999999e234 < c

   1. Initial program 22.5%

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

      1. sqr-neg 22.5%

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

      2. sqr-neg 22.5%

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

      3. associate-*l* 22.5%

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

   3. Simplified 22.5%

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

   6. Applied egg-rr 22.3%

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

      1. associate-/r* 22.5%

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

   8. Simplified 22.5%

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

      1. associate-*r/ 22.5%

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

   10. Applied egg-rr 22.5%

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

       1. associate-*r/ 22.6%

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

       2. *-rgt-identity 22.6%

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

       3. associate-*r* 22.5%

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

       4. *-commutative 22.5%

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

       5. associate-*l* 22.5%

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

   12. Simplified 22.5%

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

   13. Taylor expanded in c around 0 22.5%

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

       1. pow1/2 23.1%

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

       2. associate-*r* 23.1%

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

       3. unpow-prod-down 89.5%

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

       4. pow1/2 89.5%

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

   15. Applied egg-rr 89.5%

       \[\leadsto \frac{\frac{b - \mathsf{hypot}\left(b, \color{blue}{{\left(-3 \cdot a\right)}^{0.5} \cdot \sqrt{c}}\right)}{a}}{-3} \]
   16. Step-by-step derivation

       1. unpow1/2 89.5%

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

       2. *-commutative 89.5%

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

   17. Simplified 89.5%

       \[\leadsto \frac{\frac{b - \mathsf{hypot}\left(b, \color{blue}{\sqrt{a \cdot -3} \cdot \sqrt{c}}\right)}{a}}{-3} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;c \leq 2.65 \cdot 10^{+234}:\\ \;\;\;\;\left(b - \mathsf{hypot}\left(b, \sqrt{\left|c \cdot \left(a \cdot -3\right)\right|}\right)\right) \cdot \frac{\frac{1}{a}}{-3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{b - \mathsf{hypot}\left(b, \sqrt{a \cdot -3} \cdot \sqrt{c}\right)}{a}}{-3}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 79.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.3333333333333333 \cdot \frac{\sqrt{c \cdot \left(a \cdot -3\right)}}{a}\\ \mathbf{if}\;b \leq -1.1 \cdot 10^{-19}:\\ \;\;\;\;\frac{\frac{2}{a}}{\frac{-3}{b}}\\ \mathbf{elif}\;b \leq -3.2 \cdot 10^{-39}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq -1.65 \cdot 10^{-99}:\\ \;\;\;\;\frac{b \cdot \frac{2}{a}}{-3}\\ \mathbf{elif}\;b \leq 8.5 \cdot 10^{-88}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(b + -1.5 \cdot \left(a \cdot \frac{c}{b}\right)\right) - b}{a \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* 0.3333333333333333 (/ (sqrt (* c (* a -3.0))) a))))
   (if (<= b -1.1e-19)
     (/ (/ 2.0 a) (/ -3.0 b))
     (if (<= b -3.2e-39)
       t_0
       (if (<= b -1.65e-99)
         (/ (* b (/ 2.0 a)) -3.0)
         (if (<= b 8.5e-88)
           t_0
           (/ (- (+ b (* -1.5 (* a (/ c b)))) b) (* a 3.0))))))))

C

double code(double a, double b, double c) {
	double t_0 = 0.3333333333333333 * (sqrt((c * (a * -3.0))) / a);
	double tmp;
	if (b <= -1.1e-19) {
		tmp = (2.0 / a) / (-3.0 / b);
	} else if (b <= -3.2e-39) {
		tmp = t_0;
	} else if (b <= -1.65e-99) {
		tmp = (b * (2.0 / a)) / -3.0;
	} else if (b <= 8.5e-88) {
		tmp = t_0;
	} else {
		tmp = ((b + (-1.5 * (a * (c / b)))) - b) / (a * 3.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.3333333333333333d0 * (sqrt((c * (a * (-3.0d0)))) / a)
    if (b <= (-1.1d-19)) then
        tmp = (2.0d0 / a) / ((-3.0d0) / b)
    else if (b <= (-3.2d-39)) then
        tmp = t_0
    else if (b <= (-1.65d-99)) then
        tmp = (b * (2.0d0 / a)) / (-3.0d0)
    else if (b <= 8.5d-88) then
        tmp = t_0
    else
        tmp = ((b + ((-1.5d0) * (a * (c / b)))) - b) / (a * 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = 0.3333333333333333 * (Math.sqrt((c * (a * -3.0))) / a);
	double tmp;
	if (b <= -1.1e-19) {
		tmp = (2.0 / a) / (-3.0 / b);
	} else if (b <= -3.2e-39) {
		tmp = t_0;
	} else if (b <= -1.65e-99) {
		tmp = (b * (2.0 / a)) / -3.0;
	} else if (b <= 8.5e-88) {
		tmp = t_0;
	} else {
		tmp = ((b + (-1.5 * (a * (c / b)))) - b) / (a * 3.0);
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = 0.3333333333333333 * (math.sqrt((c * (a * -3.0))) / a)
	tmp = 0
	if b <= -1.1e-19:
		tmp = (2.0 / a) / (-3.0 / b)
	elif b <= -3.2e-39:
		tmp = t_0
	elif b <= -1.65e-99:
		tmp = (b * (2.0 / a)) / -3.0
	elif b <= 8.5e-88:
		tmp = t_0
	else:
		tmp = ((b + (-1.5 * (a * (c / b)))) - b) / (a * 3.0)
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(0.3333333333333333 * Float64(sqrt(Float64(c * Float64(a * -3.0))) / a))
	tmp = 0.0
	if (b <= -1.1e-19)
		tmp = Float64(Float64(2.0 / a) / Float64(-3.0 / b));
	elseif (b <= -3.2e-39)
		tmp = t_0;
	elseif (b <= -1.65e-99)
		tmp = Float64(Float64(b * Float64(2.0 / a)) / -3.0);
	elseif (b <= 8.5e-88)
		tmp = t_0;
	else
		tmp = Float64(Float64(Float64(b + Float64(-1.5 * Float64(a * Float64(c / b)))) - b) / Float64(a * 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = 0.3333333333333333 * (sqrt((c * (a * -3.0))) / a);
	tmp = 0.0;
	if (b <= -1.1e-19)
		tmp = (2.0 / a) / (-3.0 / b);
	elseif (b <= -3.2e-39)
		tmp = t_0;
	elseif (b <= -1.65e-99)
		tmp = (b * (2.0 / a)) / -3.0;
	elseif (b <= 8.5e-88)
		tmp = t_0;
	else
		tmp = ((b + (-1.5 * (a * (c / b)))) - b) / (a * 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(0.3333333333333333 * N[(N[Sqrt[N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -1.1e-19], N[(N[(2.0 / a), $MachinePrecision] / N[(-3.0 / b), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, -3.2e-39], t$95$0, If[LessEqual[b, -1.65e-99], N[(N[(b * N[(2.0 / a), $MachinePrecision]), $MachinePrecision] / -3.0), $MachinePrecision], If[LessEqual[b, 8.5e-88], t$95$0, N[(N[(N[(b + N[(-1.5 * N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -1.0999999999999999e-19

   1. Initial program 69.3%

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

      1. sqr-neg 69.3%

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

      2. sqr-neg 69.3%

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

      3. associate-*l* 69.3%

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

   3. Simplified 69.3%

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

   5. Taylor expanded in b around -inf 89.6%

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

      1. mul-1-neg 89.6%

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

   7. Simplified 89.6%

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

      1. count-2 89.6%

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

      2. *-commutative 89.6%

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

      3. times-frac 89.6%

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

      4. add-sqr-sqrt 89.3%

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

      5. sqrt-unprod 64.8%

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

      6. sqr-neg 64.8%

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

      7. sqrt-unprod 0.0%

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

      8. add-sqr-sqrt 1.0%

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

   9. Applied egg-rr 1.0%

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

       1. associate-*l/ 1.0%

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

       2. associate-/l* 1.0%

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

   11. Simplified 1.0%

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

       1. clear-num 1.0%

          \[\leadsto 2 \cdot \color{blue}{\frac{1}{\frac{a}{\frac{b}{3}}}} \]

       2. un-div-inv 1.0%

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

       3. div-inv 1.0%

          \[\leadsto \frac{2}{\color{blue}{a \cdot \frac{1}{\frac{b}{3}}}} \]

       4. clear-num 1.0%

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

       5. frac-2neg 1.0%

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

       6. metadata-eval 1.0%

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

       7. add-sqr-sqrt 1.0%

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

       8. sqrt-unprod 0.9%

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

       9. sqr-neg 0.9%

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

       10. sqrt-unprod 0.0%

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

       11. add-sqr-sqrt 89.6%

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

   13. Applied egg-rr 89.6%

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

       1. associate-/r* 89.7%

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

   15. Simplified 89.7%

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

   if -1.0999999999999999e-19 < b < -3.1999999999999998e-39 or -1.64999999999999993e-99 < b < 8.4999999999999996e-88

   1. Initial program 85.2%

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

      1. sqr-neg 85.2%

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

      2. sqr-neg 85.2%

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

      3. associate-*l* 85.0%

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

   3. Simplified 85.0%

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

   6. Applied egg-rr 82.4%

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

      1. associate-/r* 82.5%

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

   8. Simplified 82.5%

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

      1. associate-*r* 82.5%

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

      2. *-commutative 82.5%

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

      3. add-sqr-sqrt 82.5%

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

      4. pow1/2 82.5%

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

      5. pow1/2 82.5%

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

      6. pow-prod-down 58.6%

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

      7. pow2 58.6%

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

      8. *-commutative 58.6%

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

      9. associate-*r* 58.6%

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

   10. Applied egg-rr 58.6%

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

       1. unpow1/2 58.6%

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

       2. unpow2 58.6%

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

       3. rem-sqrt-square 84.4%

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

       4. associate-*r* 84.4%

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

       5. *-commutative 84.4%

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

       6. associate-*l* 84.3%

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

   12. Simplified 84.3%

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

   13. Taylor expanded in b around 0 82.0%

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

       1. associate-*l/ 82.0%

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

       2. *-lft-identity 82.0%

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

       3. associate-*r* 81.9%

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

       4. *-commutative 81.9%

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

       5. *-commutative 81.9%

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

       6. rem-cube-cbrt 81.2%

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

       7. sqr-pow 81.1%

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

       8. fabs-sqr 81.1%

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

       9. sqr-pow 81.0%

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

       10. rem-cube-cbrt 81.6%

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

       11. *-commutative 81.6%

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

   15. Simplified 81.6%

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

   if -3.1999999999999998e-39 < b < -1.64999999999999993e-99

   1. Initial program 99.6%

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

      1. sqr-neg 99.6%

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

      2. sqr-neg 99.6%

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

      3. associate-*l* 99.5%

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

   3. Simplified 99.5%

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

   6. Applied egg-rr 74.4%

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

      1. associate-/r* 74.5%

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

   8. Simplified 74.5%

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

      1. associate-*r/ 74.7%

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

   10. Applied egg-rr 74.7%

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

       1. associate-*r/ 74.7%

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

       2. *-rgt-identity 74.7%

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

       3. associate-*r* 74.9%

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

       4. *-commutative 74.9%

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

       5. associate-*l* 74.9%

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

   12. Simplified 74.9%

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

   13. Taylor expanded in b around -inf 76.7%

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

       1. associate-*r/ 76.7%

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

       2. associate-*l/ 76.7%

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

       3. *-commutative 76.7%

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

   15. Simplified 76.7%

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

   if 8.4999999999999996e-88 < b

   1. Initial program 39.4%

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

      1. sqr-neg 39.4%

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

      2. sqr-neg 39.4%

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

      3. associate-*l* 39.4%

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

   3. Simplified 39.4%

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

   5. Taylor expanded in b around inf 88.2%

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

      1. associate-/l* 88.2%

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

   7. Simplified 88.2%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.1 \cdot 10^{-19}:\\ \;\;\;\;\frac{\frac{2}{a}}{\frac{-3}{b}}\\ \mathbf{elif}\;b \leq -3.2 \cdot 10^{-39}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\sqrt{c \cdot \left(a \cdot -3\right)}}{a}\\ \mathbf{elif}\;b \leq -1.65 \cdot 10^{-99}:\\ \;\;\;\;\frac{b \cdot \frac{2}{a}}{-3}\\ \mathbf{elif}\;b \leq 8.5 \cdot 10^{-88}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\sqrt{c \cdot \left(a \cdot -3\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(b + -1.5 \cdot \left(a \cdot \frac{c}{b}\right)\right) - b}{a \cdot 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.1 \cdot 10^{-19}:\\ \;\;\;\;\frac{\frac{2}{a}}{\frac{-3}{b}}\\ \mathbf{elif}\;b \leq -4.4 \cdot 10^{-44}:\\ \;\;\;\;\frac{\sqrt{-3 \cdot \left(c \cdot a\right)} - b}{a \cdot 3}\\ \mathbf{elif}\;b \leq -1.9 \cdot 10^{-103}:\\ \;\;\;\;\frac{b \cdot \frac{2}{a}}{-3}\\ \mathbf{elif}\;b \leq 8.2 \cdot 10^{-88}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\sqrt{c \cdot \left(a \cdot -3\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(b + -1.5 \cdot \left(a \cdot \frac{c}{b}\right)\right) - b}{a \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.1e-19)
   (/ (/ 2.0 a) (/ -3.0 b))
   (if (<= b -4.4e-44)
     (/ (- (sqrt (* -3.0 (* c a))) b) (* a 3.0))
     (if (<= b -1.9e-103)
       (/ (* b (/ 2.0 a)) -3.0)
       (if (<= b 8.2e-88)
         (* 0.3333333333333333 (/ (sqrt (* c (* a -3.0))) a))
         (/ (- (+ b (* -1.5 (* a (/ c b)))) b) (* a 3.0)))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.1e-19) {
		tmp = (2.0 / a) / (-3.0 / b);
	} else if (b <= -4.4e-44) {
		tmp = (sqrt((-3.0 * (c * a))) - b) / (a * 3.0);
	} else if (b <= -1.9e-103) {
		tmp = (b * (2.0 / a)) / -3.0;
	} else if (b <= 8.2e-88) {
		tmp = 0.3333333333333333 * (sqrt((c * (a * -3.0))) / a);
	} else {
		tmp = ((b + (-1.5 * (a * (c / b)))) - b) / (a * 3.0);
	}
	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 <= (-1.1d-19)) then
        tmp = (2.0d0 / a) / ((-3.0d0) / b)
    else if (b <= (-4.4d-44)) then
        tmp = (sqrt(((-3.0d0) * (c * a))) - b) / (a * 3.0d0)
    else if (b <= (-1.9d-103)) then
        tmp = (b * (2.0d0 / a)) / (-3.0d0)
    else if (b <= 8.2d-88) then
        tmp = 0.3333333333333333d0 * (sqrt((c * (a * (-3.0d0)))) / a)
    else
        tmp = ((b + ((-1.5d0) * (a * (c / b)))) - b) / (a * 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.1e-19) {
		tmp = (2.0 / a) / (-3.0 / b);
	} else if (b <= -4.4e-44) {
		tmp = (Math.sqrt((-3.0 * (c * a))) - b) / (a * 3.0);
	} else if (b <= -1.9e-103) {
		tmp = (b * (2.0 / a)) / -3.0;
	} else if (b <= 8.2e-88) {
		tmp = 0.3333333333333333 * (Math.sqrt((c * (a * -3.0))) / a);
	} else {
		tmp = ((b + (-1.5 * (a * (c / b)))) - b) / (a * 3.0);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.1e-19:
		tmp = (2.0 / a) / (-3.0 / b)
	elif b <= -4.4e-44:
		tmp = (math.sqrt((-3.0 * (c * a))) - b) / (a * 3.0)
	elif b <= -1.9e-103:
		tmp = (b * (2.0 / a)) / -3.0
	elif b <= 8.2e-88:
		tmp = 0.3333333333333333 * (math.sqrt((c * (a * -3.0))) / a)
	else:
		tmp = ((b + (-1.5 * (a * (c / b)))) - b) / (a * 3.0)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.1e-19)
		tmp = Float64(Float64(2.0 / a) / Float64(-3.0 / b));
	elseif (b <= -4.4e-44)
		tmp = Float64(Float64(sqrt(Float64(-3.0 * Float64(c * a))) - b) / Float64(a * 3.0));
	elseif (b <= -1.9e-103)
		tmp = Float64(Float64(b * Float64(2.0 / a)) / -3.0);
	elseif (b <= 8.2e-88)
		tmp = Float64(0.3333333333333333 * Float64(sqrt(Float64(c * Float64(a * -3.0))) / a));
	else
		tmp = Float64(Float64(Float64(b + Float64(-1.5 * Float64(a * Float64(c / b)))) - b) / Float64(a * 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.1e-19)
		tmp = (2.0 / a) / (-3.0 / b);
	elseif (b <= -4.4e-44)
		tmp = (sqrt((-3.0 * (c * a))) - b) / (a * 3.0);
	elseif (b <= -1.9e-103)
		tmp = (b * (2.0 / a)) / -3.0;
	elseif (b <= 8.2e-88)
		tmp = 0.3333333333333333 * (sqrt((c * (a * -3.0))) / a);
	else
		tmp = ((b + (-1.5 * (a * (c / b)))) - b) / (a * 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.1e-19], N[(N[(2.0 / a), $MachinePrecision] / N[(-3.0 / b), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, -4.4e-44], N[(N[(N[Sqrt[N[(-3.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, -1.9e-103], N[(N[(b * N[(2.0 / a), $MachinePrecision]), $MachinePrecision] / -3.0), $MachinePrecision], If[LessEqual[b, 8.2e-88], N[(0.3333333333333333 * N[(N[Sqrt[N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b + N[(-1.5 * N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if b < -1.0999999999999999e-19

   1. Initial program 69.3%

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

      1. sqr-neg 69.3%

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

      2. sqr-neg 69.3%

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

      3. associate-*l* 69.3%

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

   3. Simplified 69.3%

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

   5. Taylor expanded in b around -inf 89.6%

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

      1. mul-1-neg 89.6%

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

   7. Simplified 89.6%

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

      1. count-2 89.6%

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

      2. *-commutative 89.6%

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

      3. times-frac 89.6%

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

      4. add-sqr-sqrt 89.3%

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

      5. sqrt-unprod 64.8%

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

      6. sqr-neg 64.8%

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

      7. sqrt-unprod 0.0%

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

      8. add-sqr-sqrt 1.0%

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

   9. Applied egg-rr 1.0%

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

       1. associate-*l/ 1.0%

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

       2. associate-/l* 1.0%

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

   11. Simplified 1.0%

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

       1. clear-num 1.0%

          \[\leadsto 2 \cdot \color{blue}{\frac{1}{\frac{a}{\frac{b}{3}}}} \]

       2. un-div-inv 1.0%

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

       3. div-inv 1.0%

          \[\leadsto \frac{2}{\color{blue}{a \cdot \frac{1}{\frac{b}{3}}}} \]

       4. clear-num 1.0%

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

       5. frac-2neg 1.0%

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

       6. metadata-eval 1.0%

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

       7. add-sqr-sqrt 1.0%

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

       8. sqrt-unprod 0.9%

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

       9. sqr-neg 0.9%

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

       10. sqrt-unprod 0.0%

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

       11. add-sqr-sqrt 89.6%

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

   13. Applied egg-rr 89.6%

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

       1. associate-/r* 89.7%

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

   15. Simplified 89.7%

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

   if -1.0999999999999999e-19 < b < -4.40000000000000024e-44

   1. Initial program 99.4%

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

      1. sqr-neg 99.4%

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

      2. sqr-neg 99.4%

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

      3. associate-*l* 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in b around 0 99.4%

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

   if -4.40000000000000024e-44 < b < -1.9e-103

   1. Initial program 99.6%

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

      1. sqr-neg 99.6%

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

      2. sqr-neg 99.6%

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

      3. associate-*l* 99.5%

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

   3. Simplified 99.5%

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

   6. Applied egg-rr 74.4%

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

      1. associate-/r* 74.5%

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

   8. Simplified 74.5%

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

      1. associate-*r/ 74.7%

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

   10. Applied egg-rr 74.7%

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

       1. associate-*r/ 74.7%

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

       2. *-rgt-identity 74.7%

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

       3. associate-*r* 74.9%

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

       4. *-commutative 74.9%

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

       5. associate-*l* 74.9%

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

   12. Simplified 74.9%

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

   13. Taylor expanded in b around -inf 76.7%

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

       1. associate-*r/ 76.7%

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

       2. associate-*l/ 76.7%

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

       3. *-commutative 76.7%

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

   15. Simplified 76.7%

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

   if -1.9e-103 < b < 8.2000000000000002e-88

   1. Initial program 84.1%

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

      1. sqr-neg 84.1%

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

      2. sqr-neg 84.1%

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

      3. associate-*l* 83.9%

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

   3. Simplified 83.9%

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

   6. Applied egg-rr 81.1%

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

      1. associate-/r* 81.3%

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

   8. Simplified 81.3%

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

      1. associate-*r* 81.3%

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

      2. *-commutative 81.3%

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

      3. add-sqr-sqrt 81.3%

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

      4. pow1/2 81.3%

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

      5. pow1/2 81.3%

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

      6. pow-prod-down 59.9%

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

      7. pow2 59.9%

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

      8. *-commutative 59.9%

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

      9. associate-*r* 59.8%

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

   10. Applied egg-rr 59.8%

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

       1. unpow1/2 59.8%

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

       2. unpow2 59.8%

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

       3. rem-sqrt-square 83.3%

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

       4. associate-*r* 83.3%

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

       5. *-commutative 83.3%

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

       6. associate-*l* 83.2%

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

   12. Simplified 83.2%

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

   13. Taylor expanded in b around 0 80.7%

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

       1. associate-*l/ 80.8%

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

       2. *-lft-identity 80.8%

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

       3. associate-*r* 80.7%

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

       4. *-commutative 80.7%

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

       5. *-commutative 80.7%

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

       6. rem-cube-cbrt 80.0%

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

       7. sqr-pow 79.9%

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

       8. fabs-sqr 79.9%

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

       9. sqr-pow 79.7%

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

       10. rem-cube-cbrt 80.4%

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

       11. *-commutative 80.4%

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

   15. Simplified 80.4%

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

   if 8.2000000000000002e-88 < b

   1. Initial program 39.4%

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

      1. sqr-neg 39.4%

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

      2. sqr-neg 39.4%

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

      3. associate-*l* 39.4%

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

   3. Simplified 39.4%

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

   5. Taylor expanded in b around inf 88.2%

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

      1. associate-/l* 88.2%

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

   7. Simplified 88.2%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.1 \cdot 10^{-19}:\\ \;\;\;\;\frac{\frac{2}{a}}{\frac{-3}{b}}\\ \mathbf{elif}\;b \leq -4.4 \cdot 10^{-44}:\\ \;\;\;\;\frac{\sqrt{-3 \cdot \left(c \cdot a\right)} - b}{a \cdot 3}\\ \mathbf{elif}\;b \leq -1.9 \cdot 10^{-103}:\\ \;\;\;\;\frac{b \cdot \frac{2}{a}}{-3}\\ \mathbf{elif}\;b \leq 8.2 \cdot 10^{-88}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\sqrt{c \cdot \left(a \cdot -3\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(b + -1.5 \cdot \left(a \cdot \frac{c}{b}\right)\right) - b}{a \cdot 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 79.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.4 \cdot 10^{-19}:\\ \;\;\;\;\frac{\frac{2}{a}}{\frac{-3}{b}}\\ \mathbf{elif}\;b \leq -1.4 \cdot 10^{-54}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{elif}\;b \leq -8.2 \cdot 10^{-102}:\\ \;\;\;\;\frac{b \cdot \frac{2}{a}}{-3}\\ \mathbf{elif}\;b \leq 8.5 \cdot 10^{-88}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\sqrt{c \cdot \left(a \cdot -3\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(b + -1.5 \cdot \left(a \cdot \frac{c}{b}\right)\right) - b}{a \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.4e-19)
   (/ (/ 2.0 a) (/ -3.0 b))
   (if (<= b -1.4e-54)
     (/ (- (sqrt (* a (* c -3.0))) b) (* a 3.0))
     (if (<= b -8.2e-102)
       (/ (* b (/ 2.0 a)) -3.0)
       (if (<= b 8.5e-88)
         (* 0.3333333333333333 (/ (sqrt (* c (* a -3.0))) a))
         (/ (- (+ b (* -1.5 (* a (/ c b)))) b) (* a 3.0)))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.4e-19) {
		tmp = (2.0 / a) / (-3.0 / b);
	} else if (b <= -1.4e-54) {
		tmp = (sqrt((a * (c * -3.0))) - b) / (a * 3.0);
	} else if (b <= -8.2e-102) {
		tmp = (b * (2.0 / a)) / -3.0;
	} else if (b <= 8.5e-88) {
		tmp = 0.3333333333333333 * (sqrt((c * (a * -3.0))) / a);
	} else {
		tmp = ((b + (-1.5 * (a * (c / b)))) - b) / (a * 3.0);
	}
	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 <= (-2.4d-19)) then
        tmp = (2.0d0 / a) / ((-3.0d0) / b)
    else if (b <= (-1.4d-54)) then
        tmp = (sqrt((a * (c * (-3.0d0)))) - b) / (a * 3.0d0)
    else if (b <= (-8.2d-102)) then
        tmp = (b * (2.0d0 / a)) / (-3.0d0)
    else if (b <= 8.5d-88) then
        tmp = 0.3333333333333333d0 * (sqrt((c * (a * (-3.0d0)))) / a)
    else
        tmp = ((b + ((-1.5d0) * (a * (c / b)))) - b) / (a * 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.4e-19) {
		tmp = (2.0 / a) / (-3.0 / b);
	} else if (b <= -1.4e-54) {
		tmp = (Math.sqrt((a * (c * -3.0))) - b) / (a * 3.0);
	} else if (b <= -8.2e-102) {
		tmp = (b * (2.0 / a)) / -3.0;
	} else if (b <= 8.5e-88) {
		tmp = 0.3333333333333333 * (Math.sqrt((c * (a * -3.0))) / a);
	} else {
		tmp = ((b + (-1.5 * (a * (c / b)))) - b) / (a * 3.0);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.4e-19:
		tmp = (2.0 / a) / (-3.0 / b)
	elif b <= -1.4e-54:
		tmp = (math.sqrt((a * (c * -3.0))) - b) / (a * 3.0)
	elif b <= -8.2e-102:
		tmp = (b * (2.0 / a)) / -3.0
	elif b <= 8.5e-88:
		tmp = 0.3333333333333333 * (math.sqrt((c * (a * -3.0))) / a)
	else:
		tmp = ((b + (-1.5 * (a * (c / b)))) - b) / (a * 3.0)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.4e-19)
		tmp = Float64(Float64(2.0 / a) / Float64(-3.0 / b));
	elseif (b <= -1.4e-54)
		tmp = Float64(Float64(sqrt(Float64(a * Float64(c * -3.0))) - b) / Float64(a * 3.0));
	elseif (b <= -8.2e-102)
		tmp = Float64(Float64(b * Float64(2.0 / a)) / -3.0);
	elseif (b <= 8.5e-88)
		tmp = Float64(0.3333333333333333 * Float64(sqrt(Float64(c * Float64(a * -3.0))) / a));
	else
		tmp = Float64(Float64(Float64(b + Float64(-1.5 * Float64(a * Float64(c / b)))) - b) / Float64(a * 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.4e-19)
		tmp = (2.0 / a) / (-3.0 / b);
	elseif (b <= -1.4e-54)
		tmp = (sqrt((a * (c * -3.0))) - b) / (a * 3.0);
	elseif (b <= -8.2e-102)
		tmp = (b * (2.0 / a)) / -3.0;
	elseif (b <= 8.5e-88)
		tmp = 0.3333333333333333 * (sqrt((c * (a * -3.0))) / a);
	else
		tmp = ((b + (-1.5 * (a * (c / b)))) - b) / (a * 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.4e-19], N[(N[(2.0 / a), $MachinePrecision] / N[(-3.0 / b), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, -1.4e-54], N[(N[(N[Sqrt[N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, -8.2e-102], N[(N[(b * N[(2.0 / a), $MachinePrecision]), $MachinePrecision] / -3.0), $MachinePrecision], If[LessEqual[b, 8.5e-88], N[(0.3333333333333333 * N[(N[Sqrt[N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b + N[(-1.5 * N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if b < -2.40000000000000023e-19

   1. Initial program 69.3%

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

      1. sqr-neg 69.3%

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

      2. sqr-neg 69.3%

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

      3. associate-*l* 69.3%

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

   3. Simplified 69.3%

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

   5. Taylor expanded in b around -inf 89.6%

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

      1. mul-1-neg 89.6%

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

   7. Simplified 89.6%

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

      1. count-2 89.6%

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

      2. *-commutative 89.6%

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

      3. times-frac 89.6%

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

      4. add-sqr-sqrt 89.3%

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

      5. sqrt-unprod 64.8%

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

      6. sqr-neg 64.8%

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

      7. sqrt-unprod 0.0%

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

      8. add-sqr-sqrt 1.0%

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

   9. Applied egg-rr 1.0%

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

       1. associate-*l/ 1.0%

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

       2. associate-/l* 1.0%

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

   11. Simplified 1.0%

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

       1. clear-num 1.0%

          \[\leadsto 2 \cdot \color{blue}{\frac{1}{\frac{a}{\frac{b}{3}}}} \]

       2. un-div-inv 1.0%

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

       3. div-inv 1.0%

          \[\leadsto \frac{2}{\color{blue}{a \cdot \frac{1}{\frac{b}{3}}}} \]

       4. clear-num 1.0%

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

       5. frac-2neg 1.0%

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

       6. metadata-eval 1.0%

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

       7. add-sqr-sqrt 1.0%

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

       8. sqrt-unprod 0.9%

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

       9. sqr-neg 0.9%

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

       10. sqrt-unprod 0.0%

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

       11. add-sqr-sqrt 89.6%

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

   13. Applied egg-rr 89.6%

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

       1. associate-/r* 89.7%

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

   15. Simplified 89.7%

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

   if -2.40000000000000023e-19 < b < -1.4000000000000001e-54

   1. Initial program 99.4%

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

      1. sqr-neg 99.4%

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

      2. sqr-neg 99.4%

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

      3. associate-*l* 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in b around 0 99.4%

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

      1. *-commutative 99.4%

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

      2. associate-*r* 99.7%

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

   7. Simplified 99.7%

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

   if -1.4000000000000001e-54 < b < -8.2000000000000005e-102

   1. Initial program 99.6%

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

      1. sqr-neg 99.6%

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

      2. sqr-neg 99.6%

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

      3. associate-*l* 99.5%

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

   3. Simplified 99.5%

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

   6. Applied egg-rr 74.4%

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

      1. associate-/r* 74.5%

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

   8. Simplified 74.5%

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

      1. associate-*r/ 74.7%

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

   10. Applied egg-rr 74.7%

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

       1. associate-*r/ 74.7%

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

       2. *-rgt-identity 74.7%

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

       3. associate-*r* 74.9%

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

       4. *-commutative 74.9%

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

       5. associate-*l* 74.9%

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

   12. Simplified 74.9%

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

   13. Taylor expanded in b around -inf 76.7%

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

       1. associate-*r/ 76.7%

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

       2. associate-*l/ 76.7%

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

       3. *-commutative 76.7%

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

   15. Simplified 76.7%

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

   if -8.2000000000000005e-102 < b < 8.4999999999999996e-88

   1. Initial program 84.1%

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

      1. sqr-neg 84.1%

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

      2. sqr-neg 84.1%

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

      3. associate-*l* 83.9%

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

   3. Simplified 83.9%

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

   6. Applied egg-rr 81.1%

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

      1. associate-/r* 81.3%

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

   8. Simplified 81.3%

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

      1. associate-*r* 81.3%

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

      2. *-commutative 81.3%

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

      3. add-sqr-sqrt 81.3%

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

      4. pow1/2 81.3%

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

      5. pow1/2 81.3%

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

      6. pow-prod-down 59.9%

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

      7. pow2 59.9%

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

      8. *-commutative 59.9%

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

      9. associate-*r* 59.8%

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

   10. Applied egg-rr 59.8%

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

       1. unpow1/2 59.8%

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

       2. unpow2 59.8%

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

       3. rem-sqrt-square 83.3%

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

       4. associate-*r* 83.3%

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

       5. *-commutative 83.3%

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

       6. associate-*l* 83.2%

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

   12. Simplified 83.2%

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

   13. Taylor expanded in b around 0 80.7%

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

       1. associate-*l/ 80.8%

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

       2. *-lft-identity 80.8%

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

       3. associate-*r* 80.7%

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

       4. *-commutative 80.7%

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

       5. *-commutative 80.7%

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

       6. rem-cube-cbrt 80.0%

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

       7. sqr-pow 79.9%

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

       8. fabs-sqr 79.9%

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

       9. sqr-pow 79.7%

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

       10. rem-cube-cbrt 80.4%

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

       11. *-commutative 80.4%

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

   15. Simplified 80.4%

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

   if 8.4999999999999996e-88 < b

   1. Initial program 39.4%

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

      1. sqr-neg 39.4%

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

      2. sqr-neg 39.4%

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

      3. associate-*l* 39.4%

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

   3. Simplified 39.4%

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

   5. Taylor expanded in b around inf 88.2%

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

      1. associate-/l* 88.2%

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

   7. Simplified 88.2%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.4 \cdot 10^{-19}:\\ \;\;\;\;\frac{\frac{2}{a}}{\frac{-3}{b}}\\ \mathbf{elif}\;b \leq -1.4 \cdot 10^{-54}:\\ \;\;\;\;\frac{\sqrt{a \cdot \left(c \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{elif}\;b \leq -8.2 \cdot 10^{-102}:\\ \;\;\;\;\frac{b \cdot \frac{2}{a}}{-3}\\ \mathbf{elif}\;b \leq 8.5 \cdot 10^{-88}:\\ \;\;\;\;0.3333333333333333 \cdot \frac{\sqrt{c \cdot \left(a \cdot -3\right)}}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(b + -1.5 \cdot \left(a \cdot \frac{c}{b}\right)\right) - b}{a \cdot 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 86.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -7.2 \cdot 10^{+72}:\\ \;\;\;\;\frac{\frac{b \cdot 2}{a}}{-3}\\ \mathbf{elif}\;b \leq 8.6 \cdot 10^{+32}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(c \cdot a\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(b + -1.5 \cdot \left(a \cdot \frac{c}{b}\right)\right) - b}{a \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -7.2e+72)
   (/ (/ (* b 2.0) a) -3.0)
   (if (<= b 8.6e+32)
     (/ (- (sqrt (- (* b b) (* 3.0 (* c a)))) b) (* a 3.0))
     (/ (- (+ b (* -1.5 (* a (/ c b)))) b) (* a 3.0)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -7.2e+72) {
		tmp = ((b * 2.0) / a) / -3.0;
	} else if (b <= 8.6e+32) {
		tmp = (sqrt(((b * b) - (3.0 * (c * a)))) - b) / (a * 3.0);
	} else {
		tmp = ((b + (-1.5 * (a * (c / b)))) - b) / (a * 3.0);
	}
	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 <= (-7.2d+72)) then
        tmp = ((b * 2.0d0) / a) / (-3.0d0)
    else if (b <= 8.6d+32) then
        tmp = (sqrt(((b * b) - (3.0d0 * (c * a)))) - b) / (a * 3.0d0)
    else
        tmp = ((b + ((-1.5d0) * (a * (c / b)))) - b) / (a * 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -7.2e+72) {
		tmp = ((b * 2.0) / a) / -3.0;
	} else if (b <= 8.6e+32) {
		tmp = (Math.sqrt(((b * b) - (3.0 * (c * a)))) - b) / (a * 3.0);
	} else {
		tmp = ((b + (-1.5 * (a * (c / b)))) - b) / (a * 3.0);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -7.2e+72:
		tmp = ((b * 2.0) / a) / -3.0
	elif b <= 8.6e+32:
		tmp = (math.sqrt(((b * b) - (3.0 * (c * a)))) - b) / (a * 3.0)
	else:
		tmp = ((b + (-1.5 * (a * (c / b)))) - b) / (a * 3.0)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -7.2e+72)
		tmp = Float64(Float64(Float64(b * 2.0) / a) / -3.0);
	elseif (b <= 8.6e+32)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(3.0 * Float64(c * a)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(Float64(b + Float64(-1.5 * Float64(a * Float64(c / b)))) - b) / Float64(a * 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -7.2e+72)
		tmp = ((b * 2.0) / a) / -3.0;
	elseif (b <= 8.6e+32)
		tmp = (sqrt(((b * b) - (3.0 * (c * a)))) - b) / (a * 3.0);
	else
		tmp = ((b + (-1.5 * (a * (c / b)))) - b) / (a * 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -7.2e+72], N[(N[(N[(b * 2.0), $MachinePrecision] / a), $MachinePrecision] / -3.0), $MachinePrecision], If[LessEqual[b, 8.6e+32], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(3.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b + N[(-1.5 * N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -7.20000000000000069e72

   1. Initial program 58.1%

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

      1. sqr-neg 58.1%

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

      2. sqr-neg 58.1%

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

      3. associate-*l* 58.1%

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

   3. Simplified 58.1%

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

   6. Applied egg-rr 50.0%

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

      1. associate-/r* 50.1%

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

   8. Simplified 50.1%

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

      1. associate-*r/ 50.1%

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

   10. Applied egg-rr 50.1%

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

       1. associate-*r/ 50.2%

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

       2. *-rgt-identity 50.2%

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

       3. associate-*r* 50.2%

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

       4. *-commutative 50.2%

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

       5. associate-*l* 50.2%

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

   12. Simplified 50.2%

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

   13. Taylor expanded in b around -inf 94.0%

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

       1. associate-*r/ 94.0%

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

   15. Simplified 94.0%

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

   if -7.20000000000000069e72 < b < 8.5999999999999994e32

   1. Initial program 87.8%

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

      1. sqr-neg 87.8%

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

      2. sqr-neg 87.8%

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

      3. associate-*l* 87.7%

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

   3. Simplified 87.7%

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

   if 8.5999999999999994e32 < b

   1. Initial program 29.8%

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

      1. sqr-neg 29.8%

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

      2. sqr-neg 29.8%

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

      3. associate-*l* 29.8%

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

   3. Simplified 29.8%

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

   5. Taylor expanded in b around inf 93.9%

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

      1. associate-/l* 93.9%

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

   7. Simplified 93.9%

      \[\leadsto \frac{\left(-b\right) + \color{blue}{\left(b + -1.5 \cdot \left(a \cdot \frac{c}{b}\right)\right)}}{3 \cdot a} \]
3. Recombined 3 regimes into one program.

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -7.2 \cdot 10^{+72}:\\ \;\;\;\;\frac{\frac{b \cdot 2}{a}}{-3}\\ \mathbf{elif}\;b \leq 8.6 \cdot 10^{+32}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(c \cdot a\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(b + -1.5 \cdot \left(a \cdot \frac{c}{b}\right)\right) - b}{a \cdot 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.85 \cdot 10^{+72}:\\ \;\;\;\;\frac{\frac{b \cdot 2}{a}}{-3}\\ \mathbf{elif}\;b \leq 8.6 \cdot 10^{+32}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(b + -1.5 \cdot \left(a \cdot \frac{c}{b}\right)\right) - b}{a \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.85e+72)
   (/ (/ (* b 2.0) a) -3.0)
   (if (<= b 8.6e+32)
     (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0))
     (/ (- (+ b (* -1.5 (* a (/ c b)))) b) (* a 3.0)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.85e+72) {
		tmp = ((b * 2.0) / a) / -3.0;
	} else if (b <= 8.6e+32) {
		tmp = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	} else {
		tmp = ((b + (-1.5 * (a * (c / b)))) - b) / (a * 3.0);
	}
	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 <= (-1.85d+72)) then
        tmp = ((b * 2.0d0) / a) / (-3.0d0)
    else if (b <= 8.6d+32) then
        tmp = (sqrt(((b * b) - (c * (a * 3.0d0)))) - b) / (a * 3.0d0)
    else
        tmp = ((b + ((-1.5d0) * (a * (c / b)))) - b) / (a * 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.85e+72) {
		tmp = ((b * 2.0) / a) / -3.0;
	} else if (b <= 8.6e+32) {
		tmp = (Math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	} else {
		tmp = ((b + (-1.5 * (a * (c / b)))) - b) / (a * 3.0);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.85e+72:
		tmp = ((b * 2.0) / a) / -3.0
	elif b <= 8.6e+32:
		tmp = (math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)
	else:
		tmp = ((b + (-1.5 * (a * (c / b)))) - b) / (a * 3.0)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.85e+72)
		tmp = Float64(Float64(Float64(b * 2.0) / a) / -3.0);
	elseif (b <= 8.6e+32)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(Float64(b + Float64(-1.5 * Float64(a * Float64(c / b)))) - b) / Float64(a * 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.85e+72)
		tmp = ((b * 2.0) / a) / -3.0;
	elseif (b <= 8.6e+32)
		tmp = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	else
		tmp = ((b + (-1.5 * (a * (c / b)))) - b) / (a * 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.85e+72], N[(N[(N[(b * 2.0), $MachinePrecision] / a), $MachinePrecision] / -3.0), $MachinePrecision], If[LessEqual[b, 8.6e+32], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(b + N[(-1.5 * N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if b < -1.8500000000000001e72

   1. Initial program 58.1%

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

      1. sqr-neg 58.1%

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

      2. sqr-neg 58.1%

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

      3. associate-*l* 58.1%

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

   3. Simplified 58.1%

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

   6. Applied egg-rr 50.0%

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

      1. associate-/r* 50.1%

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

   8. Simplified 50.1%

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

      1. associate-*r/ 50.1%

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

   10. Applied egg-rr 50.1%

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

       1. associate-*r/ 50.2%

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

       2. *-rgt-identity 50.2%

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

       3. associate-*r* 50.2%

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

       4. *-commutative 50.2%

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

       5. associate-*l* 50.2%

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

   12. Simplified 50.2%

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

   13. Taylor expanded in b around -inf 94.0%

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

       1. associate-*r/ 94.0%

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

   15. Simplified 94.0%

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

   if -1.8500000000000001e72 < b < 8.5999999999999994e32

   1. Initial program 87.8%

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

   if 8.5999999999999994e32 < b

   1. Initial program 29.8%

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

      1. sqr-neg 29.8%

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

      2. sqr-neg 29.8%

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

      3. associate-*l* 29.8%

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

   3. Simplified 29.8%

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

   5. Taylor expanded in b around inf 93.9%

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

      1. associate-/l* 93.9%

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

   7. Simplified 93.9%

      \[\leadsto \frac{\left(-b\right) + \color{blue}{\left(b + -1.5 \cdot \left(a \cdot \frac{c}{b}\right)\right)}}{3 \cdot a} \]
3. Recombined 3 regimes into one program.

4. Final simplification 90.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.85 \cdot 10^{+72}:\\ \;\;\;\;\frac{\frac{b \cdot 2}{a}}{-3}\\ \mathbf{elif}\;b \leq 8.6 \cdot 10^{+32}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(b + -1.5 \cdot \left(a \cdot \frac{c}{b}\right)\right) - b}{a \cdot 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 65.2% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{\frac{b \cdot 2}{a}}{-3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(b + -1.5 \cdot \left(a \cdot \frac{c}{b}\right)\right) - b}{a \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e-310)
   (/ (/ (* b 2.0) a) -3.0)
   (/ (- (+ b (* -1.5 (* a (/ c b)))) b) (* a 3.0))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = ((b * 2.0) / a) / -3.0;
	} else {
		tmp = ((b + (-1.5 * (a * (c / b)))) - b) / (a * 3.0);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-5d-310)) then
        tmp = ((b * 2.0d0) / a) / (-3.0d0)
    else
        tmp = ((b + ((-1.5d0) * (a * (c / b)))) - b) / (a * 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = ((b * 2.0) / a) / -3.0;
	} else {
		tmp = ((b + (-1.5 * (a * (c / b)))) - b) / (a * 3.0);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e-310:
		tmp = ((b * 2.0) / a) / -3.0
	else:
		tmp = ((b + (-1.5 * (a * (c / b)))) - b) / (a * 3.0)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e-310)
		tmp = Float64(Float64(Float64(b * 2.0) / a) / -3.0);
	else
		tmp = Float64(Float64(Float64(b + Float64(-1.5 * Float64(a * Float64(c / b)))) - b) / Float64(a * 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5e-310)
		tmp = ((b * 2.0) / a) / -3.0;
	else
		tmp = ((b + (-1.5 * (a * (c / b)))) - b) / (a * 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-310], N[(N[(N[(b * 2.0), $MachinePrecision] / a), $MachinePrecision] / -3.0), $MachinePrecision], N[(N[(N[(b + N[(-1.5 * N[(a * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 77.4%

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

      1. sqr-neg 77.4%

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

      2. sqr-neg 77.4%

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

      3. associate-*l* 77.4%

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

   3. Simplified 77.4%

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

   6. Applied egg-rr 63.8%

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

      1. associate-/r* 63.8%

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

   8. Simplified 63.8%

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

      1. associate-*r/ 63.8%

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

   10. Applied egg-rr 63.8%

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

       1. associate-*r/ 63.9%

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

       2. *-rgt-identity 63.9%

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

       3. associate-*r* 64.0%

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

       4. *-commutative 64.0%

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

       5. associate-*l* 63.9%

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

   12. Simplified 63.9%

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

   13. Taylor expanded in b around -inf 69.7%

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

       1. associate-*r/ 69.7%

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

   15. Simplified 69.7%

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

   if -4.999999999999985e-310 < b

   1. Initial program 53.8%

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

      1. sqr-neg 53.8%

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

      2. sqr-neg 53.8%

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

      3. associate-*l* 53.8%

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

   3. Simplified 53.8%

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

   5. Taylor expanded in b around inf 62.9%

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

      1. associate-/l* 63.4%

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

   7. Simplified 63.4%

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

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{\frac{b \cdot 2}{a}}{-3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(b + -1.5 \cdot \left(a \cdot \frac{c}{b}\right)\right) - b}{a \cdot 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 65.3% accurate, 5.8× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = ((((c / b) * (a * 1.5)) - b) - b) / (a * 3.0);
	} else {
		tmp = ((b + (-1.5 * (a * (c / b)))) - b) / (a * 3.0);
	}
	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 = ((((c / b) * (a * 1.5d0)) - b) - b) / (a * 3.0d0)
    else
        tmp = ((b + ((-1.5d0) * (a * (c / b)))) - b) / (a * 3.0d0)
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 77.4%

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

      1. sqr-neg 77.4%

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

      2. sqr-neg 77.4%

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

      3. associate-*l* 77.4%

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

   3. Simplified 77.4%

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

   5. Taylor expanded in b around -inf 66.6%

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

      1. +-commutative 66.6%

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

      2. mul-1-neg 66.6%

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

      3. unsub-neg 66.6%

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

      4. associate-/l* 69.7%

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

      5. associate-*r* 69.7%

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

   7. Simplified 69.7%

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

   if -4.999999999999985e-310 < b

   1. Initial program 53.8%

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

      1. sqr-neg 53.8%

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

      2. sqr-neg 53.8%

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

      3. associate-*l* 53.8%

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

   3. Simplified 53.8%

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

   5. Taylor expanded in b around inf 62.9%

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

      1. associate-/l* 63.4%

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

   7. Simplified 63.4%

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

4. Final simplification 66.7%

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

Alternative 9: 64.6% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.2 \cdot 10^{-231}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + \frac{c}{b} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.2e-231) (+ (* -0.6666666666666666 (/ b a)) (* (/ c b) 0.5)) 0.0))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.2e-231) {
		tmp = (-0.6666666666666666 * (b / a)) + ((c / b) * 0.5);
	} else {
		tmp = 0.0;
	}
	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 <= (-2.2d-231)) then
        tmp = ((-0.6666666666666666d0) * (b / a)) + ((c / b) * 0.5d0)
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.2e-231) {
		tmp = (-0.6666666666666666 * (b / a)) + ((c / b) * 0.5);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.2e-231:
		tmp = (-0.6666666666666666 * (b / a)) + ((c / b) * 0.5)
	else:
		tmp = 0.0
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.2e-231)
		tmp = Float64(Float64(-0.6666666666666666 * Float64(b / a)) + Float64(Float64(c / b) * 0.5));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.2e-231)
		tmp = (-0.6666666666666666 * (b / a)) + ((c / b) * 0.5);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.2e-231], N[(N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision] + N[(N[(c / b), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if b < -2.20000000000000009e-231

   1. Initial program 75.8%

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

      1. sqr-neg 75.8%

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

      2. sqr-neg 75.8%

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

      3. associate-*l* 75.8%

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

   3. Simplified 75.8%

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

   5. Taylor expanded in b around -inf 74.4%

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

   if -2.20000000000000009e-231 < b

   1. Initial program 56.9%

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

      1. sqr-neg 56.9%

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

      2. sqr-neg 56.9%

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

      3. associate-*l* 56.8%

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

   3. Simplified 56.8%

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

   5. Taylor expanded in b around -inf 2.7%

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

      1. mul-1-neg 2.7%

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

   7. Simplified 2.7%

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

      1. expm1-log1p-u 2.1%

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

      2. expm1-undefine 8.0%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\left(-b\right) + \left(-b\right)}{3 \cdot a}\right)} - 1} \]

      3. associate-/r* 8.0%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\frac{\left(-b\right) + \left(-b\right)}{3}}{a}}\right)} - 1 \]

      4. unsub-neg 8.0%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{\color{blue}{\left(-b\right) - b}}{3}}{a}\right)} - 1 \]

      5. add-sqr-sqrt 7.7%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{\left(-b\right) - \color{blue}{\sqrt{b} \cdot \sqrt{b}}}{3}}{a}\right)} - 1 \]

      6. sqrt-unprod 4.9%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{\left(-b\right) - \color{blue}{\sqrt{b \cdot b}}}{3}}{a}\right)} - 1 \]

      7. sqr-neg 4.9%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{\left(-b\right) - \sqrt{\color{blue}{\left(-b\right) \cdot \left(-b\right)}}}{3}}{a}\right)} - 1 \]

      8. sqrt-unprod 0.3%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{\left(-b\right) - \color{blue}{\sqrt{-b} \cdot \sqrt{-b}}}{3}}{a}\right)} - 1 \]

      9. add-sqr-sqrt 58.3%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{\left(-b\right) - \color{blue}{\left(-b\right)}}{3}}{a}\right)} - 1 \]

      10. +-inverses 58.3%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{\color{blue}{0}}{3}}{a}\right)} - 1 \]

      11. metadata-eval 58.3%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{0}}{a}\right)} - 1 \]

   9. Applied egg-rr 58.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{0}{a}\right)} - 1} \]
   10. Step-by-step derivation

       1. log1p-undefine 58.3%

          \[\leadsto e^{\color{blue}{\log \left(1 + \frac{0}{a}\right)}} - 1 \]

       2. div0 58.3%

          \[\leadsto e^{\log \left(1 + \color{blue}{0}\right)} - 1 \]

       3. metadata-eval 58.3%

          \[\leadsto e^{\log \color{blue}{1}} - 1 \]

       4. metadata-eval 58.3%

          \[\leadsto e^{\color{blue}{0}} - 1 \]

       5. 1-exp 58.3%

          \[\leadsto \color{blue}{1} - 1 \]

       6. metadata-eval 58.3%

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

   11. Simplified 58.3%

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

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.2 \cdot 10^{-231}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + \frac{c}{b} \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 64.6% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-304}:\\ \;\;\;\;\frac{\frac{2}{a}}{\frac{-3}{b}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2e-304) (/ (/ 2.0 a) (/ -3.0 b)) 0.0))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e-304) {
		tmp = (2.0 / a) / (-3.0 / b);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-2d-304)) then
        tmp = (2.0d0 / a) / ((-3.0d0) / b)
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -2e-304) {
		tmp = (2.0 / a) / (-3.0 / b);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2e-304:
		tmp = (2.0 / a) / (-3.0 / b)
	else:
		tmp = 0.0
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2e-304)
		tmp = Float64(Float64(2.0 / a) / Float64(-3.0 / b));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2e-304)
		tmp = (2.0 / a) / (-3.0 / b);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2e-304], N[(N[(2.0 / a), $MachinePrecision] / N[(-3.0 / b), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if b < -1.99999999999999994e-304

   1. Initial program 77.4%

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

      1. sqr-neg 77.4%

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

      2. sqr-neg 77.4%

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

      3. associate-*l* 77.4%

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

   3. Simplified 77.4%

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

   5. Taylor expanded in b around -inf 69.6%

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

      1. mul-1-neg 69.6%

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

   7. Simplified 69.6%

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

      1. count-2 69.6%

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

      2. *-commutative 69.6%

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

      3. times-frac 69.6%

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

      4. add-sqr-sqrt 69.3%

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

      5. sqrt-unprod 52.4%

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

      6. sqr-neg 52.4%

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

      7. sqrt-unprod 0.0%

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

      8. add-sqr-sqrt 1.5%

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

   9. Applied egg-rr 1.5%

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

       1. associate-*l/ 1.5%

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

       2. associate-/l* 1.5%

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

   11. Simplified 1.5%

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

       1. clear-num 1.5%

          \[\leadsto 2 \cdot \color{blue}{\frac{1}{\frac{a}{\frac{b}{3}}}} \]

       2. un-div-inv 1.5%

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

       3. div-inv 1.5%

          \[\leadsto \frac{2}{\color{blue}{a \cdot \frac{1}{\frac{b}{3}}}} \]

       4. clear-num 1.5%

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

       5. frac-2neg 1.5%

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

       6. metadata-eval 1.5%

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

       7. add-sqr-sqrt 1.5%

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

       8. sqrt-unprod 1.4%

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

       9. sqr-neg 1.4%

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

       10. sqrt-unprod 0.0%

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

       11. add-sqr-sqrt 69.6%

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

   13. Applied egg-rr 69.6%

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

       1. associate-/r* 69.6%

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

   15. Simplified 69.6%

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

   if -1.99999999999999994e-304 < b

   1. Initial program 53.8%

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

      1. sqr-neg 53.8%

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

      2. sqr-neg 53.8%

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

      3. associate-*l* 53.8%

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

   3. Simplified 53.8%

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

   5. Taylor expanded in b around -inf 2.5%

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

      1. mul-1-neg 2.5%

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

   7. Simplified 2.5%

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

      1. expm1-log1p-u 1.9%

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

      2. expm1-undefine 8.2%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\left(-b\right) + \left(-b\right)}{3 \cdot a}\right)} - 1} \]

      3. associate-/r* 8.2%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\frac{\left(-b\right) + \left(-b\right)}{3}}{a}}\right)} - 1 \]

      4. unsub-neg 8.2%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{\color{blue}{\left(-b\right) - b}}{3}}{a}\right)} - 1 \]

      5. add-sqr-sqrt 8.2%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{\left(-b\right) - \color{blue}{\sqrt{b} \cdot \sqrt{b}}}{3}}{a}\right)} - 1 \]

      6. sqrt-unprod 4.9%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{\left(-b\right) - \color{blue}{\sqrt{b \cdot b}}}{3}}{a}\right)} - 1 \]

      7. sqr-neg 4.9%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{\left(-b\right) - \sqrt{\color{blue}{\left(-b\right) \cdot \left(-b\right)}}}{3}}{a}\right)} - 1 \]

      8. sqrt-unprod 0.0%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{\left(-b\right) - \color{blue}{\sqrt{-b} \cdot \sqrt{-b}}}{3}}{a}\right)} - 1 \]

      9. add-sqr-sqrt 62.3%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{\left(-b\right) - \color{blue}{\left(-b\right)}}{3}}{a}\right)} - 1 \]

      10. +-inverses 62.3%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{\color{blue}{0}}{3}}{a}\right)} - 1 \]

      11. metadata-eval 62.3%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{0}}{a}\right)} - 1 \]

   9. Applied egg-rr 62.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{0}{a}\right)} - 1} \]
   10. Step-by-step derivation

       1. log1p-undefine 62.3%

          \[\leadsto e^{\color{blue}{\log \left(1 + \frac{0}{a}\right)}} - 1 \]

       2. div0 62.3%

          \[\leadsto e^{\log \left(1 + \color{blue}{0}\right)} - 1 \]

       3. metadata-eval 62.3%

          \[\leadsto e^{\log \color{blue}{1}} - 1 \]

       4. metadata-eval 62.3%

          \[\leadsto e^{\color{blue}{0}} - 1 \]

       5. 1-exp 62.3%

          \[\leadsto \color{blue}{1} - 1 \]

       6. metadata-eval 62.3%

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

   11. Simplified 62.3%

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

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2 \cdot 10^{-304}:\\ \;\;\;\;\frac{\frac{2}{a}}{\frac{-3}{b}}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 64.6% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{\frac{b \cdot 2}{a}}{-3}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5e-310) (/ (/ (* b 2.0) a) -3.0) 0.0))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = ((b * 2.0) / a) / -3.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= (-5d-310)) then
        tmp = ((b * 2.0d0) / a) / (-3.0d0)
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = ((b * 2.0) / a) / -3.0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -5e-310:
		tmp = ((b * 2.0) / a) / -3.0
	else:
		tmp = 0.0
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -5e-310)
		tmp = Float64(Float64(Float64(b * 2.0) / a) / -3.0);
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -5e-310)
		tmp = ((b * 2.0) / a) / -3.0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -5e-310], N[(N[(N[(b * 2.0), $MachinePrecision] / a), $MachinePrecision] / -3.0), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 77.4%

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

      1. sqr-neg 77.4%

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

      2. sqr-neg 77.4%

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

      3. associate-*l* 77.4%

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

   3. Simplified 77.4%

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

   6. Applied egg-rr 63.8%

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

      1. associate-/r* 63.8%

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

   8. Simplified 63.8%

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

      1. associate-*r/ 63.8%

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

   10. Applied egg-rr 63.8%

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

       1. associate-*r/ 63.9%

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

       2. *-rgt-identity 63.9%

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

       3. associate-*r* 64.0%

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

       4. *-commutative 64.0%

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

       5. associate-*l* 63.9%

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

   12. Simplified 63.9%

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

   13. Taylor expanded in b around -inf 69.7%

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

       1. associate-*r/ 69.7%

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

   15. Simplified 69.7%

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

   if -4.999999999999985e-310 < b

   1. Initial program 53.8%

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

      1. sqr-neg 53.8%

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

      2. sqr-neg 53.8%

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

      3. associate-*l* 53.8%

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

   3. Simplified 53.8%

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

   5. Taylor expanded in b around -inf 2.5%

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

      1. mul-1-neg 2.5%

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

   7. Simplified 2.5%

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

      1. expm1-log1p-u 1.9%

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

      2. expm1-undefine 8.2%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\left(-b\right) + \left(-b\right)}{3 \cdot a}\right)} - 1} \]

      3. associate-/r* 8.2%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\frac{\left(-b\right) + \left(-b\right)}{3}}{a}}\right)} - 1 \]

      4. unsub-neg 8.2%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{\color{blue}{\left(-b\right) - b}}{3}}{a}\right)} - 1 \]

      5. add-sqr-sqrt 8.2%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{\left(-b\right) - \color{blue}{\sqrt{b} \cdot \sqrt{b}}}{3}}{a}\right)} - 1 \]

      6. sqrt-unprod 4.9%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{\left(-b\right) - \color{blue}{\sqrt{b \cdot b}}}{3}}{a}\right)} - 1 \]

      7. sqr-neg 4.9%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{\left(-b\right) - \sqrt{\color{blue}{\left(-b\right) \cdot \left(-b\right)}}}{3}}{a}\right)} - 1 \]

      8. sqrt-unprod 0.0%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{\left(-b\right) - \color{blue}{\sqrt{-b} \cdot \sqrt{-b}}}{3}}{a}\right)} - 1 \]

      9. add-sqr-sqrt 62.3%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{\left(-b\right) - \color{blue}{\left(-b\right)}}{3}}{a}\right)} - 1 \]

      10. +-inverses 62.3%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{\color{blue}{0}}{3}}{a}\right)} - 1 \]

      11. metadata-eval 62.3%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{0}}{a}\right)} - 1 \]

   9. Applied egg-rr 62.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{0}{a}\right)} - 1} \]
   10. Step-by-step derivation

       1. log1p-undefine 62.3%

          \[\leadsto e^{\color{blue}{\log \left(1 + \frac{0}{a}\right)}} - 1 \]

       2. div0 62.3%

          \[\leadsto e^{\log \left(1 + \color{blue}{0}\right)} - 1 \]

       3. metadata-eval 62.3%

          \[\leadsto e^{\log \color{blue}{1}} - 1 \]

       4. metadata-eval 62.3%

          \[\leadsto e^{\color{blue}{0}} - 1 \]

       5. 1-exp 62.3%

          \[\leadsto \color{blue}{1} - 1 \]

       6. metadata-eval 62.3%

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

   11. Simplified 62.3%

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

4. Final simplification 66.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{\frac{b \cdot 2}{a}}{-3}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 64.6% accurate, 11.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 77.4%

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

      1. sqr-neg 77.4%

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

      2. sqr-neg 77.4%

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

      3. associate-*l* 77.4%

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

   3. Simplified 77.4%

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

   6. Applied egg-rr 63.8%

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

      1. associate-/r* 63.8%

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

   8. Simplified 63.8%

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

      1. associate-*r* 63.8%

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

      2. *-commutative 63.8%

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

      3. add-sqr-sqrt 63.8%

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

      4. pow1/2 63.8%

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

      5. pow1/2 63.9%

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

      6. pow-prod-down 76.1%

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

      7. pow2 76.1%

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

      8. *-commutative 76.1%

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

      9. associate-*r* 76.1%

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

   10. Applied egg-rr 76.1%

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

       1. unpow1/2 76.1%

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

       2. unpow2 76.1%

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

       3. rem-sqrt-square 91.1%

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

       4. associate-*r* 91.1%

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

       5. *-commutative 91.1%

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

       6. associate-*l* 91.1%

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

   12. Simplified 91.1%

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

   13. Taylor expanded in b around -inf 69.5%

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

       1. *-commutative 69.5%

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

       2. associate-*l/ 69.6%

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

       3. associate-/l* 69.5%

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

   15. Simplified 69.5%

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

   if -4.999999999999985e-310 < b

   1. Initial program 53.8%

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

      1. sqr-neg 53.8%

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

      2. sqr-neg 53.8%

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

      3. associate-*l* 53.8%

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

   3. Simplified 53.8%

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

   5. Taylor expanded in b around -inf 2.5%

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

      1. mul-1-neg 2.5%

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

   7. Simplified 2.5%

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

      1. expm1-log1p-u 1.9%

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

      2. expm1-undefine 8.2%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\left(-b\right) + \left(-b\right)}{3 \cdot a}\right)} - 1} \]

      3. associate-/r* 8.2%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\frac{\left(-b\right) + \left(-b\right)}{3}}{a}}\right)} - 1 \]

      4. unsub-neg 8.2%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{\color{blue}{\left(-b\right) - b}}{3}}{a}\right)} - 1 \]

      5. add-sqr-sqrt 8.2%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{\left(-b\right) - \color{blue}{\sqrt{b} \cdot \sqrt{b}}}{3}}{a}\right)} - 1 \]

      6. sqrt-unprod 4.9%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{\left(-b\right) - \color{blue}{\sqrt{b \cdot b}}}{3}}{a}\right)} - 1 \]

      7. sqr-neg 4.9%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{\left(-b\right) - \sqrt{\color{blue}{\left(-b\right) \cdot \left(-b\right)}}}{3}}{a}\right)} - 1 \]

      8. sqrt-unprod 0.0%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{\left(-b\right) - \color{blue}{\sqrt{-b} \cdot \sqrt{-b}}}{3}}{a}\right)} - 1 \]

      9. add-sqr-sqrt 62.3%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{\left(-b\right) - \color{blue}{\left(-b\right)}}{3}}{a}\right)} - 1 \]

      10. +-inverses 62.3%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{\color{blue}{0}}{3}}{a}\right)} - 1 \]

      11. metadata-eval 62.3%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{0}}{a}\right)} - 1 \]

   9. Applied egg-rr 62.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{0}{a}\right)} - 1} \]
   10. Step-by-step derivation

       1. log1p-undefine 62.3%

          \[\leadsto e^{\color{blue}{\log \left(1 + \frac{0}{a}\right)}} - 1 \]

       2. div0 62.3%

          \[\leadsto e^{\log \left(1 + \color{blue}{0}\right)} - 1 \]

       3. metadata-eval 62.3%

          \[\leadsto e^{\log \color{blue}{1}} - 1 \]

       4. metadata-eval 62.3%

          \[\leadsto e^{\color{blue}{0}} - 1 \]

       5. 1-exp 62.3%

          \[\leadsto \color{blue}{1} - 1 \]

       6. metadata-eval 62.3%

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

   11. Simplified 62.3%

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

4. Final simplification 66.0%

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

Alternative 13: 64.6% accurate, 11.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 77.4%

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

      1. sqr-neg 77.4%

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

      2. sqr-neg 77.4%

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

      3. associate-*l* 77.4%

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

   3. Simplified 77.4%

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

   5. Taylor expanded in b around -inf 69.5%

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

      1. *-commutative 69.5%

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

   7. Simplified 69.5%

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

   if -4.999999999999985e-310 < b

   1. Initial program 53.8%

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

      1. sqr-neg 53.8%

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

      2. sqr-neg 53.8%

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

      3. associate-*l* 53.8%

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

   3. Simplified 53.8%

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

   5. Taylor expanded in b around -inf 2.5%

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

      1. mul-1-neg 2.5%

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

   7. Simplified 2.5%

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

      1. expm1-log1p-u 1.9%

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

      2. expm1-undefine 8.2%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\left(-b\right) + \left(-b\right)}{3 \cdot a}\right)} - 1} \]

      3. associate-/r* 8.2%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\frac{\left(-b\right) + \left(-b\right)}{3}}{a}}\right)} - 1 \]

      4. unsub-neg 8.2%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{\color{blue}{\left(-b\right) - b}}{3}}{a}\right)} - 1 \]

      5. add-sqr-sqrt 8.2%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{\left(-b\right) - \color{blue}{\sqrt{b} \cdot \sqrt{b}}}{3}}{a}\right)} - 1 \]

      6. sqrt-unprod 4.9%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{\left(-b\right) - \color{blue}{\sqrt{b \cdot b}}}{3}}{a}\right)} - 1 \]

      7. sqr-neg 4.9%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{\left(-b\right) - \sqrt{\color{blue}{\left(-b\right) \cdot \left(-b\right)}}}{3}}{a}\right)} - 1 \]

      8. sqrt-unprod 0.0%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{\left(-b\right) - \color{blue}{\sqrt{-b} \cdot \sqrt{-b}}}{3}}{a}\right)} - 1 \]

      9. add-sqr-sqrt 62.3%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{\left(-b\right) - \color{blue}{\left(-b\right)}}{3}}{a}\right)} - 1 \]

      10. +-inverses 62.3%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{\color{blue}{0}}{3}}{a}\right)} - 1 \]

      11. metadata-eval 62.3%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{0}}{a}\right)} - 1 \]

   9. Applied egg-rr 62.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{0}{a}\right)} - 1} \]
   10. Step-by-step derivation

       1. log1p-undefine 62.3%

          \[\leadsto e^{\color{blue}{\log \left(1 + \frac{0}{a}\right)}} - 1 \]

       2. div0 62.3%

          \[\leadsto e^{\log \left(1 + \color{blue}{0}\right)} - 1 \]

       3. metadata-eval 62.3%

          \[\leadsto e^{\log \color{blue}{1}} - 1 \]

       4. metadata-eval 62.3%

          \[\leadsto e^{\color{blue}{0}} - 1 \]

       5. 1-exp 62.3%

          \[\leadsto \color{blue}{1} - 1 \]

       6. metadata-eval 62.3%

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

   11. Simplified 62.3%

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

4. Final simplification 66.0%

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

Alternative 14: 64.6% accurate, 11.6× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -5e-310) {
		tmp = -0.6666666666666666 / (a / b);
	} else {
		tmp = 0.0;
	}
	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) / (a / b)
    else
        tmp = 0.0d0
    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 / (a / b);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 77.4%

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

      1. sqr-neg 77.4%

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

      2. sqr-neg 77.4%

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

      3. associate-*l* 77.4%

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

   3. Simplified 77.4%

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

   5. Taylor expanded in b around -inf 69.5%

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

      1. *-commutative 69.5%

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

   7. Simplified 69.5%

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

      1. *-commutative 69.5%

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

      2. clear-num 69.4%

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

      3. un-div-inv 69.5%

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

   9. Applied egg-rr 69.5%

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

   if -4.999999999999985e-310 < b

   1. Initial program 53.8%

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

      1. sqr-neg 53.8%

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

      2. sqr-neg 53.8%

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

      3. associate-*l* 53.8%

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

   3. Simplified 53.8%

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

   5. Taylor expanded in b around -inf 2.5%

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

      1. mul-1-neg 2.5%

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

   7. Simplified 2.5%

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

      1. expm1-log1p-u 1.9%

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

      2. expm1-undefine 8.2%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\left(-b\right) + \left(-b\right)}{3 \cdot a}\right)} - 1} \]

      3. associate-/r* 8.2%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\frac{\left(-b\right) + \left(-b\right)}{3}}{a}}\right)} - 1 \]

      4. unsub-neg 8.2%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{\color{blue}{\left(-b\right) - b}}{3}}{a}\right)} - 1 \]

      5. add-sqr-sqrt 8.2%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{\left(-b\right) - \color{blue}{\sqrt{b} \cdot \sqrt{b}}}{3}}{a}\right)} - 1 \]

      6. sqrt-unprod 4.9%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{\left(-b\right) - \color{blue}{\sqrt{b \cdot b}}}{3}}{a}\right)} - 1 \]

      7. sqr-neg 4.9%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{\left(-b\right) - \sqrt{\color{blue}{\left(-b\right) \cdot \left(-b\right)}}}{3}}{a}\right)} - 1 \]

      8. sqrt-unprod 0.0%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{\left(-b\right) - \color{blue}{\sqrt{-b} \cdot \sqrt{-b}}}{3}}{a}\right)} - 1 \]

      9. add-sqr-sqrt 62.3%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{\left(-b\right) - \color{blue}{\left(-b\right)}}{3}}{a}\right)} - 1 \]

      10. +-inverses 62.3%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{\color{blue}{0}}{3}}{a}\right)} - 1 \]

      11. metadata-eval 62.3%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{0}}{a}\right)} - 1 \]

   9. Applied egg-rr 62.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{0}{a}\right)} - 1} \]
   10. Step-by-step derivation

       1. log1p-undefine 62.3%

          \[\leadsto e^{\color{blue}{\log \left(1 + \frac{0}{a}\right)}} - 1 \]

       2. div0 62.3%

          \[\leadsto e^{\log \left(1 + \color{blue}{0}\right)} - 1 \]

       3. metadata-eval 62.3%

          \[\leadsto e^{\log \color{blue}{1}} - 1 \]

       4. metadata-eval 62.3%

          \[\leadsto e^{\color{blue}{0}} - 1 \]

       5. 1-exp 62.3%

          \[\leadsto \color{blue}{1} - 1 \]

       6. metadata-eval 62.3%

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

   11. Simplified 62.3%

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

4. Final simplification 66.0%

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

Alternative 15: 64.6% accurate, 11.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < -4.999999999999985e-310

   1. Initial program 77.4%

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

      1. sqr-neg 77.4%

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

      2. sqr-neg 77.4%

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

      3. associate-*l* 77.4%

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

   3. Simplified 77.4%

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

   6. Applied egg-rr 63.8%

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

      1. associate-/r* 63.8%

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

   8. Simplified 63.8%

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

      1. associate-*r* 63.8%

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

      2. *-commutative 63.8%

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

      3. add-sqr-sqrt 63.8%

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

      4. pow1/2 63.8%

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

      5. pow1/2 63.9%

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

      6. pow-prod-down 76.1%

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

      7. pow2 76.1%

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

      8. *-commutative 76.1%

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

      9. associate-*r* 76.1%

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

   10. Applied egg-rr 76.1%

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

       1. unpow1/2 76.1%

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

       2. unpow2 76.1%

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

       3. rem-sqrt-square 91.1%

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

       4. associate-*r* 91.1%

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

       5. *-commutative 91.1%

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

       6. associate-*l* 91.1%

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

   12. Simplified 91.1%

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

   13. Taylor expanded in b around -inf 69.5%

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

       1. *-commutative 69.5%

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

       2. associate-*l/ 69.6%

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

   15. Simplified 69.6%

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

   if -4.999999999999985e-310 < b

   1. Initial program 53.8%

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

      1. sqr-neg 53.8%

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

      2. sqr-neg 53.8%

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

      3. associate-*l* 53.8%

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

   3. Simplified 53.8%

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

   5. Taylor expanded in b around -inf 2.5%

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

      1. mul-1-neg 2.5%

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

   7. Simplified 2.5%

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

      1. expm1-log1p-u 1.9%

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

      2. expm1-undefine 8.2%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\left(-b\right) + \left(-b\right)}{3 \cdot a}\right)} - 1} \]

      3. associate-/r* 8.2%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\frac{\left(-b\right) + \left(-b\right)}{3}}{a}}\right)} - 1 \]

      4. unsub-neg 8.2%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{\color{blue}{\left(-b\right) - b}}{3}}{a}\right)} - 1 \]

      5. add-sqr-sqrt 8.2%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{\left(-b\right) - \color{blue}{\sqrt{b} \cdot \sqrt{b}}}{3}}{a}\right)} - 1 \]

      6. sqrt-unprod 4.9%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{\left(-b\right) - \color{blue}{\sqrt{b \cdot b}}}{3}}{a}\right)} - 1 \]

      7. sqr-neg 4.9%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{\left(-b\right) - \sqrt{\color{blue}{\left(-b\right) \cdot \left(-b\right)}}}{3}}{a}\right)} - 1 \]

      8. sqrt-unprod 0.0%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{\left(-b\right) - \color{blue}{\sqrt{-b} \cdot \sqrt{-b}}}{3}}{a}\right)} - 1 \]

      9. add-sqr-sqrt 62.3%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{\left(-b\right) - \color{blue}{\left(-b\right)}}{3}}{a}\right)} - 1 \]

      10. +-inverses 62.3%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{\color{blue}{0}}{3}}{a}\right)} - 1 \]

      11. metadata-eval 62.3%

          \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{0}}{a}\right)} - 1 \]

   9. Applied egg-rr 62.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{0}{a}\right)} - 1} \]
   10. Step-by-step derivation

       1. log1p-undefine 62.3%

          \[\leadsto e^{\color{blue}{\log \left(1 + \frac{0}{a}\right)}} - 1 \]

       2. div0 62.3%

          \[\leadsto e^{\log \left(1 + \color{blue}{0}\right)} - 1 \]

       3. metadata-eval 62.3%

          \[\leadsto e^{\log \color{blue}{1}} - 1 \]

       4. metadata-eval 62.3%

          \[\leadsto e^{\color{blue}{0}} - 1 \]

       5. 1-exp 62.3%

          \[\leadsto \color{blue}{1} - 1 \]

       6. metadata-eval 62.3%

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

   11. Simplified 62.3%

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

4. Final simplification 66.1%

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

Alternative 16: 32.4% accurate, 116.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = 0.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[a_, b_, c_] := 0.0

Derivation

1. Initial program 66.1%

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

   1. sqr-neg 66.1%

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

   2. sqr-neg 66.1%

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

   3. associate-*l* 66.0%

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

3. Simplified 66.0%

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

5. Taylor expanded in b around -inf 37.4%

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

   1. mul-1-neg 37.4%

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

7. Simplified 37.4%

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

   1. expm1-log1p-u 21.7%

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

   2. expm1-undefine 18.3%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\left(-b\right) + \left(-b\right)}{3 \cdot a}\right)} - 1} \]

   3. associate-/r* 18.3%

      \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\frac{\left(-b\right) + \left(-b\right)}{3}}{a}}\right)} - 1 \]

   4. unsub-neg 18.3%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{\color{blue}{\left(-b\right) - b}}{3}}{a}\right)} - 1 \]

   5. add-sqr-sqrt 4.0%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{\left(-b\right) - \color{blue}{\sqrt{b} \cdot \sqrt{b}}}{3}}{a}\right)} - 1 \]

   6. sqrt-unprod 3.5%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{\left(-b\right) - \color{blue}{\sqrt{b \cdot b}}}{3}}{a}\right)} - 1 \]

   7. sqr-neg 3.5%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{\left(-b\right) - \sqrt{\color{blue}{\left(-b\right) \cdot \left(-b\right)}}}{3}}{a}\right)} - 1 \]

   8. sqrt-unprod 1.6%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{\left(-b\right) - \color{blue}{\sqrt{-b} \cdot \sqrt{-b}}}{3}}{a}\right)} - 1 \]

   9. add-sqr-sqrt 31.4%

      \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{\left(-b\right) - \color{blue}{\left(-b\right)}}{3}}{a}\right)} - 1 \]

   10. +-inverses 31.4%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{\frac{\color{blue}{0}}{3}}{a}\right)} - 1 \]

   11. metadata-eval 31.4%

       \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{0}}{a}\right)} - 1 \]

9. Applied egg-rr 31.4%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{0}{a}\right)} - 1} \]
10. Step-by-step derivation

    1. log1p-undefine 31.4%

       \[\leadsto e^{\color{blue}{\log \left(1 + \frac{0}{a}\right)}} - 1 \]

    2. div0 31.4%

       \[\leadsto e^{\log \left(1 + \color{blue}{0}\right)} - 1 \]

    3. metadata-eval 31.4%

       \[\leadsto e^{\log \color{blue}{1}} - 1 \]

    4. metadata-eval 31.4%

       \[\leadsto e^{\color{blue}{0}} - 1 \]

    5. 1-exp 31.4%

       \[\leadsto \color{blue}{1} - 1 \]

    6. metadata-eval 31.4%

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

11. Simplified 31.4%

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

12. Final simplification 31.4%

    \[\leadsto 0 \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024046 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))