Cubic critical

Percentage Accurate: 64.6% → 89.7%

Time: 23.5s

Alternatives: 15

Speedup: 11.6×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 64.6% 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: 89.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.15 \cdot 10^{+79}:\\ \;\;\;\;\frac{1}{\frac{a}{b} \cdot -1.5}\\ \mathbf{elif}\;b \leq -6.2 \cdot 10^{-75}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{elif}\;b \leq 5 \cdot 10^{-157}:\\ \;\;\;\;\left(b - \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -3\right)}\right)\right) \cdot \frac{1}{a \cdot -3}\\ \mathbf{elif}\;b \leq 6.1 \cdot 10^{+124}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.15e+79)
   (/ 1.0 (* (/ a b) -1.5))
   (if (<= b -6.2e-75)
     (/ (- (sqrt (fma b b (* a (* c -3.0)))) b) (* a 3.0))
     (if (<= b 5e-157)
       (* (- b (hypot b (sqrt (* c (* a -3.0))))) (/ 1.0 (* a -3.0)))
       (if (<= b 6.1e+124)
         (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0))
         0.0)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.15e+79) {
		tmp = 1.0 / ((a / b) * -1.5);
	} else if (b <= -6.2e-75) {
		tmp = (sqrt(fma(b, b, (a * (c * -3.0)))) - b) / (a * 3.0);
	} else if (b <= 5e-157) {
		tmp = (b - hypot(b, sqrt((c * (a * -3.0))))) * (1.0 / (a * -3.0));
	} else if (b <= 6.1e+124) {
		tmp = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.15e+79)
		tmp = Float64(1.0 / Float64(Float64(a / b) * -1.5));
	elseif (b <= -6.2e-75)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(c * -3.0)))) - b) / Float64(a * 3.0));
	elseif (b <= 5e-157)
		tmp = Float64(Float64(b - hypot(b, sqrt(Float64(c * Float64(a * -3.0))))) * Float64(1.0 / Float64(a * -3.0)));
	elseif (b <= 6.1e+124)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0));
	else
		tmp = 0.0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.15e+79], N[(1.0 / N[(N[(a / b), $MachinePrecision] * -1.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, -6.2e-75], N[(N[(N[Sqrt[N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5e-157], N[(N[(b - N[Sqrt[b ^ 2 + N[Sqrt[N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 6.1e+124], 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], 0.0]]]]

Derivation

1. Split input into 5 regimes

2. if b < -2.1500000000000002e79

   1. Initial program 67.5%

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

      1. sqr-neg 67.5%

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

      2. sqr-neg 67.5%

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

      3. associate-*l* 67.5%

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

   3. Simplified 67.5%

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

   5. Taylor expanded in b around -inf 94.7%

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

      1. mul-1-neg 94.7%

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

   7. Simplified 94.7%

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

      1. clear-num 94.7%

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

      2. inv-pow 94.7%

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

      3. count-2 94.7%

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

      4. times-frac 94.8%

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

      5. metadata-eval 94.8%

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

      6. add-sqr-sqrt 94.5%

         \[\leadsto {\left(1.5 \cdot \frac{a}{\color{blue}{\sqrt{-b} \cdot \sqrt{-b}}}\right)}^{-1} \]

      7. sqrt-unprod 67.8%

         \[\leadsto {\left(1.5 \cdot \frac{a}{\color{blue}{\sqrt{\left(-b\right) \cdot \left(-b\right)}}}\right)}^{-1} \]

      8. sqr-neg 67.8%

         \[\leadsto {\left(1.5 \cdot \frac{a}{\sqrt{\color{blue}{b \cdot b}}}\right)}^{-1} \]

      9. sqrt-unprod 0.0%

         \[\leadsto {\left(1.5 \cdot \frac{a}{\color{blue}{\sqrt{b} \cdot \sqrt{b}}}\right)}^{-1} \]

      10. add-sqr-sqrt 0.8%

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

   9. Applied egg-rr 0.8%

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

       1. unpow-1 0.8%

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

   11. Simplified 0.8%

       \[\leadsto \color{blue}{\frac{1}{1.5 \cdot \frac{a}{b}}} \]
   12. Step-by-step derivation

       1. associate-*r/ 0.8%

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

       2. add-sqr-sqrt 0.0%

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

       3. sqrt-unprod 67.8%

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

       4. sqr-neg 67.8%

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

       5. sqrt-unprod 94.5%

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

       6. add-sqr-sqrt 94.7%

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

       7. distribute-neg-frac2 94.7%

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

       8. associate-*r/ 94.8%

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

       9. neg-sub0 81.3%

          \[\leadsto \frac{1}{\color{blue}{0 - 1.5 \cdot \frac{a}{b}}} \]

   13. Applied egg-rr 81.3%

       \[\leadsto \frac{1}{\color{blue}{0 - 1.5 \cdot \frac{a}{b}}} \]
   14. Step-by-step derivation

       1. neg-sub0 94.8%

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

       2. *-commutative 94.8%

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

       3. distribute-rgt-neg-in 94.8%

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

       4. metadata-eval 94.8%

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

   15. Simplified 94.8%

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

   if -2.1500000000000002e79 < b < -6.20000000000000013e-75

   1. Initial program 99.5%

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

      1. /-rgt-identity 99.5%

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

      2. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   if -6.20000000000000013e-75 < b < 5.0000000000000002e-157

   1. Initial program 71.8%

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

      1. sqr-neg 71.8%

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

      2. sqr-neg 71.8%

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

      3. associate-*l* 71.7%

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

   3. Simplified 71.7%

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

   6. Applied egg-rr 80.3%

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

      1. *-commutative 80.3%

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

      2. associate-*r* 80.4%

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

   8. Simplified 80.4%

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

   if 5.0000000000000002e-157 < b < 6.1000000000000001e124

   1. Initial program 88.9%

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

   if 6.1000000000000001e124 < b

   1. Initial program 10.8%

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

      1. sqr-neg 10.8%

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

      2. sqr-neg 10.8%

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

      3. associate-*l* 13.2%

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

   3. Simplified 13.2%

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

   5. Taylor expanded in b around -inf 4.6%

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

      1. mul-1-neg 4.6%

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

   7. Simplified 4.6%

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

      1. unsub-neg 4.6%

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

      2. div-sub 4.6%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 10.3%

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

      5. sqr-neg 10.3%

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

      6. sqrt-unprod 43.4%

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

      7. add-sqr-sqrt 61.9%

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

      8. *-commutative 61.9%

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

      9. *-commutative 61.9%

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

   9. Applied egg-rr 61.9%

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

       1. +-inverses 100.0%

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

   11. Simplified 100.0%

       \[\leadsto \color{blue}{0} \]
3. Recombined 5 regimes into one program.

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.15 \cdot 10^{+79}:\\ \;\;\;\;\frac{1}{\frac{a}{b} \cdot -1.5}\\ \mathbf{elif}\;b \leq -6.2 \cdot 10^{-75}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{elif}\;b \leq 5 \cdot 10^{-157}:\\ \;\;\;\;\left(b - \mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot -3\right)}\right)\right) \cdot \frac{1}{a \cdot -3}\\ \mathbf{elif}\;b \leq 6.1 \cdot 10^{+124}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 89.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{if}\;b \leq -2.15 \cdot 10^{+79}:\\ \;\;\;\;\frac{1}{\frac{a}{b} \cdot -1.5}\\ \mathbf{elif}\;b \leq -3.6 \cdot 10^{-232}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 5 \cdot 10^{-147}:\\ \;\;\;\;\frac{-0.3333333333333333}{a} \cdot \left(b - \mathsf{hypot}\left(b, \sqrt{-3 \cdot \left(a \cdot c\right)}\right)\right)\\ \mathbf{elif}\;b \leq 10^{+124}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0))))
   (if (<= b -2.15e+79)
     (/ 1.0 (* (/ a b) -1.5))
     (if (<= b -3.6e-232)
       t_0
       (if (<= b 5e-147)
         (* (/ -0.3333333333333333 a) (- b (hypot b (sqrt (* -3.0 (* a c))))))
         (if (<= b 1e+124) t_0 0.0))))))

C

double code(double a, double b, double c) {
	double t_0 = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	double tmp;
	if (b <= -2.15e+79) {
		tmp = 1.0 / ((a / b) * -1.5);
	} else if (b <= -3.6e-232) {
		tmp = t_0;
	} else if (b <= 5e-147) {
		tmp = (-0.3333333333333333 / a) * (b - hypot(b, sqrt((-3.0 * (a * c)))));
	} else if (b <= 1e+124) {
		tmp = t_0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Java

public static double code(double a, double b, double c) {
	double t_0 = (Math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	double tmp;
	if (b <= -2.15e+79) {
		tmp = 1.0 / ((a / b) * -1.5);
	} else if (b <= -3.6e-232) {
		tmp = t_0;
	} else if (b <= 5e-147) {
		tmp = (-0.3333333333333333 / a) * (b - Math.hypot(b, Math.sqrt((-3.0 * (a * c)))));
	} else if (b <= 1e+124) {
		tmp = t_0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)
	tmp = 0
	if b <= -2.15e+79:
		tmp = 1.0 / ((a / b) * -1.5)
	elif b <= -3.6e-232:
		tmp = t_0
	elif b <= 5e-147:
		tmp = (-0.3333333333333333 / a) * (b - math.hypot(b, math.sqrt((-3.0 * (a * c)))))
	elif b <= 1e+124:
		tmp = t_0
	else:
		tmp = 0.0
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0))
	tmp = 0.0
	if (b <= -2.15e+79)
		tmp = Float64(1.0 / Float64(Float64(a / b) * -1.5));
	elseif (b <= -3.6e-232)
		tmp = t_0;
	elseif (b <= 5e-147)
		tmp = Float64(Float64(-0.3333333333333333 / a) * Float64(b - hypot(b, sqrt(Float64(-3.0 * Float64(a * c))))));
	elseif (b <= 1e+124)
		tmp = t_0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	tmp = 0.0;
	if (b <= -2.15e+79)
		tmp = 1.0 / ((a / b) * -1.5);
	elseif (b <= -3.6e-232)
		tmp = t_0;
	elseif (b <= 5e-147)
		tmp = (-0.3333333333333333 / a) * (b - hypot(b, sqrt((-3.0 * (a * c)))));
	elseif (b <= 1e+124)
		tmp = t_0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -2.15e+79], N[(1.0 / N[(N[(a / b), $MachinePrecision] * -1.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, -3.6e-232], t$95$0, If[LessEqual[b, 5e-147], N[(N[(-0.3333333333333333 / a), $MachinePrecision] * N[(b - N[Sqrt[b ^ 2 + N[Sqrt[N[(-3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 1e+124], t$95$0, 0.0]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -2.1500000000000002e79

   1. Initial program 67.5%

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

      1. sqr-neg 67.5%

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

      2. sqr-neg 67.5%

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

      3. associate-*l* 67.5%

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

   3. Simplified 67.5%

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

   5. Taylor expanded in b around -inf 94.7%

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

      1. mul-1-neg 94.7%

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

   7. Simplified 94.7%

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

      1. clear-num 94.7%

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

      2. inv-pow 94.7%

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

      3. count-2 94.7%

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

      4. times-frac 94.8%

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

      5. metadata-eval 94.8%

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

      6. add-sqr-sqrt 94.5%

         \[\leadsto {\left(1.5 \cdot \frac{a}{\color{blue}{\sqrt{-b} \cdot \sqrt{-b}}}\right)}^{-1} \]

      7. sqrt-unprod 67.8%

         \[\leadsto {\left(1.5 \cdot \frac{a}{\color{blue}{\sqrt{\left(-b\right) \cdot \left(-b\right)}}}\right)}^{-1} \]

      8. sqr-neg 67.8%

         \[\leadsto {\left(1.5 \cdot \frac{a}{\sqrt{\color{blue}{b \cdot b}}}\right)}^{-1} \]

      9. sqrt-unprod 0.0%

         \[\leadsto {\left(1.5 \cdot \frac{a}{\color{blue}{\sqrt{b} \cdot \sqrt{b}}}\right)}^{-1} \]

      10. add-sqr-sqrt 0.8%

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

   9. Applied egg-rr 0.8%

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

       1. unpow-1 0.8%

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

   11. Simplified 0.8%

       \[\leadsto \color{blue}{\frac{1}{1.5 \cdot \frac{a}{b}}} \]
   12. Step-by-step derivation

       1. associate-*r/ 0.8%

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

       2. add-sqr-sqrt 0.0%

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

       3. sqrt-unprod 67.8%

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

       4. sqr-neg 67.8%

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

       5. sqrt-unprod 94.5%

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

       6. add-sqr-sqrt 94.7%

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

       7. distribute-neg-frac2 94.7%

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

       8. associate-*r/ 94.8%

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

       9. neg-sub0 81.3%

          \[\leadsto \frac{1}{\color{blue}{0 - 1.5 \cdot \frac{a}{b}}} \]

   13. Applied egg-rr 81.3%

       \[\leadsto \frac{1}{\color{blue}{0 - 1.5 \cdot \frac{a}{b}}} \]
   14. Step-by-step derivation

       1. neg-sub0 94.8%

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

       2. *-commutative 94.8%

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

       3. distribute-rgt-neg-in 94.8%

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

       4. metadata-eval 94.8%

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

   15. Simplified 94.8%

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

   if -2.1500000000000002e79 < b < -3.60000000000000016e-232 or 5.00000000000000013e-147 < b < 9.99999999999999948e123

   1. Initial program 91.6%

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

   if -3.60000000000000016e-232 < b < 5.00000000000000013e-147

   1. Initial program 63.8%

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

      1. sqr-neg 63.8%

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

      2. sqr-neg 63.8%

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

      3. associate-*l* 63.7%

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

   3. Simplified 63.7%

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

   6. Applied egg-rr 78.4%

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

      1. *-commutative 78.4%

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

      2. associate-/r* 78.4%

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

      3. metadata-eval 78.4%

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

   8. Simplified 78.4%

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

   9. Taylor expanded in a around 0 78.5%

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

   if 9.99999999999999948e123 < b

   1. Initial program 10.8%

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

      1. sqr-neg 10.8%

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

      2. sqr-neg 10.8%

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

      3. associate-*l* 13.2%

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

   3. Simplified 13.2%

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

   5. Taylor expanded in b around -inf 4.6%

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

      1. mul-1-neg 4.6%

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

   7. Simplified 4.6%

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

      1. unsub-neg 4.6%

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

      2. div-sub 4.6%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 10.3%

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

      5. sqr-neg 10.3%

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

      6. sqrt-unprod 43.4%

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

      7. add-sqr-sqrt 61.9%

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

      8. *-commutative 61.9%

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

      9. *-commutative 61.9%

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

   9. Applied egg-rr 61.9%

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

       1. +-inverses 100.0%

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

   11. Simplified 100.0%

       \[\leadsto \color{blue}{0} \]
3. Recombined 4 regimes into one program.

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.15 \cdot 10^{+79}:\\ \;\;\;\;\frac{1}{\frac{a}{b} \cdot -1.5}\\ \mathbf{elif}\;b \leq -3.6 \cdot 10^{-232}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{elif}\;b \leq 5 \cdot 10^{-147}:\\ \;\;\;\;\frac{-0.3333333333333333}{a} \cdot \left(b - \mathsf{hypot}\left(b, \sqrt{-3 \cdot \left(a \cdot c\right)}\right)\right)\\ \mathbf{elif}\;b \leq 10^{+124}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{if}\;b \leq -1.2 \cdot 10^{+79}:\\ \;\;\;\;\frac{1}{\frac{a}{b} \cdot -1.5}\\ \mathbf{elif}\;b \leq -6.5 \cdot 10^{-84}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 8.5 \cdot 10^{-133}:\\ \;\;\;\;\left(b - \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -3\right)}\right)\right) \cdot \frac{-0.3333333333333333}{a}\\ \mathbf{elif}\;b \leq 5 \cdot 10^{+124}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0))))
   (if (<= b -1.2e+79)
     (/ 1.0 (* (/ a b) -1.5))
     (if (<= b -6.5e-84)
       t_0
       (if (<= b 8.5e-133)
         (* (- b (hypot b (sqrt (* a (* c -3.0))))) (/ -0.3333333333333333 a))
         (if (<= b 5e+124) t_0 0.0))))))

C

double code(double a, double b, double c) {
	double t_0 = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	double tmp;
	if (b <= -1.2e+79) {
		tmp = 1.0 / ((a / b) * -1.5);
	} else if (b <= -6.5e-84) {
		tmp = t_0;
	} else if (b <= 8.5e-133) {
		tmp = (b - hypot(b, sqrt((a * (c * -3.0))))) * (-0.3333333333333333 / a);
	} else if (b <= 5e+124) {
		tmp = t_0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Java

public static double code(double a, double b, double c) {
	double t_0 = (Math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	double tmp;
	if (b <= -1.2e+79) {
		tmp = 1.0 / ((a / b) * -1.5);
	} else if (b <= -6.5e-84) {
		tmp = t_0;
	} else if (b <= 8.5e-133) {
		tmp = (b - Math.hypot(b, Math.sqrt((a * (c * -3.0))))) * (-0.3333333333333333 / a);
	} else if (b <= 5e+124) {
		tmp = t_0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)
	tmp = 0
	if b <= -1.2e+79:
		tmp = 1.0 / ((a / b) * -1.5)
	elif b <= -6.5e-84:
		tmp = t_0
	elif b <= 8.5e-133:
		tmp = (b - math.hypot(b, math.sqrt((a * (c * -3.0))))) * (-0.3333333333333333 / a)
	elif b <= 5e+124:
		tmp = t_0
	else:
		tmp = 0.0
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0))
	tmp = 0.0
	if (b <= -1.2e+79)
		tmp = Float64(1.0 / Float64(Float64(a / b) * -1.5));
	elseif (b <= -6.5e-84)
		tmp = t_0;
	elseif (b <= 8.5e-133)
		tmp = Float64(Float64(b - hypot(b, sqrt(Float64(a * Float64(c * -3.0))))) * Float64(-0.3333333333333333 / a));
	elseif (b <= 5e+124)
		tmp = t_0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	tmp = 0.0;
	if (b <= -1.2e+79)
		tmp = 1.0 / ((a / b) * -1.5);
	elseif (b <= -6.5e-84)
		tmp = t_0;
	elseif (b <= 8.5e-133)
		tmp = (b - hypot(b, sqrt((a * (c * -3.0))))) * (-0.3333333333333333 / a);
	elseif (b <= 5e+124)
		tmp = t_0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -1.2e+79], N[(1.0 / N[(N[(a / b), $MachinePrecision] * -1.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, -6.5e-84], t$95$0, If[LessEqual[b, 8.5e-133], N[(N[(b - N[Sqrt[b ^ 2 + N[Sqrt[N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[(-0.3333333333333333 / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5e+124], t$95$0, 0.0]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -1.19999999999999993e79

   1. Initial program 67.5%

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

      1. sqr-neg 67.5%

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

      2. sqr-neg 67.5%

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

      3. associate-*l* 67.5%

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

   3. Simplified 67.5%

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

   5. Taylor expanded in b around -inf 94.7%

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

      1. mul-1-neg 94.7%

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

   7. Simplified 94.7%

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

      1. clear-num 94.7%

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

      2. inv-pow 94.7%

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

      3. count-2 94.7%

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

      4. times-frac 94.8%

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

      5. metadata-eval 94.8%

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

      6. add-sqr-sqrt 94.5%

         \[\leadsto {\left(1.5 \cdot \frac{a}{\color{blue}{\sqrt{-b} \cdot \sqrt{-b}}}\right)}^{-1} \]

      7. sqrt-unprod 67.8%

         \[\leadsto {\left(1.5 \cdot \frac{a}{\color{blue}{\sqrt{\left(-b\right) \cdot \left(-b\right)}}}\right)}^{-1} \]

      8. sqr-neg 67.8%

         \[\leadsto {\left(1.5 \cdot \frac{a}{\sqrt{\color{blue}{b \cdot b}}}\right)}^{-1} \]

      9. sqrt-unprod 0.0%

         \[\leadsto {\left(1.5 \cdot \frac{a}{\color{blue}{\sqrt{b} \cdot \sqrt{b}}}\right)}^{-1} \]

      10. add-sqr-sqrt 0.8%

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

   9. Applied egg-rr 0.8%

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

       1. unpow-1 0.8%

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

   11. Simplified 0.8%

       \[\leadsto \color{blue}{\frac{1}{1.5 \cdot \frac{a}{b}}} \]
   12. Step-by-step derivation

       1. associate-*r/ 0.8%

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

       2. add-sqr-sqrt 0.0%

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

       3. sqrt-unprod 67.8%

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

       4. sqr-neg 67.8%

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

       5. sqrt-unprod 94.5%

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

       6. add-sqr-sqrt 94.7%

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

       7. distribute-neg-frac2 94.7%

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

       8. associate-*r/ 94.8%

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

       9. neg-sub0 81.3%

          \[\leadsto \frac{1}{\color{blue}{0 - 1.5 \cdot \frac{a}{b}}} \]

   13. Applied egg-rr 81.3%

       \[\leadsto \frac{1}{\color{blue}{0 - 1.5 \cdot \frac{a}{b}}} \]
   14. Step-by-step derivation

       1. neg-sub0 94.8%

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

       2. *-commutative 94.8%

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

       3. distribute-rgt-neg-in 94.8%

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

       4. metadata-eval 94.8%

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

   15. Simplified 94.8%

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

   if -1.19999999999999993e79 < b < -6.50000000000000022e-84 or 8.49999999999999957e-133 < b < 4.9999999999999996e124

   1. Initial program 93.9%

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

   if -6.50000000000000022e-84 < b < 8.49999999999999957e-133

   1. Initial program 72.3%

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

      1. sqr-neg 72.3%

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

      2. sqr-neg 72.3%

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

      3. associate-*l* 72.1%

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

   3. Simplified 72.1%

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

   6. Applied egg-rr 80.6%

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

      1. *-commutative 80.6%

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

      2. associate-/r* 80.7%

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

      3. metadata-eval 80.7%

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

   8. Simplified 80.7%

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

   if 4.9999999999999996e124 < b

   1. Initial program 10.8%

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

      1. sqr-neg 10.8%

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

      2. sqr-neg 10.8%

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

      3. associate-*l* 13.2%

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

   3. Simplified 13.2%

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

   5. Taylor expanded in b around -inf 4.6%

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

      1. mul-1-neg 4.6%

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

   7. Simplified 4.6%

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

      1. unsub-neg 4.6%

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

      2. div-sub 4.6%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 10.3%

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

      5. sqr-neg 10.3%

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

      6. sqrt-unprod 43.4%

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

      7. add-sqr-sqrt 61.9%

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

      8. *-commutative 61.9%

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

      9. *-commutative 61.9%

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

   9. Applied egg-rr 61.9%

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

       1. +-inverses 100.0%

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

   11. Simplified 100.0%

       \[\leadsto \color{blue}{0} \]
3. Recombined 4 regimes into one program.

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.2 \cdot 10^{+79}:\\ \;\;\;\;\frac{1}{\frac{a}{b} \cdot -1.5}\\ \mathbf{elif}\;b \leq -6.5 \cdot 10^{-84}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{elif}\;b \leq 8.5 \cdot 10^{-133}:\\ \;\;\;\;\left(b - \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -3\right)}\right)\right) \cdot \frac{-0.3333333333333333}{a}\\ \mathbf{elif}\;b \leq 5 \cdot 10^{+124}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{if}\;b \leq -1.95 \cdot 10^{+79}:\\ \;\;\;\;\frac{1}{\frac{a}{b} \cdot -1.5}\\ \mathbf{elif}\;b \leq -9.2 \cdot 10^{-222}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;b \leq 2.3 \cdot 10^{-110}:\\ \;\;\;\;\frac{\frac{b - \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -3\right)}\right)}{a}}{-3}\\ \mathbf{elif}\;b \leq 10^{+124}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0))))
   (if (<= b -1.95e+79)
     (/ 1.0 (* (/ a b) -1.5))
     (if (<= b -9.2e-222)
       t_0
       (if (<= b 2.3e-110)
         (/ (/ (- b (hypot b (sqrt (* a (* c -3.0))))) a) -3.0)
         (if (<= b 1e+124) t_0 0.0))))))

C

double code(double a, double b, double c) {
	double t_0 = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	double tmp;
	if (b <= -1.95e+79) {
		tmp = 1.0 / ((a / b) * -1.5);
	} else if (b <= -9.2e-222) {
		tmp = t_0;
	} else if (b <= 2.3e-110) {
		tmp = ((b - hypot(b, sqrt((a * (c * -3.0))))) / a) / -3.0;
	} else if (b <= 1e+124) {
		tmp = t_0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Java

public static double code(double a, double b, double c) {
	double t_0 = (Math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	double tmp;
	if (b <= -1.95e+79) {
		tmp = 1.0 / ((a / b) * -1.5);
	} else if (b <= -9.2e-222) {
		tmp = t_0;
	} else if (b <= 2.3e-110) {
		tmp = ((b - Math.hypot(b, Math.sqrt((a * (c * -3.0))))) / a) / -3.0;
	} else if (b <= 1e+124) {
		tmp = t_0;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)
	tmp = 0
	if b <= -1.95e+79:
		tmp = 1.0 / ((a / b) * -1.5)
	elif b <= -9.2e-222:
		tmp = t_0
	elif b <= 2.3e-110:
		tmp = ((b - math.hypot(b, math.sqrt((a * (c * -3.0))))) / a) / -3.0
	elif b <= 1e+124:
		tmp = t_0
	else:
		tmp = 0.0
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0))
	tmp = 0.0
	if (b <= -1.95e+79)
		tmp = Float64(1.0 / Float64(Float64(a / b) * -1.5));
	elseif (b <= -9.2e-222)
		tmp = t_0;
	elseif (b <= 2.3e-110)
		tmp = Float64(Float64(Float64(b - hypot(b, sqrt(Float64(a * Float64(c * -3.0))))) / a) / -3.0);
	elseif (b <= 1e+124)
		tmp = t_0;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	tmp = 0.0;
	if (b <= -1.95e+79)
		tmp = 1.0 / ((a / b) * -1.5);
	elseif (b <= -9.2e-222)
		tmp = t_0;
	elseif (b <= 2.3e-110)
		tmp = ((b - hypot(b, sqrt((a * (c * -3.0))))) / a) / -3.0;
	elseif (b <= 1e+124)
		tmp = t_0;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -1.95e+79], N[(1.0 / N[(N[(a / b), $MachinePrecision] * -1.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, -9.2e-222], t$95$0, If[LessEqual[b, 2.3e-110], N[(N[(N[(b - N[Sqrt[b ^ 2 + N[Sqrt[N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision] / -3.0), $MachinePrecision], If[LessEqual[b, 1e+124], t$95$0, 0.0]]]]]

Derivation

1. Split input into 4 regimes

2. if b < -1.9499999999999999e79

   1. Initial program 67.5%

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

      1. sqr-neg 67.5%

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

      2. sqr-neg 67.5%

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

      3. associate-*l* 67.5%

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

   3. Simplified 67.5%

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

   5. Taylor expanded in b around -inf 94.7%

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

      1. mul-1-neg 94.7%

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

   7. Simplified 94.7%

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

      1. clear-num 94.7%

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

      2. inv-pow 94.7%

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

      3. count-2 94.7%

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

      4. times-frac 94.8%

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

      5. metadata-eval 94.8%

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

      6. add-sqr-sqrt 94.5%

         \[\leadsto {\left(1.5 \cdot \frac{a}{\color{blue}{\sqrt{-b} \cdot \sqrt{-b}}}\right)}^{-1} \]

      7. sqrt-unprod 67.8%

         \[\leadsto {\left(1.5 \cdot \frac{a}{\color{blue}{\sqrt{\left(-b\right) \cdot \left(-b\right)}}}\right)}^{-1} \]

      8. sqr-neg 67.8%

         \[\leadsto {\left(1.5 \cdot \frac{a}{\sqrt{\color{blue}{b \cdot b}}}\right)}^{-1} \]

      9. sqrt-unprod 0.0%

         \[\leadsto {\left(1.5 \cdot \frac{a}{\color{blue}{\sqrt{b} \cdot \sqrt{b}}}\right)}^{-1} \]

      10. add-sqr-sqrt 0.8%

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

   9. Applied egg-rr 0.8%

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

       1. unpow-1 0.8%

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

   11. Simplified 0.8%

       \[\leadsto \color{blue}{\frac{1}{1.5 \cdot \frac{a}{b}}} \]
   12. Step-by-step derivation

       1. associate-*r/ 0.8%

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

       2. add-sqr-sqrt 0.0%

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

       3. sqrt-unprod 67.8%

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

       4. sqr-neg 67.8%

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

       5. sqrt-unprod 94.5%

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

       6. add-sqr-sqrt 94.7%

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

       7. distribute-neg-frac2 94.7%

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

       8. associate-*r/ 94.8%

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

       9. neg-sub0 81.3%

          \[\leadsto \frac{1}{\color{blue}{0 - 1.5 \cdot \frac{a}{b}}} \]

   13. Applied egg-rr 81.3%

       \[\leadsto \frac{1}{\color{blue}{0 - 1.5 \cdot \frac{a}{b}}} \]
   14. Step-by-step derivation

       1. neg-sub0 94.8%

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

       2. *-commutative 94.8%

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

       3. distribute-rgt-neg-in 94.8%

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

       4. metadata-eval 94.8%

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

   15. Simplified 94.8%

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

   if -1.9499999999999999e79 < b < -9.2000000000000005e-222 or 2.3000000000000001e-110 < b < 9.99999999999999948e123

   1. Initial program 91.4%

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

   if -9.2000000000000005e-222 < b < 2.3000000000000001e-110

   1. Initial program 66.8%

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

      1. sqr-neg 66.8%

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

      2. sqr-neg 66.8%

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

      3. associate-*l* 66.8%

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

   3. Simplified 66.8%

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

   6. Applied egg-rr 80.0%

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

      1. associate-*l/ 80.2%

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

      2. un-div-inv 80.3%

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

   8. Applied egg-rr 80.3%

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

   if 9.99999999999999948e123 < b

   1. Initial program 10.8%

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

      1. sqr-neg 10.8%

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

      2. sqr-neg 10.8%

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

      3. associate-*l* 13.2%

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

   3. Simplified 13.2%

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

   5. Taylor expanded in b around -inf 4.6%

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

      1. mul-1-neg 4.6%

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

   7. Simplified 4.6%

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

      1. unsub-neg 4.6%

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

      2. div-sub 4.6%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 10.3%

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

      5. sqr-neg 10.3%

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

      6. sqrt-unprod 43.4%

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

      7. add-sqr-sqrt 61.9%

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

      8. *-commutative 61.9%

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

      9. *-commutative 61.9%

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

   9. Applied egg-rr 61.9%

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

       1. +-inverses 100.0%

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

   11. Simplified 100.0%

       \[\leadsto \color{blue}{0} \]
3. Recombined 4 regimes into one program.

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.95 \cdot 10^{+79}:\\ \;\;\;\;\frac{1}{\frac{a}{b} \cdot -1.5}\\ \mathbf{elif}\;b \leq -9.2 \cdot 10^{-222}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{elif}\;b \leq 2.3 \cdot 10^{-110}:\\ \;\;\;\;\frac{\frac{b - \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -3\right)}\right)}{a}}{-3}\\ \mathbf{elif}\;b \leq 10^{+124}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \left(c \cdot -3\right)\\ \mathbf{if}\;b \leq -2.1 \cdot 10^{+79}:\\ \;\;\;\;\frac{1}{\frac{a}{b} \cdot -1.5}\\ \mathbf{elif}\;b \leq -1.1 \cdot 10^{-81}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, t\_0\right)} - b}{a \cdot 3}\\ \mathbf{elif}\;b \leq 5 \cdot 10^{-147}:\\ \;\;\;\;\left(b - \mathsf{hypot}\left(b, \sqrt{t\_0}\right)\right) \cdot \frac{-0.3333333333333333}{a}\\ \mathbf{elif}\;b \leq 8.8 \cdot 10^{+124}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* a (* c -3.0))))
   (if (<= b -2.1e+79)
     (/ 1.0 (* (/ a b) -1.5))
     (if (<= b -1.1e-81)
       (/ (- (sqrt (fma b b t_0)) b) (* a 3.0))
       (if (<= b 5e-147)
         (* (- b (hypot b (sqrt t_0))) (/ -0.3333333333333333 a))
         (if (<= b 8.8e+124)
           (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0))
           0.0))))))

C

double code(double a, double b, double c) {
	double t_0 = a * (c * -3.0);
	double tmp;
	if (b <= -2.1e+79) {
		tmp = 1.0 / ((a / b) * -1.5);
	} else if (b <= -1.1e-81) {
		tmp = (sqrt(fma(b, b, t_0)) - b) / (a * 3.0);
	} else if (b <= 5e-147) {
		tmp = (b - hypot(b, sqrt(t_0))) * (-0.3333333333333333 / a);
	} else if (b <= 8.8e+124) {
		tmp = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64(a * Float64(c * -3.0))
	tmp = 0.0
	if (b <= -2.1e+79)
		tmp = Float64(1.0 / Float64(Float64(a / b) * -1.5));
	elseif (b <= -1.1e-81)
		tmp = Float64(Float64(sqrt(fma(b, b, t_0)) - b) / Float64(a * 3.0));
	elseif (b <= 5e-147)
		tmp = Float64(Float64(b - hypot(b, sqrt(t_0))) * Float64(-0.3333333333333333 / a));
	elseif (b <= 8.8e+124)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0));
	else
		tmp = 0.0;
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[b, -2.1e+79], N[(1.0 / N[(N[(a / b), $MachinePrecision] * -1.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, -1.1e-81], N[(N[(N[Sqrt[N[(b * b + t$95$0), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5e-147], N[(N[(b - N[Sqrt[b ^ 2 + N[Sqrt[t$95$0], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision] * N[(-0.3333333333333333 / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 8.8e+124], 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], 0.0]]]]]

Derivation

1. Split input into 5 regimes

2. if b < -2.10000000000000008e79

   1. Initial program 67.5%

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

      1. sqr-neg 67.5%

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

      2. sqr-neg 67.5%

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

      3. associate-*l* 67.5%

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

   3. Simplified 67.5%

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

   5. Taylor expanded in b around -inf 94.7%

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

      1. mul-1-neg 94.7%

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

   7. Simplified 94.7%

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

      1. clear-num 94.7%

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

      2. inv-pow 94.7%

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

      3. count-2 94.7%

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

      4. times-frac 94.8%

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

      5. metadata-eval 94.8%

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

      6. add-sqr-sqrt 94.5%

         \[\leadsto {\left(1.5 \cdot \frac{a}{\color{blue}{\sqrt{-b} \cdot \sqrt{-b}}}\right)}^{-1} \]

      7. sqrt-unprod 67.8%

         \[\leadsto {\left(1.5 \cdot \frac{a}{\color{blue}{\sqrt{\left(-b\right) \cdot \left(-b\right)}}}\right)}^{-1} \]

      8. sqr-neg 67.8%

         \[\leadsto {\left(1.5 \cdot \frac{a}{\sqrt{\color{blue}{b \cdot b}}}\right)}^{-1} \]

      9. sqrt-unprod 0.0%

         \[\leadsto {\left(1.5 \cdot \frac{a}{\color{blue}{\sqrt{b} \cdot \sqrt{b}}}\right)}^{-1} \]

      10. add-sqr-sqrt 0.8%

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

   9. Applied egg-rr 0.8%

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

       1. unpow-1 0.8%

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

   11. Simplified 0.8%

       \[\leadsto \color{blue}{\frac{1}{1.5 \cdot \frac{a}{b}}} \]
   12. Step-by-step derivation

       1. associate-*r/ 0.8%

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

       2. add-sqr-sqrt 0.0%

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

       3. sqrt-unprod 67.8%

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

       4. sqr-neg 67.8%

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

       5. sqrt-unprod 94.5%

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

       6. add-sqr-sqrt 94.7%

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

       7. distribute-neg-frac2 94.7%

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

       8. associate-*r/ 94.8%

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

       9. neg-sub0 81.3%

          \[\leadsto \frac{1}{\color{blue}{0 - 1.5 \cdot \frac{a}{b}}} \]

   13. Applied egg-rr 81.3%

       \[\leadsto \frac{1}{\color{blue}{0 - 1.5 \cdot \frac{a}{b}}} \]
   14. Step-by-step derivation

       1. neg-sub0 94.8%

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

       2. *-commutative 94.8%

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

       3. distribute-rgt-neg-in 94.8%

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

       4. metadata-eval 94.8%

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

   15. Simplified 94.8%

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

   if -2.10000000000000008e79 < b < -1.1e-81

   1. Initial program 99.5%

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

      1. /-rgt-identity 99.5%

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

      2. metadata-eval 99.5%

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

   3. Simplified 99.5%

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

   if -1.1e-81 < b < 5.00000000000000013e-147

   1. Initial program 72.3%

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

      1. sqr-neg 72.3%

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

      2. sqr-neg 72.3%

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

      3. associate-*l* 72.1%

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

   3. Simplified 72.1%

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

   6. Applied egg-rr 80.6%

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

      1. *-commutative 80.6%

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

      2. associate-/r* 80.7%

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

      3. metadata-eval 80.7%

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

   8. Simplified 80.7%

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

   if 5.00000000000000013e-147 < b < 8.8000000000000004e124

   1. Initial program 88.6%

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

   if 8.8000000000000004e124 < b

   1. Initial program 10.8%

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

      1. sqr-neg 10.8%

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

      2. sqr-neg 10.8%

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

      3. associate-*l* 13.2%

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

   3. Simplified 13.2%

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

   5. Taylor expanded in b around -inf 4.6%

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

      1. mul-1-neg 4.6%

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

   7. Simplified 4.6%

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

      1. unsub-neg 4.6%

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

      2. div-sub 4.6%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 10.3%

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

      5. sqr-neg 10.3%

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

      6. sqrt-unprod 43.4%

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

      7. add-sqr-sqrt 61.9%

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

      8. *-commutative 61.9%

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

      9. *-commutative 61.9%

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

   9. Applied egg-rr 61.9%

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

       1. +-inverses 100.0%

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

   11. Simplified 100.0%

       \[\leadsto \color{blue}{0} \]
3. Recombined 5 regimes into one program.

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.1 \cdot 10^{+79}:\\ \;\;\;\;\frac{1}{\frac{a}{b} \cdot -1.5}\\ \mathbf{elif}\;b \leq -1.1 \cdot 10^{-81}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{elif}\;b \leq 5 \cdot 10^{-147}:\\ \;\;\;\;\left(b - \mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot -3\right)}\right)\right) \cdot \frac{-0.3333333333333333}{a}\\ \mathbf{elif}\;b \leq 8.8 \cdot 10^{+124}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 89.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -2.15 \cdot 10^{+79}:\\ \;\;\;\;\frac{1}{\frac{a}{b} \cdot -1.5}\\ \mathbf{elif}\;b \leq 2 \cdot 10^{+124}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -2.15e+79)
   (/ 1.0 (* (/ a b) -1.5))
   (if (<= b 2e+124)
     (/ (- (sqrt (- (* b b) (* 3.0 (* a c)))) b) (* a 3.0))
     0.0)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -2.15e+79) {
		tmp = 1.0 / ((a / b) * -1.5);
	} else if (b <= 2e+124) {
		tmp = (sqrt(((b * b) - (3.0 * (a * c)))) - b) / (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 <= (-2.15d+79)) then
        tmp = 1.0d0 / ((a / b) * (-1.5d0))
    else if (b <= 2d+124) then
        tmp = (sqrt(((b * b) - (3.0d0 * (a * c)))) - b) / (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 <= -2.15e+79) {
		tmp = 1.0 / ((a / b) * -1.5);
	} else if (b <= 2e+124) {
		tmp = (Math.sqrt(((b * b) - (3.0 * (a * c)))) - b) / (a * 3.0);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -2.15e+79:
		tmp = 1.0 / ((a / b) * -1.5)
	elif b <= 2e+124:
		tmp = (math.sqrt(((b * b) - (3.0 * (a * c)))) - b) / (a * 3.0)
	else:
		tmp = 0.0
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -2.15e+79)
		tmp = Float64(1.0 / Float64(Float64(a / b) * -1.5));
	elseif (b <= 2e+124)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(3.0 * Float64(a * c)))) - b) / Float64(a * 3.0));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -2.15e+79)
		tmp = 1.0 / ((a / b) * -1.5);
	elseif (b <= 2e+124)
		tmp = (sqrt(((b * b) - (3.0 * (a * c)))) - b) / (a * 3.0);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -2.15e+79], N[(1.0 / N[(N[(a / b), $MachinePrecision] * -1.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 2e+124], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], 0.0]]

Derivation

1. Split input into 3 regimes

2. if b < -2.1500000000000002e79

   1. Initial program 67.5%

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

      1. sqr-neg 67.5%

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

      2. sqr-neg 67.5%

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

      3. associate-*l* 67.5%

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

   3. Simplified 67.5%

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

   5. Taylor expanded in b around -inf 94.7%

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

      1. mul-1-neg 94.7%

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

   7. Simplified 94.7%

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

      1. clear-num 94.7%

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

      2. inv-pow 94.7%

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

      3. count-2 94.7%

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

      4. times-frac 94.8%

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

      5. metadata-eval 94.8%

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

      6. add-sqr-sqrt 94.5%

         \[\leadsto {\left(1.5 \cdot \frac{a}{\color{blue}{\sqrt{-b} \cdot \sqrt{-b}}}\right)}^{-1} \]

      7. sqrt-unprod 67.8%

         \[\leadsto {\left(1.5 \cdot \frac{a}{\color{blue}{\sqrt{\left(-b\right) \cdot \left(-b\right)}}}\right)}^{-1} \]

      8. sqr-neg 67.8%

         \[\leadsto {\left(1.5 \cdot \frac{a}{\sqrt{\color{blue}{b \cdot b}}}\right)}^{-1} \]

      9. sqrt-unprod 0.0%

         \[\leadsto {\left(1.5 \cdot \frac{a}{\color{blue}{\sqrt{b} \cdot \sqrt{b}}}\right)}^{-1} \]

      10. add-sqr-sqrt 0.8%

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

   9. Applied egg-rr 0.8%

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

       1. unpow-1 0.8%

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

   11. Simplified 0.8%

       \[\leadsto \color{blue}{\frac{1}{1.5 \cdot \frac{a}{b}}} \]
   12. Step-by-step derivation

       1. associate-*r/ 0.8%

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

       2. add-sqr-sqrt 0.0%

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

       3. sqrt-unprod 67.8%

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

       4. sqr-neg 67.8%

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

       5. sqrt-unprod 94.5%

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

       6. add-sqr-sqrt 94.7%

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

       7. distribute-neg-frac2 94.7%

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

       8. associate-*r/ 94.8%

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

       9. neg-sub0 81.3%

          \[\leadsto \frac{1}{\color{blue}{0 - 1.5 \cdot \frac{a}{b}}} \]

   13. Applied egg-rr 81.3%

       \[\leadsto \frac{1}{\color{blue}{0 - 1.5 \cdot \frac{a}{b}}} \]
   14. Step-by-step derivation

       1. neg-sub0 94.8%

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

       2. *-commutative 94.8%

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

       3. distribute-rgt-neg-in 94.8%

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

       4. metadata-eval 94.8%

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

   15. Simplified 94.8%

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

   if -2.1500000000000002e79 < b < 1.9999999999999999e124

   1. Initial program 85.3%

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

      1. sqr-neg 85.3%

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

      2. sqr-neg 85.3%

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

      3. associate-*l* 85.2%

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

   3. Simplified 85.2%

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

   if 1.9999999999999999e124 < b

   1. Initial program 10.8%

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

      1. sqr-neg 10.8%

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

      2. sqr-neg 10.8%

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

      3. associate-*l* 13.2%

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

   3. Simplified 13.2%

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

   5. Taylor expanded in b around -inf 4.6%

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

      1. mul-1-neg 4.6%

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

   7. Simplified 4.6%

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

      1. unsub-neg 4.6%

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

      2. div-sub 4.6%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 10.3%

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

      5. sqr-neg 10.3%

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

      6. sqrt-unprod 43.4%

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

      7. add-sqr-sqrt 61.9%

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

      8. *-commutative 61.9%

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

      9. *-commutative 61.9%

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

   9. Applied egg-rr 61.9%

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

       1. +-inverses 100.0%

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

   11. Simplified 100.0%

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

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -2.15 \cdot 10^{+79}:\\ \;\;\;\;\frac{1}{\frac{a}{b} \cdot -1.5}\\ \mathbf{elif}\;b \leq 2 \cdot 10^{+124}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(a \cdot c\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 89.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.75 \cdot 10^{+79}:\\ \;\;\;\;\frac{1}{\frac{a}{b} \cdot -1.5}\\ \mathbf{elif}\;b \leq 5.5 \cdot 10^{+123}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.75e+79)
   (/ 1.0 (* (/ a b) -1.5))
   (if (<= b 5.5e+123)
     (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0))
     0.0)))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.75e+79) {
		tmp = 1.0 / ((a / b) * -1.5);
	} else if (b <= 5.5e+123) {
		tmp = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (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 <= (-1.75d+79)) then
        tmp = 1.0d0 / ((a / b) * (-1.5d0))
    else if (b <= 5.5d+123) then
        tmp = (sqrt(((b * b) - (c * (a * 3.0d0)))) - b) / (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 <= -1.75e+79) {
		tmp = 1.0 / ((a / b) * -1.5);
	} else if (b <= 5.5e+123) {
		tmp = (Math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.75e+79:
		tmp = 1.0 / ((a / b) * -1.5)
	elif b <= 5.5e+123:
		tmp = (math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)
	else:
		tmp = 0.0
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.75e+79)
		tmp = Float64(1.0 / Float64(Float64(a / b) * -1.5));
	elseif (b <= 5.5e+123)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.75e+79)
		tmp = 1.0 / ((a / b) * -1.5);
	elseif (b <= 5.5e+123)
		tmp = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.75e+79], N[(1.0 / N[(N[(a / b), $MachinePrecision] * -1.5), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 5.5e+123], 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], 0.0]]

Derivation

1. Split input into 3 regimes

2. if b < -1.7499999999999999e79

   1. Initial program 67.5%

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

      1. sqr-neg 67.5%

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

      2. sqr-neg 67.5%

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

      3. associate-*l* 67.5%

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

   3. Simplified 67.5%

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

   5. Taylor expanded in b around -inf 94.7%

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

      1. mul-1-neg 94.7%

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

   7. Simplified 94.7%

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

      1. clear-num 94.7%

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

      2. inv-pow 94.7%

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

      3. count-2 94.7%

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

      4. times-frac 94.8%

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

      5. metadata-eval 94.8%

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

      6. add-sqr-sqrt 94.5%

         \[\leadsto {\left(1.5 \cdot \frac{a}{\color{blue}{\sqrt{-b} \cdot \sqrt{-b}}}\right)}^{-1} \]

      7. sqrt-unprod 67.8%

         \[\leadsto {\left(1.5 \cdot \frac{a}{\color{blue}{\sqrt{\left(-b\right) \cdot \left(-b\right)}}}\right)}^{-1} \]

      8. sqr-neg 67.8%

         \[\leadsto {\left(1.5 \cdot \frac{a}{\sqrt{\color{blue}{b \cdot b}}}\right)}^{-1} \]

      9. sqrt-unprod 0.0%

         \[\leadsto {\left(1.5 \cdot \frac{a}{\color{blue}{\sqrt{b} \cdot \sqrt{b}}}\right)}^{-1} \]

      10. add-sqr-sqrt 0.8%

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

   9. Applied egg-rr 0.8%

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

       1. unpow-1 0.8%

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

   11. Simplified 0.8%

       \[\leadsto \color{blue}{\frac{1}{1.5 \cdot \frac{a}{b}}} \]
   12. Step-by-step derivation

       1. associate-*r/ 0.8%

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

       2. add-sqr-sqrt 0.0%

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

       3. sqrt-unprod 67.8%

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

       4. sqr-neg 67.8%

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

       5. sqrt-unprod 94.5%

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

       6. add-sqr-sqrt 94.7%

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

       7. distribute-neg-frac2 94.7%

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

       8. associate-*r/ 94.8%

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

       9. neg-sub0 81.3%

          \[\leadsto \frac{1}{\color{blue}{0 - 1.5 \cdot \frac{a}{b}}} \]

   13. Applied egg-rr 81.3%

       \[\leadsto \frac{1}{\color{blue}{0 - 1.5 \cdot \frac{a}{b}}} \]
   14. Step-by-step derivation

       1. neg-sub0 94.8%

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

       2. *-commutative 94.8%

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

       3. distribute-rgt-neg-in 94.8%

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

       4. metadata-eval 94.8%

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

   15. Simplified 94.8%

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

   if -1.7499999999999999e79 < b < 5.5000000000000002e123

   1. Initial program 85.3%

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

   if 5.5000000000000002e123 < b

   1. Initial program 10.8%

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

      1. sqr-neg 10.8%

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

      2. sqr-neg 10.8%

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

      3. associate-*l* 13.2%

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

   3. Simplified 13.2%

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

   5. Taylor expanded in b around -inf 4.6%

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

      1. mul-1-neg 4.6%

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

   7. Simplified 4.6%

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

      1. unsub-neg 4.6%

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

      2. div-sub 4.6%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 10.3%

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

      5. sqr-neg 10.3%

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

      6. sqrt-unprod 43.4%

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

      7. add-sqr-sqrt 61.9%

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

      8. *-commutative 61.9%

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

      9. *-commutative 61.9%

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

   9. Applied egg-rr 61.9%

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

       1. +-inverses 100.0%

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

   11. Simplified 100.0%

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

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.75 \cdot 10^{+79}:\\ \;\;\;\;\frac{1}{\frac{a}{b} \cdot -1.5}\\ \mathbf{elif}\;b \leq 5.5 \cdot 10^{+123}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 78.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5.4 \cdot 10^{-89}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 6 \cdot 10^{-173}:\\ \;\;\;\;\frac{\sqrt{-3 \cdot \left(a \cdot c\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(b + -1.5 \cdot \left(a \cdot \frac{c}{b}\right)\right) - b}{a \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5.4e-89)
   (+ (* -0.6666666666666666 (/ b a)) (* 0.5 (/ c b)))
   (if (<= b 6e-173)
     (/ (- (sqrt (* -3.0 (* a c))) 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 <= -5.4e-89) {
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	} else if (b <= 6e-173) {
		tmp = (sqrt((-3.0 * (a * c))) - 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 <= (-5.4d-89)) then
        tmp = ((-0.6666666666666666d0) * (b / a)) + (0.5d0 * (c / b))
    else if (b <= 6d-173) then
        tmp = (sqrt(((-3.0d0) * (a * c))) - 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 <= -5.4e-89) {
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	} else if (b <= 6e-173) {
		tmp = (Math.sqrt((-3.0 * (a * c))) - 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 <= -5.4e-89:
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b))
	elif b <= 6e-173:
		tmp = (math.sqrt((-3.0 * (a * c))) - 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 <= -5.4e-89)
		tmp = Float64(Float64(-0.6666666666666666 * Float64(b / a)) + Float64(0.5 * Float64(c / b)));
	elseif (b <= 6e-173)
		tmp = Float64(Float64(sqrt(Float64(-3.0 * Float64(a * c))) - 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 <= -5.4e-89)
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	elseif (b <= 6e-173)
		tmp = (sqrt((-3.0 * (a * c))) - 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, -5.4e-89], N[(N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 6e-173], N[(N[(N[Sqrt[N[(-3.0 * N[(a * c), $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 < -5.39999999999999975e-89

   1. Initial program 79.2%

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

      1. sqr-neg 79.2%

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

      2. sqr-neg 79.2%

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

      3. associate-*l* 79.2%

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

   3. Simplified 79.2%

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

   5. Taylor expanded in b around -inf 85.8%

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

   if -5.39999999999999975e-89 < b < 6.0000000000000002e-173

   1. Initial program 76.3%

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

      1. sqr-neg 76.3%

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

      2. sqr-neg 76.3%

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

      3. associate-*l* 76.2%

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

   3. Simplified 76.2%

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

   5. Taylor expanded in b around 0 72.0%

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

   if 6.0000000000000002e-173 < b

   1. Initial program 48.6%

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

      1. sqr-neg 48.6%

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

      2. sqr-neg 48.6%

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

      3. associate-*l* 49.7%

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

   3. Simplified 49.7%

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

   5. Taylor expanded in b around inf 79.0%

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

      1. associate-/l* 81.2%

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

   7. Simplified 81.2%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.4 \cdot 10^{-89}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 6 \cdot 10^{-173}:\\ \;\;\;\;\frac{\sqrt{-3 \cdot \left(a \cdot c\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(b + -1.5 \cdot \left(a \cdot \frac{c}{b}\right)\right) - b}{a \cdot 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 78.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -5.4 \cdot 10^{-89}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 6 \cdot 10^{-173}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(b + -1.5 \cdot \left(a \cdot \frac{c}{b}\right)\right) - b}{a \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -5.4e-89)
   (+ (* -0.6666666666666666 (/ b a)) (* 0.5 (/ c b)))
   (if (<= b 6e-173)
     (/ (- (sqrt (* 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 <= -5.4e-89) {
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	} else if (b <= 6e-173) {
		tmp = (sqrt((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 <= (-5.4d-89)) then
        tmp = ((-0.6666666666666666d0) * (b / a)) + (0.5d0 * (c / b))
    else if (b <= 6d-173) then
        tmp = (sqrt((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 <= -5.4e-89) {
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	} else if (b <= 6e-173) {
		tmp = (Math.sqrt((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 <= -5.4e-89:
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b))
	elif b <= 6e-173:
		tmp = (math.sqrt((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 <= -5.4e-89)
		tmp = Float64(Float64(-0.6666666666666666 * Float64(b / a)) + Float64(0.5 * Float64(c / b)));
	elseif (b <= 6e-173)
		tmp = Float64(Float64(sqrt(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 <= -5.4e-89)
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	elseif (b <= 6e-173)
		tmp = (sqrt((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, -5.4e-89], N[(N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[b, 6e-173], N[(N[(N[Sqrt[N[(c * N[(a * -3.0), $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 < -5.39999999999999975e-89

   1. Initial program 79.2%

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

      1. sqr-neg 79.2%

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

      2. sqr-neg 79.2%

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

      3. associate-*l* 79.2%

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

   3. Simplified 79.2%

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

   5. Taylor expanded in b around -inf 85.8%

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

   if -5.39999999999999975e-89 < b < 6.0000000000000002e-173

   1. Initial program 76.3%

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

      1. sqr-neg 76.3%

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

      2. sqr-neg 76.3%

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

      3. associate-*l* 76.2%

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

   3. Simplified 76.2%

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

   5. Taylor expanded in b around 0 72.0%

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

      1. associate-*r* 72.2%

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

      2. *-commutative 72.2%

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

   7. Simplified 72.2%

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

   if 6.0000000000000002e-173 < b

   1. Initial program 48.6%

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

      1. sqr-neg 48.6%

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

      2. sqr-neg 48.6%

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

      3. associate-*l* 49.7%

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

   3. Simplified 49.7%

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

   5. Taylor expanded in b around inf 79.0%

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

      1. associate-/l* 81.2%

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

   7. Simplified 81.2%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -5.4 \cdot 10^{-89}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{elif}\;b \leq 6 \cdot 10^{-173}:\\ \;\;\;\;\frac{\sqrt{c \cdot \left(a \cdot -3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(b + -1.5 \cdot \left(a \cdot \frac{c}{b}\right)\right) - b}{a \cdot 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 67.2% accurate, 5.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{-310}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(b + -1.5 \cdot \left(a \cdot \frac{c}{b}\right)\right) - b}{a \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1e-310)
   (+ (* -0.6666666666666666 (/ b a)) (* 0.5 (/ c b)))
   (/ (- (+ b (* -1.5 (* a (/ c b)))) b) (* a 3.0))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e-310) {
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	} 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 <= (-1d-310)) then
        tmp = ((-0.6666666666666666d0) * (b / a)) + (0.5d0 * (c / b))
    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 <= -1e-310) {
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	} else {
		tmp = ((b + (-1.5 * (a * (c / b)))) - b) / (a * 3.0);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1e-310:
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b))
	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 <= -1e-310)
		tmp = Float64(Float64(-0.6666666666666666 * Float64(b / a)) + Float64(0.5 * Float64(c / b)));
	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 <= -1e-310)
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	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, -1e-310], N[(N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b), $MachinePrecision]), $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 < -9.999999999999969e-311

   1. Initial program 78.6%

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

      1. sqr-neg 78.6%

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

      2. sqr-neg 78.6%

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

      3. associate-*l* 78.5%

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

   3. Simplified 78.5%

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

   5. Taylor expanded in b around -inf 70.3%

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

   if -9.999999999999969e-311 < b

   1. Initial program 53.1%

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

      1. sqr-neg 53.1%

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

      2. sqr-neg 53.1%

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

      3. associate-*l* 54.0%

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

   3. Simplified 54.0%

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

   5. Taylor expanded in b around inf 68.0%

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

      1. associate-/l* 69.9%

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

   7. Simplified 69.9%

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

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{-310}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(b + -1.5 \cdot \left(a \cdot \frac{c}{b}\right)\right) - b}{a \cdot 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 66.6% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.18 \cdot 10^{-243}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.18e-243)
   (+ (* -0.6666666666666666 (/ b a)) (* 0.5 (/ c b)))
   0.0))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.18e-243) {
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / 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 <= (-1.18d-243)) then
        tmp = ((-0.6666666666666666d0) * (b / a)) + (0.5d0 * (c / 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 <= -1.18e-243) {
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.18e-243:
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b))
	else:
		tmp = 0.0
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.18e-243)
		tmp = Float64(Float64(-0.6666666666666666 * Float64(b / a)) + Float64(0.5 * Float64(c / b)));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1.18e-243)
		tmp = (-0.6666666666666666 * (b / a)) + (0.5 * (c / b));
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.18e-243], N[(N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if b < -1.1800000000000001e-243

   1. Initial program 78.9%

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

      1. sqr-neg 78.9%

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

      2. sqr-neg 78.9%

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

      3. associate-*l* 78.8%

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

   3. Simplified 78.8%

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

   5. Taylor expanded in b around -inf 73.6%

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

   if -1.1800000000000001e-243 < b

   1. Initial program 54.2%

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

      1. sqr-neg 54.2%

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

      2. sqr-neg 54.2%

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

      3. associate-*l* 55.1%

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

   3. Simplified 55.1%

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

   5. Taylor expanded in b around -inf 3.6%

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

      1. mul-1-neg 3.6%

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

   7. Simplified 3.6%

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

      1. unsub-neg 3.6%

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

      2. div-sub 3.6%

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

      3. add-sqr-sqrt 0.3%

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

      4. sqrt-unprod 25.4%

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

      5. sqr-neg 25.4%

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

      6. sqrt-unprod 29.6%

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

      7. add-sqr-sqrt 47.8%

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

      8. *-commutative 47.8%

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

      9. *-commutative 47.8%

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

   9. Applied egg-rr 47.8%

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

       1. +-inverses 65.3%

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

   11. Simplified 65.3%

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

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.18 \cdot 10^{-243}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 66.4% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\frac{1}{\frac{a}{b} \cdot -1.5}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1e-310) (/ 1.0 (* (/ a b) -1.5)) 0.0))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1e-310) {
		tmp = 1.0 / ((a / b) * -1.5);
	} 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 <= (-1d-310)) then
        tmp = 1.0d0 / ((a / b) * (-1.5d0))
    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 <= -1e-310) {
		tmp = 1.0 / ((a / b) * -1.5);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1e-310:
		tmp = 1.0 / ((a / b) * -1.5)
	else:
		tmp = 0.0
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1e-310)
		tmp = Float64(1.0 / Float64(Float64(a / b) * -1.5));
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= -1e-310)
		tmp = 1.0 / ((a / b) * -1.5);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1e-310], N[(1.0 / N[(N[(a / b), $MachinePrecision] * -1.5), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if b < -9.999999999999969e-311

   1. Initial program 78.6%

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

      1. sqr-neg 78.6%

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

      2. sqr-neg 78.6%

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

      3. associate-*l* 78.5%

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

   3. Simplified 78.5%

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

   5. Taylor expanded in b around -inf 69.9%

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

      1. mul-1-neg 69.9%

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

   7. Simplified 69.9%

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

      1. clear-num 69.8%

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

      2. inv-pow 69.8%

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

      3. count-2 69.8%

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

      4. times-frac 70.0%

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

      5. metadata-eval 70.0%

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

      6. add-sqr-sqrt 69.7%

         \[\leadsto {\left(1.5 \cdot \frac{a}{\color{blue}{\sqrt{-b} \cdot \sqrt{-b}}}\right)}^{-1} \]

      7. sqrt-unprod 55.8%

         \[\leadsto {\left(1.5 \cdot \frac{a}{\color{blue}{\sqrt{\left(-b\right) \cdot \left(-b\right)}}}\right)}^{-1} \]

      8. sqr-neg 55.8%

         \[\leadsto {\left(1.5 \cdot \frac{a}{\sqrt{\color{blue}{b \cdot b}}}\right)}^{-1} \]

      9. sqrt-unprod 0.0%

         \[\leadsto {\left(1.5 \cdot \frac{a}{\color{blue}{\sqrt{b} \cdot \sqrt{b}}}\right)}^{-1} \]

      10. add-sqr-sqrt 1.5%

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

   9. Applied egg-rr 1.5%

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

       1. unpow-1 1.5%

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

   11. Simplified 1.5%

       \[\leadsto \color{blue}{\frac{1}{1.5 \cdot \frac{a}{b}}} \]
   12. Step-by-step derivation

       1. associate-*r/ 1.5%

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

       2. add-sqr-sqrt 0.0%

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

       3. sqrt-unprod 55.7%

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

       4. sqr-neg 55.7%

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

       5. sqrt-unprod 69.7%

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

       6. add-sqr-sqrt 69.8%

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

       7. distribute-neg-frac2 69.8%

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

       8. associate-*r/ 70.0%

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

       9. neg-sub0 62.6%

          \[\leadsto \frac{1}{\color{blue}{0 - 1.5 \cdot \frac{a}{b}}} \]

   13. Applied egg-rr 62.6%

       \[\leadsto \frac{1}{\color{blue}{0 - 1.5 \cdot \frac{a}{b}}} \]
   14. Step-by-step derivation

       1. neg-sub0 70.0%

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

       2. *-commutative 70.0%

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

       3. distribute-rgt-neg-in 70.0%

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

       4. metadata-eval 70.0%

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

   15. Simplified 70.0%

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

   if -9.999999999999969e-311 < b

   1. Initial program 53.1%

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

      1. sqr-neg 53.1%

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

      2. sqr-neg 53.1%

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

      3. associate-*l* 54.0%

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

   3. Simplified 54.0%

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

   5. Taylor expanded in b around -inf 3.5%

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

      1. mul-1-neg 3.5%

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

   7. Simplified 3.5%

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

      1. unsub-neg 3.5%

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

      2. div-sub 3.5%

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

      3. add-sqr-sqrt 0.0%

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

      4. sqrt-unprod 26.7%

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

      5. sqr-neg 26.7%

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

      6. sqrt-unprod 31.6%

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

      7. add-sqr-sqrt 50.7%

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

      8. *-commutative 50.7%

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

      9. *-commutative 50.7%

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

   9. Applied egg-rr 50.7%

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

       1. +-inverses 69.4%

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

   11. Simplified 69.4%

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

4. Final simplification 69.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1 \cdot 10^{-310}:\\ \;\;\;\;\frac{1}{\frac{a}{b} \cdot -1.5}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 66.4% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.95 \cdot 10^{-298}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.95e-298) (* -0.6666666666666666 (/ b a)) 0.0))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.95e-298) {
		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 <= (-1.95d-298)) 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 <= -1.95e-298) {
		tmp = -0.6666666666666666 * (b / a);
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.95e-298:
		tmp = -0.6666666666666666 * (b / a)
	else:
		tmp = 0.0
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.95e-298)
		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 <= -1.95e-298)
		tmp = -0.6666666666666666 * (b / a);
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.95e-298], N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if b < -1.95000000000000014e-298

   1. Initial program 79.6%

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

      1. sqr-neg 79.6%

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

      2. sqr-neg 79.6%

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

      3. associate-*l* 79.5%

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

   3. Simplified 79.5%

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

   5. Taylor expanded in b around -inf 70.7%

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

      1. *-commutative 70.7%

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

   7. Simplified 70.7%

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

   if -1.95000000000000014e-298 < b

   1. Initial program 52.2%

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

      1. sqr-neg 52.2%

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

      2. sqr-neg 52.2%

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

      3. associate-*l* 53.1%

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

   3. Simplified 53.1%

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

   5. Taylor expanded in b around -inf 3.5%

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

      1. mul-1-neg 3.5%

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

   7. Simplified 3.5%

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

      1. unsub-neg 3.5%

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

      2. div-sub 3.5%

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

      3. add-sqr-sqrt 0.1%

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

      4. sqrt-unprod 26.3%

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

      5. sqr-neg 26.3%

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

      6. sqrt-unprod 31.0%

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

      7. add-sqr-sqrt 49.8%

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

      8. *-commutative 49.8%

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

      9. *-commutative 49.8%

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

   9. Applied egg-rr 49.8%

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

       1. +-inverses 68.1%

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

   11. Simplified 68.1%

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

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.95 \cdot 10^{-298}:\\ \;\;\;\;-0.6666666666666666 \cdot \frac{b}{a}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 66.4% accurate, 11.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq -1.95 \cdot 10^{-298}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b -1.95e-298) (/ (* b -0.6666666666666666) a) 0.0))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= -1.95e-298) {
		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 <= (-1.95d-298)) 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 <= -1.95e-298) {
		tmp = (b * -0.6666666666666666) / a;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= -1.95e-298:
		tmp = (b * -0.6666666666666666) / a
	else:
		tmp = 0.0
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= -1.95e-298)
		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 <= -1.95e-298)
		tmp = (b * -0.6666666666666666) / a;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, -1.95e-298], N[(N[(b * -0.6666666666666666), $MachinePrecision] / a), $MachinePrecision], 0.0]

Derivation

1. Split input into 2 regimes

2. if b < -1.95000000000000014e-298

   1. Initial program 79.6%

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

      1. sqr-neg 79.6%

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

      2. sqr-neg 79.6%

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

      3. associate-*l* 79.5%

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

   3. Simplified 79.5%

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

   6. Applied egg-rr 68.8%

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

      1. associate-*l/ 68.7%

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

      2. un-div-inv 68.7%

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

   8. Applied egg-rr 68.7%

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

      1. clear-num 68.7%

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

      2. inv-pow 68.7%

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

   10. Applied egg-rr 68.7%

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

       1. unpow-1 68.7%

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

   12. Simplified 68.7%

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

   13. Taylor expanded in b around -inf 70.7%

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

       1. associate-*r/ 70.8%

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

   15. Simplified 70.8%

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

   if -1.95000000000000014e-298 < b

   1. Initial program 52.2%

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

      1. sqr-neg 52.2%

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

      2. sqr-neg 52.2%

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

      3. associate-*l* 53.1%

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

   3. Simplified 53.1%

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

   5. Taylor expanded in b around -inf 3.5%

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

      1. mul-1-neg 3.5%

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

   7. Simplified 3.5%

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

      1. unsub-neg 3.5%

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

      2. div-sub 3.5%

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

      3. add-sqr-sqrt 0.1%

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

      4. sqrt-unprod 26.3%

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

      5. sqr-neg 26.3%

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

      6. sqrt-unprod 31.0%

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

      7. add-sqr-sqrt 49.8%

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

      8. *-commutative 49.8%

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

      9. *-commutative 49.8%

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

   9. Applied egg-rr 49.8%

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

       1. +-inverses 68.1%

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

   11. Simplified 68.1%

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

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq -1.95 \cdot 10^{-298}:\\ \;\;\;\;\frac{b \cdot -0.6666666666666666}{a}\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 33.8% 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 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* 68.3%

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

3. Simplified 68.3%

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

5. Taylor expanded in b around -inf 42.2%

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

   1. mul-1-neg 42.2%

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

7. Simplified 42.2%

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

   1. unsub-neg 42.2%

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

   2. div-sub 42.2%

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

   3. add-sqr-sqrt 40.6%

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

   4. sqrt-unprod 43.7%

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

   5. sqr-neg 43.7%

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

   6. sqrt-unprod 13.2%

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

   7. add-sqr-sqrt 22.6%

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

   8. *-commutative 22.6%

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

   9. *-commutative 22.6%

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

9. Applied egg-rr 22.6%

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

    1. +-inverses 30.7%

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

11. Simplified 30.7%

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

12. Final simplification 30.7%

    \[\leadsto 0 \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024047 
(FPCore (a b c)
  :name "Cubic critical"
  :precision binary64
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))