Cubic critical, narrow range

Percentage Accurate: 55.2% → 99.3%

Time: 29.8s

Alternatives: 8

Speedup: 23.2×

Specification

\[\left(\left(1.0536712127723509 \cdot 10^{-8} < a \land a < 94906265.62425156\right) \land \left(1.0536712127723509 \cdot 10^{-8} < b \land b < 94906265.62425156\right)\right) \land \left(1.0536712127723509 \cdot 10^{-8} < c \land c < 94906265.62425156\right)\]

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: 55.2% 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: 99.3% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 52.9%

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

   1. add-sqr-sqrt 52.0%

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

   2. distribute-rgt-neg-in 52.0%

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

4. Applied egg-rr 52.0%

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

   1. flip-+ 52.1%

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

   2. pow2 52.0%

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

   3. distribute-rgt-neg-out 52.0%

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

   4. add-sqr-sqrt 52.9%

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

   5. add-sqr-sqrt 54.4%

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

   6. pow2 54.4%

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

   7. *-commutative 54.4%

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

   8. *-commutative 54.4%

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

   9. distribute-rgt-neg-out 54.4%

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

   10. add-sqr-sqrt 54.4%

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

6. Applied egg-rr 54.4%

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

   1. associate--r- 99.3%

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

   2. unpow2 99.1%

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

   3. unpow2 99.3%

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

   4. difference-of-squares 99.3%

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

   5. neg-mul-1 99.3%

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

   6. distribute-lft1-in 99.3%

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

   7. metadata-eval 99.3%

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

   8. mul0-lft 99.3%

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

   9. unpow2 99.3%

      \[\leadsto \frac{\frac{0 \cdot \left(\left(-b\right) - b\right) + c \cdot \left(a \cdot 3\right)}{\left(-b\right) - \sqrt{\color{blue}{b \cdot b} - c \cdot \left(a \cdot 3\right)}}}{3 \cdot a} \]

   10. fma-neg 99.3%

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

   11. associate-*r* 99.3%

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

   12. *-commutative 99.3%

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

   13. distribute-rgt-neg-in 99.3%

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

   14. metadata-eval 99.3%

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

8. Simplified 99.3%

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

9. Taylor expanded in a around 0 99.3%

   \[\leadsto \frac{\frac{0 \cdot \left(\left(-b\right) - b\right) + c \cdot \left(a \cdot 3\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{-3 \cdot \left(a \cdot c\right)}\right)}}}{3 \cdot a} \]
10. Step-by-step derivation

    1. associate-*r* 99.3%

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

    2. *-commutative 99.3%

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

    3. *-commutative 99.3%

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

11. Simplified 99.3%

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

12. Taylor expanded in b around 0 99.1%

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

    1. *-commutative 99.1%

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

    2. *-commutative 99.1%

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

    3. associate-*r* 99.3%

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

    4. *-commutative 99.3%

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

14. Simplified 99.3%

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

Alternative 2: 85.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - b}{3 \cdot a} \leq -0.095:\\ \;\;\;\;\frac{1}{\frac{3 \cdot a}{\sqrt{\mathsf{fma}\left(b, b, -3 \cdot \left(c \cdot a\right)\right)} - b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{b \cdot -2 + 1.5 \cdot \frac{c \cdot a}{b}}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* 3.0 a)))) b) (* 3.0 a)) -0.095)
   (/ 1.0 (/ (* 3.0 a) (- (sqrt (fma b b (* -3.0 (* c a)))) b)))
   (/ 1.0 (/ (+ (* b -2.0) (* 1.5 (/ (* c a) b))) c))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (3.0 * a)))) - b) / (3.0 * a)) <= -0.095) {
		tmp = 1.0 / ((3.0 * a) / (sqrt(fma(b, b, (-3.0 * (c * a)))) - b));
	} else {
		tmp = 1.0 / (((b * -2.0) + (1.5 * ((c * a) / b))) / c);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(3.0 * a)))) - b) / Float64(3.0 * a)) <= -0.095)
		tmp = Float64(1.0 / Float64(Float64(3.0 * a) / Float64(sqrt(fma(b, b, Float64(-3.0 * Float64(c * a)))) - b)));
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(b * -2.0) + Float64(1.5 * Float64(Float64(c * a) / b))) / c));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(3.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -0.095], N[(1.0 / N[(N[(3.0 * a), $MachinePrecision] / N[(N[Sqrt[N[(b * b + N[(-3.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(b * -2.0), $MachinePrecision] + N[(1.5 * N[(N[(c * a), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.095000000000000001

   1. Initial program 82.7%

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

      1. add-sqr-sqrt 81.0%

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

      2. distribute-rgt-neg-in 81.0%

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

   4. Applied egg-rr 81.0%

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

      1. clear-num 81.1%

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

      2. inv-pow 81.1%

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

      3. *-commutative 81.1%

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

      4. distribute-rgt-neg-out 81.1%

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

      5. add-sqr-sqrt 82.7%

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

      6. pow2 82.7%

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

      7. *-commutative 82.7%

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

      8. *-commutative 82.7%

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

   6. Applied egg-rr 82.7%

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

      1. unpow-1 82.7%

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

      2. +-commutative 82.7%

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

      3. unsub-neg 82.7%

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

      4. unpow2 82.7%

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

      5. fma-neg 82.7%

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

      6. associate-*r* 82.7%

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

      7. *-commutative 82.7%

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

      8. distribute-rgt-neg-in 82.7%

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

      9. metadata-eval 82.7%

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

   8. Simplified 82.7%

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

   if -0.095000000000000001 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 45.1%

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

      1. add-sqr-sqrt 44.5%

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

      2. distribute-rgt-neg-in 44.5%

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

   4. Applied egg-rr 44.5%

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

      1. clear-num 44.5%

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

      2. inv-pow 44.5%

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

      3. *-commutative 44.5%

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

      4. distribute-rgt-neg-out 44.5%

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

      5. add-sqr-sqrt 45.1%

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

      6. pow2 45.1%

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

      7. *-commutative 45.1%

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

      8. *-commutative 45.1%

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

   6. Applied egg-rr 45.1%

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

      1. unpow-1 45.1%

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

      2. +-commutative 45.1%

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

      3. unsub-neg 45.1%

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

      4. unpow2 45.1%

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

      5. fma-neg 45.3%

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

      6. associate-*r* 45.3%

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

      7. *-commutative 45.3%

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

      8. distribute-rgt-neg-in 45.3%

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

      9. metadata-eval 45.3%

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

   8. Simplified 45.3%

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

   9. Taylor expanded in c around 0 89.7%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - b}{3 \cdot a} \leq -0.095:\\ \;\;\;\;\frac{1}{\frac{3 \cdot a}{\sqrt{\mathsf{fma}\left(b, b, -3 \cdot \left(c \cdot a\right)\right)} - b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{b \cdot -2 + 1.5 \cdot \frac{c \cdot a}{b}}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - b}{3 \cdot a}\\ \mathbf{if}\;t\_0 \leq -0.095:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{b \cdot -2 + 1.5 \cdot \frac{c \cdot a}{b}}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* c (* 3.0 a)))) b) (* 3.0 a))))
   (if (<= t_0 -0.095)
     t_0
     (/ 1.0 (/ (+ (* b -2.0) (* 1.5 (/ (* c a) b))) c)))))

C

double code(double a, double b, double c) {
	double t_0 = (sqrt(((b * b) - (c * (3.0 * a)))) - b) / (3.0 * a);
	double tmp;
	if (t_0 <= -0.095) {
		tmp = t_0;
	} else {
		tmp = 1.0 / (((b * -2.0) + (1.5 * ((c * a) / b))) / c);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (sqrt(((b * b) - (c * (3.0d0 * a)))) - b) / (3.0d0 * a)
    if (t_0 <= (-0.095d0)) then
        tmp = t_0
    else
        tmp = 1.0d0 / (((b * (-2.0d0)) + (1.5d0 * ((c * a) / b))) / c)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (Math.sqrt(((b * b) - (c * (3.0 * a)))) - b) / (3.0 * a);
	double tmp;
	if (t_0 <= -0.095) {
		tmp = t_0;
	} else {
		tmp = 1.0 / (((b * -2.0) + (1.5 * ((c * a) / b))) / c);
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (math.sqrt(((b * b) - (c * (3.0 * a)))) - b) / (3.0 * a)
	tmp = 0
	if t_0 <= -0.095:
		tmp = t_0
	else:
		tmp = 1.0 / (((b * -2.0) + (1.5 * ((c * a) / b))) / c)
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(3.0 * a)))) - b) / Float64(3.0 * a))
	tmp = 0.0
	if (t_0 <= -0.095)
		tmp = t_0;
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(b * -2.0) + Float64(1.5 * Float64(Float64(c * a) / b))) / c));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (sqrt(((b * b) - (c * (3.0 * a)))) - b) / (3.0 * a);
	tmp = 0.0;
	if (t_0 <= -0.095)
		tmp = t_0;
	else
		tmp = 1.0 / (((b * -2.0) + (1.5 * ((c * a) / b))) / c);
	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[(3.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -0.095], t$95$0, N[(1.0 / N[(N[(N[(b * -2.0), $MachinePrecision] + N[(1.5 * N[(N[(c * a), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.095000000000000001

   1. Initial program 82.7%

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

   if -0.095000000000000001 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 45.1%

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

      1. add-sqr-sqrt 44.5%

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

      2. distribute-rgt-neg-in 44.5%

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

   4. Applied egg-rr 44.5%

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

      1. clear-num 44.5%

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

      2. inv-pow 44.5%

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

      3. *-commutative 44.5%

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

      4. distribute-rgt-neg-out 44.5%

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

      5. add-sqr-sqrt 45.1%

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

      6. pow2 45.1%

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

      7. *-commutative 45.1%

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

      8. *-commutative 45.1%

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

   6. Applied egg-rr 45.1%

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

      1. unpow-1 45.1%

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

      2. +-commutative 45.1%

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

      3. unsub-neg 45.1%

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

      4. unpow2 45.1%

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

      5. fma-neg 45.3%

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

      6. associate-*r* 45.3%

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

      7. *-commutative 45.3%

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

      8. distribute-rgt-neg-in 45.3%

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

      9. metadata-eval 45.3%

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

   8. Simplified 45.3%

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

   9. Taylor expanded in c around 0 89.7%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - b}{3 \cdot a} \leq -0.095:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{b \cdot -2 + 1.5 \cdot \frac{c \cdot a}{b}}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.1% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 52.9%

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

   1. add-sqr-sqrt 52.0%

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

   2. distribute-rgt-neg-in 52.0%

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

4. Applied egg-rr 52.0%

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

   1. flip-+ 52.1%

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

   2. pow2 52.0%

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

   3. distribute-rgt-neg-out 52.0%

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

   4. add-sqr-sqrt 52.9%

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

   5. add-sqr-sqrt 54.4%

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

   6. pow2 54.4%

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

   7. *-commutative 54.4%

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

   8. *-commutative 54.4%

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

   9. distribute-rgt-neg-out 54.4%

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

   10. add-sqr-sqrt 54.4%

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

6. Applied egg-rr 54.4%

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

   1. associate--r- 99.3%

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

   2. unpow2 99.1%

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

   3. unpow2 99.3%

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

   4. difference-of-squares 99.3%

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

   5. neg-mul-1 99.3%

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

   6. distribute-lft1-in 99.3%

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

   7. metadata-eval 99.3%

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

   8. mul0-lft 99.3%

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

   9. unpow2 99.3%

      \[\leadsto \frac{\frac{0 \cdot \left(\left(-b\right) - b\right) + c \cdot \left(a \cdot 3\right)}{\left(-b\right) - \sqrt{\color{blue}{b \cdot b} - c \cdot \left(a \cdot 3\right)}}}{3 \cdot a} \]

   10. fma-neg 99.3%

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

   11. associate-*r* 99.3%

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

   12. *-commutative 99.3%

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

   13. distribute-rgt-neg-in 99.3%

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

   14. metadata-eval 99.3%

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

8. Simplified 99.3%

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

9. Taylor expanded in a around 0 99.3%

   \[\leadsto \frac{\frac{0 \cdot \left(\left(-b\right) - b\right) + c \cdot \left(a \cdot 3\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{-3 \cdot \left(a \cdot c\right)}\right)}}}{3 \cdot a} \]
10. Step-by-step derivation

    1. associate-*r* 99.3%

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

    2. *-commutative 99.3%

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

    3. *-commutative 99.3%

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

11. Simplified 99.3%

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

12. Taylor expanded in b around 0 99.1%

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

13. Final simplification 99.1%

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

Alternative 5: 82.2% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{b \cdot -2 + 1.5 \cdot \frac{c \cdot a}{b}}{c}} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ 1.0 (/ (+ (* b -2.0) (* 1.5 (/ (* c a) b))) c)))

C

double code(double a, double b, double c) {
	return 1.0 / (((b * -2.0) + (1.5 * ((c * a) / b))) / c);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = 1.0d0 / (((b * (-2.0d0)) + (1.5d0 * ((c * a) / b))) / c)
end function

Java

public static double code(double a, double b, double c) {
	return 1.0 / (((b * -2.0) + (1.5 * ((c * a) / b))) / c);
}

Python

def code(a, b, c):
	return 1.0 / (((b * -2.0) + (1.5 * ((c * a) / b))) / c)

Julia

function code(a, b, c)
	return Float64(1.0 / Float64(Float64(Float64(b * -2.0) + Float64(1.5 * Float64(Float64(c * a) / b))) / c))
end

MATLAB

function tmp = code(a, b, c)
	tmp = 1.0 / (((b * -2.0) + (1.5 * ((c * a) / b))) / c);
end

Wolfram

code[a_, b_, c_] := N[(1.0 / N[(N[(N[(b * -2.0), $MachinePrecision] + N[(1.5 * N[(N[(c * a), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.9%

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

   1. add-sqr-sqrt 52.0%

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

   2. distribute-rgt-neg-in 52.0%

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

4. Applied egg-rr 52.0%

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

   1. clear-num 52.0%

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

   2. inv-pow 52.0%

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

   3. *-commutative 52.0%

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

   4. distribute-rgt-neg-out 52.0%

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

   5. add-sqr-sqrt 52.9%

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

   6. pow2 52.9%

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

   7. *-commutative 52.9%

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

   8. *-commutative 52.9%

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

6. Applied egg-rr 52.9%

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

   1. unpow-1 52.9%

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

   2. +-commutative 52.9%

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

   3. unsub-neg 52.9%

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

   4. unpow2 52.9%

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

   5. fma-neg 53.0%

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

   6. associate-*r* 53.0%

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

   7. *-commutative 53.0%

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

   8. distribute-rgt-neg-in 53.0%

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

   9. metadata-eval 53.0%

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

8. Simplified 53.0%

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

9. Taylor expanded in c around 0 83.7%

   \[\leadsto \frac{1}{\color{blue}{\frac{-2 \cdot b + 1.5 \cdot \frac{a \cdot c}{b}}{c}}} \]

10. Final simplification 83.7%

    \[\leadsto \frac{1}{\frac{b \cdot -2 + 1.5 \cdot \frac{c \cdot a}{b}}{c}} \]
11. Add Preprocessing

Alternative 6: 82.2% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \frac{1}{-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ 1.0 (+ (* -2.0 (/ b c)) (* 1.5 (/ a b)))))

C

double code(double a, double b, double c) {
	return 1.0 / ((-2.0 * (b / c)) + (1.5 * (a / b)));
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = 1.0d0 / (((-2.0d0) * (b / c)) + (1.5d0 * (a / b)))
end function

Java

public static double code(double a, double b, double c) {
	return 1.0 / ((-2.0 * (b / c)) + (1.5 * (a / b)));
}

Python

def code(a, b, c):
	return 1.0 / ((-2.0 * (b / c)) + (1.5 * (a / b)))

Julia

function code(a, b, c)
	return Float64(1.0 / Float64(Float64(-2.0 * Float64(b / c)) + Float64(1.5 * Float64(a / b))))
end

MATLAB

function tmp = code(a, b, c)
	tmp = 1.0 / ((-2.0 * (b / c)) + (1.5 * (a / b)));
end

Wolfram

code[a_, b_, c_] := N[(1.0 / N[(N[(-2.0 * N[(b / c), $MachinePrecision]), $MachinePrecision] + N[(1.5 * N[(a / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 52.9%

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

   1. add-sqr-sqrt 52.0%

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

   2. distribute-rgt-neg-in 52.0%

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

4. Applied egg-rr 52.0%

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

   1. clear-num 52.0%

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

   2. inv-pow 52.0%

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

   3. *-commutative 52.0%

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

   4. distribute-rgt-neg-out 52.0%

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

   5. add-sqr-sqrt 52.9%

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

   6. pow2 52.9%

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

   7. *-commutative 52.9%

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

   8. *-commutative 52.9%

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

6. Applied egg-rr 52.9%

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

   1. unpow-1 52.9%

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

   2. +-commutative 52.9%

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

   3. unsub-neg 52.9%

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

   4. unpow2 52.9%

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

   5. fma-neg 53.0%

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

   6. associate-*r* 53.0%

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

   7. *-commutative 53.0%

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

   8. distribute-rgt-neg-in 53.0%

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

   9. metadata-eval 53.0%

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

8. Simplified 53.0%

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

9. Taylor expanded in a around 0 83.7%

   \[\leadsto \frac{1}{\color{blue}{-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}}} \]
10. Add Preprocessing

Alternative 7: 64.6% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.9%

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

   1. /-rgt-identity 52.9%

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

   2. metadata-eval 52.9%

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

3. Simplified 53.0%

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

5. Taylor expanded in b around inf 66.4%

   \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
6. Add Preprocessing

Alternative 8: 3.2% accurate, 38.7× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = 0.0d0 / a
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 52.9%

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

   1. add-sqr-sqrt 52.0%

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

   2. distribute-rgt-neg-in 52.0%

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

4. Applied egg-rr 52.0%

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

5. Taylor expanded in a around 0 3.2%

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

   1. associate-*r/ 3.2%

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

   2. distribute-rgt1-in 3.2%

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

   3. metadata-eval 3.2%

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

   4. mul0-lft 3.2%

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

   5. metadata-eval 3.2%

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

7. Simplified 3.2%

   \[\leadsto \color{blue}{\frac{0}{a}} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024143 
(FPCore (a b c)
  :name "Cubic critical, narrow range"
  :precision binary64
  :pre (and (and (and (< 1.0536712127723509e-8 a) (< a 94906265.62425156)) (and (< 1.0536712127723509e-8 b) (< b 94906265.62425156))) (and (< 1.0536712127723509e-8 c) (< c 94906265.62425156)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))