Cubic critical, wide range

Percentage Accurate: 17.7% → 99.4%

Time: 13.9s

Alternatives: 8

Speedup: 23.2×

Specification

\[\left(\left(4.930380657631324 \cdot 10^{-32} < a \land a < 2.028240960365167 \cdot 10^{+31}\right) \land \left(4.930380657631324 \cdot 10^{-32} < b \land b < 2.028240960365167 \cdot 10^{+31}\right)\right) \land \left(4.930380657631324 \cdot 10^{-32} < c \land c < 2.028240960365167 \cdot 10^{+31}\right)\]

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: 17.7% 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: 99.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 16.5%

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

   1. add-sqr-sqrt 16.6%

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

   2. distribute-rgt-neg-in 16.6%

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

4. Applied egg-rr 16.6%

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

   1. flip-+ 16.7%

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

   2. pow2 16.7%

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

   3. distribute-rgt-neg-out 16.7%

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

   4. add-sqr-sqrt 16.5%

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

   5. add-sqr-sqrt 17.1%

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

   6. pow2 17.1%

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

   7. *-commutative 17.1%

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

   8. *-commutative 17.1%

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

   9. distribute-rgt-neg-out 17.1%

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

   10. add-sqr-sqrt 17.1%

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

6. Applied egg-rr 17.1%

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

   1. associate--r- 99.4%

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

   2. unpow2 99.4%

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

   3. unpow2 99.4%

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

   4. difference-of-squares 99.4%

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

   5. neg-mul-1 99.4%

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

   6. distribute-lft1-in 99.4%

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

   7. metadata-eval 99.4%

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

   8. mul0-lft 99.4%

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

   9. unpow2 99.4%

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

   10. fma-neg 99.4%

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

   11. associate-*r* 99.4%

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

   12. *-commutative 99.4%

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

   13. distribute-rgt-neg-in 99.4%

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

   14. metadata-eval 99.4%

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

8. Simplified 99.4%

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

9. Taylor expanded in c around 0 99.2%

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

10. Taylor expanded in b around 0 99.2%

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

    1. associate-*r* 99.4%

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

    2. *-commutative 99.4%

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

    3. *-commutative 99.4%

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

12. Simplified 99.4%

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

13. Final simplification 99.4%

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

Alternative 2: 99.2% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 16.5%

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

   1. add-sqr-sqrt 16.6%

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

   2. distribute-rgt-neg-in 16.6%

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

4. Applied egg-rr 16.6%

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

   1. flip-+ 16.7%

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

   2. pow2 16.7%

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

   3. distribute-rgt-neg-out 16.7%

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

   4. add-sqr-sqrt 16.5%

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

   5. add-sqr-sqrt 17.1%

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

   6. pow2 17.1%

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

   7. *-commutative 17.1%

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

   8. *-commutative 17.1%

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

   9. distribute-rgt-neg-out 17.1%

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

   10. add-sqr-sqrt 17.1%

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

6. Applied egg-rr 17.1%

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

   1. associate--r- 99.4%

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

   2. unpow2 99.4%

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

   3. unpow2 99.4%

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

   4. difference-of-squares 99.4%

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

   5. neg-mul-1 99.4%

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

   6. distribute-lft1-in 99.4%

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

   7. metadata-eval 99.4%

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

   8. mul0-lft 99.4%

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

   9. unpow2 99.4%

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

   10. fma-neg 99.4%

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

   11. associate-*r* 99.4%

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

   12. *-commutative 99.4%

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

   13. distribute-rgt-neg-in 99.4%

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

   14. metadata-eval 99.4%

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

8. Simplified 99.4%

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

9. Taylor expanded in c around 0 99.2%

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

10. Taylor expanded in b around 0 99.2%

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

11. Final simplification 99.2%

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

Alternative 3: 95.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ -0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (+ (* -0.5 (/ c b)) (* -0.375 (/ (* a (pow c 2.0)) (pow b 3.0)))))

C

double code(double a, double b, double c) {
	return (-0.5 * (c / b)) + (-0.375 * ((a * pow(c, 2.0)) / pow(b, 3.0)));
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = ((-0.5d0) * (c / b)) + ((-0.375d0) * ((a * (c ** 2.0d0)) / (b ** 3.0d0)))
end function

Java

public static double code(double a, double b, double c) {
	return (-0.5 * (c / b)) + (-0.375 * ((a * Math.pow(c, 2.0)) / Math.pow(b, 3.0)));
}

Python

def code(a, b, c):
	return (-0.5 * (c / b)) + (-0.375 * ((a * math.pow(c, 2.0)) / math.pow(b, 3.0)))

Julia

function code(a, b, c)
	return Float64(Float64(-0.5 * Float64(c / b)) + Float64(-0.375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0))))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-0.5 * (c / b)) + (-0.375 * ((a * (c ^ 2.0)) / (b ^ 3.0)));
end

Wolfram

code[a_, b_, c_] := N[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 16.5%

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

3. Taylor expanded in a around 0 96.7%

   \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}} \]
4. Add Preprocessing

Alternative 4: 95.3% accurate, 7.7× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return 1.0 / ((-2.0 * (1.0 / (c / b))) + (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) * (1.0d0 / (c / b))) + (1.5d0 * (a / b)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 16.5%

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

   1. add-sqr-sqrt 16.6%

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

   2. distribute-rgt-neg-in 16.6%

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

4. Applied egg-rr 16.6%

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

   1. clear-num 16.6%

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

   2. inv-pow 16.6%

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

   3. *-commutative 16.6%

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

   4. distribute-rgt-neg-out 16.6%

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

   5. add-sqr-sqrt 16.5%

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

   6. pow2 16.5%

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

   7. *-commutative 16.5%

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

   8. *-commutative 16.5%

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

6. Applied egg-rr 16.5%

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

   1. unpow-1 16.5%

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

   2. associate-/l* 16.5%

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

   3. +-commutative 16.5%

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

   4. unsub-neg 16.5%

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

   5. unpow2 16.5%

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

   6. fma-neg 16.6%

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

   7. associate-*r* 16.6%

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

   8. *-commutative 16.6%

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

   9. distribute-rgt-neg-in 16.6%

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

   10. metadata-eval 16.6%

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

8. Simplified 16.6%

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

9. Taylor expanded in a around 0 96.4%

   \[\leadsto \frac{1}{\color{blue}{-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}}} \]
10. Step-by-step derivation

    1. clear-num 96.6%

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

    2. inv-pow 96.6%

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

11. Applied egg-rr 96.6%

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

    1. unpow-1 96.6%

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

13. Simplified 96.6%

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

Alternative 5: 95.2% accurate, 8.9× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return 1.0 / ((1.5 * (a / b)) + (-2.0 * (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 / ((1.5d0 * (a / b)) + ((-2.0d0) * (b / c)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 16.5%

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

   1. add-sqr-sqrt 16.6%

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

   2. distribute-rgt-neg-in 16.6%

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

4. Applied egg-rr 16.6%

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

   1. clear-num 16.6%

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

   2. inv-pow 16.6%

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

   3. *-commutative 16.6%

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

   4. distribute-rgt-neg-out 16.6%

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

   5. add-sqr-sqrt 16.5%

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

   6. pow2 16.5%

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

   7. *-commutative 16.5%

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

   8. *-commutative 16.5%

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

6. Applied egg-rr 16.5%

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

   1. unpow-1 16.5%

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

   2. associate-/l* 16.5%

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

   3. +-commutative 16.5%

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

   4. unsub-neg 16.5%

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

   5. unpow2 16.5%

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

   6. fma-neg 16.6%

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

   7. associate-*r* 16.6%

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

   8. *-commutative 16.6%

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

   9. distribute-rgt-neg-in 16.6%

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

   10. metadata-eval 16.6%

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

8. Simplified 16.6%

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

9. Taylor expanded in a around 0 96.4%

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

10. Final simplification 96.4%

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

Alternative 6: 90.5% accurate, 23.2× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return (c * -0.5) / 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 = (c * (-0.5d0)) / b
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 16.5%

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

3. Taylor expanded in b around inf 91.7%

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

   1. associate-*r/ 91.7%

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

   2. *-commutative 91.7%

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

5. Simplified 91.7%

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

Alternative 7: 90.2% accurate, 23.2× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	return c * (-0.5 / 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 = c * ((-0.5d0) / b)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 16.5%

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

3. Taylor expanded in b around inf 91.7%

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

   1. associate-*r/ 91.7%

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

   2. *-commutative 91.7%

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

5. Simplified 91.7%

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

6. Taylor expanded in c around 0 91.7%

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

   1. associate-*r/ 91.7%

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

   2. *-commutative 91.7%

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

   3. associate-*r/ 91.4%

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

8. Simplified 91.4%

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

Alternative 8: 3.3% 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 16.5%

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

   1. add-sqr-sqrt 16.6%

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

   2. distribute-rgt-neg-in 16.6%

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

4. Applied egg-rr 16.6%

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

5. Taylor expanded in a around 0 3.3%

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

   1. associate-*r/ 3.3%

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

   2. distribute-rgt1-in 3.3%

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

   3. metadata-eval 3.3%

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

   4. mul0-lft 3.3%

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

   5. metadata-eval 3.3%

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

7. Simplified 3.3%

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

8. Taylor expanded in a around 0 3.3%

   \[\leadsto \color{blue}{0} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024114 
(FPCore (a b c)
  :name "Cubic critical, wide range"
  :precision binary64
  :pre (and (and (and (< 4.930380657631324e-32 a) (< a 2.028240960365167e+31)) (and (< 4.930380657631324e-32 b) (< b 2.028240960365167e+31))) (and (< 4.930380657631324e-32 c) (< c 2.028240960365167e+31)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))