Quadratic roots, medium range

Percentage Accurate: 31.9% → 99.4%

Time: 16.4s

Alternatives: 8

Speedup: 29.0×

Specification

\[\left(\left(1.1102230246251565 \cdot 10^{-16} < a \land a < 9007199254740992\right) \land \left(1.1102230246251565 \cdot 10^{-16} < b \land b < 9007199254740992\right)\right) \land \left(1.1102230246251565 \cdot 10^{-16} < c \land c < 9007199254740992\right)\]

Math

\[\begin{array}{l} \\ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * 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) - ((4.0d0 * a) * c)))) / (2.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Initial Program: 31.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * 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) - ((4.0d0 * a) * c)))) / (2.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(4.0 * a) * c)))) / Float64(2.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((4.0 * a) * c)))) / (2.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(4.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(2.0 * a), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{\left(4 \cdot c\right) \cdot a}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}\right)} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (* (* 4.0 c) a) (* (* a 2.0) (- (- b) (sqrt (fma b b (* a (* c -4.0))))))))

C

double code(double a, double b, double c) {
	return ((4.0 * c) * a) / ((a * 2.0) * (-b - sqrt(fma(b, b, (a * (c * -4.0))))));
}

Julia

function code(a, b, c)
	return Float64(Float64(Float64(4.0 * c) * a) / Float64(Float64(a * 2.0) * Float64(Float64(-b) - sqrt(fma(b, b, Float64(a * Float64(c * -4.0)))))))
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(4.0 * c), $MachinePrecision] * a), $MachinePrecision] / N[(N[(a * 2.0), $MachinePrecision] * N[((-b) - N[Sqrt[N[(b * b + N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 34.7%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Step-by-step derivation

   1. *-commutative 34.7%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

3. Simplified 34.7%

   \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. add-cbrt-cube 34.5%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\sqrt[3]{\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot b\right)}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   2. pow1/3 35.1%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left(\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot b\right)\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   3. pow3 35.1%

      \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({\left(b \cdot b\right)}^{3}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   4. pow2 35.1%

      \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({\color{blue}{\left({b}^{2}\right)}}^{3}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   5. pow-pow 35.2%

      \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({b}^{\left(2 \cdot 3\right)}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   6. metadata-eval 35.2%

      \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({b}^{\color{blue}{6}}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

6. Applied egg-rr 35.2%

   \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
7. Step-by-step derivation

   1. unpow1/3 34.1%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\sqrt[3]{{b}^{6}}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

8. Simplified 34.1%

   \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\sqrt[3]{{b}^{6}}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
9. Step-by-step derivation

   1. flip-+ 34.1%

      \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}}{a \cdot 2} \]

   2. pow2 34.1%

      \[\leadsto \frac{\frac{\color{blue}{{\left(-b\right)}^{2}} - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

   3. add-sqr-sqrt 34.6%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \color{blue}{\left(\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c\right)}}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

   4. pow1/3 35.3%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left(\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c\right)}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

   5. pow-pow 35.7%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left(\color{blue}{{b}^{\left(6 \cdot 0.3333333333333333\right)}} - \left(4 \cdot a\right) \cdot c\right)}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

   6. metadata-eval 35.7%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{\color{blue}{2}} - \left(4 \cdot a\right) \cdot c\right)}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

   7. associate-*l* 35.7%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \color{blue}{4 \cdot \left(a \cdot c\right)}\right)}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

   8. pow1/3 35.6%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - 4 \cdot \left(a \cdot c\right)\right)}{\left(-b\right) - \sqrt{\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

   9. pow-pow 35.7%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - 4 \cdot \left(a \cdot c\right)\right)}{\left(-b\right) - \sqrt{\color{blue}{{b}^{\left(6 \cdot 0.3333333333333333\right)}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

   10. metadata-eval 35.7%

       \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - 4 \cdot \left(a \cdot c\right)\right)}{\left(-b\right) - \sqrt{{b}^{\color{blue}{2}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

   11. associate-*l* 35.7%

       \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - 4 \cdot \left(a \cdot c\right)\right)}{\left(-b\right) - \sqrt{{b}^{2} - \color{blue}{4 \cdot \left(a \cdot c\right)}}}}{a \cdot 2} \]

10. Applied egg-rr 35.7%

    \[\leadsto \frac{\color{blue}{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - 4 \cdot \left(a \cdot c\right)\right)}{\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}}}{a \cdot 2} \]
11. Step-by-step derivation

    1. *-un-lft-identity 35.7%

       \[\leadsto \color{blue}{1 \cdot \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - 4 \cdot \left(a \cdot c\right)\right)}{\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}}{a \cdot 2}} \]

    2. associate-/l/ 35.7%

       \[\leadsto 1 \cdot \color{blue}{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - 4 \cdot \left(a \cdot c\right)\right)}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}\right)}} \]

    3. neg-mul-1 35.7%

       \[\leadsto 1 \cdot \frac{{\color{blue}{\left(-1 \cdot b\right)}}^{2} - \left({b}^{2} - 4 \cdot \left(a \cdot c\right)\right)}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}\right)} \]

    4. unpow-prod-down 35.7%

       \[\leadsto 1 \cdot \frac{\color{blue}{{-1}^{2} \cdot {b}^{2}} - \left({b}^{2} - 4 \cdot \left(a \cdot c\right)\right)}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}\right)} \]

    5. metadata-eval 35.7%

       \[\leadsto 1 \cdot \frac{\color{blue}{1} \cdot {b}^{2} - \left({b}^{2} - 4 \cdot \left(a \cdot c\right)\right)}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}\right)} \]

    6. *-un-lft-identity 35.7%

       \[\leadsto 1 \cdot \frac{\color{blue}{{b}^{2}} - \left({b}^{2} - 4 \cdot \left(a \cdot c\right)\right)}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}\right)} \]

    7. cancel-sign-sub-inv 35.7%

       \[\leadsto 1 \cdot \frac{{b}^{2} - \color{blue}{\left({b}^{2} + \left(-4\right) \cdot \left(a \cdot c\right)\right)}}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}\right)} \]

    8. unpow2 35.7%

       \[\leadsto 1 \cdot \frac{{b}^{2} - \left(\color{blue}{b \cdot b} + \left(-4\right) \cdot \left(a \cdot c\right)\right)}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}\right)} \]

    9. fma-define 35.6%

       \[\leadsto 1 \cdot \frac{{b}^{2} - \color{blue}{\mathsf{fma}\left(b, b, \left(-4\right) \cdot \left(a \cdot c\right)\right)}}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}\right)} \]

    10. metadata-eval 35.6%

        \[\leadsto 1 \cdot \frac{{b}^{2} - \mathsf{fma}\left(b, b, \color{blue}{-4} \cdot \left(a \cdot c\right)\right)}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}\right)} \]

    11. *-commutative 35.6%

        \[\leadsto 1 \cdot \frac{{b}^{2} - \mathsf{fma}\left(b, b, -4 \cdot \color{blue}{\left(c \cdot a\right)}\right)}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}\right)} \]

    12. *-commutative 35.6%

        \[\leadsto 1 \cdot \frac{{b}^{2} - \mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}{\color{blue}{\left(2 \cdot a\right)} \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}\right)} \]

12. Applied egg-rr 35.6%

    \[\leadsto \color{blue}{1 \cdot \frac{{b}^{2} - \mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}{\left(2 \cdot a\right) \cdot \left(\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}\right)}} \]
13. Step-by-step derivation

    1. *-lft-identity 35.6%

       \[\leadsto \color{blue}{\frac{{b}^{2} - \mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}{\left(2 \cdot a\right) \cdot \left(\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}\right)}} \]

    2. metadata-eval 35.6%

       \[\leadsto \frac{{b}^{2} - \mathsf{fma}\left(b, b, \color{blue}{\left(-4\right)} \cdot \left(c \cdot a\right)\right)}{\left(2 \cdot a\right) \cdot \left(\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}\right)} \]

    3. distribute-lft-neg-in 35.6%

       \[\leadsto \frac{{b}^{2} - \mathsf{fma}\left(b, b, \color{blue}{-4 \cdot \left(c \cdot a\right)}\right)}{\left(2 \cdot a\right) \cdot \left(\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}\right)} \]

    4. fma-neg 35.7%

       \[\leadsto \frac{{b}^{2} - \color{blue}{\left(b \cdot b - 4 \cdot \left(c \cdot a\right)\right)}}{\left(2 \cdot a\right) \cdot \left(\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}\right)} \]

    5. unpow2 35.7%

       \[\leadsto \frac{{b}^{2} - \left(\color{blue}{{b}^{2}} - 4 \cdot \left(c \cdot a\right)\right)}{\left(2 \cdot a\right) \cdot \left(\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}\right)} \]

    6. associate-+l- 99.4%

       \[\leadsto \frac{\color{blue}{\left({b}^{2} - {b}^{2}\right) + 4 \cdot \left(c \cdot a\right)}}{\left(2 \cdot a\right) \cdot \left(\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}\right)} \]

    7. +-inverses 99.4%

       \[\leadsto \frac{\color{blue}{0} + 4 \cdot \left(c \cdot a\right)}{\left(2 \cdot a\right) \cdot \left(\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}\right)} \]

    8. +-commutative 99.4%

       \[\leadsto \frac{\color{blue}{4 \cdot \left(c \cdot a\right) + 0}}{\left(2 \cdot a\right) \cdot \left(\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}\right)} \]

    9. associate-*r* 99.4%

       \[\leadsto \frac{\color{blue}{\left(4 \cdot c\right) \cdot a} + 0}{\left(2 \cdot a\right) \cdot \left(\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}\right)} \]

    10. associate-*r* 99.4%

        \[\leadsto \frac{\left(4 \cdot c\right) \cdot a + 0}{\left(2 \cdot a\right) \cdot \left(\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(-4 \cdot c\right) \cdot a}\right)}\right)} \]

14. Simplified 99.4%

    \[\leadsto \color{blue}{\frac{\left(4 \cdot c\right) \cdot a + 0}{\left(2 \cdot a\right) \cdot \left(\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot c\right) \cdot a\right)}\right)}} \]
15. Step-by-step derivation

    1. +-rgt-identity 99.4%

       \[\leadsto \frac{\color{blue}{\left(4 \cdot c\right) \cdot a}}{\left(2 \cdot a\right) \cdot \left(\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot c\right) \cdot a\right)}\right)} \]

16. Applied egg-rr 99.4%

    \[\leadsto \frac{\color{blue}{\left(4 \cdot c\right) \cdot a}}{\left(2 \cdot a\right) \cdot \left(\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot c\right) \cdot a\right)}\right)} \]

17. Final simplification 99.4%

    \[\leadsto \frac{\left(4 \cdot c\right) \cdot a}{\left(a \cdot 2\right) \cdot \left(\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}\right)} \]
18. Add Preprocessing

Alternative 2: 89.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.059:\\ \;\;\;\;\frac{1}{2 \cdot \frac{a}{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)} - b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\frac{c \cdot a}{b} - b}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.059)
   (/ 1.0 (* 2.0 (/ a (- (sqrt (fma b b (* a (* c -4.0)))) b))))
   (/ 1.0 (/ (- (/ (* c a) b) b) c))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.059) {
		tmp = 1.0 / (2.0 * (a / (sqrt(fma(b, b, (a * (c * -4.0)))) - b)));
	} else {
		tmp = 1.0 / ((((c * a) / b) - b) / c);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.059)
		tmp = Float64(1.0 / Float64(2.0 * Float64(a / Float64(sqrt(fma(b, b, Float64(a * Float64(c * -4.0)))) - b))));
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(Float64(c * a) / b) - b) / c));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.059], N[(1.0 / N[(2.0 * N[(a / N[(N[Sqrt[N[(b * b + N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(N[(c * a), $MachinePrecision] / b), $MachinePrecision] - b), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.058999999999999997

   1. Initial program 74.7%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. *-commutative 74.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

   3. Simplified 74.7%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-cbrt-cube 74.1%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\sqrt[3]{\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot b\right)}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      2. pow1/3 69.0%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left(\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot b\right)\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      3. pow3 69.0%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({\left(b \cdot b\right)}^{3}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      4. pow2 69.0%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({\color{blue}{\left({b}^{2}\right)}}^{3}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      5. pow-pow 68.9%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({b}^{\left(2 \cdot 3\right)}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      6. metadata-eval 68.9%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({b}^{\color{blue}{6}}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   6. Applied egg-rr 68.9%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
   7. Step-by-step derivation

      1. unpow1/3 73.6%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\sqrt[3]{{b}^{6}}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   8. Simplified 73.6%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\sqrt[3]{{b}^{6}}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
   9. Step-by-step derivation

      1. +-commutative 73.6%

         \[\leadsto \frac{\color{blue}{\sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}}{a \cdot 2} \]

      2. add-cube-cbrt 71.1%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt[3]{\sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}\right) \cdot \sqrt[3]{\sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}} + \left(-b\right)}{a \cdot 2} \]

      3. fma-define 70.9%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt[3]{\sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt[3]{\sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}, \sqrt[3]{\sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}, -b\right)}}{a \cdot 2} \]

   10. Applied egg-rr 72.6%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt[3]{{b}^{2} - 4 \cdot \left(a \cdot c\right)}, \sqrt[3]{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}, -b\right)}}{a \cdot 2} \]
   11. Step-by-step derivation

       1. clear-num 72.6%

          \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 2}{\mathsf{fma}\left(\sqrt[3]{{b}^{2} - 4 \cdot \left(a \cdot c\right)}, \sqrt[3]{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}, -b\right)}}} \]

       2. inv-pow 72.6%

          \[\leadsto \color{blue}{{\left(\frac{a \cdot 2}{\mathsf{fma}\left(\sqrt[3]{{b}^{2} - 4 \cdot \left(a \cdot c\right)}, \sqrt[3]{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}, -b\right)}\right)}^{-1}} \]

   12. Applied egg-rr 74.8%

       \[\leadsto \color{blue}{{\left(\frac{2 \cdot a}{\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)} + \left(-b\right)}\right)}^{-1}} \]
   13. Step-by-step derivation

       1. unpow-1 74.8%

          \[\leadsto \color{blue}{\frac{1}{\frac{2 \cdot a}{\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)} + \left(-b\right)}}} \]

       2. associate-/l* 74.8%

          \[\leadsto \frac{1}{\color{blue}{2 \cdot \frac{a}{\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)} + \left(-b\right)}}} \]

       3. unsub-neg 74.8%

          \[\leadsto \frac{1}{2 \cdot \frac{a}{\color{blue}{\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)} - b}}} \]

       4. associate-*r* 74.8%

          \[\leadsto \frac{1}{2 \cdot \frac{a}{\sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(-4 \cdot c\right) \cdot a}\right)} - b}} \]

   14. Simplified 74.8%

       \[\leadsto \color{blue}{\frac{1}{2 \cdot \frac{a}{\sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot c\right) \cdot a\right)} - b}}} \]

   if 0.058999999999999997 < b

   1. Initial program 25.4%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. *-commutative 25.4%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

   3. Simplified 25.4%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-cbrt-cube 25.4%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\sqrt[3]{\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot b\right)}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      2. pow1/3 27.3%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left(\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot b\right)\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      3. pow3 27.3%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({\left(b \cdot b\right)}^{3}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      4. pow2 27.3%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({\color{blue}{\left({b}^{2}\right)}}^{3}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      5. pow-pow 27.4%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({b}^{\left(2 \cdot 3\right)}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      6. metadata-eval 27.4%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({b}^{\color{blue}{6}}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   6. Applied egg-rr 27.4%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
   7. Step-by-step derivation

      1. unpow1/3 25.0%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\sqrt[3]{{b}^{6}}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   8. Simplified 25.0%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\sqrt[3]{{b}^{6}}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
   9. Step-by-step derivation

      1. +-commutative 25.0%

         \[\leadsto \frac{\color{blue}{\sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}}{a \cdot 2} \]

      2. add-cube-cbrt 24.7%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt[3]{\sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}\right) \cdot \sqrt[3]{\sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}} + \left(-b\right)}{a \cdot 2} \]

      3. fma-define 24.9%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt[3]{\sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt[3]{\sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}, \sqrt[3]{\sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}, -b\right)}}{a \cdot 2} \]

   10. Applied egg-rr 25.6%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt[3]{{b}^{2} - 4 \cdot \left(a \cdot c\right)}, \sqrt[3]{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}, -b\right)}}{a \cdot 2} \]
   11. Step-by-step derivation

       1. clear-num 25.6%

          \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 2}{\mathsf{fma}\left(\sqrt[3]{{b}^{2} - 4 \cdot \left(a \cdot c\right)}, \sqrt[3]{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}, -b\right)}}} \]

       2. inv-pow 25.6%

          \[\leadsto \color{blue}{{\left(\frac{a \cdot 2}{\mathsf{fma}\left(\sqrt[3]{{b}^{2} - 4 \cdot \left(a \cdot c\right)}, \sqrt[3]{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}, -b\right)}\right)}^{-1}} \]

   12. Applied egg-rr 25.5%

       \[\leadsto \color{blue}{{\left(\frac{2 \cdot a}{\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)} + \left(-b\right)}\right)}^{-1}} \]
   13. Step-by-step derivation

       1. unpow-1 25.5%

          \[\leadsto \color{blue}{\frac{1}{\frac{2 \cdot a}{\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)} + \left(-b\right)}}} \]

       2. associate-/l* 25.5%

          \[\leadsto \frac{1}{\color{blue}{2 \cdot \frac{a}{\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)} + \left(-b\right)}}} \]

       3. unsub-neg 25.5%

          \[\leadsto \frac{1}{2 \cdot \frac{a}{\color{blue}{\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)} - b}}} \]

       4. associate-*r* 25.5%

          \[\leadsto \frac{1}{2 \cdot \frac{a}{\sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(-4 \cdot c\right) \cdot a}\right)} - b}} \]

   14. Simplified 25.5%

       \[\leadsto \color{blue}{\frac{1}{2 \cdot \frac{a}{\sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot c\right) \cdot a\right)} - b}}} \]

   15. Taylor expanded in c around 0 93.7%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.059:\\ \;\;\;\;\frac{1}{2 \cdot \frac{a}{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)} - b}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\frac{c \cdot a}{b} - b}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{\left(4 \cdot c\right) \cdot a}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}} \cdot \frac{0.5}{a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (* (/ (* (* 4.0 c) a) (- (- b) (sqrt (fma b b (* a (* c -4.0)))))) (/ 0.5 a)))

C

double code(double a, double b, double c) {
	return (((4.0 * c) * a) / (-b - sqrt(fma(b, b, (a * (c * -4.0)))))) * (0.5 / a);
}

Julia

function code(a, b, c)
	return Float64(Float64(Float64(Float64(4.0 * c) * a) / Float64(Float64(-b) - sqrt(fma(b, b, Float64(a * Float64(c * -4.0)))))) * Float64(0.5 / a))
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(N[(4.0 * c), $MachinePrecision] * a), $MachinePrecision] / N[((-b) - N[Sqrt[N[(b * b + N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.5 / a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 34.7%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Step-by-step derivation

   1. *-commutative 34.7%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

3. Simplified 34.7%

   \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. add-cbrt-cube 34.5%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\sqrt[3]{\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot b\right)}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   2. pow1/3 35.1%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left(\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot b\right)\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   3. pow3 35.1%

      \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({\left(b \cdot b\right)}^{3}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   4. pow2 35.1%

      \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({\color{blue}{\left({b}^{2}\right)}}^{3}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   5. pow-pow 35.2%

      \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({b}^{\left(2 \cdot 3\right)}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   6. metadata-eval 35.2%

      \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({b}^{\color{blue}{6}}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

6. Applied egg-rr 35.2%

   \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
7. Step-by-step derivation

   1. unpow1/3 34.1%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\sqrt[3]{{b}^{6}}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

8. Simplified 34.1%

   \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\sqrt[3]{{b}^{6}}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
9. Step-by-step derivation

   1. flip-+ 34.1%

      \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}}{a \cdot 2} \]

   2. pow2 34.1%

      \[\leadsto \frac{\frac{\color{blue}{{\left(-b\right)}^{2}} - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c} \cdot \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

   3. add-sqr-sqrt 34.6%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \color{blue}{\left(\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c\right)}}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

   4. pow1/3 35.3%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left(\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c\right)}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

   5. pow-pow 35.7%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left(\color{blue}{{b}^{\left(6 \cdot 0.3333333333333333\right)}} - \left(4 \cdot a\right) \cdot c\right)}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

   6. metadata-eval 35.7%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{\color{blue}{2}} - \left(4 \cdot a\right) \cdot c\right)}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

   7. associate-*l* 35.7%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \color{blue}{4 \cdot \left(a \cdot c\right)}\right)}{\left(-b\right) - \sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

   8. pow1/3 35.6%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - 4 \cdot \left(a \cdot c\right)\right)}{\left(-b\right) - \sqrt{\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

   9. pow-pow 35.7%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - 4 \cdot \left(a \cdot c\right)\right)}{\left(-b\right) - \sqrt{\color{blue}{{b}^{\left(6 \cdot 0.3333333333333333\right)}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

   10. metadata-eval 35.7%

       \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - 4 \cdot \left(a \cdot c\right)\right)}{\left(-b\right) - \sqrt{{b}^{\color{blue}{2}} - \left(4 \cdot a\right) \cdot c}}}{a \cdot 2} \]

   11. associate-*l* 35.7%

       \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - 4 \cdot \left(a \cdot c\right)\right)}{\left(-b\right) - \sqrt{{b}^{2} - \color{blue}{4 \cdot \left(a \cdot c\right)}}}}{a \cdot 2} \]

10. Applied egg-rr 35.7%

    \[\leadsto \frac{\color{blue}{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - 4 \cdot \left(a \cdot c\right)\right)}{\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}}}{a \cdot 2} \]
11. Step-by-step derivation

    1. div-inv 35.7%

       \[\leadsto \color{blue}{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - 4 \cdot \left(a \cdot c\right)\right)}{\left(-b\right) - \sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}} \cdot \frac{1}{a \cdot 2}} \]

12. Applied egg-rr 35.6%

    \[\leadsto \color{blue}{\frac{{b}^{2} - \mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}} \cdot \frac{1}{2 \cdot a}} \]
13. Step-by-step derivation

    1. metadata-eval 35.6%

       \[\leadsto \frac{{b}^{2} - \mathsf{fma}\left(b, b, \color{blue}{\left(-4\right)} \cdot \left(c \cdot a\right)\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}} \cdot \frac{1}{2 \cdot a} \]

    2. distribute-lft-neg-in 35.6%

       \[\leadsto \frac{{b}^{2} - \mathsf{fma}\left(b, b, \color{blue}{-4 \cdot \left(c \cdot a\right)}\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}} \cdot \frac{1}{2 \cdot a} \]

    3. fma-neg 35.7%

       \[\leadsto \frac{{b}^{2} - \color{blue}{\left(b \cdot b - 4 \cdot \left(c \cdot a\right)\right)}}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}} \cdot \frac{1}{2 \cdot a} \]

    4. unpow2 35.7%

       \[\leadsto \frac{{b}^{2} - \left(\color{blue}{{b}^{2}} - 4 \cdot \left(c \cdot a\right)\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}} \cdot \frac{1}{2 \cdot a} \]

    5. associate-+l- 99.2%

       \[\leadsto \frac{\color{blue}{\left({b}^{2} - {b}^{2}\right) + 4 \cdot \left(c \cdot a\right)}}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}} \cdot \frac{1}{2 \cdot a} \]

    6. +-inverses 99.2%

       \[\leadsto \frac{\color{blue}{0} + 4 \cdot \left(c \cdot a\right)}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}} \cdot \frac{1}{2 \cdot a} \]

    7. +-commutative 99.2%

       \[\leadsto \frac{\color{blue}{4 \cdot \left(c \cdot a\right) + 0}}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}} \cdot \frac{1}{2 \cdot a} \]

    8. associate-*r* 99.2%

       \[\leadsto \frac{\color{blue}{\left(4 \cdot c\right) \cdot a} + 0}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)}} \cdot \frac{1}{2 \cdot a} \]

    9. associate-*r* 99.2%

       \[\leadsto \frac{\left(4 \cdot c\right) \cdot a + 0}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(-4 \cdot c\right) \cdot a}\right)}} \cdot \frac{1}{2 \cdot a} \]

    10. associate-/r* 99.2%

        \[\leadsto \frac{\left(4 \cdot c\right) \cdot a + 0}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot c\right) \cdot a\right)}} \cdot \color{blue}{\frac{\frac{1}{2}}{a}} \]

    11. metadata-eval 99.2%

        \[\leadsto \frac{\left(4 \cdot c\right) \cdot a + 0}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot c\right) \cdot a\right)}} \cdot \frac{\color{blue}{0.5}}{a} \]

14. Simplified 99.2%

    \[\leadsto \color{blue}{\frac{\left(4 \cdot c\right) \cdot a + 0}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot c\right) \cdot a\right)}} \cdot \frac{0.5}{a}} \]
15. Step-by-step derivation

    1. +-rgt-identity 99.4%

       \[\leadsto \frac{\color{blue}{\left(4 \cdot c\right) \cdot a}}{\left(2 \cdot a\right) \cdot \left(\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot c\right) \cdot a\right)}\right)} \]

16. Applied egg-rr 99.2%

    \[\leadsto \frac{\color{blue}{\left(4 \cdot c\right) \cdot a}}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot c\right) \cdot a\right)}} \cdot \frac{0.5}{a} \]

17. Final simplification 99.2%

    \[\leadsto \frac{\left(4 \cdot c\right) \cdot a}{\left(-b\right) - \sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)}} \cdot \frac{0.5}{a} \]
18. Add Preprocessing

Alternative 4: 89.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.059:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\frac{c \cdot a}{b} - b}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.059)
   (* (/ 0.5 a) (- (sqrt (fma b b (* a (* c -4.0)))) b))
   (/ 1.0 (/ (- (/ (* c a) b) b) c))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.059) {
		tmp = (0.5 / a) * (sqrt(fma(b, b, (a * (c * -4.0)))) - b);
	} else {
		tmp = 1.0 / ((((c * a) / b) - b) / c);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.059)
		tmp = Float64(Float64(0.5 / a) * Float64(sqrt(fma(b, b, Float64(a * Float64(c * -4.0)))) - b));
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(Float64(c * a) / b) - b) / c));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.059], N[(N[(0.5 / a), $MachinePrecision] * N[(N[Sqrt[N[(b * b + N[(a * N[(c * -4.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(N[(c * a), $MachinePrecision] / b), $MachinePrecision] - b), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.058999999999999997

   1. Initial program 74.7%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. *-commutative 74.7%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

   3. Simplified 74.7%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-cbrt-cube 74.1%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\sqrt[3]{\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot b\right)}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      2. pow1/3 69.0%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left(\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot b\right)\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      3. pow3 69.0%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({\left(b \cdot b\right)}^{3}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      4. pow2 69.0%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({\color{blue}{\left({b}^{2}\right)}}^{3}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      5. pow-pow 68.9%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({b}^{\left(2 \cdot 3\right)}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      6. metadata-eval 68.9%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({b}^{\color{blue}{6}}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   6. Applied egg-rr 68.9%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
   7. Step-by-step derivation

      1. unpow1/3 73.6%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\sqrt[3]{{b}^{6}}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   8. Simplified 73.6%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\sqrt[3]{{b}^{6}}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
   9. Step-by-step derivation

      1. +-commutative 73.6%

         \[\leadsto \frac{\color{blue}{\sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}}{a \cdot 2} \]

      2. add-cube-cbrt 71.1%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt[3]{\sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}\right) \cdot \sqrt[3]{\sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}} + \left(-b\right)}{a \cdot 2} \]

      3. fma-define 70.9%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt[3]{\sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt[3]{\sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}, \sqrt[3]{\sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}, -b\right)}}{a \cdot 2} \]

   10. Applied egg-rr 72.6%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt[3]{{b}^{2} - 4 \cdot \left(a \cdot c\right)}, \sqrt[3]{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}, -b\right)}}{a \cdot 2} \]
   11. Step-by-step derivation

       1. div-inv 72.6%

          \[\leadsto \color{blue}{\mathsf{fma}\left(\sqrt[3]{{b}^{2} - 4 \cdot \left(a \cdot c\right)}, \sqrt[3]{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}, -b\right) \cdot \frac{1}{a \cdot 2}} \]

   12. Applied egg-rr 74.8%

       \[\leadsto \color{blue}{\left(\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)} + \left(-b\right)\right) \cdot \frac{1}{2 \cdot a}} \]
   13. Step-by-step derivation

       1. unsub-neg 74.8%

          \[\leadsto \color{blue}{\left(\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)} - b\right)} \cdot \frac{1}{2 \cdot a} \]

       2. associate-*r* 74.8%

          \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(-4 \cdot c\right) \cdot a}\right)} - b\right) \cdot \frac{1}{2 \cdot a} \]

       3. associate-/r* 74.8%

          \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot c\right) \cdot a\right)} - b\right) \cdot \color{blue}{\frac{\frac{1}{2}}{a}} \]

       4. metadata-eval 74.8%

          \[\leadsto \left(\sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot c\right) \cdot a\right)} - b\right) \cdot \frac{\color{blue}{0.5}}{a} \]

   14. Simplified 74.8%

       \[\leadsto \color{blue}{\left(\sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot c\right) \cdot a\right)} - b\right) \cdot \frac{0.5}{a}} \]

   if 0.058999999999999997 < b

   1. Initial program 25.4%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. *-commutative 25.4%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

   3. Simplified 25.4%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-cbrt-cube 25.4%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\sqrt[3]{\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot b\right)}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      2. pow1/3 27.3%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left(\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot b\right)\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      3. pow3 27.3%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({\left(b \cdot b\right)}^{3}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      4. pow2 27.3%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({\color{blue}{\left({b}^{2}\right)}}^{3}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      5. pow-pow 27.4%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({b}^{\left(2 \cdot 3\right)}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      6. metadata-eval 27.4%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({b}^{\color{blue}{6}}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   6. Applied egg-rr 27.4%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
   7. Step-by-step derivation

      1. unpow1/3 25.0%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\sqrt[3]{{b}^{6}}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   8. Simplified 25.0%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\sqrt[3]{{b}^{6}}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
   9. Step-by-step derivation

      1. +-commutative 25.0%

         \[\leadsto \frac{\color{blue}{\sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}}{a \cdot 2} \]

      2. add-cube-cbrt 24.7%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt[3]{\sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}\right) \cdot \sqrt[3]{\sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}} + \left(-b\right)}{a \cdot 2} \]

      3. fma-define 24.9%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt[3]{\sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt[3]{\sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}, \sqrt[3]{\sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}, -b\right)}}{a \cdot 2} \]

   10. Applied egg-rr 25.6%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt[3]{{b}^{2} - 4 \cdot \left(a \cdot c\right)}, \sqrt[3]{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}, -b\right)}}{a \cdot 2} \]
   11. Step-by-step derivation

       1. clear-num 25.6%

          \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 2}{\mathsf{fma}\left(\sqrt[3]{{b}^{2} - 4 \cdot \left(a \cdot c\right)}, \sqrt[3]{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}, -b\right)}}} \]

       2. inv-pow 25.6%

          \[\leadsto \color{blue}{{\left(\frac{a \cdot 2}{\mathsf{fma}\left(\sqrt[3]{{b}^{2} - 4 \cdot \left(a \cdot c\right)}, \sqrt[3]{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}, -b\right)}\right)}^{-1}} \]

   12. Applied egg-rr 25.5%

       \[\leadsto \color{blue}{{\left(\frac{2 \cdot a}{\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)} + \left(-b\right)}\right)}^{-1}} \]
   13. Step-by-step derivation

       1. unpow-1 25.5%

          \[\leadsto \color{blue}{\frac{1}{\frac{2 \cdot a}{\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)} + \left(-b\right)}}} \]

       2. associate-/l* 25.5%

          \[\leadsto \frac{1}{\color{blue}{2 \cdot \frac{a}{\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)} + \left(-b\right)}}} \]

       3. unsub-neg 25.5%

          \[\leadsto \frac{1}{2 \cdot \frac{a}{\color{blue}{\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)} - b}}} \]

       4. associate-*r* 25.5%

          \[\leadsto \frac{1}{2 \cdot \frac{a}{\sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(-4 \cdot c\right) \cdot a}\right)} - b}} \]

   14. Simplified 25.5%

       \[\leadsto \color{blue}{\frac{1}{2 \cdot \frac{a}{\sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot c\right) \cdot a\right)} - b}}} \]

   15. Taylor expanded in c around 0 93.7%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.059:\\ \;\;\;\;\frac{0.5}{a} \cdot \left(\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -4\right)\right)} - b\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\frac{c \cdot a}{b} - b}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.059:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\frac{c \cdot a}{b} - b}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.059)
   (/ (- (sqrt (- (* b b) (* c (* 4.0 a)))) b) (* a 2.0))
   (/ 1.0 (/ (- (/ (* c a) b) b) c))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.059) {
		tmp = (sqrt(((b * b) - (c * (4.0 * a)))) - b) / (a * 2.0);
	} else {
		tmp = 1.0 / ((((c * a) / b) - b) / c);
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 0.059d0) then
        tmp = (sqrt(((b * b) - (c * (4.0d0 * a)))) - b) / (a * 2.0d0)
    else
        tmp = 1.0d0 / ((((c * a) / b) - b) / c)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.059) {
		tmp = (Math.sqrt(((b * b) - (c * (4.0 * a)))) - b) / (a * 2.0);
	} else {
		tmp = 1.0 / ((((c * a) / b) - b) / c);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 0.059:
		tmp = (math.sqrt(((b * b) - (c * (4.0 * a)))) - b) / (a * 2.0)
	else:
		tmp = 1.0 / ((((c * a) / b) - b) / c)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.059)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(4.0 * a)))) - b) / Float64(a * 2.0));
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(Float64(c * a) / b) - b) / c));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 0.059)
		tmp = (sqrt(((b * b) - (c * (4.0 * a)))) - b) / (a * 2.0);
	else
		tmp = 1.0 / ((((c * a) / b) - b) / c);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.059], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(4.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 2.0), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(N[(c * a), $MachinePrecision] / b), $MachinePrecision] - b), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.058999999999999997

   1. Initial program 74.7%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Add Preprocessing

   if 0.058999999999999997 < b

   1. Initial program 25.4%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
   2. Step-by-step derivation

      1. *-commutative 25.4%

         \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

   3. Simplified 25.4%

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. add-cbrt-cube 25.4%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\sqrt[3]{\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot b\right)}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      2. pow1/3 27.3%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left(\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot b\right)\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      3. pow3 27.3%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({\left(b \cdot b\right)}^{3}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      4. pow2 27.3%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({\color{blue}{\left({b}^{2}\right)}}^{3}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      5. pow-pow 27.4%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({b}^{\left(2 \cdot 3\right)}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

      6. metadata-eval 27.4%

         \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({b}^{\color{blue}{6}}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   6. Applied egg-rr 27.4%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
   7. Step-by-step derivation

      1. unpow1/3 25.0%

         \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\sqrt[3]{{b}^{6}}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   8. Simplified 25.0%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\sqrt[3]{{b}^{6}}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
   9. Step-by-step derivation

      1. +-commutative 25.0%

         \[\leadsto \frac{\color{blue}{\sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}}{a \cdot 2} \]

      2. add-cube-cbrt 24.7%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt[3]{\sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}\right) \cdot \sqrt[3]{\sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}} + \left(-b\right)}{a \cdot 2} \]

      3. fma-define 24.9%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt[3]{\sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt[3]{\sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}, \sqrt[3]{\sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}, -b\right)}}{a \cdot 2} \]

   10. Applied egg-rr 25.6%

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt[3]{{b}^{2} - 4 \cdot \left(a \cdot c\right)}, \sqrt[3]{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}, -b\right)}}{a \cdot 2} \]
   11. Step-by-step derivation

       1. clear-num 25.6%

          \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 2}{\mathsf{fma}\left(\sqrt[3]{{b}^{2} - 4 \cdot \left(a \cdot c\right)}, \sqrt[3]{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}, -b\right)}}} \]

       2. inv-pow 25.6%

          \[\leadsto \color{blue}{{\left(\frac{a \cdot 2}{\mathsf{fma}\left(\sqrt[3]{{b}^{2} - 4 \cdot \left(a \cdot c\right)}, \sqrt[3]{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}, -b\right)}\right)}^{-1}} \]

   12. Applied egg-rr 25.5%

       \[\leadsto \color{blue}{{\left(\frac{2 \cdot a}{\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)} + \left(-b\right)}\right)}^{-1}} \]
   13. Step-by-step derivation

       1. unpow-1 25.5%

          \[\leadsto \color{blue}{\frac{1}{\frac{2 \cdot a}{\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)} + \left(-b\right)}}} \]

       2. associate-/l* 25.5%

          \[\leadsto \frac{1}{\color{blue}{2 \cdot \frac{a}{\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)} + \left(-b\right)}}} \]

       3. unsub-neg 25.5%

          \[\leadsto \frac{1}{2 \cdot \frac{a}{\color{blue}{\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)} - b}}} \]

       4. associate-*r* 25.5%

          \[\leadsto \frac{1}{2 \cdot \frac{a}{\sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(-4 \cdot c\right) \cdot a}\right)} - b}} \]

   14. Simplified 25.5%

       \[\leadsto \color{blue}{\frac{1}{2 \cdot \frac{a}{\sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot c\right) \cdot a\right)} - b}}} \]

   15. Taylor expanded in c around 0 93.7%

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

4. Final simplification 90.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.059:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(4 \cdot a\right)} - b}{a \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\frac{c \cdot a}{b} - b}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 90.7% accurate, 10.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 34.7%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Step-by-step derivation

   1. *-commutative 34.7%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

3. Simplified 34.7%

   \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. add-cbrt-cube 34.5%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\sqrt[3]{\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot b\right)}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   2. pow1/3 35.1%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left(\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot b\right)\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   3. pow3 35.1%

      \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({\left(b \cdot b\right)}^{3}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   4. pow2 35.1%

      \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({\color{blue}{\left({b}^{2}\right)}}^{3}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   5. pow-pow 35.2%

      \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({b}^{\left(2 \cdot 3\right)}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   6. metadata-eval 35.2%

      \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({b}^{\color{blue}{6}}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

6. Applied egg-rr 35.2%

   \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
7. Step-by-step derivation

   1. unpow1/3 34.1%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\sqrt[3]{{b}^{6}}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

8. Simplified 34.1%

   \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\sqrt[3]{{b}^{6}}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
9. Step-by-step derivation

   1. +-commutative 34.1%

      \[\leadsto \frac{\color{blue}{\sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}}{a \cdot 2} \]

   2. add-cube-cbrt 33.4%

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt[3]{\sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}\right) \cdot \sqrt[3]{\sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}} + \left(-b\right)}{a \cdot 2} \]

   3. fma-define 33.5%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt[3]{\sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt[3]{\sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}, \sqrt[3]{\sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}, -b\right)}}{a \cdot 2} \]

10. Applied egg-rr 34.4%

    \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt[3]{{b}^{2} - 4 \cdot \left(a \cdot c\right)}, \sqrt[3]{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}, -b\right)}}{a \cdot 2} \]
11. Step-by-step derivation

    1. clear-num 34.4%

       \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 2}{\mathsf{fma}\left(\sqrt[3]{{b}^{2} - 4 \cdot \left(a \cdot c\right)}, \sqrt[3]{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}, -b\right)}}} \]

    2. inv-pow 34.4%

       \[\leadsto \color{blue}{{\left(\frac{a \cdot 2}{\mathsf{fma}\left(\sqrt[3]{{b}^{2} - 4 \cdot \left(a \cdot c\right)}, \sqrt[3]{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}, -b\right)}\right)}^{-1}} \]

12. Applied egg-rr 34.7%

    \[\leadsto \color{blue}{{\left(\frac{2 \cdot a}{\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)} + \left(-b\right)}\right)}^{-1}} \]
13. Step-by-step derivation

    1. unpow-1 34.7%

       \[\leadsto \color{blue}{\frac{1}{\frac{2 \cdot a}{\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)} + \left(-b\right)}}} \]

    2. associate-/l* 34.7%

       \[\leadsto \frac{1}{\color{blue}{2 \cdot \frac{a}{\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)} + \left(-b\right)}}} \]

    3. unsub-neg 34.7%

       \[\leadsto \frac{1}{2 \cdot \frac{a}{\color{blue}{\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)} - b}}} \]

    4. associate-*r* 34.7%

       \[\leadsto \frac{1}{2 \cdot \frac{a}{\sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(-4 \cdot c\right) \cdot a}\right)} - b}} \]

14. Simplified 34.7%

    \[\leadsto \color{blue}{\frac{1}{2 \cdot \frac{a}{\sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot c\right) \cdot a\right)} - b}}} \]

15. Taylor expanded in c around 0 88.5%

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

16. Final simplification 88.5%

    \[\leadsto \frac{1}{\frac{\frac{c \cdot a}{b} - b}{c}} \]
17. Add Preprocessing

Alternative 7: 90.7% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{a}{b} - \frac{b}{c}} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (/ 1.0 (- (/ a b) (/ b c))))

C

double code(double a, double b, double c) {
	return 1.0 / ((a / b) - (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 / ((a / b) - (b / c))
end function

Java

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

Python

def code(a, b, c):
	return 1.0 / ((a / b) - (b / c))

Julia

function code(a, b, c)
	return Float64(1.0 / Float64(Float64(a / b) - Float64(b / c)))
end

MATLAB

function tmp = code(a, b, c)
	tmp = 1.0 / ((a / b) - (b / c));
end

Wolfram

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

Derivation

1. Initial program 34.7%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Step-by-step derivation

   1. *-commutative 34.7%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

3. Simplified 34.7%

   \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. add-cbrt-cube 34.5%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\sqrt[3]{\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot b\right)}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   2. pow1/3 35.1%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left(\left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot b\right)\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   3. pow3 35.1%

      \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({\left(b \cdot b\right)}^{3}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   4. pow2 35.1%

      \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({\color{blue}{\left({b}^{2}\right)}}^{3}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   5. pow-pow 35.2%

      \[\leadsto \frac{\left(-b\right) + \sqrt{{\color{blue}{\left({b}^{\left(2 \cdot 3\right)}\right)}}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

   6. metadata-eval 35.2%

      \[\leadsto \frac{\left(-b\right) + \sqrt{{\left({b}^{\color{blue}{6}}\right)}^{0.3333333333333333} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

6. Applied egg-rr 35.2%

   \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{{\left({b}^{6}\right)}^{0.3333333333333333}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
7. Step-by-step derivation

   1. unpow1/3 34.1%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\sqrt[3]{{b}^{6}}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]

8. Simplified 34.1%

   \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\sqrt[3]{{b}^{6}}} - \left(4 \cdot a\right) \cdot c}}{a \cdot 2} \]
9. Step-by-step derivation

   1. +-commutative 34.1%

      \[\leadsto \frac{\color{blue}{\sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c} + \left(-b\right)}}{a \cdot 2} \]

   2. add-cube-cbrt 33.4%

      \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt[3]{\sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}\right) \cdot \sqrt[3]{\sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}} + \left(-b\right)}{a \cdot 2} \]

   3. fma-define 33.5%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt[3]{\sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}} \cdot \sqrt[3]{\sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}, \sqrt[3]{\sqrt{\sqrt[3]{{b}^{6}} - \left(4 \cdot a\right) \cdot c}}, -b\right)}}{a \cdot 2} \]

10. Applied egg-rr 34.4%

    \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt[3]{{b}^{2} - 4 \cdot \left(a \cdot c\right)}, \sqrt[3]{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}, -b\right)}}{a \cdot 2} \]
11. Step-by-step derivation

    1. clear-num 34.4%

       \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 2}{\mathsf{fma}\left(\sqrt[3]{{b}^{2} - 4 \cdot \left(a \cdot c\right)}, \sqrt[3]{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}, -b\right)}}} \]

    2. inv-pow 34.4%

       \[\leadsto \color{blue}{{\left(\frac{a \cdot 2}{\mathsf{fma}\left(\sqrt[3]{{b}^{2} - 4 \cdot \left(a \cdot c\right)}, \sqrt[3]{\sqrt{{b}^{2} - 4 \cdot \left(a \cdot c\right)}}, -b\right)}\right)}^{-1}} \]

12. Applied egg-rr 34.7%

    \[\leadsto \color{blue}{{\left(\frac{2 \cdot a}{\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)} + \left(-b\right)}\right)}^{-1}} \]
13. Step-by-step derivation

    1. unpow-1 34.7%

       \[\leadsto \color{blue}{\frac{1}{\frac{2 \cdot a}{\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)} + \left(-b\right)}}} \]

    2. associate-/l* 34.7%

       \[\leadsto \frac{1}{\color{blue}{2 \cdot \frac{a}{\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)} + \left(-b\right)}}} \]

    3. unsub-neg 34.7%

       \[\leadsto \frac{1}{2 \cdot \frac{a}{\color{blue}{\sqrt{\mathsf{fma}\left(b, b, -4 \cdot \left(c \cdot a\right)\right)} - b}}} \]

    4. associate-*r* 34.7%

       \[\leadsto \frac{1}{2 \cdot \frac{a}{\sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(-4 \cdot c\right) \cdot a}\right)} - b}} \]

14. Simplified 34.7%

    \[\leadsto \color{blue}{\frac{1}{2 \cdot \frac{a}{\sqrt{\mathsf{fma}\left(b, b, \left(-4 \cdot c\right) \cdot a\right)} - b}}} \]

15. Taylor expanded in a around 0 88.5%

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

16. Final simplification 88.5%

    \[\leadsto \frac{1}{\frac{a}{b} - \frac{b}{c}} \]
17. Add Preprocessing

Alternative 8: 80.9% accurate, 29.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 34.7%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{2 \cdot a} \]
2. Step-by-step derivation

   1. *-commutative 34.7%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{\color{blue}{a \cdot 2}} \]

3. Simplified 34.7%

   \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(4 \cdot a\right) \cdot c}}{a \cdot 2}} \]
4. Add Preprocessing

5. Taylor expanded in b around inf 78.7%

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

   1. associate-*r/ 78.7%

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

   2. mul-1-neg 78.7%

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

7. Simplified 78.7%

   \[\leadsto \color{blue}{\frac{-c}{b}} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024135 
(FPCore (a b c)
  :name "Quadratic roots, medium range"
  :precision binary64
  :pre (and (and (and (< 1.1102230246251565e-16 a) (< a 9007199254740992.0)) (and (< 1.1102230246251565e-16 b) (< b 9007199254740992.0))) (and (< 1.1102230246251565e-16 c) (< c 9007199254740992.0)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 4.0 a) c)))) (* 2.0 a)))