Cubic critical, medium range

Percentage Accurate: 31.1% → 99.8%

Time: 15.6s

Alternatives: 6

Speedup: 23.2×

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

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

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: 31.1% 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: 99.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \frac{c}{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot -3, a, {b}^{2}\right)}} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ c (- (- b) (sqrt (fma (* c -3.0) a (pow b 2.0))))))

C

double code(double a, double b, double c) {
	return c / (-b - sqrt(fma((c * -3.0), a, pow(b, 2.0))));
}

Julia

function code(a, b, c)
	return Float64(c / Float64(Float64(-b) - sqrt(fma(Float64(c * -3.0), a, (b ^ 2.0)))))
end

Wolfram

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

Derivation

1. Initial program 31.7%

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

   1. sqr-neg 31.7%

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

   2. sqr-neg 31.7%

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

   3. associate-*l* 31.7%

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

3. Simplified 31.7%

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

   1. add-exp-log 31.6%

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

6. Applied egg-rr 31.6%

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

   1. flip-+ 31.7%

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

   2. pow2 31.7%

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

   3. add-sqr-sqrt 32.6%

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

   4. pow2 32.6%

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

   5. rem-exp-log 32.7%

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

   6. associate-*r* 32.7%

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

   7. *-commutative 32.7%

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

   8. pow2 32.7%

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

   9. rem-exp-log 32.7%

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

   10. associate-*r* 32.7%

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

   11. *-commutative 32.7%

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

8. Applied egg-rr 32.7%

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

   1. associate--r- 99.4%

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

   2. associate-*l* 99.1%

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

   3. associate-*l* 99.1%

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

10. Simplified 99.1%

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

    1. div-inv 99.1%

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

    2. +-commutative 99.1%

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

    3. fma-define 99.1%

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

    4. neg-mul-1 99.1%

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

    5. unpow-prod-down 99.1%

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

    6. metadata-eval 99.1%

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

    7. *-un-lft-identity 99.1%

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

    8. associate-*r* 99.1%

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

    9. *-commutative 99.1%

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

12. Applied egg-rr 99.1%

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

    1. *-commutative 99.1%

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

    2. times-frac 99.1%

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

    3. associate-*r/ 99.1%

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

    4. *-lft-identity 99.1%

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

    5. associate-/r* 99.3%

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

    6. fma-undefine 99.3%

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

    7. associate-*l* 99.6%

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

    8. +-inverses 99.6%

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

    9. +-rgt-identity 99.6%

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

    10. *-commutative 99.6%

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

    11. *-commutative 99.6%

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

    12. associate-*r* 99.6%

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

    13. cancel-sign-sub-inv 99.6%

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

14. Simplified 99.6%

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

    1. div-inv 99.4%

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

    2. associate-/l* 99.5%

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

    3. pow1 99.5%

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

    4. pow1 99.5%

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

    5. pow-div 99.5%

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

    6. metadata-eval 99.5%

       \[\leadsto \left(c \cdot {\left(a \cdot 3\right)}^{\color{blue}{0}}\right) \cdot \frac{1}{\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, {b}^{2}\right)}} \]

    7. metadata-eval 99.5%

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

    8. add-log-exp 44.8%

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

    9. exp-prod 44.8%

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

    10. pow1 44.8%

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

    11. add-log-exp 99.5%

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

16. Applied egg-rr 99.5%

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

    1. associate-*r/ 99.8%

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

    2. *-rgt-identity 99.8%

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

    3. fma-define 99.8%

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

    4. associate-*r* 99.8%

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

    5. fma-define 99.8%

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

    6. *-commutative 99.8%

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

18. Simplified 99.8%

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

19. Final simplification 99.8%

    \[\leadsto \frac{c}{\left(-b\right) - \sqrt{\mathsf{fma}\left(c \cdot -3, a, {b}^{2}\right)}} \]
20. Add Preprocessing

Alternative 2: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{c \cdot \left(a \cdot 3\right)}{a \cdot 3}}{\left(-b\right) - \sqrt{{b}^{2} + -3 \cdot \left(c \cdot a\right)}} \end{array} \]

FPCore

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

C

double code(double a, double b, double c) {
	return ((c * (a * 3.0)) / (a * 3.0)) / (-b - sqrt((pow(b, 2.0) + (-3.0 * (c * 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 = ((c * (a * 3.0d0)) / (a * 3.0d0)) / (-b - sqrt(((b ** 2.0d0) + ((-3.0d0) * (c * a)))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 31.7%

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

   1. sqr-neg 31.7%

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

   2. sqr-neg 31.7%

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

   3. associate-*l* 31.7%

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

3. Simplified 31.7%

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

   1. add-exp-log 31.6%

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

6. Applied egg-rr 31.6%

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

   1. flip-+ 31.7%

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

   2. pow2 31.7%

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

   3. add-sqr-sqrt 32.6%

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

   4. pow2 32.6%

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

   5. rem-exp-log 32.7%

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

   6. associate-*r* 32.7%

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

   7. *-commutative 32.7%

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

   8. pow2 32.7%

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

   9. rem-exp-log 32.7%

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

   10. associate-*r* 32.7%

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

   11. *-commutative 32.7%

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

8. Applied egg-rr 32.7%

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

   1. associate--r- 99.4%

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

   2. associate-*l* 99.1%

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

   3. associate-*l* 99.1%

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

10. Simplified 99.1%

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

    1. div-inv 99.1%

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

    2. +-commutative 99.1%

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

    3. fma-define 99.1%

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

    4. neg-mul-1 99.1%

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

    5. unpow-prod-down 99.1%

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

    6. metadata-eval 99.1%

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

    7. *-un-lft-identity 99.1%

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

    8. associate-*r* 99.1%

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

    9. *-commutative 99.1%

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

12. Applied egg-rr 99.1%

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

    1. *-commutative 99.1%

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

    2. times-frac 99.1%

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

    3. associate-*r/ 99.1%

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

    4. *-lft-identity 99.1%

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

    5. associate-/r* 99.3%

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

    6. fma-undefine 99.3%

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

    7. associate-*l* 99.6%

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

    8. +-inverses 99.6%

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

    9. +-rgt-identity 99.6%

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

    10. *-commutative 99.6%

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

    11. *-commutative 99.6%

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

    12. associate-*r* 99.6%

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

    13. cancel-sign-sub-inv 99.6%

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

14. Simplified 99.6%

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

    1. fma-undefine 99.6%

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

16. Applied egg-rr 99.6%

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

17. Final simplification 99.6%

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

Alternative 3: 90.9% 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 31.7%

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

   1. sqr-neg 31.7%

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

   2. sqr-neg 31.7%

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

   3. associate-*l* 31.7%

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

3. Simplified 31.7%

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

5. Taylor expanded in a around 0 90.9%

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

6. Final simplification 90.9%

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

Alternative 4: 90.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ c \cdot \left(-0.375 \cdot \frac{c \cdot a}{{b}^{3}} - \frac{0.5}{b}\right) \end{array} \]

FPCore

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

C

double code(double a, double b, double c) {
	return c * ((-0.375 * ((c * a) / pow(b, 3.0))) - (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 * (((-0.375d0) * ((c * a) / (b ** 3.0d0))) - (0.5d0 / b))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 31.7%

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

   1. sqr-neg 31.7%

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

   2. sqr-neg 31.7%

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

   3. associate-*l* 31.7%

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

3. Simplified 31.7%

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

5. Taylor expanded in c around 0 90.6%

   \[\leadsto \color{blue}{c \cdot \left(-0.375 \cdot \frac{a \cdot c}{{b}^{3}} - 0.5 \cdot \frac{1}{b}\right)} \]
6. Step-by-step derivation

   1. add-cbrt-cube 90.1%

      \[\leadsto c \cdot \color{blue}{\sqrt[3]{\left(\left(-0.375 \cdot \frac{a \cdot c}{{b}^{3}} - 0.5 \cdot \frac{1}{b}\right) \cdot \left(-0.375 \cdot \frac{a \cdot c}{{b}^{3}} - 0.5 \cdot \frac{1}{b}\right)\right) \cdot \left(-0.375 \cdot \frac{a \cdot c}{{b}^{3}} - 0.5 \cdot \frac{1}{b}\right)}} \]

   2. pow3 90.2%

      \[\leadsto c \cdot \sqrt[3]{\color{blue}{{\left(-0.375 \cdot \frac{a \cdot c}{{b}^{3}} - 0.5 \cdot \frac{1}{b}\right)}^{3}}} \]

   3. associate-*r/ 90.2%

      \[\leadsto c \cdot \sqrt[3]{{\left(\color{blue}{\frac{-0.375 \cdot \left(a \cdot c\right)}{{b}^{3}}} - 0.5 \cdot \frac{1}{b}\right)}^{3}} \]

   4. un-div-inv 90.2%

      \[\leadsto c \cdot \sqrt[3]{{\left(\frac{-0.375 \cdot \left(a \cdot c\right)}{{b}^{3}} - \color{blue}{\frac{0.5}{b}}\right)}^{3}} \]

7. Applied egg-rr 90.2%

   \[\leadsto c \cdot \color{blue}{\sqrt[3]{{\left(\frac{-0.375 \cdot \left(a \cdot c\right)}{{b}^{3}} - \frac{0.5}{b}\right)}^{3}}} \]
8. Step-by-step derivation

   1. rem-cbrt-cube 90.6%

      \[\leadsto c \cdot \color{blue}{\left(\frac{-0.375 \cdot \left(a \cdot c\right)}{{b}^{3}} - \frac{0.5}{b}\right)} \]

   2. associate-/l* 90.6%

      \[\leadsto c \cdot \left(\color{blue}{-0.375 \cdot \frac{a \cdot c}{{b}^{3}}} - \frac{0.5}{b}\right) \]

9. Applied egg-rr 90.6%

   \[\leadsto c \cdot \color{blue}{\left(-0.375 \cdot \frac{a \cdot c}{{b}^{3}} - \frac{0.5}{b}\right)} \]

10. Final simplification 90.6%

    \[\leadsto c \cdot \left(-0.375 \cdot \frac{c \cdot a}{{b}^{3}} - \frac{0.5}{b}\right) \]
11. Add Preprocessing

Alternative 5: 81.2% 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 31.7%

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

   1. sqr-neg 31.7%

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

   2. sqr-neg 31.7%

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

   3. associate-*l* 31.7%

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

3. Simplified 31.7%

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

5. Taylor expanded in c around 0 90.6%

   \[\leadsto \color{blue}{c \cdot \left(-0.375 \cdot \frac{a \cdot c}{{b}^{3}} - 0.5 \cdot \frac{1}{b}\right)} \]

6. Taylor expanded in a around 0 80.8%

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

7. Final simplification 80.8%

   \[\leadsto c \cdot \frac{-0.5}{b} \]
8. Add Preprocessing

Alternative 6: 81.5% 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 31.7%

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

   1. sqr-neg 31.7%

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

   2. sqr-neg 31.7%

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

   3. associate-*l* 31.7%

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

3. Simplified 31.7%

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

5. Taylor expanded in b around inf 81.0%

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

   1. associate-*r/ 81.0%

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

   2. *-commutative 81.0%

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

7. Simplified 81.0%

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

8. Final simplification 81.0%

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

Reproduce

herbie shell --seed 2024080 
(FPCore (a b c)
  :name "Cubic critical, 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) (* (* 3.0 a) c)))) (* 3.0 a)))