Cubic critical, medium range

Percentage Accurate: 32.0% → 99.6%

Time: 15.8s

Alternatives: 5

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: 32.0% 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: 99.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 30.1%

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

   1. add-exp-log 30.0%

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

4. Applied egg-rr 30.0%

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

   1. flip-+ 30.0%

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

   2. pow2 30.0%

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

   3. add-sqr-sqrt 30.8%

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

   4. pow2 30.8%

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

   5. *-commutative 30.8%

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

   6. *-commutative 30.8%

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

   7. pow2 30.8%

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

   8. *-commutative 30.8%

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

   9. *-commutative 30.8%

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

6. Applied egg-rr 30.8%

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

   1. associate--r- 97.0%

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

8. Simplified 97.0%

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

   1. div-inv 97.0%

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

   2. +-commutative 97.0%

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

   3. *-commutative 97.0%

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

   4. rem-exp-log 99.2%

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

   5. fma-define 99.2%

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

   6. rem-exp-log 97.0%

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

   7. *-commutative 97.0%

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

   8. neg-mul-1 97.0%

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

   9. unpow-prod-down 97.0%

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

   10. metadata-eval 97.0%

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

   11. *-un-lft-identity 97.0%

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

   12. rem-exp-log 99.3%

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

   13. *-commutative 99.3%

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

10. Applied egg-rr 99.3%

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

    1. associate-*r/ 99.4%

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

    2. *-rgt-identity 99.4%

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

    3. +-inverses 99.4%

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

12. Simplified 99.4%

    \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(c, a \cdot 3, 0\right)}{\left(-b\right) - \sqrt{{b}^{2} - c \cdot \left(a \cdot 3\right)}}}{a \cdot 3}} \]
13. Step-by-step derivation

    1. div-inv 99.3%

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

14. Applied egg-rr 99.3%

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

    1. *-commutative 99.3%

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

    2. times-frac 99.4%

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

    3. *-lft-identity 99.4%

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

    4. associate-/r* 99.6%

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

    5. fma-define 99.6%

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

    6. +-rgt-identity 99.6%

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

    7. associate-*r* 99.2%

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

    8. *-commutative 99.2%

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

    9. *-commutative 99.2%

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

    10. *-commutative 99.2%

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

    11. times-frac 99.6%

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

    12. metadata-eval 99.6%

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

    13. *-commutative 99.6%

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

16. Simplified 99.6%

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

17. Final simplification 99.6%

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

Alternative 2: 90.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(-0.375, a \cdot {\left(\frac{c}{b}\right)}^{2}, c \cdot -0.5\right)}{b} \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 30.1%

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

   1. sqr-neg 30.1%

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

   2. sqr-neg 30.1%

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

   3. associate-*l* 30.1%

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

3. Simplified 30.1%

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

5. Taylor expanded in a around 0 90.8%

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

6. Taylor expanded in b around inf 90.8%

   \[\leadsto \color{blue}{\frac{-0.5 \cdot c + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}} \]
7. Step-by-step derivation

   1. +-commutative 90.8%

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

   2. *-commutative 90.8%

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

   3. fma-define 90.8%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(-0.375, \frac{a \cdot {c}^{2}}{{b}^{2}}, c \cdot -0.5\right)}}{b} \]

   4. associate-/l* 90.8%

      \[\leadsto \frac{\mathsf{fma}\left(-0.375, \color{blue}{a \cdot \frac{{c}^{2}}{{b}^{2}}}, c \cdot -0.5\right)}{b} \]

   5. unpow2 90.8%

      \[\leadsto \frac{\mathsf{fma}\left(-0.375, a \cdot \frac{\color{blue}{c \cdot c}}{{b}^{2}}, c \cdot -0.5\right)}{b} \]

   6. unpow2 90.8%

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

   7. times-frac 90.8%

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

   8. unpow1 90.8%

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

   9. pow-plus 90.8%

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

   10. metadata-eval 90.8%

       \[\leadsto \frac{\mathsf{fma}\left(-0.375, a \cdot {\left(\frac{c}{b}\right)}^{\color{blue}{2}}, c \cdot -0.5\right)}{b} \]

8. Simplified 90.8%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-0.375, a \cdot {\left(\frac{c}{b}\right)}^{2}, c \cdot -0.5\right)}{b}} \]
9. Add Preprocessing

Alternative 3: 90.3% 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 30.1%

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

   1. sqr-neg 30.1%

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

   2. sqr-neg 30.1%

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

   3. associate-*l* 30.1%

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

3. Simplified 30.1%

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

5. Taylor expanded in c around 0 94.9%

   \[\leadsto \color{blue}{c \cdot \left(c \cdot \left(-0.375 \cdot \frac{a}{{b}^{3}} + c \cdot \left(-0.5625 \cdot \frac{{a}^{2}}{{b}^{5}} + -0.16666666666666666 \cdot \frac{c \cdot \left(1.265625 \cdot \frac{{a}^{4}}{{b}^{6}} + 5.0625 \cdot \frac{{a}^{4}}{{b}^{6}}\right)}{a \cdot b}\right)\right) - 0.5 \cdot \frac{1}{b}\right)} \]
6. Step-by-step derivation

7. Simplified 94.9%

   \[\leadsto \color{blue}{c \cdot \left(c \cdot \mathsf{fma}\left(-0.375, \frac{a}{{b}^{3}}, c \cdot \mathsf{fma}\left(-0.5625, \frac{{a}^{2}}{{b}^{5}}, -0.16666666666666666 \cdot \left(c \cdot \frac{\frac{{a}^{4}}{{b}^{6}} \cdot 6.328125}{a \cdot b}\right)\right)\right) - \frac{0.5}{b}\right)} \]

8. Taylor expanded in c around 0 90.6%

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

9. Final simplification 90.6%

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

Alternative 4: 80.9% 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 30.1%

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

   1. sqr-neg 30.1%

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

   2. sqr-neg 30.1%

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

   3. associate-*l* 30.1%

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

3. Simplified 30.1%

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

5. Taylor expanded in b around inf 82.1%

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

   1. associate-*r/ 82.1%

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

   2. *-commutative 82.1%

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

7. Simplified 82.1%

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

Alternative 5: 80.7% 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 30.1%

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

   1. sqr-neg 30.1%

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

   2. sqr-neg 30.1%

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

   3. associate-*l* 30.1%

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

3. Simplified 30.1%

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

5. Taylor expanded in b around inf 82.1%

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

   1. associate-*r/ 82.1%

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

   2. *-commutative 82.1%

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

7. Simplified 82.1%

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

8. Taylor expanded in c around 0 82.1%

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

   1. *-commutative 82.1%

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

   2. associate-*l/ 82.1%

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

   3. associate-*r/ 81.9%

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

10. Simplified 81.9%

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

Reproduce

herbie shell --seed 2024119 
(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)))