Cubic critical, medium range

Percentage Accurate: 32.1% → 99.7%

Time: 14.9s

Alternatives: 9

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: 32.1% 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: 99.7% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 31.8%

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

   1. sqr-neg 31.8%

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

   2. sqr-neg 31.8%

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

   3. associate-*l* 31.8%

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

3. Simplified 31.8%

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

   1. add-cbrt-cube 31.8%

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

   2. pow1/3 31.7%

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

   3. pow3 31.7%

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

6. Applied egg-rr 31.7%

   \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{{\left({\left(3 \cdot \left(a \cdot c\right)\right)}^{3}\right)}^{0.3333333333333333}}}}{3 \cdot a} \]
7. Step-by-step derivation

   1. flip-+ 31.6%

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

8. Applied egg-rr 32.8%

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

   1. associate--r- 99.5%

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

   2. *-commutative 99.5%

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

   3. associate-*r* 99.2%

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

   4. *-commutative 99.2%

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

   5. associate-*l* 99.2%

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

   6. *-commutative 99.2%

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

   7. associate-*r* 99.2%

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

   8. *-commutative 99.2%

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

   9. associate-*l* 99.2%

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

10. Simplified 99.2%

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

    1. div-inv 99.1%

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

    2. +-commutative 99.1%

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

    3. *-commutative 99.1%

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

    4. fma-define 99.1%

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

    5. *-commutative 99.1%

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

    6. neg-mul-1 99.1%

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

    7. unpow-prod-down 99.1%

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

    8. metadata-eval 99.1%

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

    9. *-un-lft-identity 99.1%

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

    10. *-commutative 99.1%

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

12. Applied egg-rr 99.1%

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

    1. *-commutative 99.1%

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

    2. times-frac 99.2%

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

    3. associate-*r/ 99.2%

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

    4. *-lft-identity 99.2%

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

    5. associate-/r* 99.3%

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

    6. fma-undefine 99.3%

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

    7. +-inverses 99.3%

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

    8. +-rgt-identity 99.3%

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

    9. associate-*r* 99.3%

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

    10. *-commutative 99.3%

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

    11. sub-neg 99.3%

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

    12. +-commutative 99.3%

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

14. Simplified 99.3%

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

15. Taylor expanded in a around 0 99.8%

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

Alternative 2: 99.3% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 31.8%

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

   1. sqr-neg 31.8%

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

   2. sqr-neg 31.8%

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

   3. associate-*l* 31.8%

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

3. Simplified 31.8%

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

   1. add-cbrt-cube 31.8%

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

   2. pow1/3 31.7%

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

   3. pow3 31.7%

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

6. Applied egg-rr 31.7%

   \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{{\left({\left(3 \cdot \left(a \cdot c\right)\right)}^{3}\right)}^{0.3333333333333333}}}}{3 \cdot a} \]
7. Step-by-step derivation

   1. flip-+ 31.6%

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

8. Applied egg-rr 32.8%

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

   1. associate--r- 99.5%

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

   2. *-commutative 99.5%

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

   3. associate-*r* 99.2%

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

   4. *-commutative 99.2%

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

   5. associate-*l* 99.2%

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

   6. *-commutative 99.2%

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

   7. associate-*r* 99.2%

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

   8. *-commutative 99.2%

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

   9. associate-*l* 99.2%

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

10. Simplified 99.2%

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

    1. div-inv 99.1%

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

    2. +-commutative 99.1%

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

    3. *-commutative 99.1%

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

    4. fma-define 99.1%

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

    5. *-commutative 99.1%

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

    6. neg-mul-1 99.1%

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

    7. unpow-prod-down 99.1%

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

    8. metadata-eval 99.1%

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

    9. *-un-lft-identity 99.1%

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

    10. *-commutative 99.1%

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

12. Applied egg-rr 99.1%

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

    1. *-commutative 99.1%

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

    2. times-frac 99.2%

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

    3. associate-*r/ 99.2%

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

    4. *-lft-identity 99.2%

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

    5. associate-/r* 99.3%

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

    6. fma-undefine 99.3%

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

    7. +-inverses 99.3%

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

    8. +-rgt-identity 99.3%

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

    9. associate-*r* 99.3%

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

    10. *-commutative 99.3%

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

    11. sub-neg 99.3%

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

    12. +-commutative 99.3%

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

14. Simplified 99.3%

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

15. Taylor expanded in c around inf 99.3%

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

Alternative 3: 99.3% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (/ (* a (* c 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 ((a * (c * 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 = ((a * (c * 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 ((a * (c * 3.0)) / (a * 3.0)) / (-b - Math.sqrt((Math.pow(b, 2.0) + (-3.0 * (c * a)))));
}

Python

def code(a, b, c):
	return ((a * (c * 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(a * Float64(c * 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 = ((a * (c * 3.0)) / (a * 3.0)) / (-b - sqrt(((b ^ 2.0) + (-3.0 * (c * a)))));
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(a * N[(c * 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 31.8%

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

   1. sqr-neg 31.8%

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

   2. sqr-neg 31.8%

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

   3. associate-*l* 31.8%

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

3. Simplified 31.8%

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

   1. add-cbrt-cube 31.8%

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

   2. pow1/3 31.7%

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

   3. pow3 31.7%

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

6. Applied egg-rr 31.7%

   \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{{\left({\left(3 \cdot \left(a \cdot c\right)\right)}^{3}\right)}^{0.3333333333333333}}}}{3 \cdot a} \]
7. Step-by-step derivation

   1. flip-+ 31.6%

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

8. Applied egg-rr 32.8%

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

   1. associate--r- 99.5%

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

   2. *-commutative 99.5%

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

   3. associate-*r* 99.2%

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

   4. *-commutative 99.2%

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

   5. associate-*l* 99.2%

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

   6. *-commutative 99.2%

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

   7. associate-*r* 99.2%

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

   8. *-commutative 99.2%

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

   9. associate-*l* 99.2%

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

10. Simplified 99.2%

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

    1. div-inv 99.1%

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

    2. +-commutative 99.1%

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

    3. *-commutative 99.1%

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

    4. fma-define 99.1%

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

    5. *-commutative 99.1%

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

    6. neg-mul-1 99.1%

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

    7. unpow-prod-down 99.1%

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

    8. metadata-eval 99.1%

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

    9. *-un-lft-identity 99.1%

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

    10. *-commutative 99.1%

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

12. Applied egg-rr 99.1%

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

    1. *-commutative 99.1%

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

    2. times-frac 99.2%

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

    3. associate-*r/ 99.2%

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

    4. *-lft-identity 99.2%

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

    5. associate-/r* 99.3%

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

    6. fma-undefine 99.3%

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

    7. +-inverses 99.3%

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

    8. +-rgt-identity 99.3%

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

    9. associate-*r* 99.3%

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

    10. *-commutative 99.3%

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

    11. sub-neg 99.3%

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

    12. +-commutative 99.3%

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

14. Simplified 99.3%

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

15. Taylor expanded in c around 0 99.3%

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

16. Final simplification 99.3%

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

Alternative 4: 90.5% 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 31.8%

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

   1. sqr-neg 31.8%

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

   2. sqr-neg 31.8%

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

   3. associate-*l* 31.8%

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

3. Simplified 31.8%

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

5. Taylor expanded in a around 0 90.4%

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

Alternative 5: 90.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 31.8%

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

   1. sqr-neg 31.8%

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

   2. sqr-neg 31.8%

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

   3. associate-*l* 31.8%

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

3. Simplified 31.8%

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

5. Taylor expanded in c around 0 90.2%

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

   1. sub-neg 90.2%

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

   2. *-commutative 90.2%

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

   3. div-inv 90.2%

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

   4. pow-flip 90.2%

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

   5. metadata-eval 90.2%

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

   6. un-div-inv 90.2%

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

7. Applied egg-rr 90.2%

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

   1. distribute-rgt-in 90.2%

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

   2. associate-*l* 90.2%

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

   3. *-commutative 90.2%

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

   4. distribute-neg-frac 90.2%

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

   5. metadata-eval 90.2%

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

9. Applied egg-rr 90.2%

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

10. Final simplification 90.2%

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

Alternative 6: 90.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 31.8%

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

   1. sqr-neg 31.8%

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

   2. sqr-neg 31.8%

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

   3. associate-*l* 31.8%

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

3. Simplified 31.8%

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

5. Taylor expanded in c around 0 90.2%

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

   1. sub-neg 90.2%

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

   2. *-commutative 90.2%

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

   3. div-inv 90.2%

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

   4. pow-flip 90.2%

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

   5. metadata-eval 90.2%

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

   6. un-div-inv 90.2%

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

7. Applied egg-rr 90.2%

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

   1. sub-neg 90.2%

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

   2. associate-*l* 90.2%

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

9. Applied egg-rr 90.2%

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

    1. *-commutative 90.2%

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

    2. *-commutative 90.2%

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

11. Simplified 90.2%

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

Alternative 7: 80.8% 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 31.8%

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

   1. sqr-neg 31.8%

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

   2. sqr-neg 31.8%

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

   3. associate-*l* 31.8%

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

3. Simplified 31.8%

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

5. Taylor expanded in b around inf 80.9%

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

   1. associate-*r/ 80.9%

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

   2. *-commutative 80.9%

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

7. Simplified 80.9%

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

Alternative 8: 80.6% 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 31.8%

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

   1. sqr-neg 31.8%

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

   2. sqr-neg 31.8%

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

   3. associate-*l* 31.8%

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

3. Simplified 31.8%

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

5. Taylor expanded in c around 0 90.2%

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

6. Taylor expanded in a around 0 80.7%

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

Alternative 9: 3.2% accurate, 38.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 31.8%

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

   1. sqr-neg 31.8%

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

   2. sqr-neg 31.8%

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

   3. associate-*l* 31.8%

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

3. Simplified 31.8%

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

   1. add-cbrt-cube 31.8%

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

   2. pow1/3 31.7%

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

   3. pow3 31.7%

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

6. Applied egg-rr 31.7%

   \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \color{blue}{{\left({\left(3 \cdot \left(a \cdot c\right)\right)}^{3}\right)}^{0.3333333333333333}}}}{3 \cdot a} \]
7. Step-by-step derivation

   1. *-un-lft-identity 31.7%

      \[\leadsto \color{blue}{1 \cdot \frac{\left(-b\right) + \sqrt{b \cdot b - {\left({\left(3 \cdot \left(a \cdot c\right)\right)}^{3}\right)}^{0.3333333333333333}}}{3 \cdot a}} \]

   2. neg-mul-1 31.7%

      \[\leadsto 1 \cdot \frac{\color{blue}{-1 \cdot b} + \sqrt{b \cdot b - {\left({\left(3 \cdot \left(a \cdot c\right)\right)}^{3}\right)}^{0.3333333333333333}}}{3 \cdot a} \]

   3. fma-define 31.7%

      \[\leadsto 1 \cdot \frac{\color{blue}{\mathsf{fma}\left(-1, b, \sqrt{b \cdot b - {\left({\left(3 \cdot \left(a \cdot c\right)\right)}^{3}\right)}^{0.3333333333333333}}\right)}}{3 \cdot a} \]

   4. pow2 31.7%

      \[\leadsto 1 \cdot \frac{\mathsf{fma}\left(-1, b, \sqrt{\color{blue}{{b}^{2}} - {\left({\left(3 \cdot \left(a \cdot c\right)\right)}^{3}\right)}^{0.3333333333333333}}\right)}{3 \cdot a} \]

   5. pow-pow 31.8%

      \[\leadsto 1 \cdot \frac{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - \color{blue}{{\left(3 \cdot \left(a \cdot c\right)\right)}^{\left(3 \cdot 0.3333333333333333\right)}}}\right)}{3 \cdot a} \]

   6. metadata-eval 31.8%

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

   7. pow1 31.8%

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

   8. associate-*r* 31.8%

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

   9. *-commutative 31.8%

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

   10. *-commutative 31.8%

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

8. Applied egg-rr 31.8%

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

   1. associate-*r/ 31.8%

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

   2. *-commutative 31.8%

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

   3. times-frac 31.8%

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

   4. metadata-eval 31.8%

      \[\leadsto \color{blue}{0.3333333333333333} \cdot \frac{\mathsf{fma}\left(-1, b, \sqrt{{b}^{2} - \left(a \cdot 3\right) \cdot c}\right)}{a} \]

   5. unpow2 31.8%

      \[\leadsto 0.3333333333333333 \cdot \frac{\mathsf{fma}\left(-1, b, \sqrt{\color{blue}{b \cdot b} - \left(a \cdot 3\right) \cdot c}\right)}{a} \]

   6. fma-neg 31.9%

      \[\leadsto 0.3333333333333333 \cdot \frac{\mathsf{fma}\left(-1, b, \sqrt{\color{blue}{\mathsf{fma}\left(b, b, -\left(a \cdot 3\right) \cdot c\right)}}\right)}{a} \]

   7. *-commutative 31.9%

      \[\leadsto 0.3333333333333333 \cdot \frac{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(b, b, -\color{blue}{c \cdot \left(a \cdot 3\right)}\right)}\right)}{a} \]

   8. distribute-rgt-neg-in 31.9%

      \[\leadsto 0.3333333333333333 \cdot \frac{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(b, b, \color{blue}{c \cdot \left(-a \cdot 3\right)}\right)}\right)}{a} \]

   9. distribute-rgt-neg-in 31.9%

      \[\leadsto 0.3333333333333333 \cdot \frac{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(b, b, c \cdot \color{blue}{\left(a \cdot \left(-3\right)\right)}\right)}\right)}{a} \]

   10. metadata-eval 31.9%

       \[\leadsto 0.3333333333333333 \cdot \frac{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot \color{blue}{-3}\right)\right)}\right)}{a} \]

10. Simplified 31.9%

    \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{\mathsf{fma}\left(-1, b, \sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)}\right)}{a}} \]

11. Taylor expanded in c around 0 3.2%

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

    1. associate-*r/ 3.2%

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

    2. distribute-rgt1-in 3.2%

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

    3. metadata-eval 3.2%

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

    4. mul0-lft 3.2%

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

    5. metadata-eval 3.2%

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

13. Simplified 3.2%

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

Reproduce

herbie shell --seed 2024096 
(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)))