Cubic critical, medium range

Percentage Accurate: 31.2% → 99.4%

Time: 16.0s

Alternatives: 8

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: 31.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 31.3%

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

   1. sqr-neg 31.3%

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

   2. sqr-neg 31.3%

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

   3. associate-*l* 31.3%

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

3. Simplified 31.3%

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

   1. pow1 31.3%

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

   2. associate-*r* 31.3%

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

   3. *-commutative 31.3%

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

6. Applied egg-rr 31.3%

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

   1. unpow1 31.3%

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

   2. associate-*r* 31.3%

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

8. Simplified 31.3%

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

   1. neg-sub0 31.3%

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

   2. flip-- 31.5%

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

   3. metadata-eval 31.5%

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

   4. pow2 31.5%

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

   5. add-sqr-sqrt 31.6%

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

   6. sqrt-prod 31.5%

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

   7. sqr-neg 31.5%

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

   8. sqrt-unprod 0.0%

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

   9. add-sqr-sqrt 1.6%

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

   10. sub-neg 1.6%

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

   11. neg-sub0 1.6%

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

   12. add-sqr-sqrt 0.0%

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

   13. sqrt-unprod 31.5%

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

   14. sqr-neg 31.5%

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

   15. sqrt-prod 31.6%

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

   16. add-sqr-sqrt 31.5%

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

10. Applied egg-rr 31.5%

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

    1. neg-sub0 31.5%

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

12. Simplified 31.5%

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

    1. flip-+ 31.5%

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

14. Applied egg-rr 32.3%

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

    1. associate--r- 99.2%

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

16. Simplified 99.2%

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

    1. div-inv 99.1%

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

    2. +-commutative 99.1%

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

    3. fma-define 99.1%

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

    4. neg-mul-1 99.1%

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

    5. unpow-prod-down 99.1%

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

    6. metadata-eval 99.1%

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

    7. *-un-lft-identity 99.1%

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

18. Applied egg-rr 99.1%

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

    1. associate-*r/ 99.2%

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

    2. *-commutative 99.2%

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

    3. *-lft-identity 99.2%

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

    4. fma-undefine 99.2%

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

    5. +-inverses 99.2%

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

    6. +-rgt-identity 99.2%

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

    7. remove-double-div 99.1%

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

    8. distribute-frac-neg2 99.1%

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

    9. *-commutative 99.1%

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

    10. associate-*l* 99.3%

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

    11. *-commutative 99.3%

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

    12. distribute-frac-neg2 99.3%

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

    13. remove-double-div 99.4%

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

    14. associate-*r* 99.4%

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

    15. *-commutative 99.4%

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

    16. sub-neg 99.4%

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

    17. +-commutative 99.4%

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

20. Simplified 99.4%

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

21. Final simplification 99.4%

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

Alternative 2: 90.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 31.3%

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

   1. sqr-neg 31.3%

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

   2. sqr-neg 31.3%

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

   3. associate-*l* 31.3%

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

3. Simplified 31.3%

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

   1. pow1 31.3%

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

   2. associate-*r* 31.3%

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

   3. *-commutative 31.3%

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

6. Applied egg-rr 31.3%

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

   1. unpow1 31.3%

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

   2. associate-*r* 31.3%

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

8. Simplified 31.3%

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

   1. neg-sub0 31.3%

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

   2. flip-- 31.5%

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

   3. metadata-eval 31.5%

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

   4. pow2 31.5%

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

   5. add-sqr-sqrt 31.6%

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

   6. sqrt-prod 31.5%

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

   7. sqr-neg 31.5%

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

   8. sqrt-unprod 0.0%

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

   9. add-sqr-sqrt 1.6%

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

   10. sub-neg 1.6%

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

   11. neg-sub0 1.6%

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

   12. add-sqr-sqrt 0.0%

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

   13. sqrt-unprod 31.5%

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

   14. sqr-neg 31.5%

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

   15. sqrt-prod 31.6%

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

   16. add-sqr-sqrt 31.5%

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

10. Applied egg-rr 31.5%

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

    1. neg-sub0 31.5%

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

12. Simplified 31.5%

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

13. Taylor expanded in b around inf 90.5%

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

    1. fma-define 90.5%

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

    2. associate-/l* 90.5%

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

    3. unpow2 90.5%

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

    4. unpow2 90.5%

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

    5. times-frac 90.5%

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

    6. unpow1 90.5%

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

    7. pow-plus 90.5%

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

    8. metadata-eval 90.5%

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

15. Simplified 90.5%

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

Alternative 3: 90.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{b \cdot \left(1.5 \cdot \frac{a}{{b}^{2}} + 2 \cdot \frac{-1}{c}\right)} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 31.3%

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

   1. sqr-neg 31.3%

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

   2. sqr-neg 31.3%

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

   3. associate-*l* 31.3%

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

3. Simplified 31.3%

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

   1. pow1 31.3%

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

   2. associate-*r* 31.3%

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

   3. *-commutative 31.3%

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

6. Applied egg-rr 31.3%

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

   1. unpow1 31.3%

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

   2. associate-*r* 31.3%

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

8. Simplified 31.3%

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

   1. clear-num 31.3%

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

   2. inv-pow 31.3%

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

   3. *-commutative 31.3%

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

   4. neg-mul-1 31.3%

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

   5. fma-define 31.3%

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

   6. pow2 31.3%

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

   7. *-commutative 31.3%

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

10. Applied egg-rr 31.3%

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

12. Simplified 31.3%

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

13. Taylor expanded in b around inf 90.5%

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

14. Final simplification 90.5%

    \[\leadsto \frac{1}{b \cdot \left(1.5 \cdot \frac{a}{{b}^{2}} + 2 \cdot \frac{-1}{c}\right)} \]
15. Add Preprocessing

Alternative 4: 90.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1}{3 \cdot \left(b \cdot \left(\frac{a}{{b}^{2}} \cdot 0.5 - \frac{0.6666666666666666}{c}\right)\right)} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ 1.0 (* 3.0 (* b (- (* (/ a (pow b 2.0)) 0.5) (/ 0.6666666666666666 c))))))

C

double code(double a, double b, double c) {
	return 1.0 / (3.0 * (b * (((a / pow(b, 2.0)) * 0.5) - (0.6666666666666666 / c))));
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = 1.0d0 / (3.0d0 * (b * (((a / (b ** 2.0d0)) * 0.5d0) - (0.6666666666666666d0 / c))))
end function

Java

public static double code(double a, double b, double c) {
	return 1.0 / (3.0 * (b * (((a / Math.pow(b, 2.0)) * 0.5) - (0.6666666666666666 / c))));
}

Python

def code(a, b, c):
	return 1.0 / (3.0 * (b * (((a / math.pow(b, 2.0)) * 0.5) - (0.6666666666666666 / c))))

Julia

function code(a, b, c)
	return Float64(1.0 / Float64(3.0 * Float64(b * Float64(Float64(Float64(a / (b ^ 2.0)) * 0.5) - Float64(0.6666666666666666 / c)))))
end

MATLAB

function tmp = code(a, b, c)
	tmp = 1.0 / (3.0 * (b * (((a / (b ^ 2.0)) * 0.5) - (0.6666666666666666 / c))));
end

Wolfram

code[a_, b_, c_] := N[(1.0 / N[(3.0 * N[(b * N[(N[(N[(a / N[Power[b, 2.0], $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] - N[(0.6666666666666666 / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 31.3%

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

   1. sqr-neg 31.3%

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

   2. sqr-neg 31.3%

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

   3. associate-*l* 31.3%

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

3. Simplified 31.3%

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

   1. pow1 31.3%

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

   2. associate-*r* 31.3%

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

   3. *-commutative 31.3%

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

6. Applied egg-rr 31.3%

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

   1. unpow1 31.3%

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

   2. associate-*r* 31.3%

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

8. Simplified 31.3%

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

   1. clear-num 31.3%

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

   2. inv-pow 31.3%

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

   3. *-commutative 31.3%

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

   4. neg-mul-1 31.3%

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

   5. fma-define 31.3%

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

   6. pow2 31.3%

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

   7. *-commutative 31.3%

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

10. Applied egg-rr 31.3%

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

12. Simplified 31.3%

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

13. Taylor expanded in b around inf 90.2%

    \[\leadsto \frac{1}{3 \cdot \color{blue}{\left(b \cdot \left(0.5 \cdot \frac{a}{{b}^{2}} - 0.6666666666666666 \cdot \frac{1}{c}\right)\right)}} \]
14. Step-by-step derivation

    1. associate-*r/ 90.3%

       \[\leadsto \frac{1}{3 \cdot \left(b \cdot \left(0.5 \cdot \frac{a}{{b}^{2}} - \color{blue}{\frac{0.6666666666666666 \cdot 1}{c}}\right)\right)} \]

    2. metadata-eval 90.3%

       \[\leadsto \frac{1}{3 \cdot \left(b \cdot \left(0.5 \cdot \frac{a}{{b}^{2}} - \frac{\color{blue}{0.6666666666666666}}{c}\right)\right)} \]

15. Simplified 90.3%

    \[\leadsto \frac{1}{3 \cdot \color{blue}{\left(b \cdot \left(0.5 \cdot \frac{a}{{b}^{2}} - \frac{0.6666666666666666}{c}\right)\right)}} \]

16. Final simplification 90.3%

    \[\leadsto \frac{1}{3 \cdot \left(b \cdot \left(\frac{a}{{b}^{2}} \cdot 0.5 - \frac{0.6666666666666666}{c}\right)\right)} \]
17. Add Preprocessing

Alternative 5: 90.5% accurate, 1.0× speedup

Math

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

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(Float64(-0.375 * 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[(N[(-0.375 * N[(c * a), $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision] - N[(0.5 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 31.3%

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

   1. sqr-neg 31.3%

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

   2. sqr-neg 31.3%

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

   3. associate-*l* 31.3%

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

3. Simplified 31.3%

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

5. Taylor expanded in c around 0 90.3%

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

   1. associate-*r/ 90.3%

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

   2. associate-*r/ 90.3%

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

   3. metadata-eval 90.3%

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

7. Simplified 90.3%

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

8. Final simplification 90.3%

   \[\leadsto c \cdot \left(\frac{-0.375 \cdot \left(c \cdot a\right)}{{b}^{3}} - \frac{0.5}{b}\right) \]
9. 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 31.3%

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

   1. sqr-neg 31.3%

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

   2. sqr-neg 31.3%

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

   3. associate-*l* 31.3%

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

3. Simplified 31.3%

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

5. Taylor expanded in b around inf 81.1%

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

   1. associate-*r/ 81.1%

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

   2. *-commutative 81.1%

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

7. Simplified 81.1%

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

Alternative 7: 81.3% 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 31.3%

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

   1. sqr-neg 31.3%

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

   2. sqr-neg 31.3%

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

   3. associate-*l* 31.3%

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

3. Simplified 31.3%

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

5. Taylor expanded in b around inf 81.1%

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

   1. associate-*r/ 81.1%

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

   2. *-commutative 81.1%

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

7. Simplified 81.1%

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

8. Taylor expanded in c around 0 81.1%

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

   1. associate-*r/ 81.1%

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

   2. *-commutative 81.1%

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

   3. associate-*r/ 80.9%

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

10. Simplified 80.9%

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

Alternative 8: 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 31.3%

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

   1. sqr-neg 31.3%

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

   2. sqr-neg 31.3%

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

   3. associate-*l* 31.3%

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

3. Simplified 31.3%

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

   1. pow1 31.3%

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

   2. associate-*r* 31.3%

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

   3. *-commutative 31.3%

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

6. Applied egg-rr 31.3%

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

   1. unpow1 31.3%

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

   2. associate-*r* 31.3%

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

8. Simplified 31.3%

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

   1. neg-sub0 31.3%

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

   2. flip-- 31.5%

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

   3. metadata-eval 31.5%

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

   4. pow2 31.5%

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

   5. add-sqr-sqrt 31.6%

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

   6. sqrt-prod 31.5%

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

   7. sqr-neg 31.5%

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

   8. sqrt-unprod 0.0%

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

   9. add-sqr-sqrt 1.6%

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

   10. sub-neg 1.6%

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

   11. neg-sub0 1.6%

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

   12. add-sqr-sqrt 0.0%

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

   13. sqrt-unprod 31.5%

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

   14. sqr-neg 31.5%

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

   15. sqrt-prod 31.6%

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

   16. add-sqr-sqrt 31.5%

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

10. Applied egg-rr 31.5%

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

    1. neg-sub0 31.5%

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

12. Simplified 31.5%

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

    1. clear-num 31.6%

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

    2. inv-pow 31.6%

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

14. Applied egg-rr 31.6%

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

    1. unpow-1 31.6%

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

16. Simplified 31.6%

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

17. Taylor expanded in c around 0 3.2%

    \[\leadsto \color{blue}{0.3333333333333333 \cdot \frac{b + -1 \cdot b}{a}} \]
18. 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%

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

19. Simplified 3.2%

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

Reproduce

herbie shell --seed 2024137 
(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)))