Cubic critical, narrow range

Percentage Accurate: 54.9% → 99.5%

Time: 14.7s

Alternatives: 7

Speedup: 23.2×

Specification

\[\left(\left(1.0536712127723509 \cdot 10^{-8} < a \land a < 94906265.62425156\right) \land \left(1.0536712127723509 \cdot 10^{-8} < b \land b < 94906265.62425156\right)\right) \land \left(1.0536712127723509 \cdot 10^{-8} < c \land c < 94906265.62425156\right)\]

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: 54.9% 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: 99.5% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 56.2%

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

3. Taylor expanded in a around 0 56.2%

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

   1. *-commutative 56.2%

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

   2. metadata-eval 56.2%

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

   3. distribute-rgt-neg-in 56.2%

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

   4. associate-*r* 56.2%

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

   5. distribute-rgt-neg-in 56.2%

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

   6. distribute-rgt-neg-in 56.2%

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

   7. metadata-eval 56.2%

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

5. Simplified 56.2%

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

   1. flip-+ 56.0%

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

   2. pow2 56.0%

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

   3. add-sqr-sqrt 57.8%

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

   4. pow2 57.8%

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

   5. pow2 57.8%

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

7. Applied egg-rr 57.8%

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

   1. unpow2 57.8%

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

   2. sqr-neg 57.8%

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

   3. unpow2 57.8%

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

   4. sub-neg 57.8%

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

   5. +-commutative 57.8%

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

   6. distribute-rgt-neg-in 57.8%

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

   7. distribute-rgt-neg-in 57.8%

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

   8. metadata-eval 57.8%

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

   9. *-commutative 57.8%

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

   10. fma-define 57.8%

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

   11. *-commutative 57.8%

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

   12. sub-neg 57.8%

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

   13. +-commutative 57.8%

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

   14. distribute-rgt-neg-in 57.8%

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

   15. distribute-rgt-neg-in 57.8%

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

   16. metadata-eval 57.8%

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

   17. *-commutative 57.8%

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

   18. fma-define 57.8%

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

   19. *-commutative 57.8%

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

9. Simplified 57.8%

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

10. Taylor expanded in b around 0 99.1%

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

    1. *-commutative 99.1%

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

    2. *-commutative 99.1%

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

12. Simplified 99.1%

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

    1. div-inv 99.1%

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

    2. associate-*l* 99.2%

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

    3. *-commutative 99.2%

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

14. Applied egg-rr 99.2%

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

    1. *-commutative 99.2%

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

    2. times-frac 99.3%

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

    3. associate-*r/ 99.3%

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

    4. *-lft-identity 99.3%

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

    5. associate-/r* 99.5%

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

    6. *-commutative 99.5%

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

    7. *-commutative 99.5%

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

    8. associate-*r* 99.1%

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

    9. *-commutative 99.1%

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

    10. times-frac 99.4%

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

    11. metadata-eval 99.4%

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

    12. *-commutative 99.4%

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

16. Simplified 99.4%

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

17. Final simplification 99.4%

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

Alternative 2: 99.1% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.2%

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

3. Taylor expanded in a around 0 56.2%

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

   1. *-commutative 56.2%

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

   2. metadata-eval 56.2%

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

   3. distribute-rgt-neg-in 56.2%

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

   4. associate-*r* 56.2%

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

   5. distribute-rgt-neg-in 56.2%

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

   6. distribute-rgt-neg-in 56.2%

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

   7. metadata-eval 56.2%

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

5. Simplified 56.2%

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

   1. flip-+ 56.0%

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

   2. pow2 56.0%

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

   3. add-sqr-sqrt 57.8%

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

   4. pow2 57.8%

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

   5. pow2 57.8%

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

7. Applied egg-rr 57.8%

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

   1. unpow2 57.8%

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

   2. sqr-neg 57.8%

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

   3. unpow2 57.8%

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

   4. sub-neg 57.8%

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

   5. +-commutative 57.8%

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

   6. distribute-rgt-neg-in 57.8%

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

   7. distribute-rgt-neg-in 57.8%

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

   8. metadata-eval 57.8%

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

   9. *-commutative 57.8%

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

   10. fma-define 57.8%

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

   11. *-commutative 57.8%

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

   12. sub-neg 57.8%

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

   13. +-commutative 57.8%

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

   14. distribute-rgt-neg-in 57.8%

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

   15. distribute-rgt-neg-in 57.8%

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

   16. metadata-eval 57.8%

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

   17. *-commutative 57.8%

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

   18. fma-define 57.8%

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

   19. *-commutative 57.8%

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

9. Simplified 57.8%

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

10. Taylor expanded in b around 0 99.1%

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

    1. *-commutative 99.1%

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

    2. *-commutative 99.1%

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

12. Simplified 99.1%

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

    1. fma-undefine 99.1%

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

14. Applied egg-rr 99.1%

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

15. Final simplification 99.1%

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

Alternative 3: 85.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 28:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 28.0)
   (/ (- (sqrt (fma b b (* a (* c -3.0)))) b) (* a 3.0))
   (+ (* -0.5 (/ c b)) (* -0.375 (/ (* a (pow c 2.0)) (pow b 3.0))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 28.0) {
		tmp = (sqrt(fma(b, b, (a * (c * -3.0)))) - b) / (a * 3.0);
	} else {
		tmp = (-0.5 * (c / b)) + (-0.375 * ((a * pow(c, 2.0)) / pow(b, 3.0)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 28.0)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(c * -3.0)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(-0.5 * Float64(c / b)) + Float64(-0.375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 28.0], N[(N[(N[Sqrt[N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], 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. Split input into 2 regimes

2. if b < 28

   1. Initial program 81.7%

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

      1. /-rgt-identity 81.7%

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

      2. metadata-eval 81.7%

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

   3. Simplified 81.9%

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

   if 28 < b

   1. Initial program 47.7%

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

   3. Taylor expanded in a around 0 88.1%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 28:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 84.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 28:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(-0.375 \cdot \left(a \cdot \frac{c}{{b}^{3}}\right) - \frac{0.5}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 28.0)
   (/ (- (sqrt (fma b b (* a (* c -3.0)))) b) (* a 3.0))
   (* c (- (* -0.375 (* a (/ c (pow b 3.0)))) (/ 0.5 b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 28.0) {
		tmp = (sqrt(fma(b, b, (a * (c * -3.0)))) - b) / (a * 3.0);
	} else {
		tmp = c * ((-0.375 * (a * (c / pow(b, 3.0)))) - (0.5 / b));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 28.0)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(c * -3.0)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(c * Float64(Float64(-0.375 * Float64(a * Float64(c / (b ^ 3.0)))) - Float64(0.5 / b)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 28.0], N[(N[(N[Sqrt[N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(c * N[(N[(-0.375 * N[(a * N[(c / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.5 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 28

   1. Initial program 81.7%

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

      1. /-rgt-identity 81.7%

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

      2. metadata-eval 81.7%

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

   3. Simplified 81.9%

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

   if 28 < b

   1. Initial program 47.7%

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

   3. Taylor expanded in c around 0 87.8%

      \[\leadsto \frac{\color{blue}{c \cdot \left(-1.5 \cdot \frac{a}{b} + -1.125 \cdot \frac{{a}^{2} \cdot c}{{b}^{3}}\right)}}{3 \cdot a} \]
   4. Step-by-step derivation

      1. associate-*r/ 87.9%

         \[\leadsto \frac{c \cdot \left(\color{blue}{\frac{-1.5 \cdot a}{b}} + -1.125 \cdot \frac{{a}^{2} \cdot c}{{b}^{3}}\right)}{3 \cdot a} \]

   5. Applied egg-rr 87.9%

      \[\leadsto \frac{c \cdot \left(\color{blue}{\frac{-1.5 \cdot a}{b}} + -1.125 \cdot \frac{{a}^{2} \cdot c}{{b}^{3}}\right)}{3 \cdot a} \]
   6. Step-by-step derivation

      1. *-commutative 87.9%

         \[\leadsto \frac{c \cdot \left(\frac{\color{blue}{a \cdot -1.5}}{b} + -1.125 \cdot \frac{{a}^{2} \cdot c}{{b}^{3}}\right)}{3 \cdot a} \]

   7. Simplified 87.9%

      \[\leadsto \frac{c \cdot \left(\color{blue}{\frac{a \cdot -1.5}{b}} + -1.125 \cdot \frac{{a}^{2} \cdot c}{{b}^{3}}\right)}{3 \cdot a} \]

   8. Taylor expanded in c around 0 87.9%

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

      1. associate-/l* 87.9%

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

      2. associate-*r/ 87.9%

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

      3. metadata-eval 87.9%

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

   10. Simplified 87.9%

       \[\leadsto \color{blue}{c \cdot \left(-0.375 \cdot \left(a \cdot \frac{c}{{b}^{3}}\right) - \frac{0.5}{b}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 28:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(-0.375 \cdot \left(a \cdot \frac{c}{{b}^{3}}\right) - \frac{0.5}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 28:\\ \;\;\;\;\frac{\sqrt{b \cdot b - a \cdot \left(c \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(-0.375 \cdot \left(a \cdot \frac{c}{{b}^{3}}\right) - \frac{0.5}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 28.0)
   (/ (- (sqrt (- (* b b) (* a (* c 3.0)))) b) (* a 3.0))
   (* c (- (* -0.375 (* a (/ c (pow b 3.0)))) (/ 0.5 b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 28.0) {
		tmp = (sqrt(((b * b) - (a * (c * 3.0)))) - b) / (a * 3.0);
	} else {
		tmp = c * ((-0.375 * (a * (c / pow(b, 3.0)))) - (0.5 / b));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 28.0d0) then
        tmp = (sqrt(((b * b) - (a * (c * 3.0d0)))) - b) / (a * 3.0d0)
    else
        tmp = c * (((-0.375d0) * (a * (c / (b ** 3.0d0)))) - (0.5d0 / b))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 28.0) {
		tmp = (Math.sqrt(((b * b) - (a * (c * 3.0)))) - b) / (a * 3.0);
	} else {
		tmp = c * ((-0.375 * (a * (c / Math.pow(b, 3.0)))) - (0.5 / b));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 28.0:
		tmp = (math.sqrt(((b * b) - (a * (c * 3.0)))) - b) / (a * 3.0)
	else:
		tmp = c * ((-0.375 * (a * (c / math.pow(b, 3.0)))) - (0.5 / b))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 28.0)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(a * Float64(c * 3.0)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(c * Float64(Float64(-0.375 * Float64(a * Float64(c / (b ^ 3.0)))) - Float64(0.5 / b)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 28.0)
		tmp = (sqrt(((b * b) - (a * (c * 3.0)))) - b) / (a * 3.0);
	else
		tmp = c * ((-0.375 * (a * (c / (b ^ 3.0)))) - (0.5 / b));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 28.0], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(a * N[(c * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(c * N[(N[(-0.375 * N[(a * N[(c / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.5 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 28

   1. Initial program 81.7%

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

   3. Taylor expanded in a around 0 81.7%

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

      1. *-commutative 81.7%

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

      2. metadata-eval 81.7%

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

      3. distribute-rgt-neg-in 81.7%

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

      4. associate-*r* 81.8%

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

      5. distribute-rgt-neg-in 81.8%

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

      6. distribute-rgt-neg-in 81.8%

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

      7. metadata-eval 81.8%

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

   5. Simplified 81.8%

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

   if 28 < b

   1. Initial program 47.7%

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

   3. Taylor expanded in c around 0 87.8%

      \[\leadsto \frac{\color{blue}{c \cdot \left(-1.5 \cdot \frac{a}{b} + -1.125 \cdot \frac{{a}^{2} \cdot c}{{b}^{3}}\right)}}{3 \cdot a} \]
   4. Step-by-step derivation

      1. associate-*r/ 87.9%

         \[\leadsto \frac{c \cdot \left(\color{blue}{\frac{-1.5 \cdot a}{b}} + -1.125 \cdot \frac{{a}^{2} \cdot c}{{b}^{3}}\right)}{3 \cdot a} \]

   5. Applied egg-rr 87.9%

      \[\leadsto \frac{c \cdot \left(\color{blue}{\frac{-1.5 \cdot a}{b}} + -1.125 \cdot \frac{{a}^{2} \cdot c}{{b}^{3}}\right)}{3 \cdot a} \]
   6. Step-by-step derivation

      1. *-commutative 87.9%

         \[\leadsto \frac{c \cdot \left(\frac{\color{blue}{a \cdot -1.5}}{b} + -1.125 \cdot \frac{{a}^{2} \cdot c}{{b}^{3}}\right)}{3 \cdot a} \]

   7. Simplified 87.9%

      \[\leadsto \frac{c \cdot \left(\color{blue}{\frac{a \cdot -1.5}{b}} + -1.125 \cdot \frac{{a}^{2} \cdot c}{{b}^{3}}\right)}{3 \cdot a} \]

   8. Taylor expanded in c around 0 87.9%

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

      1. associate-/l* 87.9%

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

      2. associate-*r/ 87.9%

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

      3. metadata-eval 87.9%

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

   10. Simplified 87.9%

       \[\leadsto \color{blue}{c \cdot \left(-0.375 \cdot \left(a \cdot \frac{c}{{b}^{3}}\right) - \frac{0.5}{b}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 28:\\ \;\;\;\;\frac{\sqrt{b \cdot b - a \cdot \left(c \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;c \cdot \left(-0.375 \cdot \left(a \cdot \frac{c}{{b}^{3}}\right) - \frac{0.5}{b}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.2%

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

3. Taylor expanded in c around 0 80.6%

   \[\leadsto \frac{\color{blue}{c \cdot \left(-1.5 \cdot \frac{a}{b} + -1.125 \cdot \frac{{a}^{2} \cdot c}{{b}^{3}}\right)}}{3 \cdot a} \]
4. Step-by-step derivation

   1. associate-*r/ 80.7%

      \[\leadsto \frac{c \cdot \left(\color{blue}{\frac{-1.5 \cdot a}{b}} + -1.125 \cdot \frac{{a}^{2} \cdot c}{{b}^{3}}\right)}{3 \cdot a} \]

5. Applied egg-rr 80.7%

   \[\leadsto \frac{c \cdot \left(\color{blue}{\frac{-1.5 \cdot a}{b}} + -1.125 \cdot \frac{{a}^{2} \cdot c}{{b}^{3}}\right)}{3 \cdot a} \]
6. Step-by-step derivation

   1. *-commutative 80.7%

      \[\leadsto \frac{c \cdot \left(\frac{\color{blue}{a \cdot -1.5}}{b} + -1.125 \cdot \frac{{a}^{2} \cdot c}{{b}^{3}}\right)}{3 \cdot a} \]

7. Simplified 80.7%

   \[\leadsto \frac{c \cdot \left(\color{blue}{\frac{a \cdot -1.5}{b}} + -1.125 \cdot \frac{{a}^{2} \cdot c}{{b}^{3}}\right)}{3 \cdot a} \]

8. Taylor expanded in c around 0 80.8%

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

   1. associate-/l* 80.8%

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

   2. associate-*r/ 80.8%

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

   3. metadata-eval 80.8%

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

10. Simplified 80.8%

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

Alternative 7: 64.8% 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 56.2%

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

3. Taylor expanded in b around inf 63.7%

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

   1. associate-*r/ 63.7%

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

   2. *-commutative 63.7%

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

5. Simplified 63.7%

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

Reproduce

herbie shell --seed 2024090 
(FPCore (a b c)
  :name "Cubic critical, narrow range"
  :precision binary64
  :pre (and (and (and (< 1.0536712127723509e-8 a) (< a 94906265.62425156)) (and (< 1.0536712127723509e-8 b) (< b 94906265.62425156))) (and (< 1.0536712127723509e-8 c) (< c 94906265.62425156)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))