Cubic critical, medium range

Percentage Accurate: 31.3% → 99.3%

Time: 16.6s

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: 31.3% 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: 99.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \left(c \cdot 3\right)\\ \frac{\frac{c \cdot -3}{b + \frac{\sqrt{{b}^{4} - {t\_0}^{2}}}{\mathsf{hypot}\left(b, \sqrt{t\_0}\right)}}}{3} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* a (* c 3.0))))
   (/
    (/
     (* c -3.0)
     (+ b (/ (sqrt (- (pow b 4.0) (pow t_0 2.0))) (hypot b (sqrt t_0)))))
    3.0)))

C

double code(double a, double b, double c) {
	double t_0 = a * (c * 3.0);
	return ((c * -3.0) / (b + (sqrt((pow(b, 4.0) - pow(t_0, 2.0))) / hypot(b, sqrt(t_0))))) / 3.0;
}

Java

public static double code(double a, double b, double c) {
	double t_0 = a * (c * 3.0);
	return ((c * -3.0) / (b + (Math.sqrt((Math.pow(b, 4.0) - Math.pow(t_0, 2.0))) / Math.hypot(b, Math.sqrt(t_0))))) / 3.0;
}

Python

def code(a, b, c):
	t_0 = a * (c * 3.0)
	return ((c * -3.0) / (b + (math.sqrt((math.pow(b, 4.0) - math.pow(t_0, 2.0))) / math.hypot(b, math.sqrt(t_0))))) / 3.0

Julia

function code(a, b, c)
	t_0 = Float64(a * Float64(c * 3.0))
	return Float64(Float64(Float64(c * -3.0) / Float64(b + Float64(sqrt(Float64((b ^ 4.0) - (t_0 ^ 2.0))) / hypot(b, sqrt(t_0))))) / 3.0)
end

MATLAB

function tmp = code(a, b, c)
	t_0 = a * (c * 3.0);
	tmp = ((c * -3.0) / (b + (sqrt(((b ^ 4.0) - (t_0 ^ 2.0))) / hypot(b, sqrt(t_0))))) / 3.0;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(a * N[(c * 3.0), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(c * -3.0), $MachinePrecision] / N[(b + N[(N[Sqrt[N[(N[Power[b, 4.0], $MachinePrecision] - N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[b ^ 2 + N[Sqrt[t$95$0], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 3.0), $MachinePrecision]]

Derivation

1. Initial program 30.1%

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

   1. /-rgt-identity 30.1%

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

   2. metadata-eval 30.1%

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

3. Simplified 30.2%

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

   1. associate-*r* 30.2%

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

   2. *-commutative 30.2%

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

   3. metadata-eval 30.2%

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

   4. distribute-lft-neg-in 30.2%

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

   5. associate-*l* 30.2%

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

   6. fma-neg 30.1%

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

   7. flip-- 30.0%

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

   8. sqrt-div 29.9%

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

   9. pow2 29.9%

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

   10. pow2 29.9%

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

   11. pow-prod-up 30.0%

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

   12. metadata-eval 30.0%

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

   13. pow2 30.0%

       \[\leadsto \frac{\frac{\sqrt{{b}^{4} - \color{blue}{{\left(\left(3 \cdot a\right) \cdot c\right)}^{2}}}}{\sqrt{b \cdot b + \left(3 \cdot a\right) \cdot c}} - b}{3 \cdot a} \]

   14. associate-*l* 30.0%

       \[\leadsto \frac{\frac{\sqrt{{b}^{4} - {\color{blue}{\left(3 \cdot \left(a \cdot c\right)\right)}}^{2}}}{\sqrt{b \cdot b + \left(3 \cdot a\right) \cdot c}} - b}{3 \cdot a} \]

   15. fma-define 30.1%

       \[\leadsto \frac{\frac{\sqrt{{b}^{4} - {\left(3 \cdot \left(a \cdot c\right)\right)}^{2}}}{\sqrt{\color{blue}{\mathsf{fma}\left(b, b, \left(3 \cdot a\right) \cdot c\right)}}} - b}{3 \cdot a} \]

   16. associate-*l* 30.1%

       \[\leadsto \frac{\frac{\sqrt{{b}^{4} - {\left(3 \cdot \left(a \cdot c\right)\right)}^{2}}}{\sqrt{\mathsf{fma}\left(b, b, \color{blue}{3 \cdot \left(a \cdot c\right)}\right)}} - b}{3 \cdot a} \]

6. Applied egg-rr 30.1%

   \[\leadsto \frac{\color{blue}{\frac{\sqrt{{b}^{4} - {\left(3 \cdot \left(a \cdot c\right)\right)}^{2}}}{\sqrt{\mathsf{fma}\left(b, b, 3 \cdot \left(a \cdot c\right)\right)}}} - b}{3 \cdot a} \]
7. Step-by-step derivation

   1. *-commutative 30.1%

      \[\leadsto \frac{\frac{\sqrt{{b}^{4} - {\color{blue}{\left(\left(a \cdot c\right) \cdot 3\right)}}^{2}}}{\sqrt{\mathsf{fma}\left(b, b, 3 \cdot \left(a \cdot c\right)\right)}} - b}{3 \cdot a} \]

   2. *-commutative 30.1%

      \[\leadsto \frac{\frac{\sqrt{{b}^{4} - {\left(\color{blue}{\left(c \cdot a\right)} \cdot 3\right)}^{2}}}{\sqrt{\mathsf{fma}\left(b, b, 3 \cdot \left(a \cdot c\right)\right)}} - b}{3 \cdot a} \]

   3. associate-*l* 30.1%

      \[\leadsto \frac{\frac{\sqrt{{b}^{4} - {\color{blue}{\left(c \cdot \left(a \cdot 3\right)\right)}}^{2}}}{\sqrt{\mathsf{fma}\left(b, b, 3 \cdot \left(a \cdot c\right)\right)}} - b}{3 \cdot a} \]

   4. fma-undefine 30.0%

      \[\leadsto \frac{\frac{\sqrt{{b}^{4} - {\left(c \cdot \left(a \cdot 3\right)\right)}^{2}}}{\sqrt{\color{blue}{b \cdot b + 3 \cdot \left(a \cdot c\right)}}} - b}{3 \cdot a} \]

   5. unpow1 30.0%

      \[\leadsto \frac{\frac{\sqrt{{b}^{4} - {\left(c \cdot \left(a \cdot 3\right)\right)}^{2}}}{\sqrt{b \cdot b + \color{blue}{{\left(3 \cdot \left(a \cdot c\right)\right)}^{1}}}} - b}{3 \cdot a} \]

   6. sqr-pow 30.0%

      \[\leadsto \frac{\frac{\sqrt{{b}^{4} - {\left(c \cdot \left(a \cdot 3\right)\right)}^{2}}}{\sqrt{b \cdot b + \color{blue}{{\left(3 \cdot \left(a \cdot c\right)\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(3 \cdot \left(a \cdot c\right)\right)}^{\left(\frac{1}{2}\right)}}}} - b}{3 \cdot a} \]

   7. hypot-define 30.1%

      \[\leadsto \frac{\frac{\sqrt{{b}^{4} - {\left(c \cdot \left(a \cdot 3\right)\right)}^{2}}}{\color{blue}{\mathsf{hypot}\left(b, {\left(3 \cdot \left(a \cdot c\right)\right)}^{\left(\frac{1}{2}\right)}\right)}} - b}{3 \cdot a} \]

   8. metadata-eval 30.1%

      \[\leadsto \frac{\frac{\sqrt{{b}^{4} - {\left(c \cdot \left(a \cdot 3\right)\right)}^{2}}}{\mathsf{hypot}\left(b, {\left(3 \cdot \left(a \cdot c\right)\right)}^{\color{blue}{0.5}}\right)} - b}{3 \cdot a} \]

   9. unpow1/2 30.1%

      \[\leadsto \frac{\frac{\sqrt{{b}^{4} - {\left(c \cdot \left(a \cdot 3\right)\right)}^{2}}}{\mathsf{hypot}\left(b, \color{blue}{\sqrt{3 \cdot \left(a \cdot c\right)}}\right)} - b}{3 \cdot a} \]

   10. *-commutative 30.1%

       \[\leadsto \frac{\frac{\sqrt{{b}^{4} - {\left(c \cdot \left(a \cdot 3\right)\right)}^{2}}}{\mathsf{hypot}\left(b, \sqrt{\color{blue}{\left(a \cdot c\right) \cdot 3}}\right)} - b}{3 \cdot a} \]

   11. *-commutative 30.1%

       \[\leadsto \frac{\frac{\sqrt{{b}^{4} - {\left(c \cdot \left(a \cdot 3\right)\right)}^{2}}}{\mathsf{hypot}\left(b, \sqrt{\color{blue}{\left(c \cdot a\right)} \cdot 3}\right)} - b}{3 \cdot a} \]

   12. associate-*l* 30.1%

       \[\leadsto \frac{\frac{\sqrt{{b}^{4} - {\left(c \cdot \left(a \cdot 3\right)\right)}^{2}}}{\mathsf{hypot}\left(b, \sqrt{\color{blue}{c \cdot \left(a \cdot 3\right)}}\right)} - b}{3 \cdot a} \]

8. Simplified 30.1%

   \[\leadsto \frac{\color{blue}{\frac{\sqrt{{b}^{4} - {\left(c \cdot \left(a \cdot 3\right)\right)}^{2}}}{\mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot 3\right)}\right)}} - b}{3 \cdot a} \]
9. Step-by-step derivation

   1. flip-- 29.9%

      \[\leadsto \frac{\color{blue}{\frac{\frac{\sqrt{{b}^{4} - {\left(c \cdot \left(a \cdot 3\right)\right)}^{2}}}{\mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot 3\right)}\right)} \cdot \frac{\sqrt{{b}^{4} - {\left(c \cdot \left(a \cdot 3\right)\right)}^{2}}}{\mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot 3\right)}\right)} - b \cdot b}{\frac{\sqrt{{b}^{4} - {\left(c \cdot \left(a \cdot 3\right)\right)}^{2}}}{\mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot 3\right)}\right)} + b}}}{3 \cdot a} \]

10. Applied egg-rr 29.9%

    \[\leadsto \frac{\color{blue}{\frac{{\left(\frac{\sqrt{{b}^{4} - {\left(c \cdot a\right)}^{2} \cdot 9}}{\mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot 3\right)}\right)}\right)}^{2} - {b}^{2}}{\frac{\sqrt{{b}^{4} - {\left(c \cdot a\right)}^{2} \cdot 9}}{\mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot 3\right)}\right)} + b}}}{3 \cdot a} \]
11. Step-by-step derivation

12. Simplified 29.9%

    \[\leadsto \frac{\color{blue}{\frac{{\left(\frac{\sqrt{{b}^{4} - {\left(a \cdot \left(c \cdot 3\right)\right)}^{2}}}{\mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot 3\right)}\right)}\right)}^{2} - {b}^{2}}{b + \frac{\sqrt{{b}^{4} - {\left(a \cdot \left(c \cdot 3\right)\right)}^{2}}}{\mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot 3\right)}\right)}}}}{3 \cdot a} \]

13. Taylor expanded in b around 0 0.0%

    \[\leadsto \frac{\frac{\color{blue}{\frac{a \cdot \left(c \cdot {\left(\sqrt{-9}\right)}^{2}\right)}{{\left(\sqrt{3}\right)}^{2}}}}{b + \frac{\sqrt{{b}^{4} - {\left(a \cdot \left(c \cdot 3\right)\right)}^{2}}}{\mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot 3\right)}\right)}}}{3 \cdot a} \]
14. Step-by-step derivation

    1. associate-/l* 0.0%

       \[\leadsto \frac{\frac{\color{blue}{a \cdot \frac{c \cdot {\left(\sqrt{-9}\right)}^{2}}{{\left(\sqrt{3}\right)}^{2}}}}{b + \frac{\sqrt{{b}^{4} - {\left(a \cdot \left(c \cdot 3\right)\right)}^{2}}}{\mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot 3\right)}\right)}}}{3 \cdot a} \]

    2. unpow2 0.0%

       \[\leadsto \frac{\frac{a \cdot \frac{c \cdot \color{blue}{\left(\sqrt{-9} \cdot \sqrt{-9}\right)}}{{\left(\sqrt{3}\right)}^{2}}}{b + \frac{\sqrt{{b}^{4} - {\left(a \cdot \left(c \cdot 3\right)\right)}^{2}}}{\mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot 3\right)}\right)}}}{3 \cdot a} \]

    3. rem-square-sqrt 98.5%

       \[\leadsto \frac{\frac{a \cdot \frac{c \cdot \color{blue}{-9}}{{\left(\sqrt{3}\right)}^{2}}}{b + \frac{\sqrt{{b}^{4} - {\left(a \cdot \left(c \cdot 3\right)\right)}^{2}}}{\mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot 3\right)}\right)}}}{3 \cdot a} \]

    4. unpow2 98.5%

       \[\leadsto \frac{\frac{a \cdot \frac{c \cdot -9}{\color{blue}{\sqrt{3} \cdot \sqrt{3}}}}{b + \frac{\sqrt{{b}^{4} - {\left(a \cdot \left(c \cdot 3\right)\right)}^{2}}}{\mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot 3\right)}\right)}}}{3 \cdot a} \]

    5. rem-square-sqrt 99.1%

       \[\leadsto \frac{\frac{a \cdot \frac{c \cdot -9}{\color{blue}{3}}}{b + \frac{\sqrt{{b}^{4} - {\left(a \cdot \left(c \cdot 3\right)\right)}^{2}}}{\mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot 3\right)}\right)}}}{3 \cdot a} \]

15. Simplified 99.1%

    \[\leadsto \frac{\frac{\color{blue}{a \cdot \frac{c \cdot -9}{3}}}{b + \frac{\sqrt{{b}^{4} - {\left(a \cdot \left(c \cdot 3\right)\right)}^{2}}}{\mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot 3\right)}\right)}}}{3 \cdot a} \]
16. Step-by-step derivation

    1. div-inv 99.1%

       \[\leadsto \color{blue}{\frac{a \cdot \frac{c \cdot -9}{3}}{b + \frac{\sqrt{{b}^{4} - {\left(a \cdot \left(c \cdot 3\right)\right)}^{2}}}{\mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot 3\right)}\right)}} \cdot \frac{1}{3 \cdot a}} \]

    2. associate-/l* 99.1%

       \[\leadsto \color{blue}{\left(a \cdot \frac{\frac{c \cdot -9}{3}}{b + \frac{\sqrt{{b}^{4} - {\left(a \cdot \left(c \cdot 3\right)\right)}^{2}}}{\mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot 3\right)}\right)}}\right)} \cdot \frac{1}{3 \cdot a} \]

    3. associate-/l* 99.1%

       \[\leadsto \left(a \cdot \frac{\color{blue}{c \cdot \frac{-9}{3}}}{b + \frac{\sqrt{{b}^{4} - {\left(a \cdot \left(c \cdot 3\right)\right)}^{2}}}{\mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot 3\right)}\right)}}\right) \cdot \frac{1}{3 \cdot a} \]

    4. metadata-eval 99.1%

       \[\leadsto \left(a \cdot \frac{c \cdot \color{blue}{-3}}{b + \frac{\sqrt{{b}^{4} - {\left(a \cdot \left(c \cdot 3\right)\right)}^{2}}}{\mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot 3\right)}\right)}}\right) \cdot \frac{1}{3 \cdot a} \]

    5. *-commutative 99.1%

       \[\leadsto \left(a \cdot \frac{c \cdot -3}{b + \frac{\sqrt{{b}^{4} - {\left(a \cdot \left(c \cdot 3\right)\right)}^{2}}}{\mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot 3\right)}\right)}}\right) \cdot \frac{1}{\color{blue}{a \cdot 3}} \]

17. Applied egg-rr 99.1%

    \[\leadsto \color{blue}{\left(a \cdot \frac{c \cdot -3}{b + \frac{\sqrt{{b}^{4} - {\left(a \cdot \left(c \cdot 3\right)\right)}^{2}}}{\mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot 3\right)}\right)}}\right) \cdot \frac{1}{a \cdot 3}} \]
18. Step-by-step derivation

    1. associate-*r/ 99.1%

       \[\leadsto \color{blue}{\frac{\left(a \cdot \frac{c \cdot -3}{b + \frac{\sqrt{{b}^{4} - {\left(a \cdot \left(c \cdot 3\right)\right)}^{2}}}{\mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot 3\right)}\right)}}\right) \cdot 1}{a \cdot 3}} \]

    2. *-rgt-identity 99.1%

       \[\leadsto \frac{\color{blue}{a \cdot \frac{c \cdot -3}{b + \frac{\sqrt{{b}^{4} - {\left(a \cdot \left(c \cdot 3\right)\right)}^{2}}}{\mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot 3\right)}\right)}}}}{a \cdot 3} \]

    3. times-frac 99.4%

       \[\leadsto \color{blue}{\frac{a}{a} \cdot \frac{\frac{c \cdot -3}{b + \frac{\sqrt{{b}^{4} - {\left(a \cdot \left(c \cdot 3\right)\right)}^{2}}}{\mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot 3\right)}\right)}}}{3}} \]

    4. *-inverses 99.4%

       \[\leadsto \color{blue}{1} \cdot \frac{\frac{c \cdot -3}{b + \frac{\sqrt{{b}^{4} - {\left(a \cdot \left(c \cdot 3\right)\right)}^{2}}}{\mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot 3\right)}\right)}}}{3} \]

19. Simplified 99.4%

    \[\leadsto \color{blue}{1 \cdot \frac{\frac{c \cdot -3}{b + \frac{\sqrt{{b}^{4} - {\left(a \cdot \left(c \cdot 3\right)\right)}^{2}}}{\mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot 3\right)}\right)}}}{3}} \]

20. Final simplification 99.4%

    \[\leadsto \frac{\frac{c \cdot -3}{b + \frac{\sqrt{{b}^{4} - {\left(a \cdot \left(c \cdot 3\right)\right)}^{2}}}{\mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot 3\right)}\right)}}}{3} \]
21. Add Preprocessing

Alternative 2: 90.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \left(a \cdot 3\right)\\ t_1 := \sqrt{t\_0}\\ \mathbf{if}\;\frac{\sqrt{b \cdot b - t\_0} - b}{a \cdot 3} \leq -1:\\ \;\;\;\;\frac{\sqrt{\left(b + t\_1\right) \cdot \left(b - t\_1\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* c (* a 3.0))) (t_1 (sqrt t_0)))
   (if (<= (/ (- (sqrt (- (* b b) t_0)) b) (* a 3.0)) -1.0)
     (/ (- (sqrt (* (+ b t_1) (- b t_1))) 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 t_0 = c * (a * 3.0);
	double t_1 = sqrt(t_0);
	double tmp;
	if (((sqrt(((b * b) - t_0)) - b) / (a * 3.0)) <= -1.0) {
		tmp = (sqrt(((b + t_1) * (b - t_1))) - b) / (a * 3.0);
	} else {
		tmp = (-0.5 * (c / b)) + (-0.375 * ((a * pow(c, 2.0)) / pow(b, 3.0)));
	}
	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) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = c * (a * 3.0d0)
    t_1 = sqrt(t_0)
    if (((sqrt(((b * b) - t_0)) - b) / (a * 3.0d0)) <= (-1.0d0)) then
        tmp = (sqrt(((b + t_1) * (b - t_1))) - b) / (a * 3.0d0)
    else
        tmp = ((-0.5d0) * (c / b)) + ((-0.375d0) * ((a * (c ** 2.0d0)) / (b ** 3.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = c * (a * 3.0);
	double t_1 = Math.sqrt(t_0);
	double tmp;
	if (((Math.sqrt(((b * b) - t_0)) - b) / (a * 3.0)) <= -1.0) {
		tmp = (Math.sqrt(((b + t_1) * (b - t_1))) - b) / (a * 3.0);
	} else {
		tmp = (-0.5 * (c / b)) + (-0.375 * ((a * Math.pow(c, 2.0)) / Math.pow(b, 3.0)));
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = c * (a * 3.0)
	t_1 = math.sqrt(t_0)
	tmp = 0
	if ((math.sqrt(((b * b) - t_0)) - b) / (a * 3.0)) <= -1.0:
		tmp = (math.sqrt(((b + t_1) * (b - t_1))) - b) / (a * 3.0)
	else:
		tmp = (-0.5 * (c / b)) + (-0.375 * ((a * math.pow(c, 2.0)) / math.pow(b, 3.0)))
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(c * Float64(a * 3.0))
	t_1 = sqrt(t_0)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - t_0)) - b) / Float64(a * 3.0)) <= -1.0)
		tmp = Float64(Float64(sqrt(Float64(Float64(b + t_1) * Float64(b - t_1))) - 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

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = c * (a * 3.0);
	t_1 = sqrt(t_0);
	tmp = 0.0;
	if (((sqrt(((b * b) - t_0)) - b) / (a * 3.0)) <= -1.0)
		tmp = (sqrt(((b + t_1) * (b - t_1))) - b) / (a * 3.0);
	else
		tmp = (-0.5 * (c / b)) + (-0.375 * ((a * (c ^ 2.0)) / (b ^ 3.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Sqrt[t$95$0], $MachinePrecision]}, If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], -1.0], N[(N[(N[Sqrt[N[(N[(b + t$95$1), $MachinePrecision] * N[(b - t$95$1), $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 (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) < -1

   1. Initial program 75.7%

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

      1. add-sqr-sqrt 75.7%

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

      2. difference-of-squares 76.3%

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

      3. associate-*l* 76.3%

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

      4. associate-*l* 76.4%

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

   4. Applied egg-rr 76.4%

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

      1. associate-*r* 76.3%

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

      2. *-commutative 76.3%

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

      3. associate-*r* 76.3%

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

      4. *-commutative 76.3%

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

   6. Simplified 76.3%

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

   if -1 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 21.9%

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

   3. Taylor expanded in b around inf 95.6%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -1:\\ \;\;\;\;\frac{\sqrt{\left(b + \sqrt{c \cdot \left(a \cdot 3\right)}\right) \cdot \left(b - \sqrt{c \cdot \left(a \cdot 3\right)}\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := a \cdot \left(c \cdot 3\right)\\ \frac{\frac{a \cdot \frac{c \cdot -9}{3}}{b + \frac{\sqrt{{b}^{4} - t\_0 \cdot t\_0}}{\mathsf{hypot}\left(b, \sqrt{t\_0}\right)}}}{a \cdot 3} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* a (* c 3.0))))
   (/
    (/
     (* a (/ (* c -9.0) 3.0))
     (+ b (/ (sqrt (- (pow b 4.0) (* t_0 t_0))) (hypot b (sqrt t_0)))))
    (* a 3.0))))

C

double code(double a, double b, double c) {
	double t_0 = a * (c * 3.0);
	return ((a * ((c * -9.0) / 3.0)) / (b + (sqrt((pow(b, 4.0) - (t_0 * t_0))) / hypot(b, sqrt(t_0))))) / (a * 3.0);
}

Java

public static double code(double a, double b, double c) {
	double t_0 = a * (c * 3.0);
	return ((a * ((c * -9.0) / 3.0)) / (b + (Math.sqrt((Math.pow(b, 4.0) - (t_0 * t_0))) / Math.hypot(b, Math.sqrt(t_0))))) / (a * 3.0);
}

Python

def code(a, b, c):
	t_0 = a * (c * 3.0)
	return ((a * ((c * -9.0) / 3.0)) / (b + (math.sqrt((math.pow(b, 4.0) - (t_0 * t_0))) / math.hypot(b, math.sqrt(t_0))))) / (a * 3.0)

Julia

function code(a, b, c)
	t_0 = Float64(a * Float64(c * 3.0))
	return Float64(Float64(Float64(a * Float64(Float64(c * -9.0) / 3.0)) / Float64(b + Float64(sqrt(Float64((b ^ 4.0) - Float64(t_0 * t_0))) / hypot(b, sqrt(t_0))))) / Float64(a * 3.0))
end

MATLAB

function tmp = code(a, b, c)
	t_0 = a * (c * 3.0);
	tmp = ((a * ((c * -9.0) / 3.0)) / (b + (sqrt(((b ^ 4.0) - (t_0 * t_0))) / hypot(b, sqrt(t_0))))) / (a * 3.0);
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(a * N[(c * 3.0), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(a * N[(N[(c * -9.0), $MachinePrecision] / 3.0), $MachinePrecision]), $MachinePrecision] / N[(b + N[(N[Sqrt[N[(N[Power[b, 4.0], $MachinePrecision] - N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[b ^ 2 + N[Sqrt[t$95$0], $MachinePrecision] ^ 2], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 30.1%

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

   1. /-rgt-identity 30.1%

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

   2. metadata-eval 30.1%

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

3. Simplified 30.2%

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

   1. associate-*r* 30.2%

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

   2. *-commutative 30.2%

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

   3. metadata-eval 30.2%

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

   4. distribute-lft-neg-in 30.2%

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

   5. associate-*l* 30.2%

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

   6. fma-neg 30.1%

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

   7. flip-- 30.0%

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

   8. sqrt-div 29.9%

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

   9. pow2 29.9%

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

   10. pow2 29.9%

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

   11. pow-prod-up 30.0%

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

   12. metadata-eval 30.0%

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

   13. pow2 30.0%

       \[\leadsto \frac{\frac{\sqrt{{b}^{4} - \color{blue}{{\left(\left(3 \cdot a\right) \cdot c\right)}^{2}}}}{\sqrt{b \cdot b + \left(3 \cdot a\right) \cdot c}} - b}{3 \cdot a} \]

   14. associate-*l* 30.0%

       \[\leadsto \frac{\frac{\sqrt{{b}^{4} - {\color{blue}{\left(3 \cdot \left(a \cdot c\right)\right)}}^{2}}}{\sqrt{b \cdot b + \left(3 \cdot a\right) \cdot c}} - b}{3 \cdot a} \]

   15. fma-define 30.1%

       \[\leadsto \frac{\frac{\sqrt{{b}^{4} - {\left(3 \cdot \left(a \cdot c\right)\right)}^{2}}}{\sqrt{\color{blue}{\mathsf{fma}\left(b, b, \left(3 \cdot a\right) \cdot c\right)}}} - b}{3 \cdot a} \]

   16. associate-*l* 30.1%

       \[\leadsto \frac{\frac{\sqrt{{b}^{4} - {\left(3 \cdot \left(a \cdot c\right)\right)}^{2}}}{\sqrt{\mathsf{fma}\left(b, b, \color{blue}{3 \cdot \left(a \cdot c\right)}\right)}} - b}{3 \cdot a} \]

6. Applied egg-rr 30.1%

   \[\leadsto \frac{\color{blue}{\frac{\sqrt{{b}^{4} - {\left(3 \cdot \left(a \cdot c\right)\right)}^{2}}}{\sqrt{\mathsf{fma}\left(b, b, 3 \cdot \left(a \cdot c\right)\right)}}} - b}{3 \cdot a} \]
7. Step-by-step derivation

   1. *-commutative 30.1%

      \[\leadsto \frac{\frac{\sqrt{{b}^{4} - {\color{blue}{\left(\left(a \cdot c\right) \cdot 3\right)}}^{2}}}{\sqrt{\mathsf{fma}\left(b, b, 3 \cdot \left(a \cdot c\right)\right)}} - b}{3 \cdot a} \]

   2. *-commutative 30.1%

      \[\leadsto \frac{\frac{\sqrt{{b}^{4} - {\left(\color{blue}{\left(c \cdot a\right)} \cdot 3\right)}^{2}}}{\sqrt{\mathsf{fma}\left(b, b, 3 \cdot \left(a \cdot c\right)\right)}} - b}{3 \cdot a} \]

   3. associate-*l* 30.1%

      \[\leadsto \frac{\frac{\sqrt{{b}^{4} - {\color{blue}{\left(c \cdot \left(a \cdot 3\right)\right)}}^{2}}}{\sqrt{\mathsf{fma}\left(b, b, 3 \cdot \left(a \cdot c\right)\right)}} - b}{3 \cdot a} \]

   4. fma-undefine 30.0%

      \[\leadsto \frac{\frac{\sqrt{{b}^{4} - {\left(c \cdot \left(a \cdot 3\right)\right)}^{2}}}{\sqrt{\color{blue}{b \cdot b + 3 \cdot \left(a \cdot c\right)}}} - b}{3 \cdot a} \]

   5. unpow1 30.0%

      \[\leadsto \frac{\frac{\sqrt{{b}^{4} - {\left(c \cdot \left(a \cdot 3\right)\right)}^{2}}}{\sqrt{b \cdot b + \color{blue}{{\left(3 \cdot \left(a \cdot c\right)\right)}^{1}}}} - b}{3 \cdot a} \]

   6. sqr-pow 30.0%

      \[\leadsto \frac{\frac{\sqrt{{b}^{4} - {\left(c \cdot \left(a \cdot 3\right)\right)}^{2}}}{\sqrt{b \cdot b + \color{blue}{{\left(3 \cdot \left(a \cdot c\right)\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(3 \cdot \left(a \cdot c\right)\right)}^{\left(\frac{1}{2}\right)}}}} - b}{3 \cdot a} \]

   7. hypot-define 30.1%

      \[\leadsto \frac{\frac{\sqrt{{b}^{4} - {\left(c \cdot \left(a \cdot 3\right)\right)}^{2}}}{\color{blue}{\mathsf{hypot}\left(b, {\left(3 \cdot \left(a \cdot c\right)\right)}^{\left(\frac{1}{2}\right)}\right)}} - b}{3 \cdot a} \]

   8. metadata-eval 30.1%

      \[\leadsto \frac{\frac{\sqrt{{b}^{4} - {\left(c \cdot \left(a \cdot 3\right)\right)}^{2}}}{\mathsf{hypot}\left(b, {\left(3 \cdot \left(a \cdot c\right)\right)}^{\color{blue}{0.5}}\right)} - b}{3 \cdot a} \]

   9. unpow1/2 30.1%

      \[\leadsto \frac{\frac{\sqrt{{b}^{4} - {\left(c \cdot \left(a \cdot 3\right)\right)}^{2}}}{\mathsf{hypot}\left(b, \color{blue}{\sqrt{3 \cdot \left(a \cdot c\right)}}\right)} - b}{3 \cdot a} \]

   10. *-commutative 30.1%

       \[\leadsto \frac{\frac{\sqrt{{b}^{4} - {\left(c \cdot \left(a \cdot 3\right)\right)}^{2}}}{\mathsf{hypot}\left(b, \sqrt{\color{blue}{\left(a \cdot c\right) \cdot 3}}\right)} - b}{3 \cdot a} \]

   11. *-commutative 30.1%

       \[\leadsto \frac{\frac{\sqrt{{b}^{4} - {\left(c \cdot \left(a \cdot 3\right)\right)}^{2}}}{\mathsf{hypot}\left(b, \sqrt{\color{blue}{\left(c \cdot a\right)} \cdot 3}\right)} - b}{3 \cdot a} \]

   12. associate-*l* 30.1%

       \[\leadsto \frac{\frac{\sqrt{{b}^{4} - {\left(c \cdot \left(a \cdot 3\right)\right)}^{2}}}{\mathsf{hypot}\left(b, \sqrt{\color{blue}{c \cdot \left(a \cdot 3\right)}}\right)} - b}{3 \cdot a} \]

8. Simplified 30.1%

   \[\leadsto \frac{\color{blue}{\frac{\sqrt{{b}^{4} - {\left(c \cdot \left(a \cdot 3\right)\right)}^{2}}}{\mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot 3\right)}\right)}} - b}{3 \cdot a} \]
9. Step-by-step derivation

   1. flip-- 29.9%

      \[\leadsto \frac{\color{blue}{\frac{\frac{\sqrt{{b}^{4} - {\left(c \cdot \left(a \cdot 3\right)\right)}^{2}}}{\mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot 3\right)}\right)} \cdot \frac{\sqrt{{b}^{4} - {\left(c \cdot \left(a \cdot 3\right)\right)}^{2}}}{\mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot 3\right)}\right)} - b \cdot b}{\frac{\sqrt{{b}^{4} - {\left(c \cdot \left(a \cdot 3\right)\right)}^{2}}}{\mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot 3\right)}\right)} + b}}}{3 \cdot a} \]

10. Applied egg-rr 29.9%

    \[\leadsto \frac{\color{blue}{\frac{{\left(\frac{\sqrt{{b}^{4} - {\left(c \cdot a\right)}^{2} \cdot 9}}{\mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot 3\right)}\right)}\right)}^{2} - {b}^{2}}{\frac{\sqrt{{b}^{4} - {\left(c \cdot a\right)}^{2} \cdot 9}}{\mathsf{hypot}\left(b, \sqrt{c \cdot \left(a \cdot 3\right)}\right)} + b}}}{3 \cdot a} \]
11. Step-by-step derivation

12. Simplified 29.9%

    \[\leadsto \frac{\color{blue}{\frac{{\left(\frac{\sqrt{{b}^{4} - {\left(a \cdot \left(c \cdot 3\right)\right)}^{2}}}{\mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot 3\right)}\right)}\right)}^{2} - {b}^{2}}{b + \frac{\sqrt{{b}^{4} - {\left(a \cdot \left(c \cdot 3\right)\right)}^{2}}}{\mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot 3\right)}\right)}}}}{3 \cdot a} \]

13. Taylor expanded in b around 0 0.0%

    \[\leadsto \frac{\frac{\color{blue}{\frac{a \cdot \left(c \cdot {\left(\sqrt{-9}\right)}^{2}\right)}{{\left(\sqrt{3}\right)}^{2}}}}{b + \frac{\sqrt{{b}^{4} - {\left(a \cdot \left(c \cdot 3\right)\right)}^{2}}}{\mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot 3\right)}\right)}}}{3 \cdot a} \]
14. Step-by-step derivation

    1. associate-/l* 0.0%

       \[\leadsto \frac{\frac{\color{blue}{a \cdot \frac{c \cdot {\left(\sqrt{-9}\right)}^{2}}{{\left(\sqrt{3}\right)}^{2}}}}{b + \frac{\sqrt{{b}^{4} - {\left(a \cdot \left(c \cdot 3\right)\right)}^{2}}}{\mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot 3\right)}\right)}}}{3 \cdot a} \]

    2. unpow2 0.0%

       \[\leadsto \frac{\frac{a \cdot \frac{c \cdot \color{blue}{\left(\sqrt{-9} \cdot \sqrt{-9}\right)}}{{\left(\sqrt{3}\right)}^{2}}}{b + \frac{\sqrt{{b}^{4} - {\left(a \cdot \left(c \cdot 3\right)\right)}^{2}}}{\mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot 3\right)}\right)}}}{3 \cdot a} \]

    3. rem-square-sqrt 98.5%

       \[\leadsto \frac{\frac{a \cdot \frac{c \cdot \color{blue}{-9}}{{\left(\sqrt{3}\right)}^{2}}}{b + \frac{\sqrt{{b}^{4} - {\left(a \cdot \left(c \cdot 3\right)\right)}^{2}}}{\mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot 3\right)}\right)}}}{3 \cdot a} \]

    4. unpow2 98.5%

       \[\leadsto \frac{\frac{a \cdot \frac{c \cdot -9}{\color{blue}{\sqrt{3} \cdot \sqrt{3}}}}{b + \frac{\sqrt{{b}^{4} - {\left(a \cdot \left(c \cdot 3\right)\right)}^{2}}}{\mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot 3\right)}\right)}}}{3 \cdot a} \]

    5. rem-square-sqrt 99.1%

       \[\leadsto \frac{\frac{a \cdot \frac{c \cdot -9}{\color{blue}{3}}}{b + \frac{\sqrt{{b}^{4} - {\left(a \cdot \left(c \cdot 3\right)\right)}^{2}}}{\mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot 3\right)}\right)}}}{3 \cdot a} \]

15. Simplified 99.1%

    \[\leadsto \frac{\frac{\color{blue}{a \cdot \frac{c \cdot -9}{3}}}{b + \frac{\sqrt{{b}^{4} - {\left(a \cdot \left(c \cdot 3\right)\right)}^{2}}}{\mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot 3\right)}\right)}}}{3 \cdot a} \]
16. Step-by-step derivation

    1. unpow2 99.1%

       \[\leadsto \frac{\frac{a \cdot \frac{c \cdot -9}{3}}{b + \frac{\sqrt{{b}^{4} - \color{blue}{\left(a \cdot \left(c \cdot 3\right)\right) \cdot \left(a \cdot \left(c \cdot 3\right)\right)}}}{\mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot 3\right)}\right)}}}{3 \cdot a} \]

17. Applied egg-rr 99.1%

    \[\leadsto \frac{\frac{a \cdot \frac{c \cdot -9}{3}}{b + \frac{\sqrt{{b}^{4} - \color{blue}{\left(a \cdot \left(c \cdot 3\right)\right) \cdot \left(a \cdot \left(c \cdot 3\right)\right)}}}{\mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot 3\right)}\right)}}}{3 \cdot a} \]

18. Final simplification 99.1%

    \[\leadsto \frac{\frac{a \cdot \frac{c \cdot -9}{3}}{b + \frac{\sqrt{{b}^{4} - \left(a \cdot \left(c \cdot 3\right)\right) \cdot \left(a \cdot \left(c \cdot 3\right)\right)}}{\mathsf{hypot}\left(b, \sqrt{a \cdot \left(c \cdot 3\right)}\right)}}}{a \cdot 3} \]
19. Add Preprocessing

Alternative 4: 90.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -1:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, \left(c \cdot -3\right) \cdot a\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -1.0)
   (/ (- (sqrt (fma b b (* (* c -3.0) a))) 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 (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -1.0) {
		tmp = (sqrt(fma(b, b, ((c * -3.0) * a))) - 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 (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0)) <= -1.0)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(Float64(c * -3.0) * a))) - 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[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], -1.0], N[(N[(N[Sqrt[N[(b * b + N[(N[(c * -3.0), $MachinePrecision] * a), $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 (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) < -1

   1. Initial program 75.7%

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

      1. /-rgt-identity 75.7%

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

      2. metadata-eval 75.7%

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

   3. Simplified 76.3%

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

   if -1 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 21.9%

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

   3. Taylor expanded in b around inf 95.6%

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

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -1:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, \left(c \cdot -3\right) \cdot a\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -5 \cdot 10^{-8}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, \left(c \cdot -3\right) \cdot a\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -5e-8)
   (/ (- (sqrt (fma b b (* (* c -3.0) a))) b) (* a 3.0))
   (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -5e-8) {
		tmp = (sqrt(fma(b, b, ((c * -3.0) * a))) - b) / (a * 3.0);
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0)) <= -5e-8)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(Float64(c * -3.0) * a))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], -5e-8], N[(N[(N[Sqrt[N[(b * b + N[(N[(c * -3.0), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) < -4.9999999999999998e-8

   1. Initial program 69.8%

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

      1. /-rgt-identity 69.8%

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

      2. metadata-eval 69.8%

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

   3. Simplified 70.0%

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

   if -4.9999999999999998e-8 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 13.4%

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

   3. Taylor expanded in b around inf 94.0%

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

      1. *-commutative 94.0%

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

      2. associate-*l/ 94.0%

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

   5. Simplified 94.0%

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

4. Final simplification 86.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -5 \cdot 10^{-8}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, \left(c \cdot -3\right) \cdot a\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 83.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-8}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0))))
   (if (<= t_0 -5e-8) t_0 (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double t_0 = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	double tmp;
	if (t_0 <= -5e-8) {
		tmp = t_0;
	} else {
		tmp = (c * -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) :: t_0
    real(8) :: tmp
    t_0 = (sqrt(((b * b) - (c * (a * 3.0d0)))) - b) / (a * 3.0d0)
    if (t_0 <= (-5d-8)) then
        tmp = t_0
    else
        tmp = (c * (-0.5d0)) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (Math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	double tmp;
	if (t_0 <= -5e-8) {
		tmp = t_0;
	} else {
		tmp = (c * -0.5) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)
	tmp = 0
	if t_0 <= -5e-8:
		tmp = t_0
	else:
		tmp = (c * -0.5) / b
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0))
	tmp = 0.0
	if (t_0 <= -5e-8)
		tmp = t_0;
	else
		tmp = Float64(Float64(c * -0.5) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	tmp = 0.0;
	if (t_0 <= -5e-8)
		tmp = t_0;
	else
		tmp = (c * -0.5) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-8], t$95$0, N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) < -4.9999999999999998e-8

   1. Initial program 69.8%

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

   if -4.9999999999999998e-8 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 13.4%

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

   3. Taylor expanded in b around inf 94.0%

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

      1. *-commutative 94.0%

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

      2. associate-*l/ 94.0%

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

   5. Simplified 94.0%

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

4. Final simplification 86.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -5 \cdot 10^{-8}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot -0.5}{b}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 81.4% accurate, 23.2× speedup

Math

\[\begin{array}{l} \\ \frac{c \cdot -0.5}{b} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

public static double code(double a, double b, double c) {
	return (c * -0.5) / b;
}

Python

def code(a, b, c):
	return (c * -0.5) / b

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 30.1%

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

3. Taylor expanded in b around inf 81.9%

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

   1. *-commutative 81.9%

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

   2. associate-*l/ 81.9%

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

5. Simplified 81.9%

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

6. Final simplification 81.9%

   \[\leadsto \frac{c \cdot -0.5}{b} \]
7. 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 30.1%

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

   1. add-sqr-sqrt 30.1%

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

   2. difference-of-squares 30.2%

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

   3. associate-*l* 30.2%

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

   4. associate-*l* 30.2%

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

4. Applied egg-rr 30.2%

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

   1. associate-*r* 30.2%

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

   2. *-commutative 30.2%

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

   3. associate-*r* 30.2%

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

   4. *-commutative 30.2%

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

6. Simplified 30.2%

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

7. Taylor expanded in b around inf 3.2%

   \[\leadsto \color{blue}{0.16666666666666666 \cdot \frac{-1 \cdot \left(\sqrt{a \cdot c} \cdot \sqrt{3}\right) + \sqrt{a \cdot c} \cdot \sqrt{3}}{a}} \]
8. Step-by-step derivation

   1. associate-*r/ 3.2%

      \[\leadsto \color{blue}{\frac{0.16666666666666666 \cdot \left(-1 \cdot \left(\sqrt{a \cdot c} \cdot \sqrt{3}\right) + \sqrt{a \cdot c} \cdot \sqrt{3}\right)}{a}} \]

   2. distribute-lft1-in 3.2%

      \[\leadsto \frac{0.16666666666666666 \cdot \color{blue}{\left(\left(-1 + 1\right) \cdot \left(\sqrt{a \cdot c} \cdot \sqrt{3}\right)\right)}}{a} \]

   3. metadata-eval 3.2%

      \[\leadsto \frac{0.16666666666666666 \cdot \left(\color{blue}{0} \cdot \left(\sqrt{a \cdot c} \cdot \sqrt{3}\right)\right)}{a} \]

   4. mul0-lft 3.2%

      \[\leadsto \frac{0.16666666666666666 \cdot \color{blue}{0}}{a} \]

   5. metadata-eval 3.2%

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

9. Simplified 3.2%

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

10. Final simplification 3.2%

    \[\leadsto \frac{0}{a} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024040 
(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)))