Cubic critical, narrow range

Percentage Accurate: 55.3% → 99.0%

Time: 16.1s

Alternatives: 9

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: 55.3% 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: 99.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{-1}{\frac{b + \sqrt{\mathsf{fma}\left(b, b, \left(-3 \cdot c\right) \cdot a\right)}}{a \cdot \left(c \cdot 3\right)}}}{a \cdot 3} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (/ -1.0 (/ (+ b (sqrt (fma b b (* (* -3.0 c) a)))) (* a (* c 3.0))))
  (* a 3.0)))

C

double code(double a, double b, double c) {
	return (-1.0 / ((b + sqrt(fma(b, b, ((-3.0 * c) * a)))) / (a * (c * 3.0)))) / (a * 3.0);
}

Julia

function code(a, b, c)
	return Float64(Float64(-1.0 / Float64(Float64(b + sqrt(fma(b, b, Float64(Float64(-3.0 * c) * a)))) / Float64(a * Float64(c * 3.0)))) / Float64(a * 3.0))
end

Wolfram

code[a_, b_, c_] := N[(N[(-1.0 / N[(N[(b + N[Sqrt[N[(b * b + N[(N[(-3.0 * c), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(a * N[(c * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.2%

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

3. Taylor expanded in a around 0 55.2%

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

   1. *-commutative 55.2%

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

   2. metadata-eval 55.2%

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

   3. distribute-rgt-neg-in 55.2%

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

   4. associate-*r* 55.2%

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

   5. distribute-rgt-neg-in 55.2%

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

   6. distribute-rgt-neg-in 55.2%

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

   7. metadata-eval 55.2%

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

5. Simplified 55.2%

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

   1. flip-+ 55.4%

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

   2. pow2 55.4%

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

   3. add-sqr-sqrt 57.0%

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

   4. associate-*r* 57.0%

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

   5. pow2 57.0%

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

   6. *-commutative 57.0%

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

   7. *-commutative 57.0%

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

   8. associate-*r* 57.0%

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

   9. pow2 57.0%

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

   10. *-commutative 57.0%

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

   11. *-commutative 57.0%

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

7. Applied egg-rr 57.0%

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

   1. clear-num 57.0%

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

   2. inv-pow 57.0%

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

9. Applied egg-rr 56.8%

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

    1. unpow-1 56.8%

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

    2. *-commutative 56.8%

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

    3. associate-*r* 56.8%

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

    4. fma-undefine 57.0%

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

    5. unpow2 57.0%

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

    6. metadata-eval 57.0%

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

    7. cancel-sign-sub-inv 57.0%

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

    8. *-commutative 57.0%

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

    9. associate-*l* 57.0%

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

    10. associate-+l- 99.1%

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

    11. +-inverses 99.1%

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

    12. +-lft-identity 99.1%

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

    13. *-commutative 99.1%

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

    14. *-commutative 99.1%

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

11. Simplified 99.1%

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

12. Final simplification 99.1%

    \[\leadsto \frac{\frac{-1}{\frac{b + \sqrt{\mathsf{fma}\left(b, b, \left(-3 \cdot c\right) \cdot a\right)}}{a \cdot \left(c \cdot 3\right)}}}{a \cdot 3} \]
13. Add Preprocessing

Alternative 2: 89.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.9:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, \left(-3 \cdot c\right) \cdot a\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\frac{-0.6666666666666666 \cdot \frac{b}{a} + c \cdot \left(0.375 \cdot \frac{c \cdot a}{{b}^{3}} + 0.5 \cdot \frac{1}{b}\right)}{c}}}{a \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -0.9)
   (/ (- (sqrt (fma b b (* (* -3.0 c) a))) b) (* a 3.0))
   (/
    (/
     1.0
     (/
      (+
       (* -0.6666666666666666 (/ b a))
       (* c (+ (* 0.375 (/ (* c a) (pow b 3.0))) (* 0.5 (/ 1.0 b)))))
      c))
    (* a 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)) <= -0.9) {
		tmp = (sqrt(fma(b, b, ((-3.0 * c) * a))) - b) / (a * 3.0);
	} else {
		tmp = (1.0 / (((-0.6666666666666666 * (b / a)) + (c * ((0.375 * ((c * a) / pow(b, 3.0))) + (0.5 * (1.0 / b))))) / c)) / (a * 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)) <= -0.9)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(Float64(-3.0 * c) * a))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(1.0 / Float64(Float64(Float64(-0.6666666666666666 * Float64(b / a)) + Float64(c * Float64(Float64(0.375 * Float64(Float64(c * a) / (b ^ 3.0))) + Float64(0.5 * Float64(1.0 / b))))) / c)) / Float64(a * 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], -0.9], N[(N[(N[Sqrt[N[(b * b + N[(N[(-3.0 * c), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(N[(N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision] + N[(c * N[(N[(0.375 * N[(N[(c * a), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.900000000000000022

   1. Initial program 86.0%

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

      1. /-rgt-identity 86.0%

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

      2. metadata-eval 86.0%

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

   3. Simplified 86.0%

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

   if -0.900000000000000022 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 49.9%

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

   3. Taylor expanded in a around 0 49.9%

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

      1. *-commutative 49.9%

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

      2. metadata-eval 49.9%

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

      3. distribute-rgt-neg-in 49.9%

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

      4. associate-*r* 49.9%

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

      5. distribute-rgt-neg-in 49.9%

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

      6. distribute-rgt-neg-in 49.9%

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

      7. metadata-eval 49.9%

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

   5. Simplified 49.9%

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

      1. flip-+ 50.3%

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

      2. pow2 50.3%

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

      3. add-sqr-sqrt 51.9%

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

      4. associate-*r* 51.9%

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

      5. pow2 51.9%

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

      6. *-commutative 51.9%

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

      7. *-commutative 51.9%

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

      8. associate-*r* 51.9%

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

      9. pow2 51.9%

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

      10. *-commutative 51.9%

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

      11. *-commutative 51.9%

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

   7. Applied egg-rr 51.9%

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

      1. clear-num 51.9%

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

      2. inv-pow 51.9%

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

   9. Applied egg-rr 51.7%

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

       1. unpow-1 51.7%

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

       2. *-commutative 51.7%

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

       3. associate-*r* 51.7%

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

       4. fma-undefine 51.9%

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

       5. unpow2 51.9%

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

       6. metadata-eval 51.9%

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

       7. cancel-sign-sub-inv 51.9%

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

       8. *-commutative 51.9%

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

       9. associate-*l* 51.9%

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

       10. associate-+l- 99.1%

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

       11. +-inverses 99.1%

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

       12. +-lft-identity 99.1%

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

       13. *-commutative 99.1%

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

       14. *-commutative 99.1%

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

   11. Simplified 99.1%

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

   12. Taylor expanded in c around 0 90.3%

       \[\leadsto \frac{\frac{1}{\color{blue}{\frac{-0.6666666666666666 \cdot \frac{b}{a} + c \cdot \left(0.375 \cdot \frac{a \cdot c}{{b}^{3}} + 0.5 \cdot \frac{1}{b}\right)}{c}}}}{3 \cdot a} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.9:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, \left(-3 \cdot c\right) \cdot a\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\frac{-0.6666666666666666 \cdot \frac{b}{a} + c \cdot \left(0.375 \cdot \frac{c \cdot a}{{b}^{3}} + 0.5 \cdot \frac{1}{b}\right)}{c}}}{a \cdot 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{if}\;t\_0 \leq -0.9:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\frac{-0.6666666666666666 \cdot \frac{b}{a} + c \cdot \left(0.375 \cdot \frac{c \cdot a}{{b}^{3}} + 0.5 \cdot \frac{1}{b}\right)}{c}}}{a \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0))))
   (if (<= t_0 -0.9)
     t_0
     (/
      (/
       1.0
       (/
        (+
         (* -0.6666666666666666 (/ b a))
         (* c (+ (* 0.375 (/ (* c a) (pow b 3.0))) (* 0.5 (/ 1.0 b)))))
        c))
      (* a 3.0)))))

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 <= -0.9) {
		tmp = t_0;
	} else {
		tmp = (1.0 / (((-0.6666666666666666 * (b / a)) + (c * ((0.375 * ((c * a) / pow(b, 3.0))) + (0.5 * (1.0 / b))))) / c)) / (a * 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) :: tmp
    t_0 = (sqrt(((b * b) - (c * (a * 3.0d0)))) - b) / (a * 3.0d0)
    if (t_0 <= (-0.9d0)) then
        tmp = t_0
    else
        tmp = (1.0d0 / ((((-0.6666666666666666d0) * (b / a)) + (c * ((0.375d0 * ((c * a) / (b ** 3.0d0))) + (0.5d0 * (1.0d0 / b))))) / c)) / (a * 3.0d0)
    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 <= -0.9) {
		tmp = t_0;
	} else {
		tmp = (1.0 / (((-0.6666666666666666 * (b / a)) + (c * ((0.375 * ((c * a) / Math.pow(b, 3.0))) + (0.5 * (1.0 / b))))) / c)) / (a * 3.0);
	}
	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 <= -0.9:
		tmp = t_0
	else:
		tmp = (1.0 / (((-0.6666666666666666 * (b / a)) + (c * ((0.375 * ((c * a) / math.pow(b, 3.0))) + (0.5 * (1.0 / b))))) / c)) / (a * 3.0)
	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 <= -0.9)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 / Float64(Float64(Float64(-0.6666666666666666 * Float64(b / a)) + Float64(c * Float64(Float64(0.375 * Float64(Float64(c * a) / (b ^ 3.0))) + Float64(0.5 * Float64(1.0 / b))))) / c)) / Float64(a * 3.0));
	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 <= -0.9)
		tmp = t_0;
	else
		tmp = (1.0 / (((-0.6666666666666666 * (b / a)) + (c * ((0.375 * ((c * a) / (b ^ 3.0))) + (0.5 * (1.0 / b))))) / c)) / (a * 3.0);
	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, -0.9], t$95$0, N[(N[(1.0 / N[(N[(N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision] + N[(c * N[(N[(0.375 * N[(N[(c * a), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(1.0 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.900000000000000022

   1. Initial program 86.0%

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

   if -0.900000000000000022 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 49.9%

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

   3. Taylor expanded in a around 0 49.9%

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

      1. *-commutative 49.9%

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

      2. metadata-eval 49.9%

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

      3. distribute-rgt-neg-in 49.9%

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

      4. associate-*r* 49.9%

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

      5. distribute-rgt-neg-in 49.9%

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

      6. distribute-rgt-neg-in 49.9%

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

      7. metadata-eval 49.9%

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

   5. Simplified 49.9%

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

      1. flip-+ 50.3%

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

      2. pow2 50.3%

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

      3. add-sqr-sqrt 51.9%

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

      4. associate-*r* 51.9%

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

      5. pow2 51.9%

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

      6. *-commutative 51.9%

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

      7. *-commutative 51.9%

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

      8. associate-*r* 51.9%

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

      9. pow2 51.9%

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

      10. *-commutative 51.9%

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

      11. *-commutative 51.9%

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

   7. Applied egg-rr 51.9%

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

      1. clear-num 51.9%

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

      2. inv-pow 51.9%

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

   9. Applied egg-rr 51.7%

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

       1. unpow-1 51.7%

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

       2. *-commutative 51.7%

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

       3. associate-*r* 51.7%

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

       4. fma-undefine 51.9%

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

       5. unpow2 51.9%

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

       6. metadata-eval 51.9%

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

       7. cancel-sign-sub-inv 51.9%

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

       8. *-commutative 51.9%

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

       9. associate-*l* 51.9%

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

       10. associate-+l- 99.1%

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

       11. +-inverses 99.1%

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

       12. +-lft-identity 99.1%

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

       13. *-commutative 99.1%

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

       14. *-commutative 99.1%

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

   11. Simplified 99.1%

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

   12. Taylor expanded in c around 0 90.3%

       \[\leadsto \frac{\frac{1}{\color{blue}{\frac{-0.6666666666666666 \cdot \frac{b}{a} + c \cdot \left(0.375 \cdot \frac{a \cdot c}{{b}^{3}} + 0.5 \cdot \frac{1}{b}\right)}{c}}}}{3 \cdot a} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.9:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\frac{-0.6666666666666666 \cdot \frac{b}{a} + c \cdot \left(0.375 \cdot \frac{c \cdot a}{{b}^{3}} + 0.5 \cdot \frac{1}{b}\right)}{c}}}{a \cdot 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{if}\;t\_0 \leq -0.06:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\frac{-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}}{c}}}{a \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0))))
   (if (<= t_0 -0.06)
     t_0
     (/
      (/ 1.0 (/ (+ (* -0.6666666666666666 (/ b a)) (* 0.5 (/ c b))) c))
      (* a 3.0)))))

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 <= -0.06) {
		tmp = t_0;
	} else {
		tmp = (1.0 / (((-0.6666666666666666 * (b / a)) + (0.5 * (c / b))) / c)) / (a * 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) :: tmp
    t_0 = (sqrt(((b * b) - (c * (a * 3.0d0)))) - b) / (a * 3.0d0)
    if (t_0 <= (-0.06d0)) then
        tmp = t_0
    else
        tmp = (1.0d0 / ((((-0.6666666666666666d0) * (b / a)) + (0.5d0 * (c / b))) / c)) / (a * 3.0d0)
    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 <= -0.06) {
		tmp = t_0;
	} else {
		tmp = (1.0 / (((-0.6666666666666666 * (b / a)) + (0.5 * (c / b))) / c)) / (a * 3.0);
	}
	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 <= -0.06:
		tmp = t_0
	else:
		tmp = (1.0 / (((-0.6666666666666666 * (b / a)) + (0.5 * (c / b))) / c)) / (a * 3.0)
	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 <= -0.06)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 / Float64(Float64(Float64(-0.6666666666666666 * Float64(b / a)) + Float64(0.5 * Float64(c / b))) / c)) / Float64(a * 3.0));
	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 <= -0.06)
		tmp = t_0;
	else
		tmp = (1.0 / (((-0.6666666666666666 * (b / a)) + (0.5 * (c / b))) / c)) / (a * 3.0);
	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, -0.06], t$95$0, N[(N[(1.0 / N[(N[(N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.059999999999999998

   1. Initial program 81.6%

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

   if -0.059999999999999998 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 46.9%

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

   3. Taylor expanded in a around 0 46.9%

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

      1. *-commutative 46.9%

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

      2. metadata-eval 46.9%

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

      3. distribute-rgt-neg-in 46.9%

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

      4. associate-*r* 46.9%

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

      5. distribute-rgt-neg-in 46.9%

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

      6. distribute-rgt-neg-in 46.9%

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

      7. metadata-eval 46.9%

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

   5. Simplified 46.9%

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

      1. flip-+ 47.3%

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

      2. pow2 47.3%

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

      3. add-sqr-sqrt 48.8%

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

      4. associate-*r* 48.8%

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

      5. pow2 48.8%

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

      6. *-commutative 48.8%

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

      7. *-commutative 48.8%

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

      8. associate-*r* 48.8%

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

      9. pow2 48.8%

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

      10. *-commutative 48.8%

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

      11. *-commutative 48.8%

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

   7. Applied egg-rr 48.8%

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

      1. clear-num 48.8%

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

      2. inv-pow 48.8%

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

   9. Applied egg-rr 48.6%

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

       1. unpow-1 48.6%

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

       2. *-commutative 48.6%

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

       3. associate-*r* 48.6%

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

       4. fma-undefine 48.8%

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

       5. unpow2 48.8%

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

       6. metadata-eval 48.8%

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

       7. cancel-sign-sub-inv 48.8%

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

       8. *-commutative 48.8%

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

       9. associate-*l* 48.8%

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

       10. associate-+l- 99.2%

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

       11. +-inverses 99.2%

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

       12. +-lft-identity 99.2%

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

       13. *-commutative 99.2%

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

       14. *-commutative 99.2%

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

   11. Simplified 99.2%

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

   12. Taylor expanded in c around 0 86.8%

       \[\leadsto \frac{\frac{1}{\color{blue}{\frac{-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}}{c}}}}{3 \cdot a} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.06:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\frac{-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}}{c}}}{a \cdot 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.2% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 55.2%

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

3. Taylor expanded in a around 0 55.2%

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

   1. *-commutative 55.2%

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

   2. metadata-eval 55.2%

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

   3. distribute-rgt-neg-in 55.2%

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

   4. associate-*r* 55.2%

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

   5. distribute-rgt-neg-in 55.2%

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

   6. distribute-rgt-neg-in 55.2%

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

   7. metadata-eval 55.2%

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

5. Simplified 55.2%

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

   1. flip-+ 55.4%

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

   2. pow2 55.4%

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

   3. add-sqr-sqrt 57.0%

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

   4. associate-*r* 57.0%

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

   5. pow2 57.0%

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

   6. *-commutative 57.0%

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

   7. *-commutative 57.0%

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

   8. associate-*r* 57.0%

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

   9. pow2 57.0%

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

   10. *-commutative 57.0%

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

   11. *-commutative 57.0%

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

7. Applied egg-rr 57.0%

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

   1. div-inv 57.0%

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

9. Applied egg-rr 56.8%

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

    1. *-commutative 56.8%

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

    2. times-frac 56.8%

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

    3. *-lft-identity 56.8%

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

    4. associate-/r* 56.8%

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

11. Simplified 99.1%

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

12. Final simplification 99.1%

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

Alternative 6: 85.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 7:\\ \;\;\;\;\frac{\sqrt{b \cdot b - a \cdot \left(c \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\frac{-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}}{c}}}{a \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 7.0)
   (/ (- (sqrt (- (* b b) (* a (* c 3.0)))) b) (* a 3.0))
   (/
    (/ 1.0 (/ (+ (* -0.6666666666666666 (/ b a)) (* 0.5 (/ c b))) c))
    (* a 3.0))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 7.0) {
		tmp = (sqrt(((b * b) - (a * (c * 3.0)))) - b) / (a * 3.0);
	} else {
		tmp = (1.0 / (((-0.6666666666666666 * (b / a)) + (0.5 * (c / b))) / c)) / (a * 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) :: tmp
    if (b <= 7.0d0) then
        tmp = (sqrt(((b * b) - (a * (c * 3.0d0)))) - b) / (a * 3.0d0)
    else
        tmp = (1.0d0 / ((((-0.6666666666666666d0) * (b / a)) + (0.5d0 * (c / b))) / c)) / (a * 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 7.0) {
		tmp = (Math.sqrt(((b * b) - (a * (c * 3.0)))) - b) / (a * 3.0);
	} else {
		tmp = (1.0 / (((-0.6666666666666666 * (b / a)) + (0.5 * (c / b))) / c)) / (a * 3.0);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 7.0:
		tmp = (math.sqrt(((b * b) - (a * (c * 3.0)))) - b) / (a * 3.0)
	else:
		tmp = (1.0 / (((-0.6666666666666666 * (b / a)) + (0.5 * (c / b))) / c)) / (a * 3.0)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 7.0)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(a * Float64(c * 3.0)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(1.0 / Float64(Float64(Float64(-0.6666666666666666 * Float64(b / a)) + Float64(0.5 * Float64(c / b))) / c)) / Float64(a * 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 7.0)
		tmp = (sqrt(((b * b) - (a * (c * 3.0)))) - b) / (a * 3.0);
	else
		tmp = (1.0 / (((-0.6666666666666666 * (b / a)) + (0.5 * (c / b))) / c)) / (a * 3.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 7.0], 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[(N[(1.0 / N[(N[(N[(-0.6666666666666666 * N[(b / a), $MachinePrecision]), $MachinePrecision] + N[(0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 7

   1. Initial program 81.2%

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

   3. Taylor expanded in a around 0 81.1%

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

      1. *-commutative 81.1%

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

      2. metadata-eval 81.1%

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

      3. distribute-rgt-neg-in 81.1%

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

      4. associate-*r* 81.1%

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

      5. distribute-rgt-neg-in 81.1%

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

      6. distribute-rgt-neg-in 81.1%

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

      7. metadata-eval 81.1%

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

   5. Simplified 81.1%

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

   if 7 < b

   1. Initial program 47.4%

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

   3. Taylor expanded in a around 0 47.4%

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

      1. *-commutative 47.4%

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

      2. metadata-eval 47.4%

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

      3. distribute-rgt-neg-in 47.4%

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

      4. associate-*r* 47.4%

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

      5. distribute-rgt-neg-in 47.4%

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

      6. distribute-rgt-neg-in 47.4%

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

      7. metadata-eval 47.4%

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

   5. Simplified 47.4%

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

      1. flip-+ 47.6%

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

      2. pow2 47.6%

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

      3. add-sqr-sqrt 49.3%

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

      4. associate-*r* 49.3%

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

      5. pow2 49.3%

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

      6. *-commutative 49.3%

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

      7. *-commutative 49.3%

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

      8. associate-*r* 49.3%

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

      9. pow2 49.3%

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

      10. *-commutative 49.3%

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

      11. *-commutative 49.3%

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

   7. Applied egg-rr 49.3%

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

      1. clear-num 49.3%

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

      2. inv-pow 49.3%

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

   9. Applied egg-rr 49.1%

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

       1. unpow-1 49.1%

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

       2. *-commutative 49.1%

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

       3. associate-*r* 49.1%

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

       4. fma-undefine 49.3%

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

       5. unpow2 49.3%

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

       6. metadata-eval 49.3%

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

       7. cancel-sign-sub-inv 49.3%

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

       8. *-commutative 49.3%

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

       9. associate-*l* 49.3%

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

       10. associate-+l- 99.1%

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

       11. +-inverses 99.1%

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

       12. +-lft-identity 99.1%

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

       13. *-commutative 99.1%

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

       14. *-commutative 99.1%

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

   11. Simplified 99.1%

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

   12. Taylor expanded in c around 0 86.5%

       \[\leadsto \frac{\frac{1}{\color{blue}{\frac{-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}}{c}}}}{3 \cdot a} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 7:\\ \;\;\;\;\frac{\sqrt{b \cdot b - a \cdot \left(c \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\frac{-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}}{c}}}{a \cdot 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 81.8% accurate, 6.1× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{1}{\frac{-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}}{c}}}{a \cdot 3} \end{array} \]

FPCore

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

C

double code(double a, double b, double c) {
	return (1.0 / (((-0.6666666666666666 * (b / a)) + (0.5 * (c / b))) / c)) / (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 = (1.0d0 / ((((-0.6666666666666666d0) * (b / a)) + (0.5d0 * (c / b))) / c)) / (a * 3.0d0)
end function

Java

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

Python

def code(a, b, c):
	return (1.0 / (((-0.6666666666666666 * (b / a)) + (0.5 * (c / b))) / c)) / (a * 3.0)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.2%

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

3. Taylor expanded in a around 0 55.2%

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

   1. *-commutative 55.2%

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

   2. metadata-eval 55.2%

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

   3. distribute-rgt-neg-in 55.2%

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

   4. associate-*r* 55.2%

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

   5. distribute-rgt-neg-in 55.2%

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

   6. distribute-rgt-neg-in 55.2%

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

   7. metadata-eval 55.2%

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

5. Simplified 55.2%

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

   1. flip-+ 55.4%

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

   2. pow2 55.4%

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

   3. add-sqr-sqrt 57.0%

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

   4. associate-*r* 57.0%

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

   5. pow2 57.0%

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

   6. *-commutative 57.0%

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

   7. *-commutative 57.0%

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

   8. associate-*r* 57.0%

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

   9. pow2 57.0%

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

   10. *-commutative 57.0%

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

   11. *-commutative 57.0%

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

7. Applied egg-rr 57.0%

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

   1. clear-num 57.0%

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

   2. inv-pow 57.0%

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

9. Applied egg-rr 56.8%

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

    1. unpow-1 56.8%

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

    2. *-commutative 56.8%

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

    3. associate-*r* 56.8%

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

    4. fma-undefine 57.0%

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

    5. unpow2 57.0%

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

    6. metadata-eval 57.0%

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

    7. cancel-sign-sub-inv 57.0%

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

    8. *-commutative 57.0%

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

    9. associate-*l* 57.0%

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

    10. associate-+l- 99.1%

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

    11. +-inverses 99.1%

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

    12. +-lft-identity 99.1%

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

    13. *-commutative 99.1%

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

    14. *-commutative 99.1%

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

11. Simplified 99.1%

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

12. Taylor expanded in c around 0 80.7%

    \[\leadsto \frac{\frac{1}{\color{blue}{\frac{-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}}{c}}}}{3 \cdot a} \]

13. Final simplification 80.7%

    \[\leadsto \frac{\frac{1}{\frac{-0.6666666666666666 \cdot \frac{b}{a} + 0.5 \cdot \frac{c}{b}}{c}}}{a \cdot 3} \]
14. Add Preprocessing

Alternative 8: 64.5% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.2%

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

3. Taylor expanded in b around inf 64.0%

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

   1. associate-*r/ 64.0%

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

   2. *-commutative 64.0%

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

5. Simplified 64.0%

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

6. Final simplification 64.0%

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

Alternative 9: 3.2% accurate, 38.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.2%

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

   1. add-sqr-sqrt 55.1%

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

   2. difference-of-squares 55.2%

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

   3. associate-*l* 55.2%

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

   4. associate-*l* 55.2%

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

4. Applied egg-rr 55.2%

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

   1. *-commutative 55.2%

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

   2. *-commutative 55.2%

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

6. Simplified 55.2%

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

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 2024068 
(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)))