Cubic critical, narrow range

Percentage Accurate: 55.3% → 99.3%

Time: 14.6s

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: 99.3% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.1%

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

3. Taylor expanded in a around 0 55.1%

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

   1. associate-*r* 55.1%

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

   2. *-commutative 55.1%

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

   3. associate-*l* 55.1%

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

5. Simplified 55.1%

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

   1. flip-+ 54.8%

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

   2. pow2 54.8%

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

   3. add-sqr-sqrt 56.4%

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

   4. pow2 56.4%

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

   5. associate-*r* 56.4%

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

   6. pow2 56.4%

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

   7. associate-*r* 56.4%

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

7. Applied egg-rr 56.4%

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

   1. associate--r- 99.2%

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

   2. associate-*l* 99.1%

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

   3. associate-*l* 99.1%

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

9. Simplified 99.1%

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

    1. expm1-log1p-u 84.7%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{\left({\left(-b\right)}^{2} - {b}^{2}\right) + a \cdot \left(3 \cdot c\right)}{\left(-b\right) - \sqrt{{b}^{2} - a \cdot \left(3 \cdot c\right)}}}{3 \cdot a}\right)\right)} \]

    2. expm1-udef 61.9%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{\left({\left(-b\right)}^{2} - {b}^{2}\right) + a \cdot \left(3 \cdot c\right)}{\left(-b\right) - \sqrt{{b}^{2} - a \cdot \left(3 \cdot c\right)}}}{3 \cdot a}\right)} - 1} \]

11. Applied egg-rr 61.9%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\mathsf{fma}\left(a, 3 \cdot c, {b}^{2} - {b}^{2}\right)}{\left(a \cdot 3\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - \left(a \cdot 3\right) \cdot c}\right)}\right)} - 1} \]
12. Step-by-step derivation

    1. expm1-def 84.8%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left(a, 3 \cdot c, {b}^{2} - {b}^{2}\right)}{\left(a \cdot 3\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - \left(a \cdot 3\right) \cdot c}\right)}\right)\right)} \]

    2. expm1-log1p 99.1%

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

    3. fma-udef 99.1%

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

    4. associate-*l* 99.3%

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

    5. +-inverses 99.3%

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

    6. +-rgt-identity 99.3%

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

    7. *-commutative 99.3%

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

    8. sub-neg 99.3%

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

    9. +-commutative 99.3%

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

    10. distribute-lft-neg-in 99.3%

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

    11. *-commutative 99.3%

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

    12. fma-udef 99.3%

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

    13. distribute-rgt-neg-in 99.3%

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

    14. metadata-eval 99.3%

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

13. Simplified 99.3%

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

    1. fma-udef 99.3%

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

15. Applied egg-rr 99.3%

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

16. Final simplification 99.3%

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

Alternative 2: 89.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -10:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{-0.6666666666666666 \cdot \frac{b}{c \cdot a} + \left(0.375 \cdot \frac{c \cdot a}{{b}^{3}} + 0.5 \cdot \frac{1}{b}\right)}}{a \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -10.0)
   (/ (- (sqrt (fma b b (* a (* c -3.0)))) b) (* a 3.0))
   (/
    (/
     1.0
     (+
      (* -0.6666666666666666 (/ b (* c a)))
      (+ (* 0.375 (/ (* c a) (pow b 3.0))) (* 0.5 (/ 1.0 b)))))
    (* 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)) <= -10.0) {
		tmp = (sqrt(fma(b, b, (a * (c * -3.0)))) - b) / (a * 3.0);
	} else {
		tmp = (1.0 / ((-0.6666666666666666 * (b / (c * a))) + ((0.375 * ((c * a) / pow(b, 3.0))) + (0.5 * (1.0 / b))))) / (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)) <= -10.0)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(c * -3.0)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(1.0 / Float64(Float64(-0.6666666666666666 * Float64(b / Float64(c * a))) + Float64(Float64(0.375 * Float64(Float64(c * a) / (b ^ 3.0))) + Float64(0.5 * Float64(1.0 / b))))) / 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], -10.0], N[(N[(N[Sqrt[N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(N[(-0.6666666666666666 * N[(b / N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 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] / 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 3 a) c)))) (*.f64 3 a)) < -10

   1. Initial program 89.3%

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

      1. +-commutative 89.3%

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

      2. sqr-neg 89.3%

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

      3. unsub-neg 89.3%

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

      4. div-sub 87.3%

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

      5. --rgt-identity 87.3%

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

      6. div-sub 89.3%

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

   3. Simplified 89.4%

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

   if -10 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 52.2%

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

   3. Taylor expanded in a around 0 52.2%

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

      1. associate-*r* 52.2%

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

      2. *-commutative 52.2%

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

      3. associate-*l* 52.2%

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

   5. Simplified 52.2%

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

      1. flip-+ 51.9%

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

      2. pow2 51.9%

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

      3. add-sqr-sqrt 53.5%

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

      4. pow2 53.5%

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

      5. associate-*r* 53.5%

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

      6. pow2 53.5%

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

      7. associate-*r* 53.5%

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

   7. Applied egg-rr 53.5%

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

      1. associate--r- 99.2%

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

      2. associate-*l* 99.1%

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

      3. associate-*l* 99.1%

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

   9. Simplified 99.1%

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

       1. clear-num 99.0%

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

       2. inv-pow 99.0%

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

       3. associate-*r* 99.0%

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

       4. +-commutative 99.0%

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

       5. fma-def 99.0%

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

       6. neg-mul-1 99.0%

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

       7. unpow-prod-down 99.0%

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

       8. metadata-eval 99.0%

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

       9. *-un-lft-identity 99.0%

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

   11. Applied egg-rr 99.0%

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

       1. unpow-1 99.0%

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

       2. sub-neg 99.0%

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

       3. +-commutative 99.0%

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

       4. distribute-lft-neg-in 99.0%

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

       5. *-commutative 99.0%

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

       6. fma-udef 99.0%

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

       7. distribute-rgt-neg-in 99.0%

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

       8. metadata-eval 99.0%

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

       9. fma-udef 99.0%

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

       10. associate-*l* 99.2%

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

       11. +-inverses 99.2%

           \[\leadsto \frac{\frac{1}{\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c, a \cdot -3, {b}^{2}\right)}}{\left(a \cdot 3\right) \cdot c + \color{blue}{0}}}}{3 \cdot a} \]

       12. +-rgt-identity 99.2%

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

       13. *-commutative 99.2%

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

   13. Simplified 99.2%

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

   14. Taylor expanded in b around inf 90.9%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -10:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{-0.6666666666666666 \cdot \frac{b}{c \cdot a} + \left(0.375 \cdot \frac{c \cdot a}{{b}^{3}} + 0.5 \cdot \frac{1}{b}\right)}}{a \cdot 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 89.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -10:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(c \cdot a\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{-0.6666666666666666 \cdot \frac{b}{c \cdot a} + \left(0.375 \cdot \frac{c \cdot a}{{b}^{3}} + 0.5 \cdot \frac{1}{b}\right)}}{a \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -10.0)
   (/ (- (sqrt (- (* b b) (* 3.0 (* c a)))) b) (* a 3.0))
   (/
    (/
     1.0
     (+
      (* -0.6666666666666666 (/ b (* c a)))
      (+ (* 0.375 (/ (* c a) (pow b 3.0))) (* 0.5 (/ 1.0 b)))))
    (* 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)) <= -10.0) {
		tmp = (sqrt(((b * b) - (3.0 * (c * a)))) - b) / (a * 3.0);
	} else {
		tmp = (1.0 / ((-0.6666666666666666 * (b / (c * a))) + ((0.375 * ((c * a) / pow(b, 3.0))) + (0.5 * (1.0 / b))))) / (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 (((sqrt(((b * b) - (c * (a * 3.0d0)))) - b) / (a * 3.0d0)) <= (-10.0d0)) then
        tmp = (sqrt(((b * b) - (3.0d0 * (c * a)))) - b) / (a * 3.0d0)
    else
        tmp = (1.0d0 / (((-0.6666666666666666d0) * (b / (c * a))) + ((0.375d0 * ((c * a) / (b ** 3.0d0))) + (0.5d0 * (1.0d0 / b))))) / (a * 3.0d0)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (((Math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -10.0) {
		tmp = (Math.sqrt(((b * b) - (3.0 * (c * a)))) - b) / (a * 3.0);
	} else {
		tmp = (1.0 / ((-0.6666666666666666 * (b / (c * a))) + ((0.375 * ((c * a) / Math.pow(b, 3.0))) + (0.5 * (1.0 / b))))) / (a * 3.0);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if ((math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -10.0:
		tmp = (math.sqrt(((b * b) - (3.0 * (c * a)))) - b) / (a * 3.0)
	else:
		tmp = (1.0 / ((-0.6666666666666666 * (b / (c * a))) + ((0.375 * ((c * a) / math.pow(b, 3.0))) + (0.5 * (1.0 / b))))) / (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)) <= -10.0)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(3.0 * Float64(c * a)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(1.0 / Float64(Float64(-0.6666666666666666 * Float64(b / Float64(c * a))) + Float64(Float64(0.375 * Float64(Float64(c * a) / (b ^ 3.0))) + Float64(0.5 * Float64(1.0 / b))))) / Float64(a * 3.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -10.0)
		tmp = (sqrt(((b * b) - (3.0 * (c * a)))) - b) / (a * 3.0);
	else
		tmp = (1.0 / ((-0.6666666666666666 * (b / (c * a))) + ((0.375 * ((c * a) / (b ^ 3.0))) + (0.5 * (1.0 / b))))) / (a * 3.0);
	end
	tmp_2 = 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], -10.0], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(3.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / N[(N[(-0.6666666666666666 * N[(b / N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 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] / 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 3 a) c)))) (*.f64 3 a)) < -10

   1. Initial program 89.3%

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

   3. Taylor expanded in a around 0 89.4%

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

      1. *-commutative 89.4%

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

   5. Simplified 89.4%

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

   if -10 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 52.2%

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

   3. Taylor expanded in a around 0 52.2%

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

      1. associate-*r* 52.2%

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

      2. *-commutative 52.2%

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

      3. associate-*l* 52.2%

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

   5. Simplified 52.2%

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

      1. flip-+ 51.9%

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

      2. pow2 51.9%

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

      3. add-sqr-sqrt 53.5%

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

      4. pow2 53.5%

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

      5. associate-*r* 53.5%

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

      6. pow2 53.5%

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

      7. associate-*r* 53.5%

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

   7. Applied egg-rr 53.5%

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

      1. associate--r- 99.2%

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

      2. associate-*l* 99.1%

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

      3. associate-*l* 99.1%

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

   9. Simplified 99.1%

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

       1. clear-num 99.0%

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

       2. inv-pow 99.0%

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

       3. associate-*r* 99.0%

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

       4. +-commutative 99.0%

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

       5. fma-def 99.0%

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

       6. neg-mul-1 99.0%

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

       7. unpow-prod-down 99.0%

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

       8. metadata-eval 99.0%

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

       9. *-un-lft-identity 99.0%

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

   11. Applied egg-rr 99.0%

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

       1. unpow-1 99.0%

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

       2. sub-neg 99.0%

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

       3. +-commutative 99.0%

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

       4. distribute-lft-neg-in 99.0%

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

       5. *-commutative 99.0%

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

       6. fma-udef 99.0%

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

       7. distribute-rgt-neg-in 99.0%

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

       8. metadata-eval 99.0%

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

       9. fma-udef 99.0%

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

       10. associate-*l* 99.2%

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

       11. +-inverses 99.2%

           \[\leadsto \frac{\frac{1}{\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c, a \cdot -3, {b}^{2}\right)}}{\left(a \cdot 3\right) \cdot c + \color{blue}{0}}}}{3 \cdot a} \]

       12. +-rgt-identity 99.2%

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

       13. *-commutative 99.2%

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

   13. Simplified 99.2%

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

   14. Taylor expanded in b around inf 90.9%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -10:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(c \cdot a\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{-0.6666666666666666 \cdot \frac{b}{c \cdot a} + \left(0.375 \cdot \frac{c \cdot a}{{b}^{3}} + 0.5 \cdot \frac{1}{b}\right)}}{a \cdot 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := c \cdot \left(a \cdot 3\right)\\ t_1 := \frac{\sqrt{b \cdot b - t_0} - b}{a \cdot 3}\\ \mathbf{if}\;t_1 \leq -0.47:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{-6 \cdot \left(a \cdot b\right) + 4.5 \cdot \frac{c \cdot {a}^{2}}{b}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* c (* a 3.0))) (t_1 (/ (- (sqrt (- (* b b) t_0)) b) (* a 3.0))))
   (if (<= t_1 -0.47)
     t_1
     (/ t_0 (+ (* -6.0 (* a b)) (* 4.5 (/ (* c (pow a 2.0)) b)))))))

C

double code(double a, double b, double c) {
	double t_0 = c * (a * 3.0);
	double t_1 = (sqrt(((b * b) - t_0)) - b) / (a * 3.0);
	double tmp;
	if (t_1 <= -0.47) {
		tmp = t_1;
	} else {
		tmp = t_0 / ((-6.0 * (a * b)) + (4.5 * ((c * pow(a, 2.0)) / b)));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = c * (a * 3.0d0)
    t_1 = (sqrt(((b * b) - t_0)) - b) / (a * 3.0d0)
    if (t_1 <= (-0.47d0)) then
        tmp = t_1
    else
        tmp = t_0 / (((-6.0d0) * (a * b)) + (4.5d0 * ((c * (a ** 2.0d0)) / b)))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = c * (a * 3.0);
	double t_1 = (Math.sqrt(((b * b) - t_0)) - b) / (a * 3.0);
	double tmp;
	if (t_1 <= -0.47) {
		tmp = t_1;
	} else {
		tmp = t_0 / ((-6.0 * (a * b)) + (4.5 * ((c * Math.pow(a, 2.0)) / b)));
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = c * (a * 3.0)
	t_1 = (math.sqrt(((b * b) - t_0)) - b) / (a * 3.0)
	tmp = 0
	if t_1 <= -0.47:
		tmp = t_1
	else:
		tmp = t_0 / ((-6.0 * (a * b)) + (4.5 * ((c * math.pow(a, 2.0)) / b)))
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(c * Float64(a * 3.0))
	t_1 = Float64(Float64(sqrt(Float64(Float64(b * b) - t_0)) - b) / Float64(a * 3.0))
	tmp = 0.0
	if (t_1 <= -0.47)
		tmp = t_1;
	else
		tmp = Float64(t_0 / Float64(Float64(-6.0 * Float64(a * b)) + Float64(4.5 * Float64(Float64(c * (a ^ 2.0)) / b))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = c * (a * 3.0);
	t_1 = (sqrt(((b * b) - t_0)) - b) / (a * 3.0);
	tmp = 0.0;
	if (t_1 <= -0.47)
		tmp = t_1;
	else
		tmp = t_0 / ((-6.0 * (a * b)) + (4.5 * ((c * (a ^ 2.0)) / b)));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - t$95$0), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -0.47], t$95$1, N[(t$95$0 / N[(N[(-6.0 * N[(a * b), $MachinePrecision]), $MachinePrecision] + N[(4.5 * N[(N[(c * N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) < -0.46999999999999997

   1. Initial program 81.3%

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

   if -0.46999999999999997 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 49.3%

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

   3. Taylor expanded in a around 0 49.3%

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

      1. associate-*r* 49.3%

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

      2. *-commutative 49.3%

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

      3. associate-*l* 49.3%

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

   5. Simplified 49.3%

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

      1. flip-+ 49.1%

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

      2. pow2 49.1%

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

      3. add-sqr-sqrt 50.7%

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

      4. pow2 50.7%

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

      5. associate-*r* 50.7%

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

      6. pow2 50.7%

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

      7. associate-*r* 50.7%

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

   7. Applied egg-rr 50.7%

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

      1. associate--r- 99.2%

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

      2. associate-*l* 99.0%

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

      3. associate-*l* 99.0%

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

   9. Simplified 99.0%

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

       1. expm1-log1p-u 99.1%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{\left({\left(-b\right)}^{2} - {b}^{2}\right) + a \cdot \left(3 \cdot c\right)}{\left(-b\right) - \sqrt{{b}^{2} - a \cdot \left(3 \cdot c\right)}}}{3 \cdot a}\right)\right)} \]

       2. expm1-udef 71.2%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{\left({\left(-b\right)}^{2} - {b}^{2}\right) + a \cdot \left(3 \cdot c\right)}{\left(-b\right) - \sqrt{{b}^{2} - a \cdot \left(3 \cdot c\right)}}}{3 \cdot a}\right)} - 1} \]

   11. Applied egg-rr 71.2%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\mathsf{fma}\left(a, 3 \cdot c, {b}^{2} - {b}^{2}\right)}{\left(a \cdot 3\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - \left(a \cdot 3\right) \cdot c}\right)}\right)} - 1} \]
   12. Step-by-step derivation

       1. expm1-def 99.1%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left(a, 3 \cdot c, {b}^{2} - {b}^{2}\right)}{\left(a \cdot 3\right) \cdot \left(\left(-b\right) - \sqrt{{b}^{2} - \left(a \cdot 3\right) \cdot c}\right)}\right)\right)} \]

       2. expm1-log1p 99.1%

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

       3. fma-udef 99.1%

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

       4. associate-*l* 99.3%

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

       5. +-inverses 99.3%

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

       6. +-rgt-identity 99.3%

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

       7. *-commutative 99.3%

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

       8. sub-neg 99.3%

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

       9. +-commutative 99.3%

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

       10. distribute-lft-neg-in 99.3%

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

       11. *-commutative 99.3%

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

       12. fma-udef 99.3%

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

       13. distribute-rgt-neg-in 99.3%

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

       14. metadata-eval 99.3%

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

   13. Simplified 99.3%

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

   14. Taylor expanded in a around 0 87.9%

       \[\leadsto \frac{c \cdot \left(a \cdot 3\right)}{\color{blue}{-6 \cdot \left(a \cdot b\right) + 4.5 \cdot \frac{{a}^{2} \cdot c}{b}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.47:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{c \cdot \left(a \cdot 3\right)}{-6 \cdot \left(a \cdot b\right) + 4.5 \cdot \frac{c \cdot {a}^{2}}{b}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 85.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{if}\;t_0 \leq -0.47:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\mathsf{fma}\left(-0.6666666666666666, \frac{b}{c \cdot a}, \frac{0.5}{b}\right)}}{a \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0))))
   (if (<= t_0 -0.47)
     t_0
     (/ (/ 1.0 (fma -0.6666666666666666 (/ b (* c a)) (/ 0.5 b))) (* 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 <= -0.47) {
		tmp = t_0;
	} else {
		tmp = (1.0 / fma(-0.6666666666666666, (b / (c * a)), (0.5 / b))) / (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 <= -0.47)
		tmp = t_0;
	else
		tmp = Float64(Float64(1.0 / fma(-0.6666666666666666, Float64(b / Float64(c * a)), Float64(0.5 / b))) / Float64(a * 3.0));
	end
	return 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, -0.47], t$95$0, N[(N[(1.0 / N[(-0.6666666666666666 * N[(b / N[(c * a), $MachinePrecision]), $MachinePrecision] + N[(0.5 / b), $MachinePrecision]), $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 3 a) c)))) (*.f64 3 a)) < -0.46999999999999997

   1. Initial program 81.3%

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

   if -0.46999999999999997 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 49.3%

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

   3. Taylor expanded in a around 0 49.3%

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

      1. associate-*r* 49.3%

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

      2. *-commutative 49.3%

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

      3. associate-*l* 49.3%

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

   5. Simplified 49.3%

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

      1. flip-+ 49.1%

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

      2. pow2 49.1%

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

      3. add-sqr-sqrt 50.7%

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

      4. pow2 50.7%

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

      5. associate-*r* 50.7%

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

      6. pow2 50.7%

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

      7. associate-*r* 50.7%

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

   7. Applied egg-rr 50.7%

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

      1. associate--r- 99.2%

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

      2. associate-*l* 99.0%

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

      3. associate-*l* 99.0%

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

   9. Simplified 99.0%

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

       1. clear-num 99.0%

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

       2. inv-pow 99.0%

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

       3. associate-*r* 99.0%

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

       4. +-commutative 99.0%

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

       5. fma-def 99.0%

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

       6. neg-mul-1 99.0%

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

       7. unpow-prod-down 99.0%

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

       8. metadata-eval 99.0%

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

       9. *-un-lft-identity 99.0%

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

   11. Applied egg-rr 99.0%

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

       1. unpow-1 99.0%

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

       2. sub-neg 99.0%

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

       3. +-commutative 99.0%

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

       4. distribute-lft-neg-in 99.0%

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

       5. *-commutative 99.0%

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

       6. fma-udef 99.0%

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

       7. distribute-rgt-neg-in 99.0%

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

       8. metadata-eval 99.0%

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

       9. fma-udef 99.0%

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

       10. associate-*l* 99.2%

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

       11. +-inverses 99.2%

           \[\leadsto \frac{\frac{1}{\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c, a \cdot -3, {b}^{2}\right)}}{\left(a \cdot 3\right) \cdot c + \color{blue}{0}}}}{3 \cdot a} \]

       12. +-rgt-identity 99.2%

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

       13. *-commutative 99.2%

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

   13. Simplified 99.2%

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

   14. Taylor expanded in b around inf 87.8%

       \[\leadsto \frac{\frac{1}{\color{blue}{-0.6666666666666666 \cdot \frac{b}{a \cdot c} + 0.5 \cdot \frac{1}{b}}}}{3 \cdot a} \]
   15. Step-by-step derivation

       1. fma-def 87.8%

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

       2. *-commutative 87.8%

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

       3. associate-*r/ 87.8%

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

       4. metadata-eval 87.8%

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

   16. Simplified 87.8%

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

4. Final simplification 86.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.47:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\mathsf{fma}\left(-0.6666666666666666, \frac{b}{c \cdot a}, \frac{0.5}{b}\right)}}{a \cdot 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 84.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.12:\\ \;\;\;\;\frac{\sqrt{b \cdot b - a \cdot \left(c \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\mathsf{fma}\left(-0.6666666666666666, \frac{b}{c \cdot a}, \frac{0.5}{b}\right)}}{a \cdot 3}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.12)
   (/ (- (sqrt (- (* b b) (* a (* c 3.0)))) b) (* a 3.0))
   (/ (/ 1.0 (fma -0.6666666666666666 (/ b (* c a)) (/ 0.5 b))) (* a 3.0))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.12) {
		tmp = (sqrt(((b * b) - (a * (c * 3.0)))) - b) / (a * 3.0);
	} else {
		tmp = (1.0 / fma(-0.6666666666666666, (b / (c * a)), (0.5 / b))) / (a * 3.0);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.12)
		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 / fma(-0.6666666666666666, Float64(b / Float64(c * a)), Float64(0.5 / b))) / Float64(a * 3.0));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.12], 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[(-0.6666666666666666 * N[(b / N[(c * a), $MachinePrecision]), $MachinePrecision] + N[(0.5 / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.12

   1. Initial program 85.1%

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

   3. Taylor expanded in a around 0 85.0%

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

      1. associate-*r* 85.1%

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

      2. *-commutative 85.1%

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

      3. associate-*l* 85.0%

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

   5. Simplified 85.0%

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

   if 0.12 < b

   1. Initial program 51.5%

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

   3. Taylor expanded in a around 0 51.5%

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

      1. associate-*r* 51.5%

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

      2. *-commutative 51.5%

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

      3. associate-*l* 51.5%

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

   5. Simplified 51.5%

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

      1. flip-+ 51.3%

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

      2. pow2 51.3%

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

      3. add-sqr-sqrt 52.7%

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

      4. pow2 52.7%

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

      5. associate-*r* 52.7%

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

      6. pow2 52.7%

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

      7. associate-*r* 52.7%

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

   7. Applied egg-rr 52.7%

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

      1. associate--r- 99.3%

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

      2. associate-*l* 99.1%

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

      3. associate-*l* 99.1%

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

   9. Simplified 99.1%

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

       1. clear-num 99.0%

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

       2. inv-pow 99.0%

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

       3. associate-*r* 99.0%

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

       4. +-commutative 99.0%

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

       5. fma-def 99.0%

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

       6. neg-mul-1 99.0%

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

       7. unpow-prod-down 99.0%

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

       8. metadata-eval 99.0%

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

       9. *-un-lft-identity 99.0%

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

   11. Applied egg-rr 99.0%

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

       1. unpow-1 99.0%

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

       2. sub-neg 99.0%

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

       3. +-commutative 99.0%

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

       4. distribute-lft-neg-in 99.0%

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

       5. *-commutative 99.0%

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

       6. fma-udef 99.0%

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

       7. distribute-rgt-neg-in 99.0%

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

       8. metadata-eval 99.0%

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

       9. fma-udef 99.0%

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

       10. associate-*l* 99.2%

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

       11. +-inverses 99.2%

           \[\leadsto \frac{\frac{1}{\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c, a \cdot -3, {b}^{2}\right)}}{\left(a \cdot 3\right) \cdot c + \color{blue}{0}}}}{3 \cdot a} \]

       12. +-rgt-identity 99.2%

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

       13. *-commutative 99.2%

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

   13. Simplified 99.2%

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

   14. Taylor expanded in b around inf 86.3%

       \[\leadsto \frac{\frac{1}{\color{blue}{-0.6666666666666666 \cdot \frac{b}{a \cdot c} + 0.5 \cdot \frac{1}{b}}}}{3 \cdot a} \]
   15. Step-by-step derivation

       1. fma-def 86.3%

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

       2. *-commutative 86.3%

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

       3. associate-*r/ 86.3%

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

       4. metadata-eval 86.3%

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

   16. Simplified 86.3%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.12:\\ \;\;\;\;\frac{\sqrt{b \cdot b - a \cdot \left(c \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\mathsf{fma}\left(-0.6666666666666666, \frac{b}{c \cdot a}, \frac{0.5}{b}\right)}}{a \cdot 3}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 81.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 55.1%

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

3. Taylor expanded in a around 0 55.1%

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

   1. associate-*r* 55.1%

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

   2. *-commutative 55.1%

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

   3. associate-*l* 55.1%

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

5. Simplified 55.1%

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

   1. flip-+ 54.8%

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

   2. pow2 54.8%

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

   3. add-sqr-sqrt 56.4%

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

   4. pow2 56.4%

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

   5. associate-*r* 56.4%

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

   6. pow2 56.4%

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

   7. associate-*r* 56.4%

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

7. Applied egg-rr 56.4%

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

   1. associate--r- 99.2%

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

   2. associate-*l* 99.1%

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

   3. associate-*l* 99.1%

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

9. Simplified 99.1%

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

    1. clear-num 99.0%

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

    2. inv-pow 99.0%

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

    3. associate-*r* 99.0%

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

    4. +-commutative 99.0%

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

    5. fma-def 99.0%

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

    6. neg-mul-1 99.0%

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

    7. unpow-prod-down 99.0%

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

    8. metadata-eval 99.0%

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

    9. *-un-lft-identity 99.0%

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

11. Applied egg-rr 99.0%

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

    1. unpow-1 99.0%

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

    2. sub-neg 99.0%

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

    3. +-commutative 99.0%

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

    4. distribute-lft-neg-in 99.0%

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

    5. *-commutative 99.0%

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

    6. fma-udef 99.0%

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

    7. distribute-rgt-neg-in 99.0%

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

    8. metadata-eval 99.0%

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

    9. fma-udef 99.0%

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

    10. associate-*l* 99.2%

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

    11. +-inverses 99.2%

        \[\leadsto \frac{\frac{1}{\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c, a \cdot -3, {b}^{2}\right)}}{\left(a \cdot 3\right) \cdot c + \color{blue}{0}}}}{3 \cdot a} \]

    12. +-rgt-identity 99.2%

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

    13. *-commutative 99.2%

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

13. Simplified 99.2%

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

14. Taylor expanded in b around inf 83.0%

    \[\leadsto \frac{\frac{1}{\color{blue}{-0.6666666666666666 \cdot \frac{b}{a \cdot c} + 0.5 \cdot \frac{1}{b}}}}{3 \cdot a} \]
15. Step-by-step derivation

    1. fma-def 83.0%

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

    2. *-commutative 83.0%

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

    3. associate-*r/ 83.0%

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

    4. metadata-eval 83.0%

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

16. Simplified 83.0%

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

17. Final simplification 83.0%

    \[\leadsto \frac{\frac{1}{\mathsf{fma}\left(-0.6666666666666666, \frac{b}{c \cdot a}, \frac{0.5}{b}\right)}}{a \cdot 3} \]
18. Add Preprocessing

Alternative 8: 81.8% accurate, 6.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.1%

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

3. Taylor expanded in a around 0 55.1%

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

   1. associate-*r* 55.1%

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

   2. *-commutative 55.1%

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

   3. associate-*l* 55.1%

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

5. Simplified 55.1%

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

   1. flip-+ 54.8%

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

   2. pow2 54.8%

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

   3. add-sqr-sqrt 56.4%

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

   4. pow2 56.4%

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

   5. associate-*r* 56.4%

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

   6. pow2 56.4%

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

   7. associate-*r* 56.4%

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

7. Applied egg-rr 56.4%

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

   1. associate--r- 99.2%

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

   2. associate-*l* 99.1%

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

   3. associate-*l* 99.1%

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

9. Simplified 99.1%

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

    1. clear-num 99.0%

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

    2. inv-pow 99.0%

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

    3. associate-*r* 99.0%

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

    4. +-commutative 99.0%

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

    5. fma-def 99.0%

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

    6. neg-mul-1 99.0%

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

    7. unpow-prod-down 99.0%

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

    8. metadata-eval 99.0%

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

    9. *-un-lft-identity 99.0%

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

11. Applied egg-rr 99.0%

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

    1. unpow-1 99.0%

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

    2. sub-neg 99.0%

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

    3. +-commutative 99.0%

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

    4. distribute-lft-neg-in 99.0%

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

    5. *-commutative 99.0%

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

    6. fma-udef 99.0%

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

    7. distribute-rgt-neg-in 99.0%

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

    8. metadata-eval 99.0%

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

    9. fma-udef 99.0%

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

    10. associate-*l* 99.2%

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

    11. +-inverses 99.2%

        \[\leadsto \frac{\frac{1}{\frac{\left(-b\right) - \sqrt{\mathsf{fma}\left(c, a \cdot -3, {b}^{2}\right)}}{\left(a \cdot 3\right) \cdot c + \color{blue}{0}}}}{3 \cdot a} \]

    12. +-rgt-identity 99.2%

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

    13. *-commutative 99.2%

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

13. Simplified 99.2%

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

14. Taylor expanded in b around inf 83.0%

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

15. Final simplification 83.0%

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

Alternative 9: 64.5% accurate, 23.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 55.1%

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

3. Taylor expanded in b around inf 65.1%

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

4. Final simplification 65.1%

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

Reproduce

herbie shell --seed 2024011 
(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)))