Cubic critical, narrow range

Percentage Accurate: 55.3% → 99.3%

Time: 14.5s

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, 0.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.3%

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

   1. add-cbrt-cube 53.3%

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

   2. pow1/3 53.2%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{{\left(\left(\left(3 \cdot a\right) \cdot \left(3 \cdot a\right)\right) \cdot \left(3 \cdot a\right)\right)}^{0.3333333333333333}}} \]

   3. pow3 53.2%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{{\color{blue}{\left({\left(3 \cdot a\right)}^{3}\right)}}^{0.3333333333333333}} \]

4. Applied egg-rr 53.2%

   \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{{\left({\left(3 \cdot a\right)}^{3}\right)}^{0.3333333333333333}}} \]
5. Step-by-step derivation

   1. flip-+ 53.2%

      \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}}{{\left({\left(3 \cdot a\right)}^{3}\right)}^{0.3333333333333333}} \]

   2. pow2 53.2%

      \[\leadsto \frac{\frac{\color{blue}{{\left(-b\right)}^{2}} - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{{\left({\left(3 \cdot a\right)}^{3}\right)}^{0.3333333333333333}} \]

   3. add-sqr-sqrt 54.4%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \color{blue}{\left(b \cdot b - \left(3 \cdot a\right) \cdot c\right)}}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{{\left({\left(3 \cdot a\right)}^{3}\right)}^{0.3333333333333333}} \]

   4. pow2 54.4%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left(\color{blue}{{b}^{2}} - \left(3 \cdot a\right) \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{{\left({\left(3 \cdot a\right)}^{3}\right)}^{0.3333333333333333}} \]

   5. *-commutative 54.4%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \color{blue}{c \cdot \left(3 \cdot a\right)}\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{{\left({\left(3 \cdot a\right)}^{3}\right)}^{0.3333333333333333}} \]

   6. *-commutative 54.4%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - c \cdot \color{blue}{\left(a \cdot 3\right)}\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{{\left({\left(3 \cdot a\right)}^{3}\right)}^{0.3333333333333333}} \]

   7. pow2 54.4%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - c \cdot \left(a \cdot 3\right)\right)}{\left(-b\right) - \sqrt{\color{blue}{{b}^{2}} - \left(3 \cdot a\right) \cdot c}}}{{\left({\left(3 \cdot a\right)}^{3}\right)}^{0.3333333333333333}} \]

   8. *-commutative 54.4%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - c \cdot \left(a \cdot 3\right)\right)}{\left(-b\right) - \sqrt{{b}^{2} - \color{blue}{c \cdot \left(3 \cdot a\right)}}}}{{\left({\left(3 \cdot a\right)}^{3}\right)}^{0.3333333333333333}} \]

   9. *-commutative 54.4%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - c \cdot \left(a \cdot 3\right)\right)}{\left(-b\right) - \sqrt{{b}^{2} - c \cdot \color{blue}{\left(a \cdot 3\right)}}}}{{\left({\left(3 \cdot a\right)}^{3}\right)}^{0.3333333333333333}} \]

6. Applied egg-rr 54.4%

   \[\leadsto \frac{\color{blue}{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - c \cdot \left(a \cdot 3\right)\right)}{\left(-b\right) - \sqrt{{b}^{2} - c \cdot \left(a \cdot 3\right)}}}}{{\left({\left(3 \cdot a\right)}^{3}\right)}^{0.3333333333333333}} \]
7. Step-by-step derivation

   1. associate--r- 97.0%

      \[\leadsto \frac{\frac{\color{blue}{\left({\left(-b\right)}^{2} - {b}^{2}\right) + c \cdot \left(a \cdot 3\right)}}{\left(-b\right) - \sqrt{{b}^{2} - c \cdot \left(a \cdot 3\right)}}}{{\left({\left(3 \cdot a\right)}^{3}\right)}^{0.3333333333333333}} \]

8. Simplified 97.0%

   \[\leadsto \frac{\color{blue}{\frac{\left({\left(-b\right)}^{2} - {b}^{2}\right) + c \cdot \left(a \cdot 3\right)}{\left(-b\right) - \sqrt{{b}^{2} - c \cdot \left(a \cdot 3\right)}}}}{{\left({\left(3 \cdot a\right)}^{3}\right)}^{0.3333333333333333}} \]

9. Taylor expanded in a around 0 99.2%

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

    1. *-commutative 99.2%

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

11. Simplified 99.2%

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

12. Taylor expanded in c around inf 99.3%

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

13. Final simplification 99.3%

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

Alternative 2: 85.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.1:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{b \cdot -2 + 1.5 \cdot \frac{c \cdot a}{b}}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -0.1)
   (/ (- (sqrt (fma b b (* a (* c -3.0)))) b) (* a 3.0))
   (/ 1.0 (/ (+ (* b -2.0) (* 1.5 (/ (* c a) b))) c))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -0.1) {
		tmp = (sqrt(fma(b, b, (a * (c * -3.0)))) - b) / (a * 3.0);
	} else {
		tmp = 1.0 / (((b * -2.0) + (1.5 * ((c * a) / b))) / c);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0)) <= -0.1)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(c * -3.0)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(b * -2.0) + Float64(1.5 * Float64(Float64(c * a) / b))) / c));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], -0.1], 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[(1.0 / N[(N[(N[(b * -2.0), $MachinePrecision] + N[(1.5 * N[(N[(c * a), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.10000000000000001

   1. Initial program 83.7%

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

   3. Simplified 83.8%

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

   if -0.10000000000000001 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 44.6%

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

   3. Simplified 44.5%

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

   5. Taylor expanded in a around 0 89.3%

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

      1. clear-num 89.2%

         \[\leadsto \color{blue}{\frac{1}{\frac{3 \cdot a}{a \cdot \left(-1.5 \cdot \frac{c}{b} + -1.125 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right)}}} \]

      2. inv-pow 89.2%

         \[\leadsto \color{blue}{{\left(\frac{3 \cdot a}{a \cdot \left(-1.5 \cdot \frac{c}{b} + -1.125 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right)}\right)}^{-1}} \]

      3. *-commutative 89.2%

         \[\leadsto {\left(\frac{\color{blue}{a \cdot 3}}{a \cdot \left(-1.5 \cdot \frac{c}{b} + -1.125 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right)}\right)}^{-1} \]

      4. +-commutative 89.2%

         \[\leadsto {\left(\frac{a \cdot 3}{a \cdot \color{blue}{\left(-1.125 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -1.5 \cdot \frac{c}{b}\right)}}\right)}^{-1} \]

      5. fma-define 89.2%

         \[\leadsto {\left(\frac{a \cdot 3}{a \cdot \color{blue}{\mathsf{fma}\left(-1.125, \frac{a \cdot {c}^{2}}{{b}^{3}}, -1.5 \cdot \frac{c}{b}\right)}}\right)}^{-1} \]

      6. associate-/l* 89.2%

         \[\leadsto {\left(\frac{a \cdot 3}{a \cdot \mathsf{fma}\left(-1.125, \color{blue}{a \cdot \frac{{c}^{2}}{{b}^{3}}}, -1.5 \cdot \frac{c}{b}\right)}\right)}^{-1} \]

      7. div-inv 89.2%

         \[\leadsto {\left(\frac{a \cdot 3}{a \cdot \mathsf{fma}\left(-1.125, a \cdot \color{blue}{\left({c}^{2} \cdot \frac{1}{{b}^{3}}\right)}, -1.5 \cdot \frac{c}{b}\right)}\right)}^{-1} \]

      8. pow-flip 89.2%

         \[\leadsto {\left(\frac{a \cdot 3}{a \cdot \mathsf{fma}\left(-1.125, a \cdot \left({c}^{2} \cdot \color{blue}{{b}^{\left(-3\right)}}\right), -1.5 \cdot \frac{c}{b}\right)}\right)}^{-1} \]

      9. metadata-eval 89.2%

         \[\leadsto {\left(\frac{a \cdot 3}{a \cdot \mathsf{fma}\left(-1.125, a \cdot \left({c}^{2} \cdot {b}^{\color{blue}{-3}}\right), -1.5 \cdot \frac{c}{b}\right)}\right)}^{-1} \]

      10. associate-*r/ 89.2%

          \[\leadsto {\left(\frac{a \cdot 3}{a \cdot \mathsf{fma}\left(-1.125, a \cdot \left({c}^{2} \cdot {b}^{-3}\right), \color{blue}{\frac{-1.5 \cdot c}{b}}\right)}\right)}^{-1} \]

   7. Applied egg-rr 89.2%

      \[\leadsto \color{blue}{{\left(\frac{a \cdot 3}{a \cdot \mathsf{fma}\left(-1.125, a \cdot \left({c}^{2} \cdot {b}^{-3}\right), \frac{-1.5 \cdot c}{b}\right)}\right)}^{-1}} \]
   8. Step-by-step derivation

      1. unpow-1 89.2%

         \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 3}{a \cdot \mathsf{fma}\left(-1.125, a \cdot \left({c}^{2} \cdot {b}^{-3}\right), \frac{-1.5 \cdot c}{b}\right)}}} \]

      2. times-frac 89.3%

         \[\leadsto \frac{1}{\color{blue}{\frac{a}{a} \cdot \frac{3}{\mathsf{fma}\left(-1.125, a \cdot \left({c}^{2} \cdot {b}^{-3}\right), \frac{-1.5 \cdot c}{b}\right)}}} \]

      3. *-inverses 89.3%

         \[\leadsto \frac{1}{\color{blue}{1} \cdot \frac{3}{\mathsf{fma}\left(-1.125, a \cdot \left({c}^{2} \cdot {b}^{-3}\right), \frac{-1.5 \cdot c}{b}\right)}} \]

      4. associate-*r/ 89.3%

         \[\leadsto \frac{1}{\color{blue}{\frac{1 \cdot 3}{\mathsf{fma}\left(-1.125, a \cdot \left({c}^{2} \cdot {b}^{-3}\right), \frac{-1.5 \cdot c}{b}\right)}}} \]

      5. metadata-eval 89.3%

         \[\leadsto \frac{1}{\frac{\color{blue}{3}}{\mathsf{fma}\left(-1.125, a \cdot \left({c}^{2} \cdot {b}^{-3}\right), \frac{-1.5 \cdot c}{b}\right)}} \]

      6. associate-*r/ 89.2%

         \[\leadsto \frac{1}{\frac{3}{\mathsf{fma}\left(-1.125, a \cdot \left({c}^{2} \cdot {b}^{-3}\right), \color{blue}{-1.5 \cdot \frac{c}{b}}\right)}} \]

      7. *-commutative 89.2%

         \[\leadsto \frac{1}{\frac{3}{\mathsf{fma}\left(-1.125, a \cdot \left({c}^{2} \cdot {b}^{-3}\right), \color{blue}{\frac{c}{b} \cdot -1.5}\right)}} \]

   9. Simplified 89.2%

      \[\leadsto \color{blue}{\frac{1}{\frac{3}{\mathsf{fma}\left(-1.125, a \cdot \left({c}^{2} \cdot {b}^{-3}\right), \frac{c}{b} \cdot -1.5\right)}}} \]

   10. Taylor expanded in c around 0 89.9%

       \[\leadsto \frac{1}{\color{blue}{\frac{-2 \cdot b + 1.5 \cdot \frac{a \cdot c}{b}}{c}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.1:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{b \cdot -2 + 1.5 \cdot \frac{c \cdot a}{b}}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.3% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 53.3%

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

   1. add-cbrt-cube 53.3%

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

   2. pow1/3 53.2%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{{\left(\left(\left(3 \cdot a\right) \cdot \left(3 \cdot a\right)\right) \cdot \left(3 \cdot a\right)\right)}^{0.3333333333333333}}} \]

   3. pow3 53.2%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{{\color{blue}{\left({\left(3 \cdot a\right)}^{3}\right)}}^{0.3333333333333333}} \]

4. Applied egg-rr 53.2%

   \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{{\left({\left(3 \cdot a\right)}^{3}\right)}^{0.3333333333333333}}} \]
5. Step-by-step derivation

   1. flip-+ 53.2%

      \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}}{{\left({\left(3 \cdot a\right)}^{3}\right)}^{0.3333333333333333}} \]

   2. pow2 53.2%

      \[\leadsto \frac{\frac{\color{blue}{{\left(-b\right)}^{2}} - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{{\left({\left(3 \cdot a\right)}^{3}\right)}^{0.3333333333333333}} \]

   3. add-sqr-sqrt 54.4%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \color{blue}{\left(b \cdot b - \left(3 \cdot a\right) \cdot c\right)}}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{{\left({\left(3 \cdot a\right)}^{3}\right)}^{0.3333333333333333}} \]

   4. pow2 54.4%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left(\color{blue}{{b}^{2}} - \left(3 \cdot a\right) \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{{\left({\left(3 \cdot a\right)}^{3}\right)}^{0.3333333333333333}} \]

   5. *-commutative 54.4%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \color{blue}{c \cdot \left(3 \cdot a\right)}\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{{\left({\left(3 \cdot a\right)}^{3}\right)}^{0.3333333333333333}} \]

   6. *-commutative 54.4%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - c \cdot \color{blue}{\left(a \cdot 3\right)}\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{{\left({\left(3 \cdot a\right)}^{3}\right)}^{0.3333333333333333}} \]

   7. pow2 54.4%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - c \cdot \left(a \cdot 3\right)\right)}{\left(-b\right) - \sqrt{\color{blue}{{b}^{2}} - \left(3 \cdot a\right) \cdot c}}}{{\left({\left(3 \cdot a\right)}^{3}\right)}^{0.3333333333333333}} \]

   8. *-commutative 54.4%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - c \cdot \left(a \cdot 3\right)\right)}{\left(-b\right) - \sqrt{{b}^{2} - \color{blue}{c \cdot \left(3 \cdot a\right)}}}}{{\left({\left(3 \cdot a\right)}^{3}\right)}^{0.3333333333333333}} \]

   9. *-commutative 54.4%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - c \cdot \left(a \cdot 3\right)\right)}{\left(-b\right) - \sqrt{{b}^{2} - c \cdot \color{blue}{\left(a \cdot 3\right)}}}}{{\left({\left(3 \cdot a\right)}^{3}\right)}^{0.3333333333333333}} \]

6. Applied egg-rr 54.4%

   \[\leadsto \frac{\color{blue}{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - c \cdot \left(a \cdot 3\right)\right)}{\left(-b\right) - \sqrt{{b}^{2} - c \cdot \left(a \cdot 3\right)}}}}{{\left({\left(3 \cdot a\right)}^{3}\right)}^{0.3333333333333333}} \]
7. Step-by-step derivation

   1. associate--r- 97.0%

      \[\leadsto \frac{\frac{\color{blue}{\left({\left(-b\right)}^{2} - {b}^{2}\right) + c \cdot \left(a \cdot 3\right)}}{\left(-b\right) - \sqrt{{b}^{2} - c \cdot \left(a \cdot 3\right)}}}{{\left({\left(3 \cdot a\right)}^{3}\right)}^{0.3333333333333333}} \]

8. Simplified 97.0%

   \[\leadsto \frac{\color{blue}{\frac{\left({\left(-b\right)}^{2} - {b}^{2}\right) + c \cdot \left(a \cdot 3\right)}{\left(-b\right) - \sqrt{{b}^{2} - c \cdot \left(a \cdot 3\right)}}}}{{\left({\left(3 \cdot a\right)}^{3}\right)}^{0.3333333333333333}} \]

9. Taylor expanded in a around 0 99.2%

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

    1. *-commutative 99.2%

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

11. Simplified 99.2%

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

    1. div-inv 99.1%

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

    2. +-commutative 99.1%

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

    3. *-commutative 99.1%

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

    4. fma-define 99.1%

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

    5. *-commutative 99.1%

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

    6. neg-mul-1 99.1%

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

    7. unpow-prod-down 99.1%

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

    8. metadata-eval 99.1%

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

    9. *-un-lft-identity 99.1%

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

13. Applied egg-rr 99.1%

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

    1. associate-*r/ 99.2%

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

    2. *-commutative 99.2%

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

    3. *-lft-identity 99.2%

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

    4. fma-undefine 99.2%

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

    5. +-inverses 99.2%

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

    6. +-rgt-identity 99.2%

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

    7. associate-*r* 99.2%

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

    8. *-commutative 99.2%

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

    9. *-commutative 99.2%

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

    10. cancel-sign-sub-inv 99.2%

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

    11. metadata-eval 99.2%

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

    12. +-commutative 99.2%

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

    13. fma-define 99.2%

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

    14. *-commutative 99.2%

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

15. Simplified 99.2%

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

Alternative 4: 85.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.1:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(c \cdot a\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{b \cdot -2 + 1.5 \cdot \frac{c \cdot a}{b}}{c}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -0.1)
   (/ (- (sqrt (- (* b b) (* 3.0 (* c a)))) b) (* a 3.0))
   (/ 1.0 (/ (+ (* b -2.0) (* 1.5 (/ (* c a) b))) c))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -0.1) {
		tmp = (sqrt(((b * b) - (3.0 * (c * a)))) - b) / (a * 3.0);
	} else {
		tmp = 1.0 / (((b * -2.0) + (1.5 * ((c * a) / b))) / c);
	}
	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)) <= (-0.1d0)) then
        tmp = (sqrt(((b * b) - (3.0d0 * (c * a)))) - b) / (a * 3.0d0)
    else
        tmp = 1.0d0 / (((b * (-2.0d0)) + (1.5d0 * ((c * a) / b))) / c)
    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)) <= -0.1) {
		tmp = (Math.sqrt(((b * b) - (3.0 * (c * a)))) - b) / (a * 3.0);
	} else {
		tmp = 1.0 / (((b * -2.0) + (1.5 * ((c * a) / b))) / c);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if ((math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -0.1:
		tmp = (math.sqrt(((b * b) - (3.0 * (c * a)))) - b) / (a * 3.0)
	else:
		tmp = 1.0 / (((b * -2.0) + (1.5 * ((c * a) / b))) / c)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0)) <= -0.1)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(3.0 * Float64(c * a)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(b * -2.0) + Float64(1.5 * Float64(Float64(c * a) / b))) / c));
	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)) <= -0.1)
		tmp = (sqrt(((b * b) - (3.0 * (c * a)))) - b) / (a * 3.0);
	else
		tmp = 1.0 / (((b * -2.0) + (1.5 * ((c * a) / b))) / c);
	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], -0.1], 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[(1.0 / N[(N[(N[(b * -2.0), $MachinePrecision] + N[(1.5 * N[(N[(c * a), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a)) < -0.10000000000000001

   1. Initial program 83.7%

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

      1. sqr-neg 83.7%

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

      2. sqr-neg 83.7%

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

      3. associate-*l* 83.7%

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

   3. Simplified 83.7%

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

   if -0.10000000000000001 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 #s(literal 3 binary64) a) c)))) (*.f64 #s(literal 3 binary64) a))

   1. Initial program 44.6%

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

   3. Simplified 44.5%

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

   5. Taylor expanded in a around 0 89.3%

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

      1. clear-num 89.2%

         \[\leadsto \color{blue}{\frac{1}{\frac{3 \cdot a}{a \cdot \left(-1.5 \cdot \frac{c}{b} + -1.125 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right)}}} \]

      2. inv-pow 89.2%

         \[\leadsto \color{blue}{{\left(\frac{3 \cdot a}{a \cdot \left(-1.5 \cdot \frac{c}{b} + -1.125 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right)}\right)}^{-1}} \]

      3. *-commutative 89.2%

         \[\leadsto {\left(\frac{\color{blue}{a \cdot 3}}{a \cdot \left(-1.5 \cdot \frac{c}{b} + -1.125 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right)}\right)}^{-1} \]

      4. +-commutative 89.2%

         \[\leadsto {\left(\frac{a \cdot 3}{a \cdot \color{blue}{\left(-1.125 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -1.5 \cdot \frac{c}{b}\right)}}\right)}^{-1} \]

      5. fma-define 89.2%

         \[\leadsto {\left(\frac{a \cdot 3}{a \cdot \color{blue}{\mathsf{fma}\left(-1.125, \frac{a \cdot {c}^{2}}{{b}^{3}}, -1.5 \cdot \frac{c}{b}\right)}}\right)}^{-1} \]

      6. associate-/l* 89.2%

         \[\leadsto {\left(\frac{a \cdot 3}{a \cdot \mathsf{fma}\left(-1.125, \color{blue}{a \cdot \frac{{c}^{2}}{{b}^{3}}}, -1.5 \cdot \frac{c}{b}\right)}\right)}^{-1} \]

      7. div-inv 89.2%

         \[\leadsto {\left(\frac{a \cdot 3}{a \cdot \mathsf{fma}\left(-1.125, a \cdot \color{blue}{\left({c}^{2} \cdot \frac{1}{{b}^{3}}\right)}, -1.5 \cdot \frac{c}{b}\right)}\right)}^{-1} \]

      8. pow-flip 89.2%

         \[\leadsto {\left(\frac{a \cdot 3}{a \cdot \mathsf{fma}\left(-1.125, a \cdot \left({c}^{2} \cdot \color{blue}{{b}^{\left(-3\right)}}\right), -1.5 \cdot \frac{c}{b}\right)}\right)}^{-1} \]

      9. metadata-eval 89.2%

         \[\leadsto {\left(\frac{a \cdot 3}{a \cdot \mathsf{fma}\left(-1.125, a \cdot \left({c}^{2} \cdot {b}^{\color{blue}{-3}}\right), -1.5 \cdot \frac{c}{b}\right)}\right)}^{-1} \]

      10. associate-*r/ 89.2%

          \[\leadsto {\left(\frac{a \cdot 3}{a \cdot \mathsf{fma}\left(-1.125, a \cdot \left({c}^{2} \cdot {b}^{-3}\right), \color{blue}{\frac{-1.5 \cdot c}{b}}\right)}\right)}^{-1} \]

   7. Applied egg-rr 89.2%

      \[\leadsto \color{blue}{{\left(\frac{a \cdot 3}{a \cdot \mathsf{fma}\left(-1.125, a \cdot \left({c}^{2} \cdot {b}^{-3}\right), \frac{-1.5 \cdot c}{b}\right)}\right)}^{-1}} \]
   8. Step-by-step derivation

      1. unpow-1 89.2%

         \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 3}{a \cdot \mathsf{fma}\left(-1.125, a \cdot \left({c}^{2} \cdot {b}^{-3}\right), \frac{-1.5 \cdot c}{b}\right)}}} \]

      2. times-frac 89.3%

         \[\leadsto \frac{1}{\color{blue}{\frac{a}{a} \cdot \frac{3}{\mathsf{fma}\left(-1.125, a \cdot \left({c}^{2} \cdot {b}^{-3}\right), \frac{-1.5 \cdot c}{b}\right)}}} \]

      3. *-inverses 89.3%

         \[\leadsto \frac{1}{\color{blue}{1} \cdot \frac{3}{\mathsf{fma}\left(-1.125, a \cdot \left({c}^{2} \cdot {b}^{-3}\right), \frac{-1.5 \cdot c}{b}\right)}} \]

      4. associate-*r/ 89.3%

         \[\leadsto \frac{1}{\color{blue}{\frac{1 \cdot 3}{\mathsf{fma}\left(-1.125, a \cdot \left({c}^{2} \cdot {b}^{-3}\right), \frac{-1.5 \cdot c}{b}\right)}}} \]

      5. metadata-eval 89.3%

         \[\leadsto \frac{1}{\frac{\color{blue}{3}}{\mathsf{fma}\left(-1.125, a \cdot \left({c}^{2} \cdot {b}^{-3}\right), \frac{-1.5 \cdot c}{b}\right)}} \]

      6. associate-*r/ 89.2%

         \[\leadsto \frac{1}{\frac{3}{\mathsf{fma}\left(-1.125, a \cdot \left({c}^{2} \cdot {b}^{-3}\right), \color{blue}{-1.5 \cdot \frac{c}{b}}\right)}} \]

      7. *-commutative 89.2%

         \[\leadsto \frac{1}{\frac{3}{\mathsf{fma}\left(-1.125, a \cdot \left({c}^{2} \cdot {b}^{-3}\right), \color{blue}{\frac{c}{b} \cdot -1.5}\right)}} \]

   9. Simplified 89.2%

      \[\leadsto \color{blue}{\frac{1}{\frac{3}{\mathsf{fma}\left(-1.125, a \cdot \left({c}^{2} \cdot {b}^{-3}\right), \frac{c}{b} \cdot -1.5\right)}}} \]

   10. Taylor expanded in c around 0 89.9%

       \[\leadsto \frac{1}{\color{blue}{\frac{-2 \cdot b + 1.5 \cdot \frac{a \cdot c}{b}}{c}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -0.1:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(c \cdot a\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{b \cdot -2 + 1.5 \cdot \frac{c \cdot a}{b}}{c}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.3% accurate, 0.5× speedup

Math

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

FPCore

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

C

double code(double a, double b, double c) {
	double t_0 = c * (a * 3.0);
	return (t_0 / (-b - sqrt((pow(b, 2.0) - t_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
    real(8) :: t_0
    t_0 = c * (a * 3.0d0)
    code = (t_0 / (-b - sqrt(((b ** 2.0d0) - t_0)))) / (a * 3.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.3%

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

   1. add-cbrt-cube 53.3%

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

   2. pow1/3 53.2%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{{\left(\left(\left(3 \cdot a\right) \cdot \left(3 \cdot a\right)\right) \cdot \left(3 \cdot a\right)\right)}^{0.3333333333333333}}} \]

   3. pow3 53.2%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{{\color{blue}{\left({\left(3 \cdot a\right)}^{3}\right)}}^{0.3333333333333333}} \]

4. Applied egg-rr 53.2%

   \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{{\left({\left(3 \cdot a\right)}^{3}\right)}^{0.3333333333333333}}} \]
5. Step-by-step derivation

   1. flip-+ 53.2%

      \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}}{{\left({\left(3 \cdot a\right)}^{3}\right)}^{0.3333333333333333}} \]

   2. pow2 53.2%

      \[\leadsto \frac{\frac{\color{blue}{{\left(-b\right)}^{2}} - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{{\left({\left(3 \cdot a\right)}^{3}\right)}^{0.3333333333333333}} \]

   3. add-sqr-sqrt 54.4%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \color{blue}{\left(b \cdot b - \left(3 \cdot a\right) \cdot c\right)}}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{{\left({\left(3 \cdot a\right)}^{3}\right)}^{0.3333333333333333}} \]

   4. pow2 54.4%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left(\color{blue}{{b}^{2}} - \left(3 \cdot a\right) \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{{\left({\left(3 \cdot a\right)}^{3}\right)}^{0.3333333333333333}} \]

   5. *-commutative 54.4%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \color{blue}{c \cdot \left(3 \cdot a\right)}\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{{\left({\left(3 \cdot a\right)}^{3}\right)}^{0.3333333333333333}} \]

   6. *-commutative 54.4%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - c \cdot \color{blue}{\left(a \cdot 3\right)}\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{{\left({\left(3 \cdot a\right)}^{3}\right)}^{0.3333333333333333}} \]

   7. pow2 54.4%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - c \cdot \left(a \cdot 3\right)\right)}{\left(-b\right) - \sqrt{\color{blue}{{b}^{2}} - \left(3 \cdot a\right) \cdot c}}}{{\left({\left(3 \cdot a\right)}^{3}\right)}^{0.3333333333333333}} \]

   8. *-commutative 54.4%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - c \cdot \left(a \cdot 3\right)\right)}{\left(-b\right) - \sqrt{{b}^{2} - \color{blue}{c \cdot \left(3 \cdot a\right)}}}}{{\left({\left(3 \cdot a\right)}^{3}\right)}^{0.3333333333333333}} \]

   9. *-commutative 54.4%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - c \cdot \left(a \cdot 3\right)\right)}{\left(-b\right) - \sqrt{{b}^{2} - c \cdot \color{blue}{\left(a \cdot 3\right)}}}}{{\left({\left(3 \cdot a\right)}^{3}\right)}^{0.3333333333333333}} \]

6. Applied egg-rr 54.4%

   \[\leadsto \frac{\color{blue}{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - c \cdot \left(a \cdot 3\right)\right)}{\left(-b\right) - \sqrt{{b}^{2} - c \cdot \left(a \cdot 3\right)}}}}{{\left({\left(3 \cdot a\right)}^{3}\right)}^{0.3333333333333333}} \]
7. Step-by-step derivation

   1. associate--r- 97.0%

      \[\leadsto \frac{\frac{\color{blue}{\left({\left(-b\right)}^{2} - {b}^{2}\right) + c \cdot \left(a \cdot 3\right)}}{\left(-b\right) - \sqrt{{b}^{2} - c \cdot \left(a \cdot 3\right)}}}{{\left({\left(3 \cdot a\right)}^{3}\right)}^{0.3333333333333333}} \]

8. Simplified 97.0%

   \[\leadsto \frac{\color{blue}{\frac{\left({\left(-b\right)}^{2} - {b}^{2}\right) + c \cdot \left(a \cdot 3\right)}{\left(-b\right) - \sqrt{{b}^{2} - c \cdot \left(a \cdot 3\right)}}}}{{\left({\left(3 \cdot a\right)}^{3}\right)}^{0.3333333333333333}} \]

9. Taylor expanded in a around 0 99.2%

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

    1. *-commutative 99.2%

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

11. Simplified 99.2%

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

12. Taylor expanded in b around 0 99.1%

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

    1. *-commutative 99.1%

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

    2. *-commutative 99.1%

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

    3. associate-*r* 99.2%

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

14. Simplified 99.2%

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

Alternative 6: 99.1% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.3%

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

   1. add-cbrt-cube 53.3%

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

   2. pow1/3 53.2%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{{\left(\left(\left(3 \cdot a\right) \cdot \left(3 \cdot a\right)\right) \cdot \left(3 \cdot a\right)\right)}^{0.3333333333333333}}} \]

   3. pow3 53.2%

      \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{{\color{blue}{\left({\left(3 \cdot a\right)}^{3}\right)}}^{0.3333333333333333}} \]

4. Applied egg-rr 53.2%

   \[\leadsto \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\color{blue}{{\left({\left(3 \cdot a\right)}^{3}\right)}^{0.3333333333333333}}} \]
5. Step-by-step derivation

   1. flip-+ 53.2%

      \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}}{{\left({\left(3 \cdot a\right)}^{3}\right)}^{0.3333333333333333}} \]

   2. pow2 53.2%

      \[\leadsto \frac{\frac{\color{blue}{{\left(-b\right)}^{2}} - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{{\left({\left(3 \cdot a\right)}^{3}\right)}^{0.3333333333333333}} \]

   3. add-sqr-sqrt 54.4%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \color{blue}{\left(b \cdot b - \left(3 \cdot a\right) \cdot c\right)}}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{{\left({\left(3 \cdot a\right)}^{3}\right)}^{0.3333333333333333}} \]

   4. pow2 54.4%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left(\color{blue}{{b}^{2}} - \left(3 \cdot a\right) \cdot c\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{{\left({\left(3 \cdot a\right)}^{3}\right)}^{0.3333333333333333}} \]

   5. *-commutative 54.4%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - \color{blue}{c \cdot \left(3 \cdot a\right)}\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{{\left({\left(3 \cdot a\right)}^{3}\right)}^{0.3333333333333333}} \]

   6. *-commutative 54.4%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - c \cdot \color{blue}{\left(a \cdot 3\right)}\right)}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{{\left({\left(3 \cdot a\right)}^{3}\right)}^{0.3333333333333333}} \]

   7. pow2 54.4%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - c \cdot \left(a \cdot 3\right)\right)}{\left(-b\right) - \sqrt{\color{blue}{{b}^{2}} - \left(3 \cdot a\right) \cdot c}}}{{\left({\left(3 \cdot a\right)}^{3}\right)}^{0.3333333333333333}} \]

   8. *-commutative 54.4%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - c \cdot \left(a \cdot 3\right)\right)}{\left(-b\right) - \sqrt{{b}^{2} - \color{blue}{c \cdot \left(3 \cdot a\right)}}}}{{\left({\left(3 \cdot a\right)}^{3}\right)}^{0.3333333333333333}} \]

   9. *-commutative 54.4%

      \[\leadsto \frac{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - c \cdot \left(a \cdot 3\right)\right)}{\left(-b\right) - \sqrt{{b}^{2} - c \cdot \color{blue}{\left(a \cdot 3\right)}}}}{{\left({\left(3 \cdot a\right)}^{3}\right)}^{0.3333333333333333}} \]

6. Applied egg-rr 54.4%

   \[\leadsto \frac{\color{blue}{\frac{{\left(-b\right)}^{2} - \left({b}^{2} - c \cdot \left(a \cdot 3\right)\right)}{\left(-b\right) - \sqrt{{b}^{2} - c \cdot \left(a \cdot 3\right)}}}}{{\left({\left(3 \cdot a\right)}^{3}\right)}^{0.3333333333333333}} \]
7. Step-by-step derivation

   1. associate--r- 97.0%

      \[\leadsto \frac{\frac{\color{blue}{\left({\left(-b\right)}^{2} - {b}^{2}\right) + c \cdot \left(a \cdot 3\right)}}{\left(-b\right) - \sqrt{{b}^{2} - c \cdot \left(a \cdot 3\right)}}}{{\left({\left(3 \cdot a\right)}^{3}\right)}^{0.3333333333333333}} \]

8. Simplified 97.0%

   \[\leadsto \frac{\color{blue}{\frac{\left({\left(-b\right)}^{2} - {b}^{2}\right) + c \cdot \left(a \cdot 3\right)}{\left(-b\right) - \sqrt{{b}^{2} - c \cdot \left(a \cdot 3\right)}}}}{{\left({\left(3 \cdot a\right)}^{3}\right)}^{0.3333333333333333}} \]

9. Taylor expanded in a around 0 99.2%

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

    1. *-commutative 99.2%

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

11. Simplified 99.2%

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

12. Taylor expanded in b around 0 99.1%

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

13. Final simplification 99.1%

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

Alternative 7: 82.1% accurate, 7.7× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{b \cdot -2 + 1.5 \cdot \frac{c \cdot a}{b}}{c}} \end{array} \]

FPCore

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

C

double code(double a, double b, double c) {
	return 1.0 / (((b * -2.0) + (1.5 * ((c * a) / b))) / c);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = 1.0d0 / (((b * (-2.0d0)) + (1.5d0 * ((c * a) / b))) / c)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.3%

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

3. Simplified 53.3%

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

5. Taylor expanded in a around 0 81.2%

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

   1. clear-num 81.1%

      \[\leadsto \color{blue}{\frac{1}{\frac{3 \cdot a}{a \cdot \left(-1.5 \cdot \frac{c}{b} + -1.125 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right)}}} \]

   2. inv-pow 81.1%

      \[\leadsto \color{blue}{{\left(\frac{3 \cdot a}{a \cdot \left(-1.5 \cdot \frac{c}{b} + -1.125 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right)}\right)}^{-1}} \]

   3. *-commutative 81.1%

      \[\leadsto {\left(\frac{\color{blue}{a \cdot 3}}{a \cdot \left(-1.5 \cdot \frac{c}{b} + -1.125 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right)}\right)}^{-1} \]

   4. +-commutative 81.1%

      \[\leadsto {\left(\frac{a \cdot 3}{a \cdot \color{blue}{\left(-1.125 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -1.5 \cdot \frac{c}{b}\right)}}\right)}^{-1} \]

   5. fma-define 81.1%

      \[\leadsto {\left(\frac{a \cdot 3}{a \cdot \color{blue}{\mathsf{fma}\left(-1.125, \frac{a \cdot {c}^{2}}{{b}^{3}}, -1.5 \cdot \frac{c}{b}\right)}}\right)}^{-1} \]

   6. associate-/l* 81.1%

      \[\leadsto {\left(\frac{a \cdot 3}{a \cdot \mathsf{fma}\left(-1.125, \color{blue}{a \cdot \frac{{c}^{2}}{{b}^{3}}}, -1.5 \cdot \frac{c}{b}\right)}\right)}^{-1} \]

   7. div-inv 81.1%

      \[\leadsto {\left(\frac{a \cdot 3}{a \cdot \mathsf{fma}\left(-1.125, a \cdot \color{blue}{\left({c}^{2} \cdot \frac{1}{{b}^{3}}\right)}, -1.5 \cdot \frac{c}{b}\right)}\right)}^{-1} \]

   8. pow-flip 81.1%

      \[\leadsto {\left(\frac{a \cdot 3}{a \cdot \mathsf{fma}\left(-1.125, a \cdot \left({c}^{2} \cdot \color{blue}{{b}^{\left(-3\right)}}\right), -1.5 \cdot \frac{c}{b}\right)}\right)}^{-1} \]

   9. metadata-eval 81.1%

      \[\leadsto {\left(\frac{a \cdot 3}{a \cdot \mathsf{fma}\left(-1.125, a \cdot \left({c}^{2} \cdot {b}^{\color{blue}{-3}}\right), -1.5 \cdot \frac{c}{b}\right)}\right)}^{-1} \]

   10. associate-*r/ 81.1%

       \[\leadsto {\left(\frac{a \cdot 3}{a \cdot \mathsf{fma}\left(-1.125, a \cdot \left({c}^{2} \cdot {b}^{-3}\right), \color{blue}{\frac{-1.5 \cdot c}{b}}\right)}\right)}^{-1} \]

7. Applied egg-rr 81.1%

   \[\leadsto \color{blue}{{\left(\frac{a \cdot 3}{a \cdot \mathsf{fma}\left(-1.125, a \cdot \left({c}^{2} \cdot {b}^{-3}\right), \frac{-1.5 \cdot c}{b}\right)}\right)}^{-1}} \]
8. Step-by-step derivation

   1. unpow-1 81.1%

      \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 3}{a \cdot \mathsf{fma}\left(-1.125, a \cdot \left({c}^{2} \cdot {b}^{-3}\right), \frac{-1.5 \cdot c}{b}\right)}}} \]

   2. times-frac 81.1%

      \[\leadsto \frac{1}{\color{blue}{\frac{a}{a} \cdot \frac{3}{\mathsf{fma}\left(-1.125, a \cdot \left({c}^{2} \cdot {b}^{-3}\right), \frac{-1.5 \cdot c}{b}\right)}}} \]

   3. *-inverses 81.1%

      \[\leadsto \frac{1}{\color{blue}{1} \cdot \frac{3}{\mathsf{fma}\left(-1.125, a \cdot \left({c}^{2} \cdot {b}^{-3}\right), \frac{-1.5 \cdot c}{b}\right)}} \]

   4. associate-*r/ 81.1%

      \[\leadsto \frac{1}{\color{blue}{\frac{1 \cdot 3}{\mathsf{fma}\left(-1.125, a \cdot \left({c}^{2} \cdot {b}^{-3}\right), \frac{-1.5 \cdot c}{b}\right)}}} \]

   5. metadata-eval 81.1%

      \[\leadsto \frac{1}{\frac{\color{blue}{3}}{\mathsf{fma}\left(-1.125, a \cdot \left({c}^{2} \cdot {b}^{-3}\right), \frac{-1.5 \cdot c}{b}\right)}} \]

   6. associate-*r/ 81.1%

      \[\leadsto \frac{1}{\frac{3}{\mathsf{fma}\left(-1.125, a \cdot \left({c}^{2} \cdot {b}^{-3}\right), \color{blue}{-1.5 \cdot \frac{c}{b}}\right)}} \]

   7. *-commutative 81.1%

      \[\leadsto \frac{1}{\frac{3}{\mathsf{fma}\left(-1.125, a \cdot \left({c}^{2} \cdot {b}^{-3}\right), \color{blue}{\frac{c}{b} \cdot -1.5}\right)}} \]

9. Simplified 81.1%

   \[\leadsto \color{blue}{\frac{1}{\frac{3}{\mathsf{fma}\left(-1.125, a \cdot \left({c}^{2} \cdot {b}^{-3}\right), \frac{c}{b} \cdot -1.5\right)}}} \]

10. Taylor expanded in c around 0 81.9%

    \[\leadsto \frac{1}{\color{blue}{\frac{-2 \cdot b + 1.5 \cdot \frac{a \cdot c}{b}}{c}}} \]

11. Final simplification 81.9%

    \[\leadsto \frac{1}{\frac{b \cdot -2 + 1.5 \cdot \frac{c \cdot a}{b}}{c}} \]
12. Add Preprocessing

Alternative 8: 82.1% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \frac{1}{-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.3%

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

3. Simplified 53.3%

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

5. Taylor expanded in a around 0 81.2%

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

   1. clear-num 81.1%

      \[\leadsto \color{blue}{\frac{1}{\frac{3 \cdot a}{a \cdot \left(-1.5 \cdot \frac{c}{b} + -1.125 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right)}}} \]

   2. inv-pow 81.1%

      \[\leadsto \color{blue}{{\left(\frac{3 \cdot a}{a \cdot \left(-1.5 \cdot \frac{c}{b} + -1.125 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right)}\right)}^{-1}} \]

   3. *-commutative 81.1%

      \[\leadsto {\left(\frac{\color{blue}{a \cdot 3}}{a \cdot \left(-1.5 \cdot \frac{c}{b} + -1.125 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right)}\right)}^{-1} \]

   4. +-commutative 81.1%

      \[\leadsto {\left(\frac{a \cdot 3}{a \cdot \color{blue}{\left(-1.125 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -1.5 \cdot \frac{c}{b}\right)}}\right)}^{-1} \]

   5. fma-define 81.1%

      \[\leadsto {\left(\frac{a \cdot 3}{a \cdot \color{blue}{\mathsf{fma}\left(-1.125, \frac{a \cdot {c}^{2}}{{b}^{3}}, -1.5 \cdot \frac{c}{b}\right)}}\right)}^{-1} \]

   6. associate-/l* 81.1%

      \[\leadsto {\left(\frac{a \cdot 3}{a \cdot \mathsf{fma}\left(-1.125, \color{blue}{a \cdot \frac{{c}^{2}}{{b}^{3}}}, -1.5 \cdot \frac{c}{b}\right)}\right)}^{-1} \]

   7. div-inv 81.1%

      \[\leadsto {\left(\frac{a \cdot 3}{a \cdot \mathsf{fma}\left(-1.125, a \cdot \color{blue}{\left({c}^{2} \cdot \frac{1}{{b}^{3}}\right)}, -1.5 \cdot \frac{c}{b}\right)}\right)}^{-1} \]

   8. pow-flip 81.1%

      \[\leadsto {\left(\frac{a \cdot 3}{a \cdot \mathsf{fma}\left(-1.125, a \cdot \left({c}^{2} \cdot \color{blue}{{b}^{\left(-3\right)}}\right), -1.5 \cdot \frac{c}{b}\right)}\right)}^{-1} \]

   9. metadata-eval 81.1%

      \[\leadsto {\left(\frac{a \cdot 3}{a \cdot \mathsf{fma}\left(-1.125, a \cdot \left({c}^{2} \cdot {b}^{\color{blue}{-3}}\right), -1.5 \cdot \frac{c}{b}\right)}\right)}^{-1} \]

   10. associate-*r/ 81.1%

       \[\leadsto {\left(\frac{a \cdot 3}{a \cdot \mathsf{fma}\left(-1.125, a \cdot \left({c}^{2} \cdot {b}^{-3}\right), \color{blue}{\frac{-1.5 \cdot c}{b}}\right)}\right)}^{-1} \]

7. Applied egg-rr 81.1%

   \[\leadsto \color{blue}{{\left(\frac{a \cdot 3}{a \cdot \mathsf{fma}\left(-1.125, a \cdot \left({c}^{2} \cdot {b}^{-3}\right), \frac{-1.5 \cdot c}{b}\right)}\right)}^{-1}} \]
8. Step-by-step derivation

   1. unpow-1 81.1%

      \[\leadsto \color{blue}{\frac{1}{\frac{a \cdot 3}{a \cdot \mathsf{fma}\left(-1.125, a \cdot \left({c}^{2} \cdot {b}^{-3}\right), \frac{-1.5 \cdot c}{b}\right)}}} \]

   2. times-frac 81.1%

      \[\leadsto \frac{1}{\color{blue}{\frac{a}{a} \cdot \frac{3}{\mathsf{fma}\left(-1.125, a \cdot \left({c}^{2} \cdot {b}^{-3}\right), \frac{-1.5 \cdot c}{b}\right)}}} \]

   3. *-inverses 81.1%

      \[\leadsto \frac{1}{\color{blue}{1} \cdot \frac{3}{\mathsf{fma}\left(-1.125, a \cdot \left({c}^{2} \cdot {b}^{-3}\right), \frac{-1.5 \cdot c}{b}\right)}} \]

   4. associate-*r/ 81.1%

      \[\leadsto \frac{1}{\color{blue}{\frac{1 \cdot 3}{\mathsf{fma}\left(-1.125, a \cdot \left({c}^{2} \cdot {b}^{-3}\right), \frac{-1.5 \cdot c}{b}\right)}}} \]

   5. metadata-eval 81.1%

      \[\leadsto \frac{1}{\frac{\color{blue}{3}}{\mathsf{fma}\left(-1.125, a \cdot \left({c}^{2} \cdot {b}^{-3}\right), \frac{-1.5 \cdot c}{b}\right)}} \]

   6. associate-*r/ 81.1%

      \[\leadsto \frac{1}{\frac{3}{\mathsf{fma}\left(-1.125, a \cdot \left({c}^{2} \cdot {b}^{-3}\right), \color{blue}{-1.5 \cdot \frac{c}{b}}\right)}} \]

   7. *-commutative 81.1%

      \[\leadsto \frac{1}{\frac{3}{\mathsf{fma}\left(-1.125, a \cdot \left({c}^{2} \cdot {b}^{-3}\right), \color{blue}{\frac{c}{b} \cdot -1.5}\right)}} \]

9. Simplified 81.1%

   \[\leadsto \color{blue}{\frac{1}{\frac{3}{\mathsf{fma}\left(-1.125, a \cdot \left({c}^{2} \cdot {b}^{-3}\right), \frac{c}{b} \cdot -1.5\right)}}} \]

10. Taylor expanded in a around 0 81.9%

    \[\leadsto \frac{1}{\color{blue}{-2 \cdot \frac{b}{c} + 1.5 \cdot \frac{a}{b}}} \]
11. 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 53.3%

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

3. Simplified 53.3%

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

5. Taylor expanded in b around inf 65.8%

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

Reproduce

herbie shell --seed 2024188 
(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)))