Cubic critical, narrow range

Percentage Accurate: 55.4% → 90.8%

Time: 16.5s

Alternatives: 8

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: 55.4% 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: 90.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(c \cdot a\right)}^{2}\\ t_1 := t_0 \cdot 2.25\\ \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.16666666666666666, \frac{\mathsf{fma}\left(1.5 \cdot a, c \cdot \mathsf{fma}\left(1.5, \left(c \cdot a\right) \cdot t_1, -3 \cdot \left(a \cdot \left(c \cdot 0\right)\right)\right), {\left(-0.5 \cdot t_1\right)}^{2}\right)}{a \cdot {b}^{7}}, \frac{t_0 \cdot -0.375}{a \cdot {b}^{3}} + \frac{\left(2.25 \cdot {\left(c \cdot a\right)}^{3}\right) \cdot -0.25}{a \cdot {b}^{5}}\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (pow (* c a) 2.0)) (t_1 (* t_0 2.25)))
   (fma
    -0.5
    (/ c b)
    (fma
     -0.16666666666666666
     (/
      (fma
       (* 1.5 a)
       (* c (fma 1.5 (* (* c a) t_1) (* -3.0 (* a (* c 0.0)))))
       (pow (* -0.5 t_1) 2.0))
      (* a (pow b 7.0)))
     (+
      (/ (* t_0 -0.375) (* a (pow b 3.0)))
      (/ (* (* 2.25 (pow (* c a) 3.0)) -0.25) (* a (pow b 5.0))))))))

C

double code(double a, double b, double c) {
	double t_0 = pow((c * a), 2.0);
	double t_1 = t_0 * 2.25;
	return fma(-0.5, (c / b), fma(-0.16666666666666666, (fma((1.5 * a), (c * fma(1.5, ((c * a) * t_1), (-3.0 * (a * (c * 0.0))))), pow((-0.5 * t_1), 2.0)) / (a * pow(b, 7.0))), (((t_0 * -0.375) / (a * pow(b, 3.0))) + (((2.25 * pow((c * a), 3.0)) * -0.25) / (a * pow(b, 5.0))))));
}

Julia

function code(a, b, c)
	t_0 = Float64(c * a) ^ 2.0
	t_1 = Float64(t_0 * 2.25)
	return fma(-0.5, Float64(c / b), fma(-0.16666666666666666, Float64(fma(Float64(1.5 * a), Float64(c * fma(1.5, Float64(Float64(c * a) * t_1), Float64(-3.0 * Float64(a * Float64(c * 0.0))))), (Float64(-0.5 * t_1) ^ 2.0)) / Float64(a * (b ^ 7.0))), Float64(Float64(Float64(t_0 * -0.375) / Float64(a * (b ^ 3.0))) + Float64(Float64(Float64(2.25 * (Float64(c * a) ^ 3.0)) * -0.25) / Float64(a * (b ^ 5.0))))))
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[Power[N[(c * a), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 * 2.25), $MachinePrecision]}, N[(-0.5 * N[(c / b), $MachinePrecision] + N[(-0.16666666666666666 * N[(N[(N[(1.5 * a), $MachinePrecision] * N[(c * N[(1.5 * N[(N[(c * a), $MachinePrecision] * t$95$1), $MachinePrecision] + N[(-3.0 * N[(a * N[(c * 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[Power[N[(-0.5 * t$95$1), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / N[(a * N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(t$95$0 * -0.375), $MachinePrecision] / N[(a * N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(2.25 * N[Power[N[(c * a), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] * -0.25), $MachinePrecision] / N[(a * N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 56.0%

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

   1. flip3-- 55.7%

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

   2. clear-num 55.5%

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

   3. pow2 55.5%

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

   4. pow2 55.5%

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

   5. pow-prod-up 55.1%

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

   6. metadata-eval 55.1%

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

   7. distribute-rgt-out 55.1%

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

   8. associate-*l* 55.1%

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

   9. +-commutative 55.1%

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

   10. fma-def 55.1%

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

   11. associate-*l* 55.1%

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

3. Applied egg-rr 55.7%

   \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\frac{1}{\frac{{b}^{4} + \left(3 \cdot \left(a \cdot c\right)\right) \cdot \mathsf{fma}\left(b, b, 3 \cdot \left(a \cdot c\right)\right)}{{b}^{6} - 27 \cdot {\left(a \cdot c\right)}^{3}}}}}}{3 \cdot a} \]

4. Taylor expanded in b around inf 91.2%

   \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b} + \left(-0.16666666666666666 \cdot \frac{-9 \cdot \left({a}^{2} \cdot \left({c}^{2} \cdot \left(-9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 9 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)\right)\right) + \left(1.5 \cdot \left(a \cdot \left(c \cdot \left(-3 \cdot \left(a \cdot \left(c \cdot \left(-9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 9 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)\right)\right) + 1.5 \cdot \left(a \cdot \left(c \cdot \left(-9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + \left(9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + {\left(-1.5 \cdot \left(a \cdot c\right)\right)}^{2}\right)\right)\right)\right)\right)\right)\right) + \left(9 \cdot \left({a}^{2} \cdot \left({c}^{2} \cdot \left(-9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 9 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)\right)\right) + {\left(-0.5 \cdot \left(-9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + \left(9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + {\left(-1.5 \cdot \left(a \cdot c\right)\right)}^{2}\right)\right)\right)}^{2}\right)\right)}{a \cdot {b}^{7}} + \left(-0.16666666666666666 \cdot \frac{-9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + \left(9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + {\left(-1.5 \cdot \left(a \cdot c\right)\right)}^{2}\right)}{a \cdot {b}^{3}} + -0.16666666666666666 \cdot \frac{-3 \cdot \left(a \cdot \left(c \cdot \left(-9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 9 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)\right)\right) + 1.5 \cdot \left(a \cdot \left(c \cdot \left(-9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + \left(9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + {\left(-1.5 \cdot \left(a \cdot c\right)\right)}^{2}\right)\right)\right)\right)}{a \cdot {b}^{5}}\right)\right)} \]

6. Simplified 91.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.16666666666666666, \frac{\mathsf{fma}\left(1.5 \cdot a, c \cdot \mathsf{fma}\left(1.5, \left(a \cdot c\right) \cdot \left(0 + {\left(a \cdot c\right)}^{2} \cdot 2.25\right), -3 \cdot \left(a \cdot \left(c \cdot 0\right)\right)\right), {\left(0 + -0.5 \cdot \left({\left(a \cdot c\right)}^{2} \cdot 2.25\right)\right)}^{2} + 0\right) + 0}{a \cdot {b}^{7}}, -0.16666666666666666 \cdot \left(\frac{0 + {\left(a \cdot c\right)}^{2} \cdot 2.25}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(1.5, \left(a \cdot c\right) \cdot \left(0 + {\left(a \cdot c\right)}^{2} \cdot 2.25\right), -3 \cdot \left(a \cdot \left(c \cdot 0\right)\right)\right)}{a \cdot {b}^{5}}\right)\right)\right)} \]
7. Step-by-step derivation

   1. distribute-lft-in 91.2%

      \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.16666666666666666, \frac{\mathsf{fma}\left(1.5 \cdot a, c \cdot \mathsf{fma}\left(1.5, \left(a \cdot c\right) \cdot \left(0 + {\left(a \cdot c\right)}^{2} \cdot 2.25\right), -3 \cdot \left(a \cdot \left(c \cdot 0\right)\right)\right), {\left(0 + -0.5 \cdot \left({\left(a \cdot c\right)}^{2} \cdot 2.25\right)\right)}^{2} + 0\right) + 0}{a \cdot {b}^{7}}, \color{blue}{-0.16666666666666666 \cdot \frac{0 + {\left(a \cdot c\right)}^{2} \cdot 2.25}{a \cdot {b}^{3}} + -0.16666666666666666 \cdot \frac{\mathsf{fma}\left(1.5, \left(a \cdot c\right) \cdot \left(0 + {\left(a \cdot c\right)}^{2} \cdot 2.25\right), -3 \cdot \left(a \cdot \left(c \cdot 0\right)\right)\right)}{a \cdot {b}^{5}}}\right)\right) \]

   2. +-lft-identity 91.2%

      \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.16666666666666666, \frac{\mathsf{fma}\left(1.5 \cdot a, c \cdot \mathsf{fma}\left(1.5, \left(a \cdot c\right) \cdot \left(0 + {\left(a \cdot c\right)}^{2} \cdot 2.25\right), -3 \cdot \left(a \cdot \left(c \cdot 0\right)\right)\right), {\left(0 + -0.5 \cdot \left({\left(a \cdot c\right)}^{2} \cdot 2.25\right)\right)}^{2} + 0\right) + 0}{a \cdot {b}^{7}}, -0.16666666666666666 \cdot \frac{\color{blue}{{\left(a \cdot c\right)}^{2} \cdot 2.25}}{a \cdot {b}^{3}} + -0.16666666666666666 \cdot \frac{\mathsf{fma}\left(1.5, \left(a \cdot c\right) \cdot \left(0 + {\left(a \cdot c\right)}^{2} \cdot 2.25\right), -3 \cdot \left(a \cdot \left(c \cdot 0\right)\right)\right)}{a \cdot {b}^{5}}\right)\right) \]

8. Applied egg-rr 91.2%

   \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.16666666666666666, \frac{\mathsf{fma}\left(1.5 \cdot a, c \cdot \mathsf{fma}\left(1.5, \left(a \cdot c\right) \cdot \left(0 + {\left(a \cdot c\right)}^{2} \cdot 2.25\right), -3 \cdot \left(a \cdot \left(c \cdot 0\right)\right)\right), {\left(0 + -0.5 \cdot \left({\left(a \cdot c\right)}^{2} \cdot 2.25\right)\right)}^{2} + 0\right) + 0}{a \cdot {b}^{7}}, \color{blue}{-0.16666666666666666 \cdot \frac{{\left(a \cdot c\right)}^{2} \cdot 2.25}{a \cdot {b}^{3}} + -0.16666666666666666 \cdot \frac{\mathsf{fma}\left(1.5, a \cdot \left(c \cdot \left({\left(a \cdot c\right)}^{2} \cdot 2.25\right)\right), 0\right)}{a \cdot {b}^{5}}}\right)\right) \]
9. Step-by-step derivation

   1. associate-*r/ 91.2%

      \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.16666666666666666, \frac{\mathsf{fma}\left(1.5 \cdot a, c \cdot \mathsf{fma}\left(1.5, \left(a \cdot c\right) \cdot \left(0 + {\left(a \cdot c\right)}^{2} \cdot 2.25\right), -3 \cdot \left(a \cdot \left(c \cdot 0\right)\right)\right), {\left(0 + -0.5 \cdot \left({\left(a \cdot c\right)}^{2} \cdot 2.25\right)\right)}^{2} + 0\right) + 0}{a \cdot {b}^{7}}, \color{blue}{\frac{-0.16666666666666666 \cdot \left({\left(a \cdot c\right)}^{2} \cdot 2.25\right)}{a \cdot {b}^{3}}} + -0.16666666666666666 \cdot \frac{\mathsf{fma}\left(1.5, a \cdot \left(c \cdot \left({\left(a \cdot c\right)}^{2} \cdot 2.25\right)\right), 0\right)}{a \cdot {b}^{5}}\right)\right) \]

   2. *-commutative 91.2%

      \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.16666666666666666, \frac{\mathsf{fma}\left(1.5 \cdot a, c \cdot \mathsf{fma}\left(1.5, \left(a \cdot c\right) \cdot \left(0 + {\left(a \cdot c\right)}^{2} \cdot 2.25\right), -3 \cdot \left(a \cdot \left(c \cdot 0\right)\right)\right), {\left(0 + -0.5 \cdot \left({\left(a \cdot c\right)}^{2} \cdot 2.25\right)\right)}^{2} + 0\right) + 0}{a \cdot {b}^{7}}, \frac{-0.16666666666666666 \cdot \color{blue}{\left(2.25 \cdot {\left(a \cdot c\right)}^{2}\right)}}{a \cdot {b}^{3}} + -0.16666666666666666 \cdot \frac{\mathsf{fma}\left(1.5, a \cdot \left(c \cdot \left({\left(a \cdot c\right)}^{2} \cdot 2.25\right)\right), 0\right)}{a \cdot {b}^{5}}\right)\right) \]

   3. associate-*r* 91.2%

      \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.16666666666666666, \frac{\mathsf{fma}\left(1.5 \cdot a, c \cdot \mathsf{fma}\left(1.5, \left(a \cdot c\right) \cdot \left(0 + {\left(a \cdot c\right)}^{2} \cdot 2.25\right), -3 \cdot \left(a \cdot \left(c \cdot 0\right)\right)\right), {\left(0 + -0.5 \cdot \left({\left(a \cdot c\right)}^{2} \cdot 2.25\right)\right)}^{2} + 0\right) + 0}{a \cdot {b}^{7}}, \frac{\color{blue}{\left(-0.16666666666666666 \cdot 2.25\right) \cdot {\left(a \cdot c\right)}^{2}}}{a \cdot {b}^{3}} + -0.16666666666666666 \cdot \frac{\mathsf{fma}\left(1.5, a \cdot \left(c \cdot \left({\left(a \cdot c\right)}^{2} \cdot 2.25\right)\right), 0\right)}{a \cdot {b}^{5}}\right)\right) \]

   4. metadata-eval 91.2%

      \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.16666666666666666, \frac{\mathsf{fma}\left(1.5 \cdot a, c \cdot \mathsf{fma}\left(1.5, \left(a \cdot c\right) \cdot \left(0 + {\left(a \cdot c\right)}^{2} \cdot 2.25\right), -3 \cdot \left(a \cdot \left(c \cdot 0\right)\right)\right), {\left(0 + -0.5 \cdot \left({\left(a \cdot c\right)}^{2} \cdot 2.25\right)\right)}^{2} + 0\right) + 0}{a \cdot {b}^{7}}, \frac{\color{blue}{-0.375} \cdot {\left(a \cdot c\right)}^{2}}{a \cdot {b}^{3}} + -0.16666666666666666 \cdot \frac{\mathsf{fma}\left(1.5, a \cdot \left(c \cdot \left({\left(a \cdot c\right)}^{2} \cdot 2.25\right)\right), 0\right)}{a \cdot {b}^{5}}\right)\right) \]

   5. *-commutative 91.2%

      \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.16666666666666666, \frac{\mathsf{fma}\left(1.5 \cdot a, c \cdot \mathsf{fma}\left(1.5, \left(a \cdot c\right) \cdot \left(0 + {\left(a \cdot c\right)}^{2} \cdot 2.25\right), -3 \cdot \left(a \cdot \left(c \cdot 0\right)\right)\right), {\left(0 + -0.5 \cdot \left({\left(a \cdot c\right)}^{2} \cdot 2.25\right)\right)}^{2} + 0\right) + 0}{a \cdot {b}^{7}}, \frac{-0.375 \cdot {\left(a \cdot c\right)}^{2}}{a \cdot {b}^{3}} + \color{blue}{\frac{\mathsf{fma}\left(1.5, a \cdot \left(c \cdot \left({\left(a \cdot c\right)}^{2} \cdot 2.25\right)\right), 0\right)}{a \cdot {b}^{5}} \cdot -0.16666666666666666}\right)\right) \]

   6. associate-*l/ 91.2%

      \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.16666666666666666, \frac{\mathsf{fma}\left(1.5 \cdot a, c \cdot \mathsf{fma}\left(1.5, \left(a \cdot c\right) \cdot \left(0 + {\left(a \cdot c\right)}^{2} \cdot 2.25\right), -3 \cdot \left(a \cdot \left(c \cdot 0\right)\right)\right), {\left(0 + -0.5 \cdot \left({\left(a \cdot c\right)}^{2} \cdot 2.25\right)\right)}^{2} + 0\right) + 0}{a \cdot {b}^{7}}, \frac{-0.375 \cdot {\left(a \cdot c\right)}^{2}}{a \cdot {b}^{3}} + \color{blue}{\frac{\mathsf{fma}\left(1.5, a \cdot \left(c \cdot \left({\left(a \cdot c\right)}^{2} \cdot 2.25\right)\right), 0\right) \cdot -0.16666666666666666}{a \cdot {b}^{5}}}\right)\right) \]

10. Simplified 91.2%

    \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.16666666666666666, \frac{\mathsf{fma}\left(1.5 \cdot a, c \cdot \mathsf{fma}\left(1.5, \left(a \cdot c\right) \cdot \left(0 + {\left(a \cdot c\right)}^{2} \cdot 2.25\right), -3 \cdot \left(a \cdot \left(c \cdot 0\right)\right)\right), {\left(0 + -0.5 \cdot \left({\left(a \cdot c\right)}^{2} \cdot 2.25\right)\right)}^{2} + 0\right) + 0}{a \cdot {b}^{7}}, \color{blue}{\frac{-0.375 \cdot {\left(a \cdot c\right)}^{2}}{a \cdot {b}^{3}} + \frac{\left({\left(a \cdot c\right)}^{3} \cdot 2.25\right) \cdot -0.25}{a \cdot {b}^{5}}}\right)\right) \]

11. Final simplification 91.2%

    \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.16666666666666666, \frac{\mathsf{fma}\left(1.5 \cdot a, c \cdot \mathsf{fma}\left(1.5, \left(c \cdot a\right) \cdot \left({\left(c \cdot a\right)}^{2} \cdot 2.25\right), -3 \cdot \left(a \cdot \left(c \cdot 0\right)\right)\right), {\left(-0.5 \cdot \left({\left(c \cdot a\right)}^{2} \cdot 2.25\right)\right)}^{2}\right)}{a \cdot {b}^{7}}, \frac{{\left(c \cdot a\right)}^{2} \cdot -0.375}{a \cdot {b}^{3}} + \frac{\left(2.25 \cdot {\left(c \cdot a\right)}^{3}\right) \cdot -0.25}{a \cdot {b}^{5}}\right)\right) \]

Alternative 2: 90.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \left(\frac{{\left(c \cdot a\right)}^{4}}{a} \cdot \frac{6.328125}{{b}^{7}}\right)\right)\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (+
  (* -0.5625 (/ (* (pow a 2.0) (pow c 3.0)) (pow b 5.0)))
  (+
   (* -0.5 (/ c b))
   (+
    (* -0.375 (/ (* a (pow c 2.0)) (pow b 3.0)))
    (*
     -0.16666666666666666
     (* (/ (pow (* c a) 4.0) a) (/ 6.328125 (pow b 7.0))))))))

C

double code(double a, double b, double c) {
	return (-0.5625 * ((pow(a, 2.0) * pow(c, 3.0)) / pow(b, 5.0))) + ((-0.5 * (c / b)) + ((-0.375 * ((a * pow(c, 2.0)) / pow(b, 3.0))) + (-0.16666666666666666 * ((pow((c * a), 4.0) / a) * (6.328125 / pow(b, 7.0))))));
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = ((-0.5625d0) * (((a ** 2.0d0) * (c ** 3.0d0)) / (b ** 5.0d0))) + (((-0.5d0) * (c / b)) + (((-0.375d0) * ((a * (c ** 2.0d0)) / (b ** 3.0d0))) + ((-0.16666666666666666d0) * ((((c * a) ** 4.0d0) / a) * (6.328125d0 / (b ** 7.0d0))))))
end function

Java

public static double code(double a, double b, double c) {
	return (-0.5625 * ((Math.pow(a, 2.0) * Math.pow(c, 3.0)) / Math.pow(b, 5.0))) + ((-0.5 * (c / b)) + ((-0.375 * ((a * Math.pow(c, 2.0)) / Math.pow(b, 3.0))) + (-0.16666666666666666 * ((Math.pow((c * a), 4.0) / a) * (6.328125 / Math.pow(b, 7.0))))));
}

Python

def code(a, b, c):
	return (-0.5625 * ((math.pow(a, 2.0) * math.pow(c, 3.0)) / math.pow(b, 5.0))) + ((-0.5 * (c / b)) + ((-0.375 * ((a * math.pow(c, 2.0)) / math.pow(b, 3.0))) + (-0.16666666666666666 * ((math.pow((c * a), 4.0) / a) * (6.328125 / math.pow(b, 7.0))))))

Julia

function code(a, b, c)
	return Float64(Float64(-0.5625 * Float64(Float64((a ^ 2.0) * (c ^ 3.0)) / (b ^ 5.0))) + Float64(Float64(-0.5 * Float64(c / b)) + Float64(Float64(-0.375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0))) + Float64(-0.16666666666666666 * Float64(Float64((Float64(c * a) ^ 4.0) / a) * Float64(6.328125 / (b ^ 7.0)))))))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-0.5625 * (((a ^ 2.0) * (c ^ 3.0)) / (b ^ 5.0))) + ((-0.5 * (c / b)) + ((-0.375 * ((a * (c ^ 2.0)) / (b ^ 3.0))) + (-0.16666666666666666 * ((((c * a) ^ 4.0) / a) * (6.328125 / (b ^ 7.0))))));
end

Wolfram

code[a_, b_, c_] := N[(N[(-0.5625 * N[(N[(N[Power[a, 2.0], $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.375 * N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.16666666666666666 * N[(N[(N[Power[N[(c * a), $MachinePrecision], 4.0], $MachinePrecision] / a), $MachinePrecision] * N[(6.328125 / N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 56.0%

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

2. Taylor expanded in b around inf 91.2%

   \[\leadsto \color{blue}{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \frac{5.0625 \cdot \left({a}^{4} \cdot {c}^{4}\right) + {\left(-1.125 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)}^{2}}{a \cdot {b}^{7}}\right)\right)} \]

3. Taylor expanded in c around 0 91.2%

   \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \color{blue}{-0.16666666666666666 \cdot \frac{{c}^{4} \cdot \left(1.265625 \cdot {a}^{4} + 5.0625 \cdot {a}^{4}\right)}{a \cdot {b}^{7}}}\right)\right) \]
4. Step-by-step derivation

   1. distribute-rgt-out 91.2%

      \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \frac{{c}^{4} \cdot \color{blue}{\left({a}^{4} \cdot \left(1.265625 + 5.0625\right)\right)}}{a \cdot {b}^{7}}\right)\right) \]

   2. associate-*r* 91.2%

      \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \frac{\color{blue}{\left({c}^{4} \cdot {a}^{4}\right) \cdot \left(1.265625 + 5.0625\right)}}{a \cdot {b}^{7}}\right)\right) \]

   3. *-commutative 91.2%

      \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \frac{\color{blue}{\left({a}^{4} \cdot {c}^{4}\right)} \cdot \left(1.265625 + 5.0625\right)}{a \cdot {b}^{7}}\right)\right) \]

   4. times-frac 91.2%

      \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \color{blue}{\left(\frac{{a}^{4} \cdot {c}^{4}}{a} \cdot \frac{1.265625 + 5.0625}{{b}^{7}}\right)}\right)\right) \]

5. Simplified 91.2%

   \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \color{blue}{-0.16666666666666666 \cdot \left(\frac{{\left(a \cdot c\right)}^{4}}{a} \cdot \frac{6.328125}{{b}^{7}}\right)}\right)\right) \]

6. Final simplification 91.2%

   \[\leadsto -0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + \left(-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.16666666666666666 \cdot \left(\frac{{\left(c \cdot a\right)}^{4}}{a} \cdot \frac{6.328125}{{b}^{7}}\right)\right)\right) \]

Alternative 3: 89.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(c \cdot a\right)}^{2} \cdot 2.25\\ \mathbf{if}\;b \leq 0.74:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, \frac{t_0}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(1.5, \left(c \cdot a\right) \cdot t_0, -3 \cdot \left(a \cdot \left(c \cdot 0\right)\right)\right)}{a \cdot {b}^{5}}, -0.5 \cdot \frac{c}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* (pow (* c a) 2.0) 2.25)))
   (if (<= b 0.74)
     (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0))
     (fma
      -0.16666666666666666
      (+
       (/ t_0 (* a (pow b 3.0)))
       (/
        (fma 1.5 (* (* c a) t_0) (* -3.0 (* a (* c 0.0))))
        (* a (pow b 5.0))))
      (* -0.5 (/ c b))))))

C

double code(double a, double b, double c) {
	double t_0 = pow((c * a), 2.0) * 2.25;
	double tmp;
	if (b <= 0.74) {
		tmp = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	} else {
		tmp = fma(-0.16666666666666666, ((t_0 / (a * pow(b, 3.0))) + (fma(1.5, ((c * a) * t_0), (-3.0 * (a * (c * 0.0)))) / (a * pow(b, 5.0)))), (-0.5 * (c / b)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	t_0 = Float64((Float64(c * a) ^ 2.0) * 2.25)
	tmp = 0.0
	if (b <= 0.74)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0));
	else
		tmp = fma(-0.16666666666666666, Float64(Float64(t_0 / Float64(a * (b ^ 3.0))) + Float64(fma(1.5, Float64(Float64(c * a) * t_0), Float64(-3.0 * Float64(a * Float64(c * 0.0)))) / Float64(a * (b ^ 5.0)))), Float64(-0.5 * Float64(c / b)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[Power[N[(c * a), $MachinePrecision], 2.0], $MachinePrecision] * 2.25), $MachinePrecision]}, If[LessEqual[b, 0.74], 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], N[(-0.16666666666666666 * N[(N[(t$95$0 / N[(a * N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.5 * N[(N[(c * a), $MachinePrecision] * t$95$0), $MachinePrecision] + N[(-3.0 * N[(a * N[(c * 0.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if b < 0.73999999999999999

   1. Initial program 84.6%

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

   if 0.73999999999999999 < 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. Step-by-step derivation

      1. flip3-- 51.1%

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

      2. clear-num 51.0%

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

      3. pow2 51.0%

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

      4. pow2 51.0%

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

      5. pow-prod-up 50.5%

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

      6. metadata-eval 50.5%

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

      7. distribute-rgt-out 50.6%

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

      8. associate-*l* 50.6%

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

      9. +-commutative 50.6%

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

      10. fma-def 50.6%

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

      11. associate-*l* 50.6%

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

   3. Applied egg-rr 51.1%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\color{blue}{\frac{1}{\frac{{b}^{4} + \left(3 \cdot \left(a \cdot c\right)\right) \cdot \mathsf{fma}\left(b, b, 3 \cdot \left(a \cdot c\right)\right)}{{b}^{6} - 27 \cdot {\left(a \cdot c\right)}^{3}}}}}}{3 \cdot a} \]

   4. Taylor expanded in b around inf 91.0%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b} + \left(-0.16666666666666666 \cdot \frac{-9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + \left(9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + {\left(-1.5 \cdot \left(a \cdot c\right)\right)}^{2}\right)}{a \cdot {b}^{3}} + -0.16666666666666666 \cdot \frac{-3 \cdot \left(a \cdot \left(c \cdot \left(-9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + 9 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)\right)\right) + 1.5 \cdot \left(a \cdot \left(c \cdot \left(-9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + \left(9 \cdot \left({a}^{2} \cdot {c}^{2}\right) + {\left(-1.5 \cdot \left(a \cdot c\right)\right)}^{2}\right)\right)\right)\right)}{a \cdot {b}^{5}}\right)} \]

   6. Simplified 91.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.16666666666666666, \frac{0 + {\left(a \cdot c\right)}^{2} \cdot 2.25}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(1.5, \left(a \cdot c\right) \cdot \left(0 + {\left(a \cdot c\right)}^{2} \cdot 2.25\right), -3 \cdot \left(a \cdot \left(c \cdot 0\right)\right)\right)}{a \cdot {b}^{5}}, -0.5 \cdot \frac{c}{b}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.74:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.16666666666666666, \frac{{\left(c \cdot a\right)}^{2} \cdot 2.25}{a \cdot {b}^{3}} + \frac{\mathsf{fma}\left(1.5, \left(c \cdot a\right) \cdot \left({\left(c \cdot a\right)}^{2} \cdot 2.25\right), -3 \cdot \left(a \cdot \left(c \cdot 0\right)\right)\right)}{a \cdot {b}^{5}}, -0.5 \cdot \frac{c}{b}\right)\\ \end{array} \]

Alternative 4: 89.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 0.74:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 0.74)
   (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0))
   (+
    (* -0.5625 (/ (* (pow a 2.0) (pow c 3.0)) (pow b 5.0)))
    (+ (* -0.5 (/ c b)) (* -0.375 (/ (* a (pow c 2.0)) (pow b 3.0)))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.74) {
		tmp = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	} else {
		tmp = (-0.5625 * ((pow(a, 2.0) * pow(c, 3.0)) / pow(b, 5.0))) + ((-0.5 * (c / b)) + (-0.375 * ((a * pow(c, 2.0)) / pow(b, 3.0))));
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: tmp
    if (b <= 0.74d0) then
        tmp = (sqrt(((b * b) - (c * (a * 3.0d0)))) - b) / (a * 3.0d0)
    else
        tmp = ((-0.5625d0) * (((a ** 2.0d0) * (c ** 3.0d0)) / (b ** 5.0d0))) + (((-0.5d0) * (c / b)) + ((-0.375d0) * ((a * (c ** 2.0d0)) / (b ** 3.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 0.74) {
		tmp = (Math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	} else {
		tmp = (-0.5625 * ((Math.pow(a, 2.0) * Math.pow(c, 3.0)) / Math.pow(b, 5.0))) + ((-0.5 * (c / b)) + (-0.375 * ((a * Math.pow(c, 2.0)) / Math.pow(b, 3.0))));
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 0.74:
		tmp = (math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)
	else:
		tmp = (-0.5625 * ((math.pow(a, 2.0) * math.pow(c, 3.0)) / math.pow(b, 5.0))) + ((-0.5 * (c / b)) + (-0.375 * ((a * math.pow(c, 2.0)) / math.pow(b, 3.0))))
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 0.74)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(-0.5625 * Float64(Float64((a ^ 2.0) * (c ^ 3.0)) / (b ^ 5.0))) + Float64(Float64(-0.5 * Float64(c / b)) + Float64(-0.375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 0.74)
		tmp = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	else
		tmp = (-0.5625 * (((a ^ 2.0) * (c ^ 3.0)) / (b ^ 5.0))) + ((-0.5 * (c / b)) + (-0.375 * ((a * (c ^ 2.0)) / (b ^ 3.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 0.74], 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], N[(N[(-0.5625 * N[(N[(N[Power[a, 2.0], $MachinePrecision] * N[Power[c, 3.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 0.73999999999999999

   1. Initial program 84.6%

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

   if 0.73999999999999999 < 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. Taylor expanded in b around inf 90.9%

      \[\leadsto \color{blue}{-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 0.74:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5625 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5}} + \left(-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\right)\\ \end{array} \]

Alternative 5: 84.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -5.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -5.5e-5)
   (/ (- (sqrt (fma b b (* c (* a -3.0)))) b) (* a 3.0))
   (+ (* -0.5 (/ c b)) (* -0.375 (/ (* a (pow c 2.0)) (pow b 3.0))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -5.5e-5) {
		tmp = (sqrt(fma(b, b, (c * (a * -3.0)))) - b) / (a * 3.0);
	} else {
		tmp = (-0.5 * (c / b)) + (-0.375 * ((a * pow(c, 2.0)) / pow(b, 3.0)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0)) <= -5.5e-5)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -3.0)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(-0.5 * Float64(c / b)) + Float64(-0.375 * Float64(Float64(a * (c ^ 2.0)) / (b ^ 3.0))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[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], -5.5e-5], N[(N[(N[Sqrt[N[(b * b + N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(N[(a * N[Power[c, 2.0], $MachinePrecision]), $MachinePrecision] / N[Power[b, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) < -5.5000000000000002e-5

   1. Initial program 77.9%

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

   3. Simplified 78.1%

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

   if -5.5000000000000002e-5 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 39.8%

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

   2. Taylor expanded in b around inf 92.3%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -5.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}\\ \end{array} \]

Alternative 6: 76.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -1.2 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0)) -1.2e-7)
   (/ (- (sqrt (fma b b (* c (* a -3.0)))) b) (* a 3.0))
   (/ (* -0.5 c) b)))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)) <= -1.2e-7) {
		tmp = (sqrt(fma(b, b, (c * (a * -3.0)))) - b) / (a * 3.0);
	} else {
		tmp = (-0.5 * c) / b;
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0)) <= -1.2e-7)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -3.0)))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(-0.5 * c) / b);
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], -1.2e-7], N[(N[(N[Sqrt[N[(b * b + N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[(-0.5 * c), $MachinePrecision] / b), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) < -1.19999999999999989e-7

   1. Initial program 74.2%

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

   3. Simplified 74.4%

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

   if -1.19999999999999989e-7 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 31.0%

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

   2. Taylor expanded in b around inf 84.3%

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

      1. *-commutative 84.3%

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

      2. associate-*l/ 84.3%

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

   4. Simplified 84.3%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -1.2 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b}\\ \end{array} \]

Alternative 7: 75.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{if}\;t_0 \leq -1.2 \cdot 10^{-7}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (/ (- (sqrt (- (* b b) (* c (* a 3.0)))) b) (* a 3.0))))
   (if (<= t_0 -1.2e-7) t_0 (/ (* -0.5 c) b))))

C

double code(double a, double b, double c) {
	double t_0 = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	double tmp;
	if (t_0 <= -1.2e-7) {
		tmp = t_0;
	} else {
		tmp = (-0.5 * c) / b;
	}
	return tmp;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (sqrt(((b * b) - (c * (a * 3.0d0)))) - b) / (a * 3.0d0)
    if (t_0 <= (-1.2d-7)) then
        tmp = t_0
    else
        tmp = ((-0.5d0) * c) / b
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double t_0 = (Math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	double tmp;
	if (t_0 <= -1.2e-7) {
		tmp = t_0;
	} else {
		tmp = (-0.5 * c) / b;
	}
	return tmp;
}

Python

def code(a, b, c):
	t_0 = (math.sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0)
	tmp = 0
	if t_0 <= -1.2e-7:
		tmp = t_0
	else:
		tmp = (-0.5 * c) / b
	return tmp

Julia

function code(a, b, c)
	t_0 = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(c * Float64(a * 3.0)))) - b) / Float64(a * 3.0))
	tmp = 0.0
	if (t_0 <= -1.2e-7)
		tmp = t_0;
	else
		tmp = Float64(Float64(-0.5 * c) / b);
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	t_0 = (sqrt(((b * b) - (c * (a * 3.0)))) - b) / (a * 3.0);
	tmp = 0.0;
	if (t_0 <= -1.2e-7)
		tmp = t_0;
	else
		tmp = (-0.5 * c) / b;
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -1.2e-7], t$95$0, N[(N[(-0.5 * c), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a)) < -1.19999999999999989e-7

   1. Initial program 74.2%

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

   if -1.19999999999999989e-7 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 31.0%

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

   2. Taylor expanded in b around inf 84.3%

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

      1. *-commutative 84.3%

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

      2. associate-*l/ 84.3%

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

   4. Simplified 84.3%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \leq -1.2 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - c \cdot \left(a \cdot 3\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot c}{b}\\ \end{array} \]

Alternative 8: 64.4% accurate, 23.2× speedup

Math

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

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(Float64(-0.5 * c) / b)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.0%

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

2. Taylor expanded in b around inf 63.6%

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

   1. *-commutative 63.6%

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

   2. associate-*l/ 63.6%

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

4. Simplified 63.6%

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

5. Final simplification 63.6%

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

Reproduce

herbie shell --seed 2023319 
(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)))