Cubic critical, narrow range

Percentage Accurate: 55.2% → 91.2%

Time: 21.4s

Alternatives: 13

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: 55.2% 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: 91.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(-0.375, \frac{a}{\frac{\frac{{b}^{3}}{c}}{c}}, \frac{{\left(c \cdot a\right)}^{4}}{a \cdot {b}^{7}} \cdot -1.0546875\right)\right)\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (fma
  -0.5
  (/ c b)
  (fma
   -0.5625
   (/ (pow c 3.0) (/ (pow b 5.0) (* a a)))
   (fma
    -0.375
    (/ a (/ (/ (pow b 3.0) c) c))
    (* (/ (pow (* c a) 4.0) (* a (pow b 7.0))) -1.0546875)))))

C

double code(double a, double b, double c) {
	return fma(-0.5, (c / b), fma(-0.5625, (pow(c, 3.0) / (pow(b, 5.0) / (a * a))), fma(-0.375, (a / ((pow(b, 3.0) / c) / c)), ((pow((c * a), 4.0) / (a * pow(b, 7.0))) * -1.0546875))));
}

Julia

function code(a, b, c)
	return fma(-0.5, Float64(c / b), fma(-0.5625, Float64((c ^ 3.0) / Float64((b ^ 5.0) / Float64(a * a))), fma(-0.375, Float64(a / Float64(Float64((b ^ 3.0) / c) / c)), Float64(Float64((Float64(c * a) ^ 4.0) / Float64(a * (b ^ 7.0))) * -1.0546875))))
end

Wolfram

code[a_, b_, c_] := N[(-0.5 * N[(c / b), $MachinePrecision] + N[(-0.5625 * N[(N[Power[c, 3.0], $MachinePrecision] / N[(N[Power[b, 5.0], $MachinePrecision] / N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.375 * N[(a / N[(N[(N[Power[b, 3.0], $MachinePrecision] / c), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision] + N[(N[(N[Power[N[(c * a), $MachinePrecision], 4.0], $MachinePrecision] / N[(a * N[Power[b, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -1.0546875), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 56.1%

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

   1. sqr-neg 56.1%

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

   2. sqr-neg 56.1%

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

   3. associate-*l* 56.1%

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

3. Simplified 56.1%

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

   1. associate-*r* 56.1%

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

   2. flip3-- 55.7%

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

   3. pow2 55.7%

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

   4. pow-pow 55.7%

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

   5. metadata-eval 55.7%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\frac{{b}^{\color{blue}{6}} - {\left(\left(3 \cdot a\right) \cdot c\right)}^{3}}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(3 \cdot a\right) \cdot c\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right)\right)}}}{3 \cdot a} \]

   6. associate-*r* 55.7%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\frac{{b}^{6} - {\color{blue}{\left(3 \cdot \left(a \cdot c\right)\right)}}^{3}}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(3 \cdot a\right) \cdot c\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right)\right)}}}{3 \cdot a} \]

   7. *-commutative 55.7%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\frac{{b}^{6} - {\color{blue}{\left(\left(a \cdot c\right) \cdot 3\right)}}^{3}}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(3 \cdot a\right) \cdot c\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right)\right)}}}{3 \cdot a} \]

   8. unpow-prod-down 55.7%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\frac{{b}^{6} - \color{blue}{{\left(a \cdot c\right)}^{3} \cdot {3}^{3}}}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(3 \cdot a\right) \cdot c\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right)\right)}}}{3 \cdot a} \]

   9. metadata-eval 55.7%

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

   10. pow2 55.7%

       \[\leadsto \frac{\left(-b\right) + \sqrt{\frac{{b}^{6} - {\left(a \cdot c\right)}^{3} \cdot 27}{\color{blue}{{b}^{2}} \cdot \left(b \cdot b\right) + \left(\left(\left(3 \cdot a\right) \cdot c\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right)\right)}}}{3 \cdot a} \]

   11. pow2 55.7%

       \[\leadsto \frac{\left(-b\right) + \sqrt{\frac{{b}^{6} - {\left(a \cdot c\right)}^{3} \cdot 27}{{b}^{2} \cdot \color{blue}{{b}^{2}} + \left(\left(\left(3 \cdot a\right) \cdot c\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right)\right)}}}{3 \cdot a} \]

   12. pow-prod-up 55.9%

       \[\leadsto \frac{\left(-b\right) + \sqrt{\frac{{b}^{6} - {\left(a \cdot c\right)}^{3} \cdot 27}{\color{blue}{{b}^{\left(2 + 2\right)}} + \left(\left(\left(3 \cdot a\right) \cdot c\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right)\right)}}}{3 \cdot a} \]

   13. metadata-eval 55.9%

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

   14. associate-*r* 55.9%

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

   15. associate-*r* 55.9%

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

   16. associate-*r* 55.9%

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

5. Applied egg-rr 55.9%

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

6. Taylor expanded in b around inf 91.6%

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

8. Simplified 91.6%

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

9. Taylor expanded in b around -inf 91.6%

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

    1. fma-def 91.6%

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

    2. *-commutative 91.6%

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

    3. associate-/l* 91.6%

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

    4. unpow2 91.6%

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

    5. fma-def 91.6%

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

    6. associate-/l* 91.6%

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

    7. unpow2 91.6%

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

    8. associate-/r* 91.6%

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

    9. associate-*r/ 91.6%

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

11. Simplified 91.6%

    \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, \color{blue}{\mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(-0.375, \frac{a}{\frac{\frac{{b}^{3}}{c}}{c}}, \frac{-0.16666666666666666}{{b}^{7}} \cdot \frac{{\left(a \cdot c\right)}^{4}}{\frac{a}{6.328125}}\right)\right)}\right) \]
12. Step-by-step derivation

    1. frac-times 91.6%

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

    2. div-inv 91.6%

       \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(-0.375, \frac{a}{\frac{\frac{{b}^{3}}{c}}{c}}, \frac{-0.16666666666666666 \cdot {\left(a \cdot c\right)}^{4}}{{b}^{7} \cdot \color{blue}{\left(a \cdot \frac{1}{6.328125}\right)}}\right)\right)\right) \]

    3. metadata-eval 91.6%

       \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(-0.375, \frac{a}{\frac{\frac{{b}^{3}}{c}}{c}}, \frac{-0.16666666666666666 \cdot {\left(a \cdot c\right)}^{4}}{{b}^{7} \cdot \left(a \cdot \color{blue}{0.1580246913580247}\right)}\right)\right)\right) \]

13. Applied egg-rr 91.6%

    \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(-0.375, \frac{a}{\frac{\frac{{b}^{3}}{c}}{c}}, \color{blue}{\frac{-0.16666666666666666 \cdot {\left(a \cdot c\right)}^{4}}{{b}^{7} \cdot \left(a \cdot 0.1580246913580247\right)}}\right)\right)\right) \]
14. Step-by-step derivation

    1. *-commutative 91.6%

       \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(-0.375, \frac{a}{\frac{\frac{{b}^{3}}{c}}{c}}, \frac{\color{blue}{{\left(a \cdot c\right)}^{4} \cdot -0.16666666666666666}}{{b}^{7} \cdot \left(a \cdot 0.1580246913580247\right)}\right)\right)\right) \]

    2. associate-*r* 91.6%

       \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(-0.375, \frac{a}{\frac{\frac{{b}^{3}}{c}}{c}}, \frac{{\left(a \cdot c\right)}^{4} \cdot -0.16666666666666666}{\color{blue}{\left({b}^{7} \cdot a\right) \cdot 0.1580246913580247}}\right)\right)\right) \]

    3. *-commutative 91.6%

       \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(-0.375, \frac{a}{\frac{\frac{{b}^{3}}{c}}{c}}, \frac{{\left(a \cdot c\right)}^{4} \cdot -0.16666666666666666}{\color{blue}{\left(a \cdot {b}^{7}\right)} \cdot 0.1580246913580247}\right)\right)\right) \]

    4. times-frac 91.6%

       \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(-0.375, \frac{a}{\frac{\frac{{b}^{3}}{c}}{c}}, \color{blue}{\frac{{\left(a \cdot c\right)}^{4}}{a \cdot {b}^{7}} \cdot \frac{-0.16666666666666666}{0.1580246913580247}}\right)\right)\right) \]

    5. metadata-eval 91.6%

       \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(-0.375, \frac{a}{\frac{\frac{{b}^{3}}{c}}{c}}, \frac{{\left(a \cdot c\right)}^{4}}{a \cdot {b}^{7}} \cdot \color{blue}{-1.0546875}\right)\right)\right) \]

15. Simplified 91.6%

    \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(-0.375, \frac{a}{\frac{\frac{{b}^{3}}{c}}{c}}, \color{blue}{\frac{{\left(a \cdot c\right)}^{4}}{a \cdot {b}^{7}} \cdot -1.0546875}\right)\right)\right) \]

16. Final simplification 91.6%

    \[\leadsto \mathsf{fma}\left(-0.5, \frac{c}{b}, \mathsf{fma}\left(-0.5625, \frac{{c}^{3}}{\frac{{b}^{5}}{a \cdot a}}, \mathsf{fma}\left(-0.375, \frac{a}{\frac{\frac{{b}^{3}}{c}}{c}}, \frac{{\left(c \cdot a\right)}^{4}}{a \cdot {b}^{7}} \cdot -1.0546875\right)\right)\right) \]

Alternative 2: 91.1% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 56.1%

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

   1. neg-sub0 56.1%

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

   2. sqr-neg 56.1%

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

   3. associate-+l- 56.1%

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

   4. sub0-neg 56.1%

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

   5. neg-mul-1 56.1%

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

3. Simplified 56.2%

   \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{\frac{a}{0.3333333333333333}}} \]
4. Step-by-step derivation

   1. div-inv 56.2%

      \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{\color{blue}{a \cdot \frac{1}{0.3333333333333333}}} \]

   2. metadata-eval 56.2%

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

   3. *-commutative 56.2%

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

   4. add-sqr-sqrt 56.2%

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

   5. pow2 56.2%

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

5. Applied egg-rr 56.2%

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

6. Taylor expanded in b around inf 90.8%

   \[\leadsto \color{blue}{-1.6875 \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{5} \cdot {\left(\sqrt{3}\right)}^{2}} + \left(-1.5 \cdot \frac{c}{b \cdot {\left(\sqrt{3}\right)}^{2}} + \left(-1.125 \cdot \frac{a \cdot {c}^{2}}{{b}^{3} \cdot {\left(\sqrt{3}\right)}^{2}} + -0.5 \cdot \frac{5.0625 \cdot \left({a}^{4} \cdot {c}^{4}\right) + {\left(-1.125 \cdot \left({a}^{2} \cdot {c}^{2}\right)\right)}^{2}}{a \cdot \left({b}^{7} \cdot {\left(\sqrt{3}\right)}^{2}\right)}\right)\right)} \]

8. Simplified 91.5%

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

9. Final simplification 91.5%

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

Alternative 3: 88.1% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (pow (* (/ c b) -1.5) 2.0)))
   (fma
    -0.16666666666666666
    (+
     (/ a (/ b t_0))
     (/
      (* a a)
      (/ b (fma 1.5 (* (/ c b) (/ t_0 b)) (* (/ -3.0 b) (/ (* c 0.0) b))))))
    (* -0.5 (/ c b)))))

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 56.1%

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

   1. sqr-neg 56.1%

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

   2. sqr-neg 56.1%

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

   3. associate-*l* 56.1%

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

3. Simplified 56.1%

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

   1. associate-*r* 56.1%

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

   2. flip3-- 55.7%

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

   3. pow2 55.7%

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

   4. pow-pow 55.7%

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

   5. metadata-eval 55.7%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\frac{{b}^{\color{blue}{6}} - {\left(\left(3 \cdot a\right) \cdot c\right)}^{3}}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(3 \cdot a\right) \cdot c\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right)\right)}}}{3 \cdot a} \]

   6. associate-*r* 55.7%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\frac{{b}^{6} - {\color{blue}{\left(3 \cdot \left(a \cdot c\right)\right)}}^{3}}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(3 \cdot a\right) \cdot c\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right)\right)}}}{3 \cdot a} \]

   7. *-commutative 55.7%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\frac{{b}^{6} - {\color{blue}{\left(\left(a \cdot c\right) \cdot 3\right)}}^{3}}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(3 \cdot a\right) \cdot c\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right)\right)}}}{3 \cdot a} \]

   8. unpow-prod-down 55.7%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\frac{{b}^{6} - \color{blue}{{\left(a \cdot c\right)}^{3} \cdot {3}^{3}}}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(3 \cdot a\right) \cdot c\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right)\right)}}}{3 \cdot a} \]

   9. metadata-eval 55.7%

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

   10. pow2 55.7%

       \[\leadsto \frac{\left(-b\right) + \sqrt{\frac{{b}^{6} - {\left(a \cdot c\right)}^{3} \cdot 27}{\color{blue}{{b}^{2}} \cdot \left(b \cdot b\right) + \left(\left(\left(3 \cdot a\right) \cdot c\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right)\right)}}}{3 \cdot a} \]

   11. pow2 55.7%

       \[\leadsto \frac{\left(-b\right) + \sqrt{\frac{{b}^{6} - {\left(a \cdot c\right)}^{3} \cdot 27}{{b}^{2} \cdot \color{blue}{{b}^{2}} + \left(\left(\left(3 \cdot a\right) \cdot c\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right)\right)}}}{3 \cdot a} \]

   12. pow-prod-up 55.9%

       \[\leadsto \frac{\left(-b\right) + \sqrt{\frac{{b}^{6} - {\left(a \cdot c\right)}^{3} \cdot 27}{\color{blue}{{b}^{\left(2 + 2\right)}} + \left(\left(\left(3 \cdot a\right) \cdot c\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right)\right)}}}{3 \cdot a} \]

   13. metadata-eval 55.9%

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

   14. associate-*r* 55.9%

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

   15. associate-*r* 55.9%

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

   16. associate-*r* 55.9%

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

5. Applied egg-rr 55.9%

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

6. Taylor expanded in a around 0 88.9%

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

8. Simplified 88.9%

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

9. Final simplification 88.9%

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

Alternative 4: 87.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(-0.5, \frac{c \cdot c}{\frac{b}{{\left(\frac{a \cdot -1.5}{b}\right)}^{2}}} + \left(a \cdot \frac{1.5}{\frac{b \cdot b}{{\left(-1.5 \cdot \frac{a}{b}\right)}^{2}}}\right) \cdot \frac{{c}^{3}}{b}, -1.5 \cdot \frac{a}{\frac{b}{c}}\right)}{3 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  (fma
   -0.5
   (+
    (/ (* c c) (/ b (pow (/ (* a -1.5) b) 2.0)))
    (* (* a (/ 1.5 (/ (* b b) (pow (* -1.5 (/ a b)) 2.0)))) (/ (pow c 3.0) b)))
   (* -1.5 (/ a (/ b c))))
  (* 3.0 a)))

C

double code(double a, double b, double c) {
	return fma(-0.5, (((c * c) / (b / pow(((a * -1.5) / b), 2.0))) + ((a * (1.5 / ((b * b) / pow((-1.5 * (a / b)), 2.0)))) * (pow(c, 3.0) / b))), (-1.5 * (a / (b / c)))) / (3.0 * a);
}

Julia

function code(a, b, c)
	return Float64(fma(-0.5, Float64(Float64(Float64(c * c) / Float64(b / (Float64(Float64(a * -1.5) / b) ^ 2.0))) + Float64(Float64(a * Float64(1.5 / Float64(Float64(b * b) / (Float64(-1.5 * Float64(a / b)) ^ 2.0)))) * Float64((c ^ 3.0) / b))), Float64(-1.5 * Float64(a / Float64(b / c)))) / Float64(3.0 * a))
end

Wolfram

code[a_, b_, c_] := N[(N[(-0.5 * N[(N[(N[(c * c), $MachinePrecision] / N[(b / N[Power[N[(N[(a * -1.5), $MachinePrecision] / b), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(a * N[(1.5 / N[(N[(b * b), $MachinePrecision] / N[Power[N[(-1.5 * N[(a / b), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[Power[c, 3.0], $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-1.5 * N[(a / N[(b / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 56.1%

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

   1. sqr-neg 56.1%

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

   2. sqr-neg 56.1%

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

   3. associate-*l* 56.1%

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

3. Simplified 56.1%

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

   1. associate-*r* 56.1%

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

   2. flip3-- 55.7%

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

   3. pow2 55.7%

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

   4. pow-pow 55.7%

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

   5. metadata-eval 55.7%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\frac{{b}^{\color{blue}{6}} - {\left(\left(3 \cdot a\right) \cdot c\right)}^{3}}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(3 \cdot a\right) \cdot c\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right)\right)}}}{3 \cdot a} \]

   6. associate-*r* 55.7%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\frac{{b}^{6} - {\color{blue}{\left(3 \cdot \left(a \cdot c\right)\right)}}^{3}}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(3 \cdot a\right) \cdot c\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right)\right)}}}{3 \cdot a} \]

   7. *-commutative 55.7%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\frac{{b}^{6} - {\color{blue}{\left(\left(a \cdot c\right) \cdot 3\right)}}^{3}}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(3 \cdot a\right) \cdot c\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right)\right)}}}{3 \cdot a} \]

   8. unpow-prod-down 55.7%

      \[\leadsto \frac{\left(-b\right) + \sqrt{\frac{{b}^{6} - \color{blue}{{\left(a \cdot c\right)}^{3} \cdot {3}^{3}}}{\left(b \cdot b\right) \cdot \left(b \cdot b\right) + \left(\left(\left(3 \cdot a\right) \cdot c\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right)\right)}}}{3 \cdot a} \]

   9. metadata-eval 55.7%

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

   10. pow2 55.7%

       \[\leadsto \frac{\left(-b\right) + \sqrt{\frac{{b}^{6} - {\left(a \cdot c\right)}^{3} \cdot 27}{\color{blue}{{b}^{2}} \cdot \left(b \cdot b\right) + \left(\left(\left(3 \cdot a\right) \cdot c\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right)\right)}}}{3 \cdot a} \]

   11. pow2 55.7%

       \[\leadsto \frac{\left(-b\right) + \sqrt{\frac{{b}^{6} - {\left(a \cdot c\right)}^{3} \cdot 27}{{b}^{2} \cdot \color{blue}{{b}^{2}} + \left(\left(\left(3 \cdot a\right) \cdot c\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right)\right)}}}{3 \cdot a} \]

   12. pow-prod-up 55.9%

       \[\leadsto \frac{\left(-b\right) + \sqrt{\frac{{b}^{6} - {\left(a \cdot c\right)}^{3} \cdot 27}{\color{blue}{{b}^{\left(2 + 2\right)}} + \left(\left(\left(3 \cdot a\right) \cdot c\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right) + \left(b \cdot b\right) \cdot \left(\left(3 \cdot a\right) \cdot c\right)\right)}}}{3 \cdot a} \]

   13. metadata-eval 55.9%

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

   14. associate-*r* 55.9%

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

   15. associate-*r* 55.9%

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

   16. associate-*r* 55.9%

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

5. Applied egg-rr 55.9%

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

6. Taylor expanded in c around 0 88.4%

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

8. Simplified 88.5%

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

   1. div-inv 88.5%

      \[\leadsto \frac{\mathsf{fma}\left(-0.5, \frac{c \cdot c}{\frac{b}{{\left(\frac{-1.5 \cdot a}{b}\right)}^{2} + 0}} + \color{blue}{{c}^{3} \cdot \frac{1}{\frac{b}{\mathsf{fma}\left(-3, \frac{a}{b} \cdot \frac{0}{b}, \frac{1.5 \cdot a}{\frac{b \cdot b}{{\left(\frac{-1.5 \cdot a}{b}\right)}^{2} + 0}}\right)}}}, -1.5 \cdot \frac{a}{\frac{b}{c}}\right)}{3 \cdot a} \]

   2. div0 88.5%

      \[\leadsto \frac{\mathsf{fma}\left(-0.5, \frac{c \cdot c}{\frac{b}{{\left(\frac{-1.5 \cdot a}{b}\right)}^{2} + 0}} + {c}^{3} \cdot \frac{1}{\frac{b}{\mathsf{fma}\left(-3, \frac{a}{b} \cdot \color{blue}{0}, \frac{1.5 \cdot a}{\frac{b \cdot b}{{\left(\frac{-1.5 \cdot a}{b}\right)}^{2} + 0}}\right)}}, -1.5 \cdot \frac{a}{\frac{b}{c}}\right)}{3 \cdot a} \]

   3. associate-*l/ 88.5%

      \[\leadsto \frac{\mathsf{fma}\left(-0.5, \frac{c \cdot c}{\frac{b}{{\left(\frac{-1.5 \cdot a}{b}\right)}^{2} + 0}} + {c}^{3} \cdot \frac{1}{\frac{b}{\mathsf{fma}\left(-3, \color{blue}{\frac{a \cdot 0}{b}}, \frac{1.5 \cdot a}{\frac{b \cdot b}{{\left(\frac{-1.5 \cdot a}{b}\right)}^{2} + 0}}\right)}}, -1.5 \cdot \frac{a}{\frac{b}{c}}\right)}{3 \cdot a} \]

   4. associate-/r/ 88.5%

      \[\leadsto \frac{\mathsf{fma}\left(-0.5, \frac{c \cdot c}{\frac{b}{{\left(\frac{-1.5 \cdot a}{b}\right)}^{2} + 0}} + {c}^{3} \cdot \frac{1}{\frac{b}{\mathsf{fma}\left(-3, \frac{a \cdot 0}{b}, \color{blue}{\frac{1.5 \cdot a}{b \cdot b} \cdot \left({\left(\frac{-1.5 \cdot a}{b}\right)}^{2} + 0\right)}\right)}}, -1.5 \cdot \frac{a}{\frac{b}{c}}\right)}{3 \cdot a} \]

   5. *-commutative 88.5%

      \[\leadsto \frac{\mathsf{fma}\left(-0.5, \frac{c \cdot c}{\frac{b}{{\left(\frac{-1.5 \cdot a}{b}\right)}^{2} + 0}} + {c}^{3} \cdot \frac{1}{\frac{b}{\mathsf{fma}\left(-3, \frac{a \cdot 0}{b}, \frac{\color{blue}{a \cdot 1.5}}{b \cdot b} \cdot \left({\left(\frac{-1.5 \cdot a}{b}\right)}^{2} + 0\right)\right)}}, -1.5 \cdot \frac{a}{\frac{b}{c}}\right)}{3 \cdot a} \]

   6. +-rgt-identity 88.5%

      \[\leadsto \frac{\mathsf{fma}\left(-0.5, \frac{c \cdot c}{\frac{b}{{\left(\frac{-1.5 \cdot a}{b}\right)}^{2} + 0}} + {c}^{3} \cdot \frac{1}{\frac{b}{\mathsf{fma}\left(-3, \frac{a \cdot 0}{b}, \frac{a \cdot 1.5}{b \cdot b} \cdot \color{blue}{{\left(\frac{-1.5 \cdot a}{b}\right)}^{2}}\right)}}, -1.5 \cdot \frac{a}{\frac{b}{c}}\right)}{3 \cdot a} \]

   7. *-commutative 88.5%

      \[\leadsto \frac{\mathsf{fma}\left(-0.5, \frac{c \cdot c}{\frac{b}{{\left(\frac{-1.5 \cdot a}{b}\right)}^{2} + 0}} + {c}^{3} \cdot \frac{1}{\frac{b}{\mathsf{fma}\left(-3, \frac{a \cdot 0}{b}, \frac{a \cdot 1.5}{b \cdot b} \cdot {\left(\frac{\color{blue}{a \cdot -1.5}}{b}\right)}^{2}\right)}}, -1.5 \cdot \frac{a}{\frac{b}{c}}\right)}{3 \cdot a} \]

10. Applied egg-rr 88.5%

    \[\leadsto \frac{\mathsf{fma}\left(-0.5, \frac{c \cdot c}{\frac{b}{{\left(\frac{-1.5 \cdot a}{b}\right)}^{2} + 0}} + \color{blue}{{c}^{3} \cdot \frac{1}{\frac{b}{\mathsf{fma}\left(-3, \frac{a \cdot 0}{b}, \frac{a \cdot 1.5}{b \cdot b} \cdot {\left(\frac{a \cdot -1.5}{b}\right)}^{2}\right)}}}, -1.5 \cdot \frac{a}{\frac{b}{c}}\right)}{3 \cdot a} \]
11. Step-by-step derivation

    1. associate-*r/ 88.5%

       \[\leadsto \frac{\mathsf{fma}\left(-0.5, \frac{c \cdot c}{\frac{b}{{\left(\frac{-1.5 \cdot a}{b}\right)}^{2} + 0}} + \color{blue}{\frac{{c}^{3} \cdot 1}{\frac{b}{\mathsf{fma}\left(-3, \frac{a \cdot 0}{b}, \frac{a \cdot 1.5}{b \cdot b} \cdot {\left(\frac{a \cdot -1.5}{b}\right)}^{2}\right)}}}, -1.5 \cdot \frac{a}{\frac{b}{c}}\right)}{3 \cdot a} \]

    2. *-rgt-identity 88.5%

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

    3. associate-/r/ 88.5%

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

    4. *-commutative 88.5%

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

12. Simplified 88.5%

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

13. Final simplification 88.5%

    \[\leadsto \frac{\mathsf{fma}\left(-0.5, \frac{c \cdot c}{\frac{b}{{\left(\frac{a \cdot -1.5}{b}\right)}^{2}}} + \left(a \cdot \frac{1.5}{\frac{b \cdot b}{{\left(-1.5 \cdot \frac{a}{b}\right)}^{2}}}\right) \cdot \frac{{c}^{3}}{b}, -1.5 \cdot \frac{a}{\frac{b}{c}}\right)}{3 \cdot a} \]

Alternative 5: 85.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 5.2:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b}{{\left(\sqrt{3 \cdot a}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.375, \frac{a}{\frac{{b}^{3}}{c \cdot c}}, -0.5 \cdot \frac{c}{b}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 5.2)
   (/ (- (sqrt (fma b b (* c (* a -3.0)))) b) (pow (sqrt (* 3.0 a)) 2.0))
   (fma -0.375 (/ a (/ (pow b 3.0) (* c c))) (* -0.5 (/ c b)))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 5.2) {
		tmp = (sqrt(fma(b, b, (c * (a * -3.0)))) - b) / pow(sqrt((3.0 * a)), 2.0);
	} else {
		tmp = fma(-0.375, (a / (pow(b, 3.0) / (c * c))), (-0.5 * (c / b)));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 5.2)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(c * Float64(a * -3.0)))) - b) / (sqrt(Float64(3.0 * a)) ^ 2.0));
	else
		tmp = fma(-0.375, Float64(a / Float64((b ^ 3.0) / Float64(c * c))), Float64(-0.5 * Float64(c / b)));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 5.2], N[(N[(N[Sqrt[N[(b * b + N[(c * N[(a * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[Power[N[Sqrt[N[(3.0 * a), $MachinePrecision]], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], N[(-0.375 * N[(a / N[(N[Power[b, 3.0], $MachinePrecision] / N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 5.20000000000000018

   1. Initial program 79.2%

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

      1. neg-sub0 79.2%

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

      2. sqr-neg 79.2%

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

      3. associate-+l- 79.2%

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

      4. sub0-neg 79.2%

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

      5. neg-mul-1 79.2%

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

   3. Simplified 79.3%

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{\frac{a}{0.3333333333333333}}} \]
   4. Step-by-step derivation

      1. div-inv 79.3%

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{\color{blue}{a \cdot \frac{1}{0.3333333333333333}}} \]

      2. metadata-eval 79.3%

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

      3. *-commutative 79.3%

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

      4. add-sqr-sqrt 79.3%

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

      5. pow2 79.3%

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

   5. Applied egg-rr 79.3%

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

   6. Taylor expanded in a around 0 79.4%

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

      1. *-commutative 79.4%

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

      2. associate-*r* 79.3%

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

      3. *-commutative 79.3%

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

      4. associate-*l* 79.4%

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

   8. Simplified 79.4%

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

   if 5.20000000000000018 < b

   1. Initial program 50.5%

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

      1. sqr-neg 50.5%

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

      2. sqr-neg 50.5%

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

      3. associate-*l* 50.5%

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

   3. Simplified 50.5%

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

   4. Taylor expanded in b around inf 86.7%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}} \]
   5. Step-by-step derivation

      1. +-commutative 86.7%

         \[\leadsto \color{blue}{-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.5 \cdot \frac{c}{b}} \]

      2. fma-def 86.7%

         \[\leadsto \color{blue}{\mathsf{fma}\left(-0.375, \frac{a \cdot {c}^{2}}{{b}^{3}}, -0.5 \cdot \frac{c}{b}\right)} \]

      3. associate-/l* 86.7%

         \[\leadsto \mathsf{fma}\left(-0.375, \color{blue}{\frac{a}{\frac{{b}^{3}}{{c}^{2}}}}, -0.5 \cdot \frac{c}{b}\right) \]

      4. unpow2 86.7%

         \[\leadsto \mathsf{fma}\left(-0.375, \frac{a}{\frac{{b}^{3}}{\color{blue}{c \cdot c}}}, -0.5 \cdot \frac{c}{b}\right) \]

   6. Simplified 86.7%

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

4. Final simplification 85.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 5.2:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, c \cdot \left(a \cdot -3\right)\right)} - b}{{\left(\sqrt{3 \cdot a}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.375, \frac{a}{\frac{{b}^{3}}{c \cdot c}}, -0.5 \cdot \frac{c}{b}\right)\\ \end{array} \]

Alternative 6: 76.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - b}{3 \cdot a} \leq -3.92 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{a} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* 3.0 a)))) b) (* 3.0 a)) -3.92e-7)
   (* (/ (- (sqrt (fma b b (* a (* c -3.0)))) b) a) 0.3333333333333333)
   (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (3.0 * a)))) - b) / (3.0 * a)) <= -3.92e-7) {
		tmp = ((sqrt(fma(b, b, (a * (c * -3.0)))) - b) / a) * 0.3333333333333333;
	} 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(3.0 * a)))) - b) / Float64(3.0 * a)) <= -3.92e-7)
		tmp = Float64(Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(c * -3.0)))) - b) / a) * 0.3333333333333333);
	else
		tmp = Float64(-0.5 * Float64(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[(3.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -3.92e-7], N[(N[(N[(N[Sqrt[N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / a), $MachinePrecision] * 0.3333333333333333), $MachinePrecision], N[(-0.5 * N[(c / b), $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)) < -3.9200000000000002e-7

   1. Initial program 71.6%

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

      1. neg-sub0 71.6%

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

      2. sqr-neg 71.6%

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

      3. associate-+l- 71.6%

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

      4. sub0-neg 71.6%

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

      5. neg-mul-1 71.6%

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

   3. Simplified 71.7%

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{\frac{a}{0.3333333333333333}}} \]
   4. Step-by-step derivation

      1. div-inv 71.7%

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{\color{blue}{a \cdot \frac{1}{0.3333333333333333}}} \]

      2. metadata-eval 71.7%

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

      3. *-commutative 71.7%

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

      4. add-sqr-sqrt 71.7%

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

      5. pow2 71.7%

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

   5. Applied egg-rr 71.7%

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

      1. unpow2 71.7%

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

      2. add-sqr-sqrt 71.7%

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

      3. add-cbrt-cube 71.7%

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

      4. *-commutative 71.7%

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

      5. *-commutative 71.7%

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

      6. *-commutative 71.7%

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

   7. Applied egg-rr 71.7%

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

      1. swap-sqr 71.7%

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

      2. metadata-eval 71.7%

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

      3. associate-*r* 71.7%

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

   9. Simplified 71.7%

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

       1. div-sub 70.7%

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

       2. associate-*l* 70.2%

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

       3. associate-*l* 70.6%

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

   11. Applied egg-rr 70.6%

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

       1. div-sub 71.7%

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

       2. associate-*r* 71.7%

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

       3. associate-*r* 71.7%

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

       4. metadata-eval 71.7%

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

       5. swap-sqr 71.7%

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

       6. unpow3 71.7%

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

       7. rem-cbrt-cube 71.7%

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

       8. *-lft-identity 71.7%

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

       9. *-commutative 71.7%

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

       10. times-frac 71.7%

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

       11. metadata-eval 71.7%

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

   13. Simplified 71.7%

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

   if -3.9200000000000002e-7 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 34.2%

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

      1. sqr-neg 34.2%

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

      2. sqr-neg 34.2%

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

      3. associate-*l* 34.2%

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

   3. Simplified 34.2%

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

   4. Taylor expanded in b around inf 81.6%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - b}{3 \cdot a} \leq -3.92 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{a} \cdot 0.3333333333333333\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 7: 76.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - b}{3 \cdot a} \leq -3.92 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{\frac{a}{0.3333333333333333}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* 3.0 a)))) b) (* 3.0 a)) -3.92e-7)
   (/ (- (sqrt (fma b b (* a (* c -3.0)))) b) (/ a 0.3333333333333333))
   (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (3.0 * a)))) - b) / (3.0 * a)) <= -3.92e-7) {
		tmp = (sqrt(fma(b, b, (a * (c * -3.0)))) - b) / (a / 0.3333333333333333);
	} 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(3.0 * a)))) - b) / Float64(3.0 * a)) <= -3.92e-7)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(c * -3.0)))) - b) / Float64(a / 0.3333333333333333));
	else
		tmp = Float64(-0.5 * Float64(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[(3.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -3.92e-7], N[(N[(N[Sqrt[N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a / 0.3333333333333333), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $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)) < -3.9200000000000002e-7

   1. Initial program 71.6%

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

      1. neg-sub0 71.6%

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

      2. sqr-neg 71.6%

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

      3. associate-+l- 71.6%

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

      4. sub0-neg 71.6%

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

      5. neg-mul-1 71.6%

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

   3. Simplified 71.7%

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

   if -3.9200000000000002e-7 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 34.2%

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

      1. sqr-neg 34.2%

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

      2. sqr-neg 34.2%

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

      3. associate-*l* 34.2%

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

   3. Simplified 34.2%

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

   4. Taylor expanded in b around inf 81.6%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - b}{3 \cdot a} \leq -3.92 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{\frac{a}{0.3333333333333333}}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 8: 76.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - b}{3 \cdot a} \leq -3.92 \cdot 10^{-7}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{a}}{3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* 3.0 a)))) b) (* 3.0 a)) -3.92e-7)
   (/ (/ (- (sqrt (fma b b (* a (* c -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 * (3.0 * a)))) - b) / (3.0 * a)) <= -3.92e-7) {
		tmp = ((sqrt(fma(b, b, (a * (c * -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(3.0 * a)))) - b) / Float64(3.0 * a)) <= -3.92e-7)
		tmp = Float64(Float64(Float64(sqrt(fma(b, b, Float64(a * Float64(c * -3.0)))) - b) / a) / 3.0);
	else
		tmp = Float64(-0.5 * Float64(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[(3.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -3.92e-7], N[(N[(N[(N[Sqrt[N[(b * b + N[(a * N[(c * -3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / a), $MachinePrecision] / 3.0), $MachinePrecision], N[(-0.5 * N[(c / b), $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)) < -3.9200000000000002e-7

   1. Initial program 71.6%

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

      1. neg-sub0 71.6%

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

      2. sqr-neg 71.6%

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

      3. associate-+l- 71.6%

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

      4. sub0-neg 71.6%

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

      5. neg-mul-1 71.6%

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

   3. Simplified 71.7%

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{\frac{a}{0.3333333333333333}}} \]
   4. Step-by-step derivation

      1. div-inv 71.7%

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{\color{blue}{a \cdot \frac{1}{0.3333333333333333}}} \]

      2. metadata-eval 71.7%

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

      3. *-commutative 71.7%

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

      4. add-sqr-sqrt 71.7%

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

      5. pow2 71.7%

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

   5. Applied egg-rr 71.7%

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

      1. add-log-exp 67.2%

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

      2. unpow2 67.2%

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

      3. add-sqr-sqrt 67.2%

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

      4. *-commutative 67.2%

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

   7. Applied egg-rr 67.2%

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

      1. add-log-exp 71.7%

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

      2. div-sub 70.7%

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

   9. Applied egg-rr 70.7%

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

       1. div-sub 71.7%

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

       2. associate-/r* 71.7%

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

   11. Simplified 71.7%

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

   if -3.9200000000000002e-7 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 34.2%

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

      1. sqr-neg 34.2%

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

      2. sqr-neg 34.2%

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

      3. associate-*l* 34.2%

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

   3. Simplified 34.2%

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

   4. Taylor expanded in b around inf 81.6%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - b}{3 \cdot a} \leq -3.92 \cdot 10^{-7}:\\ \;\;\;\;\frac{\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{a}}{3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 9: 76.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - b}{3 \cdot a} \leq -3.92 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - a \cdot \left(c \cdot 3\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= (/ (- (sqrt (- (* b b) (* c (* 3.0 a)))) b) (* 3.0 a)) -3.92e-7)
   (/ (- (sqrt (- (* b b) (* a (* c 3.0)))) b) (* 3.0 a))
   (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (((sqrt(((b * b) - (c * (3.0 * a)))) - b) / (3.0 * a)) <= -3.92e-7) {
		tmp = (sqrt(((b * b) - (a * (c * 3.0)))) - b) / (3.0 * a);
	} 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) :: tmp
    if (((sqrt(((b * b) - (c * (3.0d0 * a)))) - b) / (3.0d0 * a)) <= (-3.92d-7)) then
        tmp = (sqrt(((b * b) - (a * (c * 3.0d0)))) - b) / (3.0d0 * a)
    else
        tmp = (-0.5d0) * (c / b)
    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 * (3.0 * a)))) - b) / (3.0 * a)) <= -3.92e-7) {
		tmp = (Math.sqrt(((b * b) - (a * (c * 3.0)))) - b) / (3.0 * a);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if ((math.sqrt(((b * b) - (c * (3.0 * a)))) - b) / (3.0 * a)) <= -3.92e-7:
		tmp = (math.sqrt(((b * b) - (a * (c * 3.0)))) - b) / (3.0 * a)
	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(3.0 * a)))) - b) / Float64(3.0 * a)) <= -3.92e-7)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(a * Float64(c * 3.0)))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (((sqrt(((b * b) - (c * (3.0 * a)))) - b) / (3.0 * a)) <= -3.92e-7)
		tmp = (sqrt(((b * b) - (a * (c * 3.0)))) - b) / (3.0 * a);
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(c * N[(3.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], -3.92e-7], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(a * N[(c * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $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)) < -3.9200000000000002e-7

   1. Initial program 71.6%

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

   2. Taylor expanded in a around 0 71.6%

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

      1. *-commutative 71.6%

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

      2. associate-*l* 71.6%

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

   4. Simplified 71.6%

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

   if -3.9200000000000002e-7 < (/.f64 (+.f64 (neg.f64 b) (sqrt.f64 (-.f64 (*.f64 b b) (*.f64 (*.f64 3 a) c)))) (*.f64 3 a))

   1. Initial program 34.2%

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

      1. sqr-neg 34.2%

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

      2. sqr-neg 34.2%

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

      3. associate-*l* 34.2%

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

   3. Simplified 34.2%

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

   4. Taylor expanded in b around inf 81.6%

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

4. Final simplification 75.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sqrt{b \cdot b - c \cdot \left(3 \cdot a\right)} - b}{3 \cdot a} \leq -3.92 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sqrt{b \cdot b - a \cdot \left(c \cdot 3\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 10: 85.3% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < 5.5

   1. Initial program 79.2%

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

      1. neg-sub0 79.2%

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

      2. sqr-neg 79.2%

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

      3. associate-+l- 79.2%

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

      4. sub0-neg 79.2%

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

      5. neg-mul-1 79.2%

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

   3. Simplified 79.3%

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{\frac{a}{0.3333333333333333}}} \]
   4. Step-by-step derivation

      1. div-inv 79.3%

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{\color{blue}{a \cdot \frac{1}{0.3333333333333333}}} \]

      2. metadata-eval 79.3%

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

      3. *-commutative 79.3%

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

      4. add-sqr-sqrt 79.3%

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

      5. pow2 79.3%

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

   5. Applied egg-rr 79.3%

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

      1. add-log-exp 74.3%

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

      2. unpow2 74.3%

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

      3. add-sqr-sqrt 74.3%

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

      4. *-commutative 74.3%

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

   7. Applied egg-rr 74.3%

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

      1. add-log-exp 79.3%

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

      2. div-sub 78.0%

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

   9. Applied egg-rr 78.0%

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

       1. div-sub 79.3%

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

       2. associate-/r* 79.4%

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

   11. Simplified 79.4%

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

   if 5.5 < b

   1. Initial program 50.5%

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

      1. neg-sub0 50.5%

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

      2. sqr-neg 50.5%

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

      3. associate-+l- 50.5%

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

      4. sub0-neg 50.5%

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

      5. neg-mul-1 50.5%

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

   3. Simplified 50.6%

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{\frac{a}{0.3333333333333333}}} \]
   4. Step-by-step derivation

      1. div-inv 50.6%

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{\color{blue}{a \cdot \frac{1}{0.3333333333333333}}} \]

      2. metadata-eval 50.6%

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

      3. *-commutative 50.6%

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

      4. add-sqr-sqrt 50.6%

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

      5. pow2 50.6%

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

   5. Applied egg-rr 50.6%

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

      1. unpow2 50.6%

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

      2. add-sqr-sqrt 50.6%

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

      3. add-cbrt-cube 50.6%

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

      4. *-commutative 50.6%

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

      5. *-commutative 50.6%

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

      6. *-commutative 50.6%

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

   7. Applied egg-rr 50.6%

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

      1. swap-sqr 50.6%

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

      2. metadata-eval 50.6%

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

      3. associate-*r* 50.6%

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

   9. Simplified 50.6%

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

   10. Taylor expanded in b around inf 86.7%

       \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}} \]
   11. Step-by-step derivation

       1. +-commutative 86.7%

          \[\leadsto \color{blue}{-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.5 \cdot \frac{c}{b}} \]

       2. fma-def 86.7%

          \[\leadsto \color{blue}{\mathsf{fma}\left(-0.375, \frac{a \cdot {c}^{2}}{{b}^{3}}, -0.5 \cdot \frac{c}{b}\right)} \]

       3. associate-/l* 86.7%

          \[\leadsto \mathsf{fma}\left(-0.375, \color{blue}{\frac{a}{\frac{{b}^{3}}{{c}^{2}}}}, -0.5 \cdot \frac{c}{b}\right) \]

       4. unpow2 86.7%

          \[\leadsto \mathsf{fma}\left(-0.375, \frac{a}{\frac{{b}^{3}}{\color{blue}{c \cdot c}}}, -0.5 \cdot \frac{c}{b}\right) \]

       5. associate-/r* 86.7%

          \[\leadsto \mathsf{fma}\left(-0.375, \frac{a}{\color{blue}{\frac{\frac{{b}^{3}}{c}}{c}}}, -0.5 \cdot \frac{c}{b}\right) \]

       6. associate-*r/ 86.7%

          \[\leadsto \mathsf{fma}\left(-0.375, \frac{a}{\frac{\frac{{b}^{3}}{c}}{c}}, \color{blue}{\frac{-0.5 \cdot c}{b}}\right) \]

       7. associate-/l* 86.5%

          \[\leadsto \mathsf{fma}\left(-0.375, \frac{a}{\frac{\frac{{b}^{3}}{c}}{c}}, \color{blue}{\frac{-0.5}{\frac{b}{c}}}\right) \]

   12. Simplified 86.5%

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

4. Final simplification 85.1%

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

Alternative 11: 85.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if b < 5.5

   1. Initial program 79.2%

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

      1. neg-sub0 79.2%

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

      2. sqr-neg 79.2%

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

      3. associate-+l- 79.2%

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

      4. sub0-neg 79.2%

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

      5. neg-mul-1 79.2%

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

   3. Simplified 79.3%

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{\frac{a}{0.3333333333333333}}} \]
   4. Step-by-step derivation

      1. div-inv 79.3%

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, a \cdot \left(c \cdot -3\right)\right)} - b}{\color{blue}{a \cdot \frac{1}{0.3333333333333333}}} \]

      2. metadata-eval 79.3%

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

      3. *-commutative 79.3%

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

      4. add-sqr-sqrt 79.3%

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

      5. pow2 79.3%

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

   5. Applied egg-rr 79.3%

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

      1. add-log-exp 74.3%

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

      2. unpow2 74.3%

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

      3. add-sqr-sqrt 74.3%

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

      4. *-commutative 74.3%

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

   7. Applied egg-rr 74.3%

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

      1. add-log-exp 79.3%

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

      2. div-sub 78.0%

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

   9. Applied egg-rr 78.0%

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

       1. div-sub 79.3%

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

       2. associate-/r* 79.4%

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

   11. Simplified 79.4%

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

   if 5.5 < b

   1. Initial program 50.5%

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

      1. sqr-neg 50.5%

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

      2. sqr-neg 50.5%

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

      3. associate-*l* 50.5%

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

   3. Simplified 50.5%

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

   4. Taylor expanded in b around inf 86.7%

      \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b} + -0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}} \]
   5. Step-by-step derivation

      1. +-commutative 86.7%

         \[\leadsto \color{blue}{-0.375 \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + -0.5 \cdot \frac{c}{b}} \]

      2. fma-def 86.7%

         \[\leadsto \color{blue}{\mathsf{fma}\left(-0.375, \frac{a \cdot {c}^{2}}{{b}^{3}}, -0.5 \cdot \frac{c}{b}\right)} \]

      3. associate-/l* 86.7%

         \[\leadsto \mathsf{fma}\left(-0.375, \color{blue}{\frac{a}{\frac{{b}^{3}}{{c}^{2}}}}, -0.5 \cdot \frac{c}{b}\right) \]

      4. unpow2 86.7%

         \[\leadsto \mathsf{fma}\left(-0.375, \frac{a}{\frac{{b}^{3}}{\color{blue}{c \cdot c}}}, -0.5 \cdot \frac{c}{b}\right) \]

   6. Simplified 86.7%

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

4. Final simplification 85.2%

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

Alternative 12: 74.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 850.8:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(c \cdot a\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 850.8)
   (/ (- (sqrt (- (* b b) (* 3.0 (* c a)))) b) (* 3.0 a))
   (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 850.8) {
		tmp = (sqrt(((b * b) - (3.0 * (c * a)))) - b) / (3.0 * a);
	} 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) :: tmp
    if (b <= 850.8d0) then
        tmp = (sqrt(((b * b) - (3.0d0 * (c * a)))) - b) / (3.0d0 * a)
    else
        tmp = (-0.5d0) * (c / b)
    end if
    code = tmp
end function

Java

public static double code(double a, double b, double c) {
	double tmp;
	if (b <= 850.8) {
		tmp = (Math.sqrt(((b * b) - (3.0 * (c * a)))) - b) / (3.0 * a);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Python

def code(a, b, c):
	tmp = 0
	if b <= 850.8:
		tmp = (math.sqrt(((b * b) - (3.0 * (c * a)))) - b) / (3.0 * a)
	else:
		tmp = -0.5 * (c / b)
	return tmp

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 850.8)
		tmp = Float64(Float64(sqrt(Float64(Float64(b * b) - Float64(3.0 * Float64(c * a)))) - b) / Float64(3.0 * a));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

MATLAB

function tmp_2 = code(a, b, c)
	tmp = 0.0;
	if (b <= 850.8)
		tmp = (sqrt(((b * b) - (3.0 * (c * a)))) - b) / (3.0 * a);
	else
		tmp = -0.5 * (c / b);
	end
	tmp_2 = tmp;
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 850.8], N[(N[(N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(3.0 * N[(c * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 850.79999999999995

   1. Initial program 73.3%

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

      1. sqr-neg 73.3%

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

      2. sqr-neg 73.3%

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

      3. associate-*l* 73.3%

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

   3. Simplified 73.3%

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

   if 850.79999999999995 < b

   1. Initial program 44.7%

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

      1. sqr-neg 44.7%

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

      2. sqr-neg 44.7%

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

      3. associate-*l* 44.7%

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

   3. Simplified 44.7%

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

   4. Taylor expanded in b around inf 73.5%

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

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;b \leq 850.8:\\ \;\;\;\;\frac{\sqrt{b \cdot b - 3 \cdot \left(c \cdot a\right)} - b}{3 \cdot a}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \]

Alternative 13: 64.6% 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 56.1%

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

   1. sqr-neg 56.1%

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

   2. sqr-neg 56.1%

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

   3. associate-*l* 56.1%

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

3. Simplified 56.1%

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

4. Taylor expanded in b around inf 64.1%

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

5. Final simplification 64.1%

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

Reproduce

herbie shell --seed 2023277 
(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)))