Cubic critical, wide range

Percentage Accurate: 17.1% → 97.9%

Time: 14.2s

Alternatives: 8

Speedup: 2.9×

Specification

\[\left(\left(4.930380657631324 \cdot 10^{-32} < a \land a < 2.028240960365167 \cdot 10^{+31}\right) \land \left(4.930380657631324 \cdot 10^{-32} < b \land b < 2.028240960365167 \cdot 10^{+31}\right)\right) \land \left(4.930380657631324 \cdot 10^{-32} < c \land c < 2.028240960365167 \cdot 10^{+31}\right)\]

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: 17.1% 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: 97.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot \left(b \cdot b\right)\\ t_1 := b \cdot t\_0\\ \mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{\frac{\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot \left(a \cdot 6.328125\right)}{\left(b \cdot b\right) \cdot t\_1}}{b}, -0.16666666666666666, \frac{\left(c \cdot c\right) \cdot \left(c \cdot -0.5625\right)}{b \cdot t\_1}\right), \frac{\left(c \cdot c\right) \cdot -0.375}{t\_0}\right), \frac{c \cdot -0.5}{b}\right) \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* b (* b b))) (t_1 (* b t_0)))
   (fma
    a
    (fma
     a
     (fma
      (/ (/ (* (* c (* c (* c c))) (* a 6.328125)) (* (* b b) t_1)) b)
      -0.16666666666666666
      (/ (* (* c c) (* c -0.5625)) (* b t_1)))
     (/ (* (* c c) -0.375) t_0))
    (/ (* c -0.5) b))))

C

double code(double a, double b, double c) {
	double t_0 = b * (b * b);
	double t_1 = b * t_0;
	return fma(a, fma(a, fma(((((c * (c * (c * c))) * (a * 6.328125)) / ((b * b) * t_1)) / b), -0.16666666666666666, (((c * c) * (c * -0.5625)) / (b * t_1))), (((c * c) * -0.375) / t_0)), ((c * -0.5) / b));
}

Julia

function code(a, b, c)
	t_0 = Float64(b * Float64(b * b))
	t_1 = Float64(b * t_0)
	return fma(a, fma(a, fma(Float64(Float64(Float64(Float64(c * Float64(c * Float64(c * c))) * Float64(a * 6.328125)) / Float64(Float64(b * b) * t_1)) / b), -0.16666666666666666, Float64(Float64(Float64(c * c) * Float64(c * -0.5625)) / Float64(b * t_1))), Float64(Float64(Float64(c * c) * -0.375) / t_0)), Float64(Float64(c * -0.5) / b))
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(b * t$95$0), $MachinePrecision]}, N[(a * N[(a * N[(N[(N[(N[(N[(c * N[(c * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(a * 6.328125), $MachinePrecision]), $MachinePrecision] / N[(N[(b * b), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision] * -0.16666666666666666 + N[(N[(N[(c * c), $MachinePrecision] * N[(c * -0.5625), $MachinePrecision]), $MachinePrecision] / N[(b * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(c * c), $MachinePrecision] * -0.375), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 20.0%

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

3. Taylor expanded in a around 0

   \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b} + a \cdot \left(\frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}} + a \cdot \left(\frac{-9}{16} \cdot \frac{{c}^{3}}{{b}^{5}} + \frac{-1}{6} \cdot \frac{a \cdot \left(\frac{81}{64} \cdot \frac{{c}^{4}}{{b}^{6}} + \frac{81}{16} \cdot \frac{{c}^{4}}{{b}^{6}}\right)}{b}\right)\right)} \]

5. Simplified 97.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{\frac{{c}^{4}}{{b}^{6}} \cdot \left(6.328125 \cdot a\right)}{b}, -0.16666666666666666, \frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot -0.5625}{{b}^{5}}\right), \frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot \left(b \cdot b\right)}\right), \frac{c \cdot -0.5}{b}\right)} \]
6. Step-by-step derivation

   1. lift-pow.f64 N/A

      \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{\frac{\color{blue}{{c}^{4}}}{{b}^{6}} \cdot \left(\frac{405}{64} \cdot a\right)}{b}, \frac{-1}{6}, \frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot \frac{-9}{16}}{{b}^{5}}\right), \frac{\left(c \cdot c\right) \cdot \frac{-3}{8}}{b \cdot \left(b \cdot b\right)}\right), \frac{c \cdot \frac{-1}{2}}{b}\right) \]

   2. lift-pow.f64 N/A

      \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{\frac{{c}^{4}}{\color{blue}{{b}^{6}}} \cdot \left(\frac{405}{64} \cdot a\right)}{b}, \frac{-1}{6}, \frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot \frac{-9}{16}}{{b}^{5}}\right), \frac{\left(c \cdot c\right) \cdot \frac{-3}{8}}{b \cdot \left(b \cdot b\right)}\right), \frac{c \cdot \frac{-1}{2}}{b}\right) \]

   3. lift-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{\frac{{c}^{4}}{{b}^{6}} \cdot \color{blue}{\left(\frac{405}{64} \cdot a\right)}}{b}, \frac{-1}{6}, \frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot \frac{-9}{16}}{{b}^{5}}\right), \frac{\left(c \cdot c\right) \cdot \frac{-3}{8}}{b \cdot \left(b \cdot b\right)}\right), \frac{c \cdot \frac{-1}{2}}{b}\right) \]

   4. associate-*l/ N/A

      \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{\color{blue}{\frac{{c}^{4} \cdot \left(\frac{405}{64} \cdot a\right)}{{b}^{6}}}}{b}, \frac{-1}{6}, \frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot \frac{-9}{16}}{{b}^{5}}\right), \frac{\left(c \cdot c\right) \cdot \frac{-3}{8}}{b \cdot \left(b \cdot b\right)}\right), \frac{c \cdot \frac{-1}{2}}{b}\right) \]

   5. lower-/.f64 N/A

      \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{\color{blue}{\frac{{c}^{4} \cdot \left(\frac{405}{64} \cdot a\right)}{{b}^{6}}}}{b}, \frac{-1}{6}, \frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot \frac{-9}{16}}{{b}^{5}}\right), \frac{\left(c \cdot c\right) \cdot \frac{-3}{8}}{b \cdot \left(b \cdot b\right)}\right), \frac{c \cdot \frac{-1}{2}}{b}\right) \]

   6. lower-*.f64 97.9

      \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{\frac{\color{blue}{{c}^{4} \cdot \left(6.328125 \cdot a\right)}}{{b}^{6}}}{b}, -0.16666666666666666, \frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot -0.5625}{{b}^{5}}\right), \frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot \left(b \cdot b\right)}\right), \frac{c \cdot -0.5}{b}\right) \]

   7. lift-pow.f64 N/A

      \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{\frac{\color{blue}{{c}^{4}} \cdot \left(\frac{405}{64} \cdot a\right)}{{b}^{6}}}{b}, \frac{-1}{6}, \frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot \frac{-9}{16}}{{b}^{5}}\right), \frac{\left(c \cdot c\right) \cdot \frac{-3}{8}}{b \cdot \left(b \cdot b\right)}\right), \frac{c \cdot \frac{-1}{2}}{b}\right) \]

   8. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{\frac{{c}^{\color{blue}{\left(2 + 2\right)}} \cdot \left(\frac{405}{64} \cdot a\right)}{{b}^{6}}}{b}, \frac{-1}{6}, \frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot \frac{-9}{16}}{{b}^{5}}\right), \frac{\left(c \cdot c\right) \cdot \frac{-3}{8}}{b \cdot \left(b \cdot b\right)}\right), \frac{c \cdot \frac{-1}{2}}{b}\right) \]

   9. pow-prod-up N/A

      \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{\frac{\color{blue}{\left({c}^{2} \cdot {c}^{2}\right)} \cdot \left(\frac{405}{64} \cdot a\right)}{{b}^{6}}}{b}, \frac{-1}{6}, \frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot \frac{-9}{16}}{{b}^{5}}\right), \frac{\left(c \cdot c\right) \cdot \frac{-3}{8}}{b \cdot \left(b \cdot b\right)}\right), \frac{c \cdot \frac{-1}{2}}{b}\right) \]

   10. pow2 N/A

       \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{\frac{\left(\color{blue}{\left(c \cdot c\right)} \cdot {c}^{2}\right) \cdot \left(\frac{405}{64} \cdot a\right)}{{b}^{6}}}{b}, \frac{-1}{6}, \frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot \frac{-9}{16}}{{b}^{5}}\right), \frac{\left(c \cdot c\right) \cdot \frac{-3}{8}}{b \cdot \left(b \cdot b\right)}\right), \frac{c \cdot \frac{-1}{2}}{b}\right) \]

   11. pow2 N/A

       \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{\frac{\left(\left(c \cdot c\right) \cdot \color{blue}{\left(c \cdot c\right)}\right) \cdot \left(\frac{405}{64} \cdot a\right)}{{b}^{6}}}{b}, \frac{-1}{6}, \frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot \frac{-9}{16}}{{b}^{5}}\right), \frac{\left(c \cdot c\right) \cdot \frac{-3}{8}}{b \cdot \left(b \cdot b\right)}\right), \frac{c \cdot \frac{-1}{2}}{b}\right) \]

   12. lift-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{\frac{\left(\left(c \cdot c\right) \cdot \color{blue}{\left(c \cdot c\right)}\right) \cdot \left(\frac{405}{64} \cdot a\right)}{{b}^{6}}}{b}, \frac{-1}{6}, \frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot \frac{-9}{16}}{{b}^{5}}\right), \frac{\left(c \cdot c\right) \cdot \frac{-3}{8}}{b \cdot \left(b \cdot b\right)}\right), \frac{c \cdot \frac{-1}{2}}{b}\right) \]

   13. associate-*l* N/A

       \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{\frac{\color{blue}{\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right)} \cdot \left(\frac{405}{64} \cdot a\right)}{{b}^{6}}}{b}, \frac{-1}{6}, \frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot \frac{-9}{16}}{{b}^{5}}\right), \frac{\left(c \cdot c\right) \cdot \frac{-3}{8}}{b \cdot \left(b \cdot b\right)}\right), \frac{c \cdot \frac{-1}{2}}{b}\right) \]

   14. lift-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{\frac{\left(c \cdot \color{blue}{\left(c \cdot \left(c \cdot c\right)\right)}\right) \cdot \left(\frac{405}{64} \cdot a\right)}{{b}^{6}}}{b}, \frac{-1}{6}, \frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot \frac{-9}{16}}{{b}^{5}}\right), \frac{\left(c \cdot c\right) \cdot \frac{-3}{8}}{b \cdot \left(b \cdot b\right)}\right), \frac{c \cdot \frac{-1}{2}}{b}\right) \]

   15. lower-*.f64 97.9

       \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{\frac{\color{blue}{\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right)} \cdot \left(6.328125 \cdot a\right)}{{b}^{6}}}{b}, -0.16666666666666666, \frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot -0.5625}{{b}^{5}}\right), \frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot \left(b \cdot b\right)}\right), \frac{c \cdot -0.5}{b}\right) \]

   16. lift-pow.f64 N/A

       \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{\frac{\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot \left(\frac{405}{64} \cdot a\right)}{\color{blue}{{b}^{6}}}}{b}, \frac{-1}{6}, \frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot \frac{-9}{16}}{{b}^{5}}\right), \frac{\left(c \cdot c\right) \cdot \frac{-3}{8}}{b \cdot \left(b \cdot b\right)}\right), \frac{c \cdot \frac{-1}{2}}{b}\right) \]

   17. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{\frac{\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot \left(\frac{405}{64} \cdot a\right)}{{b}^{\color{blue}{\left(2 \cdot 3\right)}}}}{b}, \frac{-1}{6}, \frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot \frac{-9}{16}}{{b}^{5}}\right), \frac{\left(c \cdot c\right) \cdot \frac{-3}{8}}{b \cdot \left(b \cdot b\right)}\right), \frac{c \cdot \frac{-1}{2}}{b}\right) \]

   18. pow-pow N/A

       \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{\frac{\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot \left(\frac{405}{64} \cdot a\right)}{\color{blue}{{\left({b}^{2}\right)}^{3}}}}{b}, \frac{-1}{6}, \frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot \frac{-9}{16}}{{b}^{5}}\right), \frac{\left(c \cdot c\right) \cdot \frac{-3}{8}}{b \cdot \left(b \cdot b\right)}\right), \frac{c \cdot \frac{-1}{2}}{b}\right) \]

   19. pow2 N/A

       \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{\frac{\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot \left(\frac{405}{64} \cdot a\right)}{{\color{blue}{\left(b \cdot b\right)}}^{3}}}{b}, \frac{-1}{6}, \frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot \frac{-9}{16}}{{b}^{5}}\right), \frac{\left(c \cdot c\right) \cdot \frac{-3}{8}}{b \cdot \left(b \cdot b\right)}\right), \frac{c \cdot \frac{-1}{2}}{b}\right) \]

   20. lift-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{\frac{\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot \left(\frac{405}{64} \cdot a\right)}{{\color{blue}{\left(b \cdot b\right)}}^{3}}}{b}, \frac{-1}{6}, \frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot \frac{-9}{16}}{{b}^{5}}\right), \frac{\left(c \cdot c\right) \cdot \frac{-3}{8}}{b \cdot \left(b \cdot b\right)}\right), \frac{c \cdot \frac{-1}{2}}{b}\right) \]

   21. cube-mult N/A

       \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{\frac{\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot \left(\frac{405}{64} \cdot a\right)}{\color{blue}{\left(b \cdot b\right) \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot b\right)\right)}}}{b}, \frac{-1}{6}, \frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot \frac{-9}{16}}{{b}^{5}}\right), \frac{\left(c \cdot c\right) \cdot \frac{-3}{8}}{b \cdot \left(b \cdot b\right)}\right), \frac{c \cdot \frac{-1}{2}}{b}\right) \]

   22. lift-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{\frac{\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot \left(\frac{405}{64} \cdot a\right)}{\left(b \cdot b\right) \cdot \left(\color{blue}{\left(b \cdot b\right)} \cdot \left(b \cdot b\right)\right)}}{b}, \frac{-1}{6}, \frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot \frac{-9}{16}}{{b}^{5}}\right), \frac{\left(c \cdot c\right) \cdot \frac{-3}{8}}{b \cdot \left(b \cdot b\right)}\right), \frac{c \cdot \frac{-1}{2}}{b}\right) \]

   23. associate-*r* N/A

       \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{\frac{\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot \left(\frac{405}{64} \cdot a\right)}{\left(b \cdot b\right) \cdot \color{blue}{\left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}}}{b}, \frac{-1}{6}, \frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot \frac{-9}{16}}{{b}^{5}}\right), \frac{\left(c \cdot c\right) \cdot \frac{-3}{8}}{b \cdot \left(b \cdot b\right)}\right), \frac{c \cdot \frac{-1}{2}}{b}\right) \]

   24. lift-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{\frac{\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot \left(\frac{405}{64} \cdot a\right)}{\left(b \cdot b\right) \cdot \left(b \cdot \color{blue}{\left(b \cdot \left(b \cdot b\right)\right)}\right)}}{b}, \frac{-1}{6}, \frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot \frac{-9}{16}}{{b}^{5}}\right), \frac{\left(c \cdot c\right) \cdot \frac{-3}{8}}{b \cdot \left(b \cdot b\right)}\right), \frac{c \cdot \frac{-1}{2}}{b}\right) \]

   25. lift-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{\frac{\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot \left(\frac{405}{64} \cdot a\right)}{\left(b \cdot b\right) \cdot \color{blue}{\left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}}}{b}, \frac{-1}{6}, \frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot \frac{-9}{16}}{{b}^{5}}\right), \frac{\left(c \cdot c\right) \cdot \frac{-3}{8}}{b \cdot \left(b \cdot b\right)}\right), \frac{c \cdot \frac{-1}{2}}{b}\right) \]

   26. lower-*.f64 97.9

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

7. Applied egg-rr 97.9%

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

   1. lift-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{\frac{\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot \left(\frac{405}{64} \cdot a\right)}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}}{b}, \frac{-1}{6}, \frac{\left(c \cdot \color{blue}{\left(c \cdot c\right)}\right) \cdot \frac{-9}{16}}{{b}^{5}}\right), \frac{\left(c \cdot c\right) \cdot \frac{-3}{8}}{b \cdot \left(b \cdot b\right)}\right), \frac{c \cdot \frac{-1}{2}}{b}\right) \]

   2. lift-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{\frac{\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot \left(\frac{405}{64} \cdot a\right)}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}}{b}, \frac{-1}{6}, \frac{\color{blue}{\left(c \cdot \left(c \cdot c\right)\right)} \cdot \frac{-9}{16}}{{b}^{5}}\right), \frac{\left(c \cdot c\right) \cdot \frac{-3}{8}}{b \cdot \left(b \cdot b\right)}\right), \frac{c \cdot \frac{-1}{2}}{b}\right) \]

   3. lift-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{\frac{\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot \left(\frac{405}{64} \cdot a\right)}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}}{b}, \frac{-1}{6}, \frac{\color{blue}{\left(c \cdot \left(c \cdot c\right)\right) \cdot \frac{-9}{16}}}{{b}^{5}}\right), \frac{\left(c \cdot c\right) \cdot \frac{-3}{8}}{b \cdot \left(b \cdot b\right)}\right), \frac{c \cdot \frac{-1}{2}}{b}\right) \]

   4. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{\frac{\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot \left(\frac{405}{64} \cdot a\right)}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}}{b}, \frac{-1}{6}, \frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot \frac{-9}{16}}{{b}^{\color{blue}{\left(2 + 3\right)}}}\right), \frac{\left(c \cdot c\right) \cdot \frac{-3}{8}}{b \cdot \left(b \cdot b\right)}\right), \frac{c \cdot \frac{-1}{2}}{b}\right) \]

   5. pow-prod-up N/A

      \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{\frac{\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot \left(\frac{405}{64} \cdot a\right)}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}}{b}, \frac{-1}{6}, \frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot \frac{-9}{16}}{\color{blue}{{b}^{2} \cdot {b}^{3}}}\right), \frac{\left(c \cdot c\right) \cdot \frac{-3}{8}}{b \cdot \left(b \cdot b\right)}\right), \frac{c \cdot \frac{-1}{2}}{b}\right) \]

   6. pow2 N/A

      \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{\frac{\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot \left(\frac{405}{64} \cdot a\right)}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}}{b}, \frac{-1}{6}, \frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot \frac{-9}{16}}{\color{blue}{\left(b \cdot b\right)} \cdot {b}^{3}}\right), \frac{\left(c \cdot c\right) \cdot \frac{-3}{8}}{b \cdot \left(b \cdot b\right)}\right), \frac{c \cdot \frac{-1}{2}}{b}\right) \]

   7. cube-unmult N/A

      \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{\frac{\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot \left(\frac{405}{64} \cdot a\right)}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}}{b}, \frac{-1}{6}, \frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot \frac{-9}{16}}{\left(b \cdot b\right) \cdot \color{blue}{\left(b \cdot \left(b \cdot b\right)\right)}}\right), \frac{\left(c \cdot c\right) \cdot \frac{-3}{8}}{b \cdot \left(b \cdot b\right)}\right), \frac{c \cdot \frac{-1}{2}}{b}\right) \]

   8. lift-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{\frac{\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot \left(\frac{405}{64} \cdot a\right)}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}}{b}, \frac{-1}{6}, \frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot \frac{-9}{16}}{\left(b \cdot b\right) \cdot \left(b \cdot \color{blue}{\left(b \cdot b\right)}\right)}\right), \frac{\left(c \cdot c\right) \cdot \frac{-3}{8}}{b \cdot \left(b \cdot b\right)}\right), \frac{c \cdot \frac{-1}{2}}{b}\right) \]

   9. lift-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{\frac{\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot \left(\frac{405}{64} \cdot a\right)}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}}{b}, \frac{-1}{6}, \frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot \frac{-9}{16}}{\left(b \cdot b\right) \cdot \color{blue}{\left(b \cdot \left(b \cdot b\right)\right)}}\right), \frac{\left(c \cdot c\right) \cdot \frac{-3}{8}}{b \cdot \left(b \cdot b\right)}\right), \frac{c \cdot \frac{-1}{2}}{b}\right) \]

   10. associate-*r* N/A

       \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{\frac{\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot \left(\frac{405}{64} \cdot a\right)}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}}{b}, \frac{-1}{6}, \frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot \frac{-9}{16}}{\color{blue}{b \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}}\right), \frac{\left(c \cdot c\right) \cdot \frac{-3}{8}}{b \cdot \left(b \cdot b\right)}\right), \frac{c \cdot \frac{-1}{2}}{b}\right) \]

   11. lift-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{\frac{\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot \left(\frac{405}{64} \cdot a\right)}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}}{b}, \frac{-1}{6}, \frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot \frac{-9}{16}}{b \cdot \color{blue}{\left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}}\right), \frac{\left(c \cdot c\right) \cdot \frac{-3}{8}}{b \cdot \left(b \cdot b\right)}\right), \frac{c \cdot \frac{-1}{2}}{b}\right) \]

   12. lower-/.f64 N/A

       \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{\frac{\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot \left(\frac{405}{64} \cdot a\right)}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}}{b}, \frac{-1}{6}, \color{blue}{\frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot \frac{-9}{16}}{b \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}}\right), \frac{\left(c \cdot c\right) \cdot \frac{-3}{8}}{b \cdot \left(b \cdot b\right)}\right), \frac{c \cdot \frac{-1}{2}}{b}\right) \]

   13. lift-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{\frac{\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot \left(\frac{405}{64} \cdot a\right)}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}}{b}, \frac{-1}{6}, \frac{\color{blue}{\left(c \cdot \left(c \cdot c\right)\right) \cdot \frac{-9}{16}}}{b \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}\right), \frac{\left(c \cdot c\right) \cdot \frac{-3}{8}}{b \cdot \left(b \cdot b\right)}\right), \frac{c \cdot \frac{-1}{2}}{b}\right) \]

   14. lift-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{\frac{\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot \left(\frac{405}{64} \cdot a\right)}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}}{b}, \frac{-1}{6}, \frac{\color{blue}{\left(c \cdot \left(c \cdot c\right)\right)} \cdot \frac{-9}{16}}{b \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}\right), \frac{\left(c \cdot c\right) \cdot \frac{-3}{8}}{b \cdot \left(b \cdot b\right)}\right), \frac{c \cdot \frac{-1}{2}}{b}\right) \]

   15. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{\frac{\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot \left(\frac{405}{64} \cdot a\right)}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}}{b}, \frac{-1}{6}, \frac{\color{blue}{\left(\left(c \cdot c\right) \cdot c\right)} \cdot \frac{-9}{16}}{b \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}\right), \frac{\left(c \cdot c\right) \cdot \frac{-3}{8}}{b \cdot \left(b \cdot b\right)}\right), \frac{c \cdot \frac{-1}{2}}{b}\right) \]

   16. associate-*l* N/A

       \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{\frac{\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot \left(\frac{405}{64} \cdot a\right)}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}}{b}, \frac{-1}{6}, \frac{\color{blue}{\left(c \cdot c\right) \cdot \left(c \cdot \frac{-9}{16}\right)}}{b \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}\right), \frac{\left(c \cdot c\right) \cdot \frac{-3}{8}}{b \cdot \left(b \cdot b\right)}\right), \frac{c \cdot \frac{-1}{2}}{b}\right) \]

   17. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{\frac{\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot \left(\frac{405}{64} \cdot a\right)}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}}{b}, \frac{-1}{6}, \frac{\color{blue}{\left(c \cdot c\right) \cdot \left(c \cdot \frac{-9}{16}\right)}}{b \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}\right), \frac{\left(c \cdot c\right) \cdot \frac{-3}{8}}{b \cdot \left(b \cdot b\right)}\right), \frac{c \cdot \frac{-1}{2}}{b}\right) \]

   18. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{\frac{\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot \left(\frac{405}{64} \cdot a\right)}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}}{b}, \frac{-1}{6}, \frac{\left(c \cdot c\right) \cdot \color{blue}{\left(c \cdot \frac{-9}{16}\right)}}{b \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}\right), \frac{\left(c \cdot c\right) \cdot \frac{-3}{8}}{b \cdot \left(b \cdot b\right)}\right), \frac{c \cdot \frac{-1}{2}}{b}\right) \]

   19. lower-*.f64 97.9

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

9. Applied egg-rr 97.9%

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

10. Final simplification 97.9%

    \[\leadsto \mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(\frac{\frac{\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot \left(a \cdot 6.328125\right)}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}}{b}, -0.16666666666666666, \frac{\left(c \cdot c\right) \cdot \left(c \cdot -0.5625\right)}{b \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)}\right), \frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot \left(b \cdot b\right)}\right), \frac{c \cdot -0.5}{b}\right) \]
11. Add Preprocessing

Alternative 2: 96.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\frac{\mathsf{fma}\left(c, \mathsf{fma}\left(c \cdot 3, \frac{a \cdot a}{b \cdot \left(b \cdot b\right)} \cdot 0.375, \frac{a \cdot 1.5}{b}\right), b \cdot -2\right)}{c}} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  1.0
  (/
   (fma
    c
    (fma (* c 3.0) (* (/ (* a a) (* b (* b b))) 0.375) (/ (* a 1.5) b))
    (* b -2.0))
   c)))

C

double code(double a, double b, double c) {
	return 1.0 / (fma(c, fma((c * 3.0), (((a * a) / (b * (b * b))) * 0.375), ((a * 1.5) / b)), (b * -2.0)) / c);
}

Julia

function code(a, b, c)
	return Float64(1.0 / Float64(fma(c, fma(Float64(c * 3.0), Float64(Float64(Float64(a * a) / Float64(b * Float64(b * b))) * 0.375), Float64(Float64(a * 1.5) / b)), Float64(b * -2.0)) / c))
end

Wolfram

code[a_, b_, c_] := N[(1.0 / N[(N[(c * N[(N[(c * 3.0), $MachinePrecision] * N[(N[(N[(a * a), $MachinePrecision] / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 0.375), $MachinePrecision] + N[(N[(a * 1.5), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision] + N[(b * -2.0), $MachinePrecision]), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 20.0%

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

4. Applied egg-rr 20.0%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-fma.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. lift--.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. clear-num N/A

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

   9. lower-/.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. lift-/.f64 N/A

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

   12. un-div-inv N/A

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

   13. associate-/r/ N/A

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

6. Applied egg-rr 20.0%

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

7. Taylor expanded in c around 0

   \[\leadsto \frac{1}{\color{blue}{\frac{-2 \cdot b + c \cdot \left(\frac{3}{2} \cdot \frac{a}{b} + 3 \cdot \left(c \cdot \left(\frac{-3}{8} \cdot \frac{{a}^{2}}{{b}^{3}} + \frac{3}{4} \cdot \frac{{a}^{2}}{{b}^{3}}\right)\right)\right)}{c}}} \]
8. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{-2 \cdot b + c \cdot \left(\frac{3}{2} \cdot \frac{a}{b} + 3 \cdot \left(c \cdot \left(\frac{-3}{8} \cdot \frac{{a}^{2}}{{b}^{3}} + \frac{3}{4} \cdot \frac{{a}^{2}}{{b}^{3}}\right)\right)\right)}{c}}} \]

9. Simplified 96.8%

   \[\leadsto \frac{1}{\color{blue}{\frac{\mathsf{fma}\left(c, \mathsf{fma}\left(3 \cdot c, \frac{a \cdot a}{b \cdot \left(b \cdot b\right)} \cdot 0.375, \frac{1.5 \cdot a}{b}\right), -2 \cdot b\right)}{c}}} \]

10. Final simplification 96.8%

    \[\leadsto \frac{1}{\frac{\mathsf{fma}\left(c, \mathsf{fma}\left(c \cdot 3, \frac{a \cdot a}{b \cdot \left(b \cdot b\right)} \cdot 0.375, \frac{a \cdot 1.5}{b}\right), b \cdot -2\right)}{c}} \]
11. Add Preprocessing

Alternative 3: 96.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(a \cdot 3, 0.375 \cdot \frac{c}{b \cdot \left(b \cdot b\right)}, \frac{1.5}{b}\right), \frac{b \cdot -2}{c}\right)} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  1.0
  (fma
   a
   (fma (* a 3.0) (* 0.375 (/ c (* b (* b b)))) (/ 1.5 b))
   (/ (* b -2.0) c))))

C

double code(double a, double b, double c) {
	return 1.0 / fma(a, fma((a * 3.0), (0.375 * (c / (b * (b * b)))), (1.5 / b)), ((b * -2.0) / c));
}

Julia

function code(a, b, c)
	return Float64(1.0 / fma(a, fma(Float64(a * 3.0), Float64(0.375 * Float64(c / Float64(b * Float64(b * b)))), Float64(1.5 / b)), Float64(Float64(b * -2.0) / c)))
end

Wolfram

code[a_, b_, c_] := N[(1.0 / N[(a * N[(N[(a * 3.0), $MachinePrecision] * N[(0.375 * N[(c / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.5 / b), $MachinePrecision]), $MachinePrecision] + N[(N[(b * -2.0), $MachinePrecision] / c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 20.0%

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

4. Applied egg-rr 20.0%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-fma.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. lift--.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. clear-num N/A

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

   9. lower-/.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. lift-/.f64 N/A

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

   12. un-div-inv N/A

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

   13. associate-/r/ N/A

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

6. Applied egg-rr 20.0%

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

7. Taylor expanded in a around 0

   \[\leadsto \frac{1}{\color{blue}{-2 \cdot \frac{b}{c} + a \cdot \left(3 \cdot \left(a \cdot \left(\frac{-3}{8} \cdot \frac{c}{{b}^{3}} + \frac{3}{4} \cdot \frac{c}{{b}^{3}}\right)\right) + \frac{3}{2} \cdot \frac{1}{b}\right)}} \]
8. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{1}{\color{blue}{a \cdot \left(3 \cdot \left(a \cdot \left(\frac{-3}{8} \cdot \frac{c}{{b}^{3}} + \frac{3}{4} \cdot \frac{c}{{b}^{3}}\right)\right) + \frac{3}{2} \cdot \frac{1}{b}\right) + -2 \cdot \frac{b}{c}}} \]

   2. lower-fma.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(a, 3 \cdot \left(a \cdot \left(\frac{-3}{8} \cdot \frac{c}{{b}^{3}} + \frac{3}{4} \cdot \frac{c}{{b}^{3}}\right)\right) + \frac{3}{2} \cdot \frac{1}{b}, -2 \cdot \frac{b}{c}\right)}} \]

9. Simplified 96.8%

   \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(a, \mathsf{fma}\left(3 \cdot a, \frac{c}{b \cdot \left(b \cdot b\right)} \cdot 0.375, \frac{1.5}{b}\right), \frac{-2 \cdot b}{c}\right)}} \]

10. Final simplification 96.8%

    \[\leadsto \frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(a \cdot 3, 0.375 \cdot \frac{c}{b \cdot \left(b \cdot b\right)}, \frac{1.5}{b}\right), \frac{b \cdot -2}{c}\right)} \]
11. Add Preprocessing

Alternative 4: 95.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(a, \frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot \left(b \cdot b\right)}, \frac{c \cdot -0.5}{b}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (fma a (/ (* (* c c) -0.375) (* b (* b b))) (/ (* c -0.5) b)))

C

double code(double a, double b, double c) {
	return fma(a, (((c * c) * -0.375) / (b * (b * b))), ((c * -0.5) / b));
}

Julia

function code(a, b, c)
	return fma(a, Float64(Float64(Float64(c * c) * -0.375) / Float64(b * Float64(b * b))), Float64(Float64(c * -0.5) / b))
end

Wolfram

code[a_, b_, c_] := N[(a * N[(N[(N[(c * c), $MachinePrecision] * -0.375), $MachinePrecision] / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 20.0%

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

3. Taylor expanded in a around 0

   \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b} + \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{3}}} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{\frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{3}} + \frac{-1}{2} \cdot \frac{c}{b}} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\frac{a \cdot {c}^{2}}{{b}^{3}} \cdot \frac{-3}{8}} + \frac{-1}{2} \cdot \frac{c}{b} \]

   3. associate-/l* N/A

      \[\leadsto \color{blue}{\left(a \cdot \frac{{c}^{2}}{{b}^{3}}\right)} \cdot \frac{-3}{8} + \frac{-1}{2} \cdot \frac{c}{b} \]

   4. associate-*r* N/A

      \[\leadsto \color{blue}{a \cdot \left(\frac{{c}^{2}}{{b}^{3}} \cdot \frac{-3}{8}\right)} + \frac{-1}{2} \cdot \frac{c}{b} \]

   5. *-commutative N/A

      \[\leadsto a \cdot \color{blue}{\left(\frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}}\right)} + \frac{-1}{2} \cdot \frac{c}{b} \]

   6. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, \frac{-3}{8} \cdot \frac{{c}^{2}}{{b}^{3}}, \frac{-1}{2} \cdot \frac{c}{b}\right)} \]

   7. associate-*r/ N/A

      \[\leadsto \mathsf{fma}\left(a, \color{blue}{\frac{\frac{-3}{8} \cdot {c}^{2}}{{b}^{3}}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

   8. lower-/.f64 N/A

      \[\leadsto \mathsf{fma}\left(a, \color{blue}{\frac{\frac{-3}{8} \cdot {c}^{2}}{{b}^{3}}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

   9. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(a, \frac{\color{blue}{{c}^{2} \cdot \frac{-3}{8}}}{{b}^{3}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

   10. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(a, \frac{\color{blue}{{c}^{2} \cdot \frac{-3}{8}}}{{b}^{3}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

   11. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(a, \frac{\color{blue}{\left(c \cdot c\right)} \cdot \frac{-3}{8}}{{b}^{3}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

   12. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(a, \frac{\color{blue}{\left(c \cdot c\right)} \cdot \frac{-3}{8}}{{b}^{3}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

   13. cube-mult N/A

       \[\leadsto \mathsf{fma}\left(a, \frac{\left(c \cdot c\right) \cdot \frac{-3}{8}}{\color{blue}{b \cdot \left(b \cdot b\right)}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

   14. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(a, \frac{\left(c \cdot c\right) \cdot \frac{-3}{8}}{b \cdot \color{blue}{{b}^{2}}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

   15. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(a, \frac{\left(c \cdot c\right) \cdot \frac{-3}{8}}{\color{blue}{b \cdot {b}^{2}}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

   16. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(a, \frac{\left(c \cdot c\right) \cdot \frac{-3}{8}}{b \cdot \color{blue}{\left(b \cdot b\right)}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

   17. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(a, \frac{\left(c \cdot c\right) \cdot \frac{-3}{8}}{b \cdot \color{blue}{\left(b \cdot b\right)}}, \frac{-1}{2} \cdot \frac{c}{b}\right) \]

   18. associate-*r/ N/A

       \[\leadsto \mathsf{fma}\left(a, \frac{\left(c \cdot c\right) \cdot \frac{-3}{8}}{b \cdot \left(b \cdot b\right)}, \color{blue}{\frac{\frac{-1}{2} \cdot c}{b}}\right) \]

   19. lower-/.f64 N/A

       \[\leadsto \mathsf{fma}\left(a, \frac{\left(c \cdot c\right) \cdot \frac{-3}{8}}{b \cdot \left(b \cdot b\right)}, \color{blue}{\frac{\frac{-1}{2} \cdot c}{b}}\right) \]

   20. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(a, \frac{\left(c \cdot c\right) \cdot \frac{-3}{8}}{b \cdot \left(b \cdot b\right)}, \frac{\color{blue}{c \cdot \frac{-1}{2}}}{b}\right) \]

   21. lower-*.f64 94.8

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

5. Simplified 94.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(a, \frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot \left(b \cdot b\right)}, \frac{c \cdot -0.5}{b}\right)} \]
6. Add Preprocessing

Alternative 5: 95.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\mathsf{fma}\left(a, -0.375 \cdot \left(c \cdot \frac{c}{b \cdot b}\right), c \cdot -0.5\right)}{b} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (fma a (* -0.375 (* c (/ c (* b b)))) (* c -0.5)) b))

C

double code(double a, double b, double c) {
	return fma(a, (-0.375 * (c * (c / (b * b)))), (c * -0.5)) / b;
}

Julia

function code(a, b, c)
	return Float64(fma(a, Float64(-0.375 * Float64(c * Float64(c / Float64(b * b)))), Float64(c * -0.5)) / b)
end

Wolfram

code[a_, b_, c_] := N[(N[(a * N[(-0.375 * N[(c * N[(c / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(c * -0.5), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]

Derivation

1. Initial program 20.0%

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

3. Taylor expanded in b around inf

   \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c + \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot c + \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}}{b}} \]

5. Simplified 94.8%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(a, -0.375 \cdot \left(c \cdot \frac{c}{b \cdot b}\right), c \cdot -0.5\right)}{b}} \]
6. Add Preprocessing

Alternative 6: 95.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\mathsf{fma}\left(-2, \frac{b}{c}, \frac{a \cdot 1.5}{b}\right)} \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 20.0%

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

4. Applied egg-rr 20.0%

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-fma.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. lift--.f64 N/A

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

   6. lift-/.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. clear-num N/A

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

   9. lower-/.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. lift-/.f64 N/A

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

   12. un-div-inv N/A

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

   13. associate-/r/ N/A

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

6. Applied egg-rr 20.0%

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

7. Taylor expanded in a around 0

   \[\leadsto \frac{1}{\color{blue}{-2 \cdot \frac{b}{c} + \frac{3}{2} \cdot \frac{a}{b}}} \]
8. Step-by-step derivation

   1. lower-fma.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. lower-/.f64 N/A

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

   5. lower-*.f64 94.7

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

9. Simplified 94.7%

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

10. Final simplification 94.7%

    \[\leadsto \frac{1}{\mathsf{fma}\left(-2, \frac{b}{c}, \frac{a \cdot 1.5}{b}\right)} \]
11. Add Preprocessing

Alternative 7: 90.9% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 20.0%

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

3. Taylor expanded in b around inf

   \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b}} \]
4. Step-by-step derivation

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 88.7

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

5. Simplified 88.7%

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

Alternative 8: 90.6% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 20.0%

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

3. Taylor expanded in b around inf

   \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b}} \]
4. Step-by-step derivation

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 88.7

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

5. Simplified 88.7%

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

   1. *-commutative N/A

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

   2. associate-*l/ N/A

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

   3. lift-/.f64 N/A

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

   4. lower-*.f64 88.3

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

7. Applied egg-rr 88.3%

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

8. Final simplification 88.3%

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

Reproduce

herbie shell --seed 2024215 
(FPCore (a b c)
  :name "Cubic critical, wide range"
  :precision binary64
  :pre (and (and (and (< 4.930380657631324e-32 a) (< a 2.028240960365167e+31)) (and (< 4.930380657631324e-32 b) (< b 2.028240960365167e+31))) (and (< 4.930380657631324e-32 c) (< c 2.028240960365167e+31)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))