Cubic critical, medium range

Percentage Accurate: 31.8% → 95.4%

Time: 13.9s

Alternatives: 6

Speedup: 2.9×

Specification

\[\left(\left(1.1102230246251565 \cdot 10^{-16} < a \land a < 9007199254740992\right) \land \left(1.1102230246251565 \cdot 10^{-16} < b \land b < 9007199254740992\right)\right) \land \left(1.1102230246251565 \cdot 10^{-16} < c \land c < 9007199254740992\right)\]

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: 31.8% 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: 95.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot \left(b \cdot b\right)\\ t_1 := c \cdot \left(c \cdot c\right)\\ \frac{\mathsf{fma}\left(c, -0.5, \mathsf{fma}\left(a, \frac{\mathsf{fma}\left(-0.5625, \frac{a \cdot t\_1}{b \cdot b}, \left(c \cdot c\right) \cdot -0.375\right)}{b \cdot b}, \frac{-1.0546875 \cdot \left(a \cdot \left(\left(c \cdot t\_1\right) \cdot \left(a \cdot a\right)\right)\right)}{t\_0 \cdot t\_0}\right)\right)}{b} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* b (* b b))) (t_1 (* c (* c c))))
   (/
    (fma
     c
     -0.5
     (fma
      a
      (/ (fma -0.5625 (/ (* a t_1) (* b b)) (* (* c c) -0.375)) (* b b))
      (/ (* -1.0546875 (* a (* (* c t_1) (* a a)))) (* t_0 t_0))))
    b)))

C

double code(double a, double b, double c) {
	double t_0 = b * (b * b);
	double t_1 = c * (c * c);
	return fma(c, -0.5, fma(a, (fma(-0.5625, ((a * t_1) / (b * b)), ((c * c) * -0.375)) / (b * b)), ((-1.0546875 * (a * ((c * t_1) * (a * a)))) / (t_0 * t_0)))) / b;
}

Julia

function code(a, b, c)
	t_0 = Float64(b * Float64(b * b))
	t_1 = Float64(c * Float64(c * c))
	return Float64(fma(c, -0.5, fma(a, Float64(fma(-0.5625, Float64(Float64(a * t_1) / Float64(b * b)), Float64(Float64(c * c) * -0.375)) / Float64(b * b)), Float64(Float64(-1.0546875 * Float64(a * Float64(Float64(c * t_1) * Float64(a * a)))) / Float64(t_0 * t_0)))) / b)
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(c * N[(c * c), $MachinePrecision]), $MachinePrecision]}, N[(N[(c * -0.5 + N[(a * N[(N[(-0.5625 * N[(N[(a * t$95$1), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(N[(c * c), $MachinePrecision] * -0.375), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(N[(-1.0546875 * N[(a * N[(N[(c * t$95$1), $MachinePrecision] * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / b), $MachinePrecision]]]

Derivation

1. Initial program 32.9%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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. Applied rewrites 95.0%

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

6. Taylor expanded in b around inf

   \[\leadsto \color{blue}{\frac{\frac{-135}{128} \cdot \frac{{a}^{3} \cdot {c}^{4}}{{b}^{6}} + \left(\frac{-9}{16} \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\frac{-1}{2} \cdot c + \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}\right)\right)}{b}} \]

8. Applied rewrites 95.0%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-1.0546875, \frac{{c}^{4} \cdot \left(a \cdot \left(a \cdot a\right)\right)}{{b}^{6}}, \mathsf{fma}\left(a, \mathsf{fma}\left(a, \frac{-0.5625 \cdot \left(c \cdot \left(c \cdot c\right)\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, \frac{c \cdot \left(c \cdot -0.375\right)}{b \cdot b}\right), c \cdot -0.5\right)\right)}{b}} \]

10. Applied rewrites 95.0%

    \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(c, -0.5, \mathsf{fma}\left(a, \mathsf{fma}\left(a \cdot -0.5625, \frac{c \cdot \left(c \cdot c\right)}{b \cdot \left(b \cdot \left(b \cdot b\right)\right)}, \frac{c \cdot c}{\left(b \cdot b\right) \cdot -2.6666666666666665}\right), \frac{-1.0546875 \cdot \left(a \cdot \left(\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot \left(a \cdot a\right)\right)\right)}{\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)}\right)\right)}}{b} \]

11. Taylor expanded in b around inf

    \[\leadsto \frac{\mathsf{fma}\left(c, \frac{-1}{2}, \mathsf{fma}\left(a, \color{blue}{\frac{\frac{-9}{16} \cdot \frac{a \cdot {c}^{3}}{{b}^{2}} + \frac{-3}{8} \cdot {c}^{2}}{{b}^{2}}}, \frac{\frac{-135}{128} \cdot \left(a \cdot \left(\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot \left(a \cdot a\right)\right)\right)}{\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)}\right)\right)}{b} \]
12. Step-by-step derivation

    1. lower-/.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(c, \frac{-1}{2}, \mathsf{fma}\left(a, \color{blue}{\frac{\frac{-9}{16} \cdot \frac{a \cdot {c}^{3}}{{b}^{2}} + \frac{-3}{8} \cdot {c}^{2}}{{b}^{2}}}, \frac{\frac{-135}{128} \cdot \left(a \cdot \left(\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot \left(a \cdot a\right)\right)\right)}{\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)}\right)\right)}{b} \]

    2. lower-fma.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(c, \frac{-1}{2}, \mathsf{fma}\left(a, \frac{\color{blue}{\mathsf{fma}\left(\frac{-9}{16}, \frac{a \cdot {c}^{3}}{{b}^{2}}, \frac{-3}{8} \cdot {c}^{2}\right)}}{{b}^{2}}, \frac{\frac{-135}{128} \cdot \left(a \cdot \left(\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot \left(a \cdot a\right)\right)\right)}{\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)}\right)\right)}{b} \]

    3. lower-/.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(c, \frac{-1}{2}, \mathsf{fma}\left(a, \frac{\mathsf{fma}\left(\frac{-9}{16}, \color{blue}{\frac{a \cdot {c}^{3}}{{b}^{2}}}, \frac{-3}{8} \cdot {c}^{2}\right)}{{b}^{2}}, \frac{\frac{-135}{128} \cdot \left(a \cdot \left(\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot \left(a \cdot a\right)\right)\right)}{\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)}\right)\right)}{b} \]

    4. lower-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(c, \frac{-1}{2}, \mathsf{fma}\left(a, \frac{\mathsf{fma}\left(\frac{-9}{16}, \frac{\color{blue}{a \cdot {c}^{3}}}{{b}^{2}}, \frac{-3}{8} \cdot {c}^{2}\right)}{{b}^{2}}, \frac{\frac{-135}{128} \cdot \left(a \cdot \left(\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot \left(a \cdot a\right)\right)\right)}{\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)}\right)\right)}{b} \]

    5. cube-mult N/A

       \[\leadsto \frac{\mathsf{fma}\left(c, \frac{-1}{2}, \mathsf{fma}\left(a, \frac{\mathsf{fma}\left(\frac{-9}{16}, \frac{a \cdot \color{blue}{\left(c \cdot \left(c \cdot c\right)\right)}}{{b}^{2}}, \frac{-3}{8} \cdot {c}^{2}\right)}{{b}^{2}}, \frac{\frac{-135}{128} \cdot \left(a \cdot \left(\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot \left(a \cdot a\right)\right)\right)}{\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)}\right)\right)}{b} \]

    6. unpow2 N/A

       \[\leadsto \frac{\mathsf{fma}\left(c, \frac{-1}{2}, \mathsf{fma}\left(a, \frac{\mathsf{fma}\left(\frac{-9}{16}, \frac{a \cdot \left(c \cdot \color{blue}{{c}^{2}}\right)}{{b}^{2}}, \frac{-3}{8} \cdot {c}^{2}\right)}{{b}^{2}}, \frac{\frac{-135}{128} \cdot \left(a \cdot \left(\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot \left(a \cdot a\right)\right)\right)}{\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)}\right)\right)}{b} \]

    7. lower-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(c, \frac{-1}{2}, \mathsf{fma}\left(a, \frac{\mathsf{fma}\left(\frac{-9}{16}, \frac{a \cdot \color{blue}{\left(c \cdot {c}^{2}\right)}}{{b}^{2}}, \frac{-3}{8} \cdot {c}^{2}\right)}{{b}^{2}}, \frac{\frac{-135}{128} \cdot \left(a \cdot \left(\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot \left(a \cdot a\right)\right)\right)}{\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)}\right)\right)}{b} \]

    8. unpow2 N/A

       \[\leadsto \frac{\mathsf{fma}\left(c, \frac{-1}{2}, \mathsf{fma}\left(a, \frac{\mathsf{fma}\left(\frac{-9}{16}, \frac{a \cdot \left(c \cdot \color{blue}{\left(c \cdot c\right)}\right)}{{b}^{2}}, \frac{-3}{8} \cdot {c}^{2}\right)}{{b}^{2}}, \frac{\frac{-135}{128} \cdot \left(a \cdot \left(\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot \left(a \cdot a\right)\right)\right)}{\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)}\right)\right)}{b} \]

    9. lower-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(c, \frac{-1}{2}, \mathsf{fma}\left(a, \frac{\mathsf{fma}\left(\frac{-9}{16}, \frac{a \cdot \left(c \cdot \color{blue}{\left(c \cdot c\right)}\right)}{{b}^{2}}, \frac{-3}{8} \cdot {c}^{2}\right)}{{b}^{2}}, \frac{\frac{-135}{128} \cdot \left(a \cdot \left(\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot \left(a \cdot a\right)\right)\right)}{\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)}\right)\right)}{b} \]

    10. unpow2 N/A

        \[\leadsto \frac{\mathsf{fma}\left(c, \frac{-1}{2}, \mathsf{fma}\left(a, \frac{\mathsf{fma}\left(\frac{-9}{16}, \frac{a \cdot \left(c \cdot \left(c \cdot c\right)\right)}{\color{blue}{b \cdot b}}, \frac{-3}{8} \cdot {c}^{2}\right)}{{b}^{2}}, \frac{\frac{-135}{128} \cdot \left(a \cdot \left(\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot \left(a \cdot a\right)\right)\right)}{\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)}\right)\right)}{b} \]

    11. lower-*.f64 N/A

        \[\leadsto \frac{\mathsf{fma}\left(c, \frac{-1}{2}, \mathsf{fma}\left(a, \frac{\mathsf{fma}\left(\frac{-9}{16}, \frac{a \cdot \left(c \cdot \left(c \cdot c\right)\right)}{\color{blue}{b \cdot b}}, \frac{-3}{8} \cdot {c}^{2}\right)}{{b}^{2}}, \frac{\frac{-135}{128} \cdot \left(a \cdot \left(\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot \left(a \cdot a\right)\right)\right)}{\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)}\right)\right)}{b} \]

    12. lower-*.f64 N/A

        \[\leadsto \frac{\mathsf{fma}\left(c, \frac{-1}{2}, \mathsf{fma}\left(a, \frac{\mathsf{fma}\left(\frac{-9}{16}, \frac{a \cdot \left(c \cdot \left(c \cdot c\right)\right)}{b \cdot b}, \color{blue}{\frac{-3}{8} \cdot {c}^{2}}\right)}{{b}^{2}}, \frac{\frac{-135}{128} \cdot \left(a \cdot \left(\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot \left(a \cdot a\right)\right)\right)}{\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)}\right)\right)}{b} \]

    13. unpow2 N/A

        \[\leadsto \frac{\mathsf{fma}\left(c, \frac{-1}{2}, \mathsf{fma}\left(a, \frac{\mathsf{fma}\left(\frac{-9}{16}, \frac{a \cdot \left(c \cdot \left(c \cdot c\right)\right)}{b \cdot b}, \frac{-3}{8} \cdot \color{blue}{\left(c \cdot c\right)}\right)}{{b}^{2}}, \frac{\frac{-135}{128} \cdot \left(a \cdot \left(\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot \left(a \cdot a\right)\right)\right)}{\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)}\right)\right)}{b} \]

    14. lower-*.f64 N/A

        \[\leadsto \frac{\mathsf{fma}\left(c, \frac{-1}{2}, \mathsf{fma}\left(a, \frac{\mathsf{fma}\left(\frac{-9}{16}, \frac{a \cdot \left(c \cdot \left(c \cdot c\right)\right)}{b \cdot b}, \frac{-3}{8} \cdot \color{blue}{\left(c \cdot c\right)}\right)}{{b}^{2}}, \frac{\frac{-135}{128} \cdot \left(a \cdot \left(\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot \left(a \cdot a\right)\right)\right)}{\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)}\right)\right)}{b} \]

    15. unpow2 N/A

        \[\leadsto \frac{\mathsf{fma}\left(c, \frac{-1}{2}, \mathsf{fma}\left(a, \frac{\mathsf{fma}\left(\frac{-9}{16}, \frac{a \cdot \left(c \cdot \left(c \cdot c\right)\right)}{b \cdot b}, \frac{-3}{8} \cdot \left(c \cdot c\right)\right)}{\color{blue}{b \cdot b}}, \frac{\frac{-135}{128} \cdot \left(a \cdot \left(\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot \left(a \cdot a\right)\right)\right)}{\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)}\right)\right)}{b} \]

    16. lower-*.f64 95.0

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

13. Applied rewrites 95.0%

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

14. Final simplification 95.0%

    \[\leadsto \frac{\mathsf{fma}\left(c, -0.5, \mathsf{fma}\left(a, \frac{\mathsf{fma}\left(-0.5625, \frac{a \cdot \left(c \cdot \left(c \cdot c\right)\right)}{b \cdot b}, \left(c \cdot c\right) \cdot -0.375\right)}{b \cdot b}, \frac{-1.0546875 \cdot \left(a \cdot \left(\left(c \cdot \left(c \cdot \left(c \cdot c\right)\right)\right) \cdot \left(a \cdot a\right)\right)\right)}{\left(b \cdot \left(b \cdot b\right)\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)}\right)\right)}{b} \]
15. Add Preprocessing

Alternative 2: 93.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \frac{c \cdot 1.125}{b \cdot \left(b \cdot b\right)}, \frac{1.5}{b}\right), \frac{b}{c} \cdot -2\right)} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/
  1.0
  (fma a (fma a (/ (* c 1.125) (* b (* b b))) (/ 1.5 b)) (* (/ b c) -2.0))))

C

double code(double a, double b, double c) {
	return 1.0 / fma(a, fma(a, ((c * 1.125) / (b * (b * b))), (1.5 / b)), ((b / c) * -2.0));
}

Julia

function code(a, b, c)
	return Float64(1.0 / fma(a, fma(a, Float64(Float64(c * 1.125) / Float64(b * Float64(b * b))), Float64(1.5 / b)), Float64(Float64(b / c) * -2.0)))
end

Wolfram

code[a_, b_, c_] := N[(1.0 / N[(a * N[(a * N[(N[(c * 1.125), $MachinePrecision] / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.5 / b), $MachinePrecision]), $MachinePrecision] + N[(N[(b / c), $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 32.9%

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

3. Taylor expanded in b around inf

   \[\leadsto \color{blue}{\frac{\frac{-9}{16} \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\frac{-1}{2} \cdot c + \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}\right)}{b}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

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

5. Applied rewrites 93.1%

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

7. Applied rewrites 92.8%

   \[\leadsto \color{blue}{\frac{1}{\frac{b}{\mathsf{fma}\left(a, c \cdot \left(c \cdot \frac{-0.375}{b \cdot b}\right), \mathsf{fma}\left(c, -0.5, \frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot \left(-0.5625 \cdot \left(a \cdot a\right)\right)}{b \cdot \left(b \cdot \left(b \cdot b\right)\right)}\right)\right)}}} \]

8. Taylor expanded in a around 0

   \[\leadsto \frac{1}{\color{blue}{-2 \cdot \frac{b}{c} + a \cdot \left(-1 \cdot \left(a \cdot \left(\frac{-9}{4} \cdot \frac{c}{{b}^{3}} + \frac{9}{8} \cdot \frac{c}{{b}^{3}}\right)\right) + \frac{3}{2} \cdot \frac{1}{b}\right)}} \]
9. Step-by-step derivation

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

10. Applied rewrites 93.2%

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

11. Final simplification 93.2%

    \[\leadsto \frac{1}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \frac{c \cdot 1.125}{b \cdot \left(b \cdot b\right)}, \frac{1.5}{b}\right), \frac{b}{c} \cdot -2\right)} \]
12. Add Preprocessing

Alternative 3: 90.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 32.9%

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

3. Taylor expanded in b around inf

   \[\leadsto \color{blue}{\frac{\frac{-9}{16} \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\frac{-1}{2} \cdot c + \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}\right)}{b}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

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

5. Applied rewrites 93.1%

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

7. Applied rewrites 92.8%

   \[\leadsto \color{blue}{\frac{1}{\frac{b}{\mathsf{fma}\left(a, c \cdot \left(c \cdot \frac{-0.375}{b \cdot b}\right), \mathsf{fma}\left(c, -0.5, \frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot \left(-0.5625 \cdot \left(a \cdot a\right)\right)}{b \cdot \left(b \cdot \left(b \cdot b\right)\right)}\right)\right)}}} \]

8. Taylor expanded in b around inf

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

   1. lower-*.f64 N/A

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

   2. sub-neg N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 N/A

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

   8. associate-*r/ N/A

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

   9. metadata-eval N/A

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

   10. distribute-neg-frac N/A

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

   11. metadata-eval N/A

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

   12. lower-/.f64 89.7

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

10. Applied rewrites 89.7%

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

    1. lift-*.f64 N/A

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

    2. lift-/.f64 N/A

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

    3. lift-/.f64 N/A

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

    4. lift-fma.f64 N/A

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

    5. *-commutative N/A

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

    6. associate-/r* N/A

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

    7. lower-/.f64 N/A

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

    8. lower-/.f64 89.9

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

    9. lift-fma.f64 N/A

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

    10. lift-/.f64 N/A

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

    11. associate-*l/ N/A

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

    12. associate-/l* N/A

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

    13. lower-fma.f64 N/A

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

    14. lower-/.f64 89.9

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

12. Applied rewrites 89.9%

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

Alternative 4: 90.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 32.9%

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

3. Taylor expanded in b around inf

   \[\leadsto \color{blue}{\frac{\frac{-9}{16} \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\frac{-1}{2} \cdot c + \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}\right)}{b}} \]
4. Step-by-step derivation

   1. lower-/.f64 N/A

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

5. Applied rewrites 93.1%

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

7. Applied rewrites 92.8%

   \[\leadsto \color{blue}{\frac{1}{\frac{b}{\mathsf{fma}\left(a, c \cdot \left(c \cdot \frac{-0.375}{b \cdot b}\right), \mathsf{fma}\left(c, -0.5, \frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot \left(-0.5625 \cdot \left(a \cdot a\right)\right)}{b \cdot \left(b \cdot \left(b \cdot b\right)\right)}\right)\right)}}} \]

8. Taylor expanded in b around inf

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

   1. lower-*.f64 N/A

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

   2. sub-neg N/A

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

   3. *-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 N/A

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

   8. associate-*r/ N/A

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

   9. metadata-eval N/A

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

   10. distribute-neg-frac N/A

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

   11. metadata-eval N/A

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

   12. lower-/.f64 89.7

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

10. Applied rewrites 89.7%

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

11. Taylor expanded in a around 0

    \[\leadsto \frac{1}{\color{blue}{-2 \cdot \frac{b}{c} + \frac{3}{2} \cdot \frac{a}{b}}} \]
12. 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. lower-*.f64 N/A

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

    4. lower-/.f64 89.8

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

13. Applied rewrites 89.8%

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

Alternative 5: 90.6% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 32.9%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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. Applied rewrites 95.0%

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

6. Taylor expanded in b around inf

   \[\leadsto \color{blue}{\frac{\frac{-135}{128} \cdot \frac{{a}^{3} \cdot {c}^{4}}{{b}^{6}} + \left(\frac{-9}{16} \cdot \frac{{a}^{2} \cdot {c}^{3}}{{b}^{4}} + \left(\frac{-1}{2} \cdot c + \frac{-3}{8} \cdot \frac{a \cdot {c}^{2}}{{b}^{2}}\right)\right)}{b}} \]

8. Applied rewrites 95.0%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(-1.0546875, \frac{{c}^{4} \cdot \left(a \cdot \left(a \cdot a\right)\right)}{{b}^{6}}, \mathsf{fma}\left(a, \mathsf{fma}\left(a, \frac{-0.5625 \cdot \left(c \cdot \left(c \cdot c\right)\right)}{\left(b \cdot b\right) \cdot \left(b \cdot b\right)}, \frac{c \cdot \left(c \cdot -0.375\right)}{b \cdot b}\right), c \cdot -0.5\right)\right)}{b}} \]

9. Taylor expanded in c around 0

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

    1. lower-*.f64 N/A

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

    2. sub-neg N/A

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

    3. metadata-eval N/A

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

    4. lower-fma.f64 N/A

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

    5. lower-/.f64 N/A

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

    6. *-commutative N/A

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

    7. lower-*.f64 N/A

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

    8. unpow2 N/A

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

    9. lower-*.f64 89.6

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

11. Applied rewrites 89.6%

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

Alternative 6: 81.0% accurate, 2.9× 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 32.9%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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. lower-*.f64 N/A

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

   2. lower-/.f64 80.3

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

5. Applied rewrites 80.3%

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

Reproduce

herbie shell --seed 2024216 
(FPCore (a b c)
  :name "Cubic critical, medium range"
  :precision binary64
  :pre (and (and (and (< 1.1102230246251565e-16 a) (< a 9007199254740992.0)) (and (< 1.1102230246251565e-16 b) (< b 9007199254740992.0))) (and (< 1.1102230246251565e-16 c) (< c 9007199254740992.0)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))