Cubic critical, medium range

Percentage Accurate: 31.1% → 96.2%

Time: 15.2s

Alternatives: 9

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: 31.1% 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: 96.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot \left(b \cdot b\right)\\ t_1 := b \cdot t\_0\\ \frac{{c}^{4} \cdot \mathsf{fma}\left(0.140625, \frac{a \cdot a}{{b}^{6}}, \frac{0.421875 \cdot \left(c \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)}{{b}^{8}}\right) - \frac{\left(c \cdot c\right) \cdot 0.25}{b \cdot b}}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(c, \frac{c \cdot c}{\left(b \cdot t\_1\right) \cdot -1.7777777777777777}, \frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot \left(c \cdot \left(a \cdot 6.328125\right)\right)}{b \cdot \left(\left(b \cdot b\right) \cdot t\_1\right)} \cdot -0.16666666666666666\right), \frac{\left(c \cdot c\right) \cdot -0.375}{t\_0}\right), -c \cdot \frac{-0.5}{b}\right)} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* b (* b b))) (t_1 (* b t_0)))
   (/
    (-
     (*
      (pow c 4.0)
      (fma
       0.140625
       (/ (* a a) (pow b 6.0))
       (/ (* 0.421875 (* c (* a (* a a)))) (pow b 8.0))))
     (/ (* (* c c) 0.25) (* b b)))
    (fma
     a
     (fma
      a
      (fma
       c
       (/ (* c c) (* (* b t_1) -1.7777777777777777))
       (*
        (/ (* (* c (* c c)) (* c (* a 6.328125))) (* b (* (* b b) t_1)))
        -0.16666666666666666))
      (/ (* (* 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 ((pow(c, 4.0) * fma(0.140625, ((a * a) / pow(b, 6.0)), ((0.421875 * (c * (a * (a * a)))) / pow(b, 8.0)))) - (((c * c) * 0.25) / (b * b))) / fma(a, fma(a, fma(c, ((c * c) / ((b * t_1) * -1.7777777777777777)), ((((c * (c * c)) * (c * (a * 6.328125))) / (b * ((b * b) * t_1))) * -0.16666666666666666)), (((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 Float64(Float64(Float64((c ^ 4.0) * fma(0.140625, Float64(Float64(a * a) / (b ^ 6.0)), Float64(Float64(0.421875 * Float64(c * Float64(a * Float64(a * a)))) / (b ^ 8.0)))) - Float64(Float64(Float64(c * c) * 0.25) / Float64(b * b))) / fma(a, fma(a, fma(c, Float64(Float64(c * c) / Float64(Float64(b * t_1) * -1.7777777777777777)), Float64(Float64(Float64(Float64(c * Float64(c * c)) * Float64(c * Float64(a * 6.328125))) / Float64(b * Float64(Float64(b * b) * t_1))) * -0.16666666666666666)), Float64(Float64(Float64(c * c) * -0.375) / t_0)), Float64(-Float64(c * Float64(-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[(N[(N[(N[Power[c, 4.0], $MachinePrecision] * N[(0.140625 * N[(N[(a * a), $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.421875 * N[(c * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 8.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(N[(N[(c * c), $MachinePrecision] * 0.25), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * N[(a * N[(c * N[(N[(c * c), $MachinePrecision] / N[(N[(b * t$95$1), $MachinePrecision] * -1.7777777777777777), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(c * N[(c * c), $MachinePrecision]), $MachinePrecision] * N[(c * N[(a * 6.328125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * N[(N[(b * b), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(c * c), $MachinePrecision] * -0.375), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] + (-N[(c * N[(-0.5 / b), $MachinePrecision]), $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 27.7%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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 95.3%

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

7. Applied egg-rr 95.3%

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

9. Applied egg-rr 94.6%

   \[\leadsto \color{blue}{\frac{{\left(a \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(c, \frac{c \cdot c}{\left(b \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right) \cdot -1.7777777777777777}, \frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot \left(c \cdot \left(6.328125 \cdot a\right)\right)}{b \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)} \cdot -0.16666666666666666\right), \frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot \left(b \cdot b\right)}\right)\right)}^{2} - \frac{\left(c \cdot c\right) \cdot 0.25}{b \cdot b}}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(c, \frac{c \cdot c}{\left(b \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right) \cdot -1.7777777777777777}, \frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot \left(c \cdot \left(6.328125 \cdot a\right)\right)}{b \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)} \cdot -0.16666666666666666\right), \frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot \left(b \cdot b\right)}\right), -c \cdot \frac{-0.5}{b}\right)}} \]

10. Taylor expanded in c around 0

    \[\leadsto \frac{\color{blue}{{c}^{4} \cdot \left(\frac{9}{64} \cdot \frac{{a}^{2}}{{b}^{6}} + \frac{27}{64} \cdot \frac{{a}^{3} \cdot c}{{b}^{8}}\right)} - \frac{\left(c \cdot c\right) \cdot \frac{1}{4}}{b \cdot b}}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(c, \frac{c \cdot c}{\left(b \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right) \cdot \frac{-16}{9}}, \frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot \left(c \cdot \left(\frac{405}{64} \cdot a\right)\right)}{b \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)} \cdot \frac{-1}{6}\right), \frac{\left(c \cdot c\right) \cdot \frac{-3}{8}}{b \cdot \left(b \cdot b\right)}\right), \mathsf{neg}\left(c \cdot \frac{\frac{-1}{2}}{b}\right)\right)} \]
11. Step-by-step derivation

    1. *-lowering-*.f64 N/A

       \[\leadsto \frac{\color{blue}{{c}^{4} \cdot \left(\frac{9}{64} \cdot \frac{{a}^{2}}{{b}^{6}} + \frac{27}{64} \cdot \frac{{a}^{3} \cdot c}{{b}^{8}}\right)} - \frac{\left(c \cdot c\right) \cdot \frac{1}{4}}{b \cdot b}}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(c, \frac{c \cdot c}{\left(b \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right) \cdot \frac{-16}{9}}, \frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot \left(c \cdot \left(\frac{405}{64} \cdot a\right)\right)}{b \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)} \cdot \frac{-1}{6}\right), \frac{\left(c \cdot c\right) \cdot \frac{-3}{8}}{b \cdot \left(b \cdot b\right)}\right), \mathsf{neg}\left(c \cdot \frac{\frac{-1}{2}}{b}\right)\right)} \]

    2. pow-lowering-pow.f64 N/A

       \[\leadsto \frac{\color{blue}{{c}^{4}} \cdot \left(\frac{9}{64} \cdot \frac{{a}^{2}}{{b}^{6}} + \frac{27}{64} \cdot \frac{{a}^{3} \cdot c}{{b}^{8}}\right) - \frac{\left(c \cdot c\right) \cdot \frac{1}{4}}{b \cdot b}}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(c, \frac{c \cdot c}{\left(b \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right) \cdot \frac{-16}{9}}, \frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot \left(c \cdot \left(\frac{405}{64} \cdot a\right)\right)}{b \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)} \cdot \frac{-1}{6}\right), \frac{\left(c \cdot c\right) \cdot \frac{-3}{8}}{b \cdot \left(b \cdot b\right)}\right), \mathsf{neg}\left(c \cdot \frac{\frac{-1}{2}}{b}\right)\right)} \]

    3. accelerator-lowering-fma.f64 N/A

       \[\leadsto \frac{{c}^{4} \cdot \color{blue}{\mathsf{fma}\left(\frac{9}{64}, \frac{{a}^{2}}{{b}^{6}}, \frac{27}{64} \cdot \frac{{a}^{3} \cdot c}{{b}^{8}}\right)} - \frac{\left(c \cdot c\right) \cdot \frac{1}{4}}{b \cdot b}}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(c, \frac{c \cdot c}{\left(b \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right) \cdot \frac{-16}{9}}, \frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot \left(c \cdot \left(\frac{405}{64} \cdot a\right)\right)}{b \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)} \cdot \frac{-1}{6}\right), \frac{\left(c \cdot c\right) \cdot \frac{-3}{8}}{b \cdot \left(b \cdot b\right)}\right), \mathsf{neg}\left(c \cdot \frac{\frac{-1}{2}}{b}\right)\right)} \]

    4. /-lowering-/.f64 N/A

       \[\leadsto \frac{{c}^{4} \cdot \mathsf{fma}\left(\frac{9}{64}, \color{blue}{\frac{{a}^{2}}{{b}^{6}}}, \frac{27}{64} \cdot \frac{{a}^{3} \cdot c}{{b}^{8}}\right) - \frac{\left(c \cdot c\right) \cdot \frac{1}{4}}{b \cdot b}}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(c, \frac{c \cdot c}{\left(b \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right) \cdot \frac{-16}{9}}, \frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot \left(c \cdot \left(\frac{405}{64} \cdot a\right)\right)}{b \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)} \cdot \frac{-1}{6}\right), \frac{\left(c \cdot c\right) \cdot \frac{-3}{8}}{b \cdot \left(b \cdot b\right)}\right), \mathsf{neg}\left(c \cdot \frac{\frac{-1}{2}}{b}\right)\right)} \]

    5. unpow2 N/A

       \[\leadsto \frac{{c}^{4} \cdot \mathsf{fma}\left(\frac{9}{64}, \frac{\color{blue}{a \cdot a}}{{b}^{6}}, \frac{27}{64} \cdot \frac{{a}^{3} \cdot c}{{b}^{8}}\right) - \frac{\left(c \cdot c\right) \cdot \frac{1}{4}}{b \cdot b}}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(c, \frac{c \cdot c}{\left(b \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right) \cdot \frac{-16}{9}}, \frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot \left(c \cdot \left(\frac{405}{64} \cdot a\right)\right)}{b \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)} \cdot \frac{-1}{6}\right), \frac{\left(c \cdot c\right) \cdot \frac{-3}{8}}{b \cdot \left(b \cdot b\right)}\right), \mathsf{neg}\left(c \cdot \frac{\frac{-1}{2}}{b}\right)\right)} \]

    6. *-lowering-*.f64 N/A

       \[\leadsto \frac{{c}^{4} \cdot \mathsf{fma}\left(\frac{9}{64}, \frac{\color{blue}{a \cdot a}}{{b}^{6}}, \frac{27}{64} \cdot \frac{{a}^{3} \cdot c}{{b}^{8}}\right) - \frac{\left(c \cdot c\right) \cdot \frac{1}{4}}{b \cdot b}}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(c, \frac{c \cdot c}{\left(b \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right) \cdot \frac{-16}{9}}, \frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot \left(c \cdot \left(\frac{405}{64} \cdot a\right)\right)}{b \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)} \cdot \frac{-1}{6}\right), \frac{\left(c \cdot c\right) \cdot \frac{-3}{8}}{b \cdot \left(b \cdot b\right)}\right), \mathsf{neg}\left(c \cdot \frac{\frac{-1}{2}}{b}\right)\right)} \]

    7. pow-lowering-pow.f64 N/A

       \[\leadsto \frac{{c}^{4} \cdot \mathsf{fma}\left(\frac{9}{64}, \frac{a \cdot a}{\color{blue}{{b}^{6}}}, \frac{27}{64} \cdot \frac{{a}^{3} \cdot c}{{b}^{8}}\right) - \frac{\left(c \cdot c\right) \cdot \frac{1}{4}}{b \cdot b}}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(c, \frac{c \cdot c}{\left(b \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right) \cdot \frac{-16}{9}}, \frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot \left(c \cdot \left(\frac{405}{64} \cdot a\right)\right)}{b \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)} \cdot \frac{-1}{6}\right), \frac{\left(c \cdot c\right) \cdot \frac{-3}{8}}{b \cdot \left(b \cdot b\right)}\right), \mathsf{neg}\left(c \cdot \frac{\frac{-1}{2}}{b}\right)\right)} \]

    8. associate-*r/ N/A

       \[\leadsto \frac{{c}^{4} \cdot \mathsf{fma}\left(\frac{9}{64}, \frac{a \cdot a}{{b}^{6}}, \color{blue}{\frac{\frac{27}{64} \cdot \left({a}^{3} \cdot c\right)}{{b}^{8}}}\right) - \frac{\left(c \cdot c\right) \cdot \frac{1}{4}}{b \cdot b}}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(c, \frac{c \cdot c}{\left(b \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right) \cdot \frac{-16}{9}}, \frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot \left(c \cdot \left(\frac{405}{64} \cdot a\right)\right)}{b \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)} \cdot \frac{-1}{6}\right), \frac{\left(c \cdot c\right) \cdot \frac{-3}{8}}{b \cdot \left(b \cdot b\right)}\right), \mathsf{neg}\left(c \cdot \frac{\frac{-1}{2}}{b}\right)\right)} \]

    9. /-lowering-/.f64 N/A

       \[\leadsto \frac{{c}^{4} \cdot \mathsf{fma}\left(\frac{9}{64}, \frac{a \cdot a}{{b}^{6}}, \color{blue}{\frac{\frac{27}{64} \cdot \left({a}^{3} \cdot c\right)}{{b}^{8}}}\right) - \frac{\left(c \cdot c\right) \cdot \frac{1}{4}}{b \cdot b}}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(c, \frac{c \cdot c}{\left(b \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right) \cdot \frac{-16}{9}}, \frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot \left(c \cdot \left(\frac{405}{64} \cdot a\right)\right)}{b \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)} \cdot \frac{-1}{6}\right), \frac{\left(c \cdot c\right) \cdot \frac{-3}{8}}{b \cdot \left(b \cdot b\right)}\right), \mathsf{neg}\left(c \cdot \frac{\frac{-1}{2}}{b}\right)\right)} \]

    10. *-lowering-*.f64 N/A

        \[\leadsto \frac{{c}^{4} \cdot \mathsf{fma}\left(\frac{9}{64}, \frac{a \cdot a}{{b}^{6}}, \frac{\color{blue}{\frac{27}{64} \cdot \left({a}^{3} \cdot c\right)}}{{b}^{8}}\right) - \frac{\left(c \cdot c\right) \cdot \frac{1}{4}}{b \cdot b}}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(c, \frac{c \cdot c}{\left(b \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right) \cdot \frac{-16}{9}}, \frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot \left(c \cdot \left(\frac{405}{64} \cdot a\right)\right)}{b \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)} \cdot \frac{-1}{6}\right), \frac{\left(c \cdot c\right) \cdot \frac{-3}{8}}{b \cdot \left(b \cdot b\right)}\right), \mathsf{neg}\left(c \cdot \frac{\frac{-1}{2}}{b}\right)\right)} \]

    11. *-commutative N/A

        \[\leadsto \frac{{c}^{4} \cdot \mathsf{fma}\left(\frac{9}{64}, \frac{a \cdot a}{{b}^{6}}, \frac{\frac{27}{64} \cdot \color{blue}{\left(c \cdot {a}^{3}\right)}}{{b}^{8}}\right) - \frac{\left(c \cdot c\right) \cdot \frac{1}{4}}{b \cdot b}}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(c, \frac{c \cdot c}{\left(b \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right) \cdot \frac{-16}{9}}, \frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot \left(c \cdot \left(\frac{405}{64} \cdot a\right)\right)}{b \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)} \cdot \frac{-1}{6}\right), \frac{\left(c \cdot c\right) \cdot \frac{-3}{8}}{b \cdot \left(b \cdot b\right)}\right), \mathsf{neg}\left(c \cdot \frac{\frac{-1}{2}}{b}\right)\right)} \]

    12. *-lowering-*.f64 N/A

        \[\leadsto \frac{{c}^{4} \cdot \mathsf{fma}\left(\frac{9}{64}, \frac{a \cdot a}{{b}^{6}}, \frac{\frac{27}{64} \cdot \color{blue}{\left(c \cdot {a}^{3}\right)}}{{b}^{8}}\right) - \frac{\left(c \cdot c\right) \cdot \frac{1}{4}}{b \cdot b}}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(c, \frac{c \cdot c}{\left(b \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right) \cdot \frac{-16}{9}}, \frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot \left(c \cdot \left(\frac{405}{64} \cdot a\right)\right)}{b \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)} \cdot \frac{-1}{6}\right), \frac{\left(c \cdot c\right) \cdot \frac{-3}{8}}{b \cdot \left(b \cdot b\right)}\right), \mathsf{neg}\left(c \cdot \frac{\frac{-1}{2}}{b}\right)\right)} \]

    13. cube-mult N/A

        \[\leadsto \frac{{c}^{4} \cdot \mathsf{fma}\left(\frac{9}{64}, \frac{a \cdot a}{{b}^{6}}, \frac{\frac{27}{64} \cdot \left(c \cdot \color{blue}{\left(a \cdot \left(a \cdot a\right)\right)}\right)}{{b}^{8}}\right) - \frac{\left(c \cdot c\right) \cdot \frac{1}{4}}{b \cdot b}}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(c, \frac{c \cdot c}{\left(b \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right) \cdot \frac{-16}{9}}, \frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot \left(c \cdot \left(\frac{405}{64} \cdot a\right)\right)}{b \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)} \cdot \frac{-1}{6}\right), \frac{\left(c \cdot c\right) \cdot \frac{-3}{8}}{b \cdot \left(b \cdot b\right)}\right), \mathsf{neg}\left(c \cdot \frac{\frac{-1}{2}}{b}\right)\right)} \]

    14. unpow2 N/A

        \[\leadsto \frac{{c}^{4} \cdot \mathsf{fma}\left(\frac{9}{64}, \frac{a \cdot a}{{b}^{6}}, \frac{\frac{27}{64} \cdot \left(c \cdot \left(a \cdot \color{blue}{{a}^{2}}\right)\right)}{{b}^{8}}\right) - \frac{\left(c \cdot c\right) \cdot \frac{1}{4}}{b \cdot b}}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(c, \frac{c \cdot c}{\left(b \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right) \cdot \frac{-16}{9}}, \frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot \left(c \cdot \left(\frac{405}{64} \cdot a\right)\right)}{b \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)} \cdot \frac{-1}{6}\right), \frac{\left(c \cdot c\right) \cdot \frac{-3}{8}}{b \cdot \left(b \cdot b\right)}\right), \mathsf{neg}\left(c \cdot \frac{\frac{-1}{2}}{b}\right)\right)} \]

    15. *-lowering-*.f64 N/A

        \[\leadsto \frac{{c}^{4} \cdot \mathsf{fma}\left(\frac{9}{64}, \frac{a \cdot a}{{b}^{6}}, \frac{\frac{27}{64} \cdot \left(c \cdot \color{blue}{\left(a \cdot {a}^{2}\right)}\right)}{{b}^{8}}\right) - \frac{\left(c \cdot c\right) \cdot \frac{1}{4}}{b \cdot b}}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(c, \frac{c \cdot c}{\left(b \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right) \cdot \frac{-16}{9}}, \frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot \left(c \cdot \left(\frac{405}{64} \cdot a\right)\right)}{b \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)} \cdot \frac{-1}{6}\right), \frac{\left(c \cdot c\right) \cdot \frac{-3}{8}}{b \cdot \left(b \cdot b\right)}\right), \mathsf{neg}\left(c \cdot \frac{\frac{-1}{2}}{b}\right)\right)} \]

    16. unpow2 N/A

        \[\leadsto \frac{{c}^{4} \cdot \mathsf{fma}\left(\frac{9}{64}, \frac{a \cdot a}{{b}^{6}}, \frac{\frac{27}{64} \cdot \left(c \cdot \left(a \cdot \color{blue}{\left(a \cdot a\right)}\right)\right)}{{b}^{8}}\right) - \frac{\left(c \cdot c\right) \cdot \frac{1}{4}}{b \cdot b}}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(c, \frac{c \cdot c}{\left(b \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right) \cdot \frac{-16}{9}}, \frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot \left(c \cdot \left(\frac{405}{64} \cdot a\right)\right)}{b \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)} \cdot \frac{-1}{6}\right), \frac{\left(c \cdot c\right) \cdot \frac{-3}{8}}{b \cdot \left(b \cdot b\right)}\right), \mathsf{neg}\left(c \cdot \frac{\frac{-1}{2}}{b}\right)\right)} \]

    17. *-lowering-*.f64 N/A

        \[\leadsto \frac{{c}^{4} \cdot \mathsf{fma}\left(\frac{9}{64}, \frac{a \cdot a}{{b}^{6}}, \frac{\frac{27}{64} \cdot \left(c \cdot \left(a \cdot \color{blue}{\left(a \cdot a\right)}\right)\right)}{{b}^{8}}\right) - \frac{\left(c \cdot c\right) \cdot \frac{1}{4}}{b \cdot b}}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(c, \frac{c \cdot c}{\left(b \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right) \cdot \frac{-16}{9}}, \frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot \left(c \cdot \left(\frac{405}{64} \cdot a\right)\right)}{b \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)} \cdot \frac{-1}{6}\right), \frac{\left(c \cdot c\right) \cdot \frac{-3}{8}}{b \cdot \left(b \cdot b\right)}\right), \mathsf{neg}\left(c \cdot \frac{\frac{-1}{2}}{b}\right)\right)} \]

    18. pow-lowering-pow.f64 95.7

        \[\leadsto \frac{{c}^{4} \cdot \mathsf{fma}\left(0.140625, \frac{a \cdot a}{{b}^{6}}, \frac{0.421875 \cdot \left(c \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)}{\color{blue}{{b}^{8}}}\right) - \frac{\left(c \cdot c\right) \cdot 0.25}{b \cdot b}}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(c, \frac{c \cdot c}{\left(b \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right) \cdot -1.7777777777777777}, \frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot \left(c \cdot \left(6.328125 \cdot a\right)\right)}{b \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)} \cdot -0.16666666666666666\right), \frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot \left(b \cdot b\right)}\right), -c \cdot \frac{-0.5}{b}\right)} \]

12. Simplified 95.7%

    \[\leadsto \frac{\color{blue}{{c}^{4} \cdot \mathsf{fma}\left(0.140625, \frac{a \cdot a}{{b}^{6}}, \frac{0.421875 \cdot \left(c \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)}{{b}^{8}}\right)} - \frac{\left(c \cdot c\right) \cdot 0.25}{b \cdot b}}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(c, \frac{c \cdot c}{\left(b \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right) \cdot -1.7777777777777777}, \frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot \left(c \cdot \left(6.328125 \cdot a\right)\right)}{b \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)} \cdot -0.16666666666666666\right), \frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot \left(b \cdot b\right)}\right), -c \cdot \frac{-0.5}{b}\right)} \]

13. Final simplification 95.7%

    \[\leadsto \frac{{c}^{4} \cdot \mathsf{fma}\left(0.140625, \frac{a \cdot a}{{b}^{6}}, \frac{0.421875 \cdot \left(c \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)}{{b}^{8}}\right) - \frac{\left(c \cdot c\right) \cdot 0.25}{b \cdot b}}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(c, \frac{c \cdot c}{\left(b \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right) \cdot -1.7777777777777777}, \frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot \left(c \cdot \left(a \cdot 6.328125\right)\right)}{b \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)} \cdot -0.16666666666666666\right), \frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot \left(b \cdot b\right)}\right), -c \cdot \frac{-0.5}{b}\right)} \]
14. Add Preprocessing

Alternative 2: 96.1% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot \left(b \cdot b\right)\\ t_1 := b \cdot t\_0\\ \frac{\left(c \cdot c\right) \cdot \mathsf{fma}\left(c \cdot c, \mathsf{fma}\left(0.140625, \frac{a \cdot a}{{b}^{6}}, \frac{0.421875 \cdot \left(c \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)}{{b}^{8}}\right), \frac{-0.25}{b \cdot b}\right)}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(c, \frac{c \cdot c}{\left(b \cdot t\_1\right) \cdot -1.7777777777777777}, \frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot \left(c \cdot \left(a \cdot 6.328125\right)\right)}{b \cdot \left(\left(b \cdot b\right) \cdot t\_1\right)} \cdot -0.16666666666666666\right), \frac{\left(c \cdot c\right) \cdot -0.375}{t\_0}\right), -c \cdot \frac{-0.5}{b}\right)} \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* b (* b b))) (t_1 (* b t_0)))
   (/
    (*
     (* c c)
     (fma
      (* c c)
      (fma
       0.140625
       (/ (* a a) (pow b 6.0))
       (/ (* 0.421875 (* c (* a (* a a)))) (pow b 8.0)))
      (/ -0.25 (* b b))))
    (fma
     a
     (fma
      a
      (fma
       c
       (/ (* c c) (* (* b t_1) -1.7777777777777777))
       (*
        (/ (* (* c (* c c)) (* c (* a 6.328125))) (* b (* (* b b) t_1)))
        -0.16666666666666666))
      (/ (* (* 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 ((c * c) * fma((c * c), fma(0.140625, ((a * a) / pow(b, 6.0)), ((0.421875 * (c * (a * (a * a)))) / pow(b, 8.0))), (-0.25 / (b * b)))) / fma(a, fma(a, fma(c, ((c * c) / ((b * t_1) * -1.7777777777777777)), ((((c * (c * c)) * (c * (a * 6.328125))) / (b * ((b * b) * t_1))) * -0.16666666666666666)), (((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 Float64(Float64(Float64(c * c) * fma(Float64(c * c), fma(0.140625, Float64(Float64(a * a) / (b ^ 6.0)), Float64(Float64(0.421875 * Float64(c * Float64(a * Float64(a * a)))) / (b ^ 8.0))), Float64(-0.25 / Float64(b * b)))) / fma(a, fma(a, fma(c, Float64(Float64(c * c) / Float64(Float64(b * t_1) * -1.7777777777777777)), Float64(Float64(Float64(Float64(c * Float64(c * c)) * Float64(c * Float64(a * 6.328125))) / Float64(b * Float64(Float64(b * b) * t_1))) * -0.16666666666666666)), Float64(Float64(Float64(c * c) * -0.375) / t_0)), Float64(-Float64(c * Float64(-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[(N[(N[(c * c), $MachinePrecision] * N[(N[(c * c), $MachinePrecision] * N[(0.140625 * N[(N[(a * a), $MachinePrecision] / N[Power[b, 6.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.421875 * N[(c * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[b, 8.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.25 / N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(a * N[(a * N[(c * N[(N[(c * c), $MachinePrecision] / N[(N[(b * t$95$1), $MachinePrecision] * -1.7777777777777777), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(c * N[(c * c), $MachinePrecision]), $MachinePrecision] * N[(c * N[(a * 6.328125), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(b * N[(N[(b * b), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(c * c), $MachinePrecision] * -0.375), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] + (-N[(c * N[(-0.5 / b), $MachinePrecision]), $MachinePrecision])), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 27.7%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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 95.3%

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

7. Applied egg-rr 95.3%

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

9. Applied egg-rr 94.6%

   \[\leadsto \color{blue}{\frac{{\left(a \cdot \mathsf{fma}\left(a, \mathsf{fma}\left(c, \frac{c \cdot c}{\left(b \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right) \cdot -1.7777777777777777}, \frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot \left(c \cdot \left(6.328125 \cdot a\right)\right)}{b \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)} \cdot -0.16666666666666666\right), \frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot \left(b \cdot b\right)}\right)\right)}^{2} - \frac{\left(c \cdot c\right) \cdot 0.25}{b \cdot b}}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(c, \frac{c \cdot c}{\left(b \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right) \cdot -1.7777777777777777}, \frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot \left(c \cdot \left(6.328125 \cdot a\right)\right)}{b \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)} \cdot -0.16666666666666666\right), \frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot \left(b \cdot b\right)}\right), -c \cdot \frac{-0.5}{b}\right)}} \]

10. Taylor expanded in c around 0

    \[\leadsto \frac{\color{blue}{{c}^{2} \cdot \left({c}^{2} \cdot \left(\frac{9}{64} \cdot \frac{{a}^{2}}{{b}^{6}} + \frac{27}{64} \cdot \frac{{a}^{3} \cdot c}{{b}^{8}}\right) - \frac{1}{4} \cdot \frac{1}{{b}^{2}}\right)}}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(c, \frac{c \cdot c}{\left(b \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right) \cdot \frac{-16}{9}}, \frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot \left(c \cdot \left(\frac{405}{64} \cdot a\right)\right)}{b \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)} \cdot \frac{-1}{6}\right), \frac{\left(c \cdot c\right) \cdot \frac{-3}{8}}{b \cdot \left(b \cdot b\right)}\right), \mathsf{neg}\left(c \cdot \frac{\frac{-1}{2}}{b}\right)\right)} \]
11. Step-by-step derivation

    1. *-lowering-*.f64 N/A

       \[\leadsto \frac{\color{blue}{{c}^{2} \cdot \left({c}^{2} \cdot \left(\frac{9}{64} \cdot \frac{{a}^{2}}{{b}^{6}} + \frac{27}{64} \cdot \frac{{a}^{3} \cdot c}{{b}^{8}}\right) - \frac{1}{4} \cdot \frac{1}{{b}^{2}}\right)}}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(c, \frac{c \cdot c}{\left(b \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right) \cdot \frac{-16}{9}}, \frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot \left(c \cdot \left(\frac{405}{64} \cdot a\right)\right)}{b \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)} \cdot \frac{-1}{6}\right), \frac{\left(c \cdot c\right) \cdot \frac{-3}{8}}{b \cdot \left(b \cdot b\right)}\right), \mathsf{neg}\left(c \cdot \frac{\frac{-1}{2}}{b}\right)\right)} \]

    2. unpow2 N/A

       \[\leadsto \frac{\color{blue}{\left(c \cdot c\right)} \cdot \left({c}^{2} \cdot \left(\frac{9}{64} \cdot \frac{{a}^{2}}{{b}^{6}} + \frac{27}{64} \cdot \frac{{a}^{3} \cdot c}{{b}^{8}}\right) - \frac{1}{4} \cdot \frac{1}{{b}^{2}}\right)}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(c, \frac{c \cdot c}{\left(b \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right) \cdot \frac{-16}{9}}, \frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot \left(c \cdot \left(\frac{405}{64} \cdot a\right)\right)}{b \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)} \cdot \frac{-1}{6}\right), \frac{\left(c \cdot c\right) \cdot \frac{-3}{8}}{b \cdot \left(b \cdot b\right)}\right), \mathsf{neg}\left(c \cdot \frac{\frac{-1}{2}}{b}\right)\right)} \]

    3. *-lowering-*.f64 N/A

       \[\leadsto \frac{\color{blue}{\left(c \cdot c\right)} \cdot \left({c}^{2} \cdot \left(\frac{9}{64} \cdot \frac{{a}^{2}}{{b}^{6}} + \frac{27}{64} \cdot \frac{{a}^{3} \cdot c}{{b}^{8}}\right) - \frac{1}{4} \cdot \frac{1}{{b}^{2}}\right)}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(c, \frac{c \cdot c}{\left(b \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right) \cdot \frac{-16}{9}}, \frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot \left(c \cdot \left(\frac{405}{64} \cdot a\right)\right)}{b \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)} \cdot \frac{-1}{6}\right), \frac{\left(c \cdot c\right) \cdot \frac{-3}{8}}{b \cdot \left(b \cdot b\right)}\right), \mathsf{neg}\left(c \cdot \frac{\frac{-1}{2}}{b}\right)\right)} \]

    4. sub-neg N/A

       \[\leadsto \frac{\left(c \cdot c\right) \cdot \color{blue}{\left({c}^{2} \cdot \left(\frac{9}{64} \cdot \frac{{a}^{2}}{{b}^{6}} + \frac{27}{64} \cdot \frac{{a}^{3} \cdot c}{{b}^{8}}\right) + \left(\mathsf{neg}\left(\frac{1}{4} \cdot \frac{1}{{b}^{2}}\right)\right)\right)}}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(c, \frac{c \cdot c}{\left(b \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right) \cdot \frac{-16}{9}}, \frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot \left(c \cdot \left(\frac{405}{64} \cdot a\right)\right)}{b \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)} \cdot \frac{-1}{6}\right), \frac{\left(c \cdot c\right) \cdot \frac{-3}{8}}{b \cdot \left(b \cdot b\right)}\right), \mathsf{neg}\left(c \cdot \frac{\frac{-1}{2}}{b}\right)\right)} \]

    5. accelerator-lowering-fma.f64 N/A

       \[\leadsto \frac{\left(c \cdot c\right) \cdot \color{blue}{\mathsf{fma}\left({c}^{2}, \frac{9}{64} \cdot \frac{{a}^{2}}{{b}^{6}} + \frac{27}{64} \cdot \frac{{a}^{3} \cdot c}{{b}^{8}}, \mathsf{neg}\left(\frac{1}{4} \cdot \frac{1}{{b}^{2}}\right)\right)}}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(c, \frac{c \cdot c}{\left(b \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right) \cdot \frac{-16}{9}}, \frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot \left(c \cdot \left(\frac{405}{64} \cdot a\right)\right)}{b \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)} \cdot \frac{-1}{6}\right), \frac{\left(c \cdot c\right) \cdot \frac{-3}{8}}{b \cdot \left(b \cdot b\right)}\right), \mathsf{neg}\left(c \cdot \frac{\frac{-1}{2}}{b}\right)\right)} \]

12. Simplified 95.6%

    \[\leadsto \frac{\color{blue}{\left(c \cdot c\right) \cdot \mathsf{fma}\left(c \cdot c, \mathsf{fma}\left(0.140625, \frac{a \cdot a}{{b}^{6}}, \frac{0.421875 \cdot \left(c \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)}{{b}^{8}}\right), \frac{-0.25}{b \cdot b}\right)}}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(c, \frac{c \cdot c}{\left(b \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right) \cdot -1.7777777777777777}, \frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot \left(c \cdot \left(6.328125 \cdot a\right)\right)}{b \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)} \cdot -0.16666666666666666\right), \frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot \left(b \cdot b\right)}\right), -c \cdot \frac{-0.5}{b}\right)} \]

13. Final simplification 95.6%

    \[\leadsto \frac{\left(c \cdot c\right) \cdot \mathsf{fma}\left(c \cdot c, \mathsf{fma}\left(0.140625, \frac{a \cdot a}{{b}^{6}}, \frac{0.421875 \cdot \left(c \cdot \left(a \cdot \left(a \cdot a\right)\right)\right)}{{b}^{8}}\right), \frac{-0.25}{b \cdot b}\right)}{\mathsf{fma}\left(a, \mathsf{fma}\left(a, \mathsf{fma}\left(c, \frac{c \cdot c}{\left(b \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right) \cdot -1.7777777777777777}, \frac{\left(c \cdot \left(c \cdot c\right)\right) \cdot \left(c \cdot \left(a \cdot 6.328125\right)\right)}{b \cdot \left(\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right)} \cdot -0.16666666666666666\right), \frac{\left(c \cdot c\right) \cdot -0.375}{b \cdot \left(b \cdot b\right)}\right), -c \cdot \frac{-0.5}{b}\right)} \]
14. Add Preprocessing

Alternative 3: 95.6% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 27.7%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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 95.3%

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

7. Applied egg-rr 95.3%

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

8. Final simplification 95.3%

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

Alternative 4: 94.1% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 27.7%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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 95.3%

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

7. Applied egg-rr 95.3%

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

8. Taylor expanded in b around inf

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

   1. /-lowering-/.f64 N/A

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

   2. accelerator-lowering-fma.f64 N/A

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

   3. /-lowering-/.f64 N/A

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

   4. *-commutative N/A

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

   5. *-lowering-*.f64 N/A

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

   6. cube-mult N/A

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

   7. unpow2 N/A

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

   8. *-lowering-*.f64 N/A

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

   9. unpow2 N/A

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

   10. *-lowering-*.f64 N/A

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

   11. unpow2 N/A

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

   12. *-lowering-*.f64 N/A

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

   13. *-lowering-*.f64 N/A

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

   14. unpow2 N/A

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

   15. *-lowering-*.f64 N/A

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

   16. cube-mult N/A

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

   17. unpow2 N/A

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

   18. *-lowering-*.f64 N/A

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

   19. unpow2 N/A

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

   20. *-lowering-*.f64 94.1

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

10. Simplified 94.1%

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

11. Taylor expanded in c around 0

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

    1. *-lowering-*.f64 N/A

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

    2. unpow2 N/A

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

    3. *-lowering-*.f64 N/A

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

    4. sub-neg N/A

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

    5. metadata-eval N/A

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

    6. accelerator-lowering-fma.f64 N/A

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

    7. /-lowering-/.f64 N/A

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

    8. *-commutative N/A

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

    9. *-lowering-*.f64 N/A

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

    10. unpow2 N/A

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

    11. *-lowering-*.f64 94.1

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

13. Simplified 94.1%

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

Alternative 5: 91.0% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

function code(a, b, c)
	return fma(a, Float64(Float64(Float64(c * c) * -0.375) / Float64(b * Float64(b * b))), Float64(-0.5 * Float64(c / 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[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 27.7%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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 95.3%

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

6. Taylor expanded in a around 0

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

   1. associate-*r/ N/A

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

   2. /-lowering-/.f64 N/A

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

   3. *-lowering-*.f64 N/A

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

   4. unpow2 N/A

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

   5. *-lowering-*.f64 N/A

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

   6. cube-mult N/A

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

   7. unpow2 N/A

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

   8. *-lowering-*.f64 N/A

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

   9. unpow2 N/A

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

   10. *-lowering-*.f64 91.5

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

8. Simplified 91.5%

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

9. Final simplification 91.5%

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

Alternative 6: 91.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 27.7%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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. /-lowering-/.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 91.4%

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

Alternative 7: 91.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 27.7%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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. /-lowering-/.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 91.4%

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

6. Taylor expanded in c around 0

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

   1. *-lowering-*.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. accelerator-lowering-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. associate-/l* N/A

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

   6. *-lowering-*.f64 N/A

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

   7. /-lowering-/.f64 N/A

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

   8. unpow2 N/A

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

   9. *-lowering-*.f64 91.4

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

8. Simplified 91.4%

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

Alternative 8: 81.6% 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 27.7%

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

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

   2. /-lowering-/.f64 84.2

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

5. Simplified 84.2%

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

Alternative 9: 3.2% accurate, 50.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 0.0)

C

double code(double a, double b, double c) {
	return 0.0;
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = 0.0d0
end function

Java

public static double code(double a, double b, double c) {
	return 0.0;
}

Python

def code(a, b, c):
	return 0.0

Julia

function code(a, b, c)
	return 0.0
end

MATLAB

function tmp = code(a, b, c)
	tmp = 0.0;
end

Wolfram

code[a_, b_, c_] := 0.0

Derivation

1. Initial program 27.7%

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

4. Applied egg-rr 27.7%

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

   1. div-sub N/A

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

   2. --lowering--.f64 N/A

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

   3. /-lowering-/.f64 N/A

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

   4. /-lowering-/.f64 N/A

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

   5. sqrt-lowering-sqrt.f64 N/A

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

   6. *-commutative N/A

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

   7. *-commutative N/A

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

   8. associate-*l* N/A

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

   9. metadata-eval N/A

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

   10. distribute-lft-neg-in N/A

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

   11. accelerator-lowering-fma.f64 N/A

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

   12. *-commutative N/A

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

   13. distribute-rgt-neg-in N/A

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

   14. metadata-eval N/A

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

   15. *-lowering-*.f64 N/A

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

   16. *-lowering-*.f64 27.1

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

6. Applied egg-rr 27.1%

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. div-inv N/A

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

   4. distribute-rgt-neg-in N/A

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

   5. accelerator-lowering-fma.f64 N/A

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

8. Applied egg-rr 29.4%

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

9. Taylor expanded in c around 0

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

    1. distribute-lft1-in N/A

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

    2. metadata-eval N/A

       \[\leadsto \frac{-1}{3} \cdot \left(\color{blue}{0} \cdot \frac{b}{a}\right) \]

    3. mul0-lft N/A

       \[\leadsto \frac{-1}{3} \cdot \color{blue}{0} \]

    4. metadata-eval 3.2

       \[\leadsto \color{blue}{0} \]

11. Simplified 3.2%

    \[\leadsto \color{blue}{0} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024205 
(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)))