Cubic critical, wide range

Percentage Accurate: 17.8% → 97.7%

Time: 16.5s

Alternatives: 10

Speedup: 2.9×

Specification

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

Math

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

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
}

Fortran

real(8) function code(a, b, c)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-b + sqrt(((b * b) - ((3.0d0 * a) * c)))) / (3.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
end

Wolfram

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

Initial Program: 17.8% 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: 97.7% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

function code(a, b, c)
	t_0 = Float64(b * Float64(b * b))
	return fma(fma(c, Float64(c * Float64(-0.375 / t_0)), Float64(a * fma(c, Float64(Float64(c * c) * Float64(-0.5625 / Float64(Float64(b * b) * t_0))), Float64(Float64(Float64(Float64(c * Float64(c * Float64(c * c))) * Float64(a * 6.328125)) / Float64(b * Float64(Float64(b * b) * Float64(b * t_0)))) * -0.16666666666666666)))), 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[(c * N[(c * N[(-0.375 / t$95$0), $MachinePrecision]), $MachinePrecision] + N[(a * N[(c * N[(N[(c * c), $MachinePrecision] * N[(-0.5625 / N[(N[(b * b), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(c * N[(c * N[(c * c), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(a * 6.328125), $MachinePrecision]), $MachinePrecision] / N[(b * N[(N[(b * b), $MachinePrecision] * N[(b * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * a + N[(N[(c * -0.5), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 17.6%

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

3. Taylor expanded in a around 0

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

5. Simplified 97.8%

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

7. Applied egg-rr 97.8%

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

8. Final simplification 97.8%

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

Alternative 2: 97.7% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 17.6%

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

3. Taylor expanded in a around 0

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

5. Simplified 97.8%

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

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. Simplified 97.8%

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

   1. associate-+r+ N/A

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

   2. +-commutative N/A

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

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

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

   4. +-commutative N/A

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

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

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

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

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

   7. associate-*r* N/A

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

   8. *-commutative N/A

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

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

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

   10. *-commutative N/A

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

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

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

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

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

   13. associate-*r/ N/A

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

   14. /-lowering-/.f64 N/A

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

10. Applied egg-rr 97.8%

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

Alternative 3: 96.9% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 17.6%

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

3. Taylor expanded in a around 0

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

5. Simplified 97.8%

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

6. Taylor expanded in b around inf

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

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

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

8. Simplified 96.9%

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

Alternative 4: 96.7% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 17.6%

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

4. Applied egg-rr 17.6%

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

   1. clear-num N/A

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

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

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

   3. un-div-inv N/A

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

   4. associate-/r/ N/A

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

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

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

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

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

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

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

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

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

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

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

   10. *-commutative N/A

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

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

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

   12. *-lowering-*.f64 17.6

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

6. Applied egg-rr 17.6%

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

7. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

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

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

9. Simplified 96.8%

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

10. Final simplification 96.8%

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

Alternative 5: 96.6% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 17.6%

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

3. Taylor expanded in c around 0

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

5. Simplified 96.6%

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

6. Taylor expanded in b around inf

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

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

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

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

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

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

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

   4. *-commutative N/A

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

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

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

   6. unpow2 N/A

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

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

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

   8. unpow2 N/A

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

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

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

   10. *-commutative N/A

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

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

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

   12. cube-mult N/A

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

   13. unpow2 N/A

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

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

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

   15. unpow2 N/A

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

   16. *-lowering-*.f64 96.6

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

8. Simplified 96.6%

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

9. Final simplification 96.6%

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

Alternative 6: 95.4% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 17.6%

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

3. Taylor expanded in a around 0

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

5. Simplified 97.8%

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

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. Simplified 97.8%

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

9. Taylor expanded in a around 0

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

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

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

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

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

    3. associate-*r/ N/A

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

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

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

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

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

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

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

    7. unpow2 N/A

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

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

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

    9. cube-mult N/A

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

    10. unpow2 N/A

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

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

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

    12. unpow2 N/A

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

    13. *-lowering-*.f64 95.4

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

11. Simplified 95.4%

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

Alternative 7: 95.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 17.6%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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 95.4%

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

6. Final simplification 95.4%

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

Alternative 8: 95.4% 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 17.6%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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 95.4%

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

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 95.4

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

8. Simplified 95.4%

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

Alternative 9: 95.0% accurate, 1.1× speedup

Math

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

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

Wolfram

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

Derivation

1. Initial program 17.6%

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

3. Taylor expanded in c around 0

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

5. Simplified 96.6%

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

6. Taylor expanded in b around inf

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

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

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

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

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

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

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

   6. *-commutative N/A

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

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

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

   8. unpow2 N/A

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

   9. *-lowering-*.f64 95.1

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

8. Simplified 95.1%

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

Alternative 10: 90.5% 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 17.6%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
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 90.3

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

5. Simplified 90.3%

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

Reproduce

herbie shell --seed 2024205 
(FPCore (a b c)
  :name "Cubic critical, wide range"
  :precision binary64
  :pre (and (and (and (< 4.930380657631324e-32 a) (< a 2.028240960365167e+31)) (and (< 4.930380657631324e-32 b) (< b 2.028240960365167e+31))) (and (< 4.930380657631324e-32 c) (< c 2.028240960365167e+31)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))