Cubic critical, medium range

Percentage Accurate: 31.5% → 95.4%

Time: 13.5s

Alternatives: 7

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: 31.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 95.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := b \cdot \left(b \cdot b\right)\\ \mathsf{fma}\left(\mathsf{fma}\left(a, \frac{-0.375}{t\_0}, c \cdot \mathsf{fma}\left(a, a \cdot \frac{-0.5625}{\left(b \cdot b\right) \cdot t\_0}, \frac{\left(a \cdot \left(a \cdot \left(a \cdot a\right)\right)\right) \cdot \left(c \cdot 6.328125\right)}{\left(\left(b \cdot b\right) \cdot \left(b \cdot t\_0\right)\right) \cdot \left(a \cdot b\right)} \cdot -0.16666666666666666\right)\right), c \cdot c, \frac{c}{b \cdot -2}\right) \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (let* ((t_0 (* b (* b b))))
   (fma
    (fma
     a
     (/ -0.375 t_0)
     (*
      c
      (fma
       a
       (* a (/ -0.5625 (* (* b b) t_0)))
       (*
        (/
         (* (* a (* a (* a a))) (* c 6.328125))
         (* (* (* b b) (* b t_0)) (* a b)))
        -0.16666666666666666))))
    (* c c)
    (/ c (* b -2.0)))))

C

double code(double a, double b, double c) {
	double t_0 = b * (b * b);
	return fma(fma(a, (-0.375 / t_0), (c * fma(a, (a * (-0.5625 / ((b * b) * t_0))), ((((a * (a * (a * a))) * (c * 6.328125)) / (((b * b) * (b * t_0)) * (a * b))) * -0.16666666666666666)))), (c * c), (c / (b * -2.0)));
}

Julia

function code(a, b, c)
	t_0 = Float64(b * Float64(b * b))
	return fma(fma(a, Float64(-0.375 / t_0), Float64(c * fma(a, Float64(a * Float64(-0.5625 / Float64(Float64(b * b) * t_0))), Float64(Float64(Float64(Float64(a * Float64(a * Float64(a * a))) * Float64(c * 6.328125)) / Float64(Float64(Float64(b * b) * Float64(b * t_0)) * Float64(a * b))) * -0.16666666666666666)))), Float64(c * c), Float64(c / Float64(b * -2.0)))
end

Wolfram

code[a_, b_, c_] := Block[{t$95$0 = N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]}, N[(N[(a * N[(-0.375 / t$95$0), $MachinePrecision] + N[(c * N[(a * N[(a * N[(-0.5625 / N[(N[(b * b), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(N[(a * N[(a * N[(a * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(c * 6.328125), $MachinePrecision]), $MachinePrecision] / N[(N[(N[(b * b), $MachinePrecision] * N[(b * t$95$0), $MachinePrecision]), $MachinePrecision] * N[(a * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(c * c), $MachinePrecision] + N[(c / N[(b * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 29.8%

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

3. Taylor expanded in c around 0

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

5. Simplified 94.6%

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

7. Applied egg-rr 94.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(a, \frac{-0.375}{b \cdot \left(b \cdot b\right)}, c \cdot \mathsf{fma}\left(a, a \cdot \frac{-0.5625}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)}, \frac{\left(a \cdot \left(a \cdot \left(a \cdot a\right)\right)\right) \cdot \left(6.328125 \cdot c\right)}{\left(\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right) \cdot \left(b \cdot a\right)} \cdot -0.16666666666666666\right)\right), c \cdot c, \frac{c}{b \cdot -2}\right)} \]

8. Final simplification 94.8%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(a, \frac{-0.375}{b \cdot \left(b \cdot b\right)}, c \cdot \mathsf{fma}\left(a, a \cdot \frac{-0.5625}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)}, \frac{\left(a \cdot \left(a \cdot \left(a \cdot a\right)\right)\right) \cdot \left(c \cdot 6.328125\right)}{\left(\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right) \cdot \left(a \cdot b\right)} \cdot -0.16666666666666666\right)\right), c \cdot c, \frac{c}{b \cdot -2}\right) \]
9. Add Preprocessing

Alternative 2: 95.4% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 29.8%

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

3. Taylor expanded in c around 0

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

5. Simplified 94.6%

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

6. Taylor expanded in b around inf

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

   1. lower-/.f64 N/A

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

8. Simplified 94.8%

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

10. Applied egg-rr 94.8%

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

11. Final simplification 94.8%

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

Alternative 3: 95.1% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 29.8%

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

3. Taylor expanded in c around 0

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

5. Simplified 94.6%

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

7. Applied egg-rr 94.6%

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

8. Final simplification 94.6%

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

Alternative 4: 93.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{\mathsf{fma}\left(-0.5625, \frac{c \cdot \left(a \cdot a\right)}{b \cdot b}, a \cdot -0.375\right)}{b \cdot \left(b \cdot b\right)}, c \cdot c, \frac{c}{b \cdot -2}\right) \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (fma
  (/ (fma -0.5625 (/ (* c (* a a)) (* b b)) (* a -0.375)) (* b (* b b)))
  (* c c)
  (/ c (* b -2.0))))

C

double code(double a, double b, double c) {
	return fma((fma(-0.5625, ((c * (a * a)) / (b * b)), (a * -0.375)) / (b * (b * b))), (c * c), (c / (b * -2.0)));
}

Julia

function code(a, b, c)
	return fma(Float64(fma(-0.5625, Float64(Float64(c * Float64(a * a)) / Float64(b * b)), Float64(a * -0.375)) / Float64(b * Float64(b * b))), Float64(c * c), Float64(c / Float64(b * -2.0)))
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(-0.5625 * N[(N[(c * N[(a * a), $MachinePrecision]), $MachinePrecision] / N[(b * b), $MachinePrecision]), $MachinePrecision] + N[(a * -0.375), $MachinePrecision]), $MachinePrecision] / N[(b * N[(b * b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(c * c), $MachinePrecision] + N[(c / N[(b * -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 29.8%

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

3. Taylor expanded in c around 0

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

5. Simplified 94.6%

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

7. Applied egg-rr 94.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(a, \frac{-0.375}{b \cdot \left(b \cdot b\right)}, c \cdot \mathsf{fma}\left(a, a \cdot \frac{-0.5625}{\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot b\right)\right)}, \frac{\left(a \cdot \left(a \cdot \left(a \cdot a\right)\right)\right) \cdot \left(6.328125 \cdot c\right)}{\left(\left(b \cdot b\right) \cdot \left(b \cdot \left(b \cdot \left(b \cdot b\right)\right)\right)\right) \cdot \left(b \cdot a\right)} \cdot -0.16666666666666666\right)\right), c \cdot c, \frac{c}{b \cdot -2}\right)} \]

8. Taylor expanded in b around inf

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

   1. lower-/.f64 N/A

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

   2. lower-fma.f64 N/A

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

   3. lower-/.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. *-commutative N/A

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

   11. lower-*.f64 N/A

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

   12. cube-mult N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 N/A

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

   15. unpow2 N/A

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

   16. lower-*.f64 93.6

       \[\leadsto \mathsf{fma}\left(\frac{\mathsf{fma}\left(-0.5625, \frac{c \cdot \left(a \cdot a\right)}{b \cdot b}, a \cdot -0.375\right)}{b \cdot \color{blue}{\left(b \cdot b\right)}}, c \cdot c, \frac{c}{b \cdot -2}\right) \]

10. Simplified 93.6%

    \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\mathsf{fma}\left(-0.5625, \frac{c \cdot \left(a \cdot a\right)}{b \cdot b}, a \cdot -0.375\right)}{b \cdot \left(b \cdot b\right)}}, c \cdot c, \frac{c}{b \cdot -2}\right) \]
11. Add Preprocessing

Alternative 5: 90.6% 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 29.8%

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

3. Taylor expanded in b around inf

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

   1. lower-/.f64 N/A

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

5. Simplified 90.6%

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

6. Final simplification 90.6%

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

Alternative 6: 90.6% 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 29.8%

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

3. Taylor expanded in b around inf

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

   1. lower-/.f64 N/A

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

5. Simplified 90.6%

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

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. lower-*.f64 N/A

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

   2. sub-neg N/A

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

   3. metadata-eval N/A

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

   4. lower-fma.f64 N/A

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

   5. 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. lower-*.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. lower-/.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. lower-*.f64 90.6

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

8. Simplified 90.6%

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

Alternative 7: 81.1% 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 29.8%

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

3. Taylor expanded in b around inf

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 82.0

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

5. Simplified 82.0%

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

Reproduce

herbie shell --seed 2024207 
(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)))