Jmat.Real.erfi, branch x less than or equal to 0.5

Percentage Accurate: 99.8% → 99.8%

Time: 6.1s

Alternatives: 17

Speedup: 2.6×

• 28.7% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[x \leq 0.5\]

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\\ t_1 := \left(t\_0 \cdot \left|x\right|\right) \cdot \left|x\right|\\ \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot t\_0\right) + \frac{1}{5} \cdot t\_1\right) + \frac{1}{21} \cdot \left(\left(t\_1 \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (* (fabs x) (fabs x)) (fabs x)))
        (t_1 (* (* t_0 (fabs x)) (fabs x))))
   (fabs
    (*
     (/ 1.0 (sqrt PI))
     (+
      (+ (+ (* 2.0 (fabs x)) (* (/ 2.0 3.0) t_0)) (* (/ 1.0 5.0) t_1))
      (* (/ 1.0 21.0) (* (* t_1 (fabs x)) (fabs x))))))))

C

double code(double x) {
	double t_0 = (fabs(x) * fabs(x)) * fabs(x);
	double t_1 = (t_0 * fabs(x)) * fabs(x);
	return fabs(((1.0 / sqrt(((double) M_PI))) * ((((2.0 * fabs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * fabs(x)) * fabs(x))))));
}

Java

public static double code(double x) {
	double t_0 = (Math.abs(x) * Math.abs(x)) * Math.abs(x);
	double t_1 = (t_0 * Math.abs(x)) * Math.abs(x);
	return Math.abs(((1.0 / Math.sqrt(Math.PI)) * ((((2.0 * Math.abs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * Math.abs(x)) * Math.abs(x))))));
}

Python

def code(x):
	t_0 = (math.fabs(x) * math.fabs(x)) * math.fabs(x)
	t_1 = (t_0 * math.fabs(x)) * math.fabs(x)
	return math.fabs(((1.0 / math.sqrt(math.pi)) * ((((2.0 * math.fabs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * math.fabs(x)) * math.fabs(x))))))

Julia

function code(x)
	t_0 = Float64(Float64(abs(x) * abs(x)) * abs(x))
	t_1 = Float64(Float64(t_0 * abs(x)) * abs(x))
	return abs(Float64(Float64(1.0 / sqrt(pi)) * Float64(Float64(Float64(Float64(2.0 * abs(x)) + Float64(Float64(2.0 / 3.0) * t_0)) + Float64(Float64(1.0 / 5.0) * t_1)) + Float64(Float64(1.0 / 21.0) * Float64(Float64(t_1 * abs(x)) * abs(x))))))
end

MATLAB

function tmp = code(x)
	t_0 = (abs(x) * abs(x)) * abs(x);
	t_1 = (t_0 * abs(x)) * abs(x);
	tmp = abs(((1.0 / sqrt(pi)) * ((((2.0 * abs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * abs(x)) * abs(x))))));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, N[Abs[N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(2.0 * N[Abs[x], $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 / 3.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / 5.0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / 21.0), $MachinePrecision] * N[(N[(t$95$1 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Initial Program: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\\ t_1 := \left(t\_0 \cdot \left|x\right|\right) \cdot \left|x\right|\\ \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot t\_0\right) + \frac{1}{5} \cdot t\_1\right) + \frac{1}{21} \cdot \left(\left(t\_1 \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (* (fabs x) (fabs x)) (fabs x)))
        (t_1 (* (* t_0 (fabs x)) (fabs x))))
   (fabs
    (*
     (/ 1.0 (sqrt PI))
     (+
      (+ (+ (* 2.0 (fabs x)) (* (/ 2.0 3.0) t_0)) (* (/ 1.0 5.0) t_1))
      (* (/ 1.0 21.0) (* (* t_1 (fabs x)) (fabs x))))))))

C

double code(double x) {
	double t_0 = (fabs(x) * fabs(x)) * fabs(x);
	double t_1 = (t_0 * fabs(x)) * fabs(x);
	return fabs(((1.0 / sqrt(((double) M_PI))) * ((((2.0 * fabs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * fabs(x)) * fabs(x))))));
}

Java

public static double code(double x) {
	double t_0 = (Math.abs(x) * Math.abs(x)) * Math.abs(x);
	double t_1 = (t_0 * Math.abs(x)) * Math.abs(x);
	return Math.abs(((1.0 / Math.sqrt(Math.PI)) * ((((2.0 * Math.abs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * Math.abs(x)) * Math.abs(x))))));
}

Python

def code(x):
	t_0 = (math.fabs(x) * math.fabs(x)) * math.fabs(x)
	t_1 = (t_0 * math.fabs(x)) * math.fabs(x)
	return math.fabs(((1.0 / math.sqrt(math.pi)) * ((((2.0 * math.fabs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * math.fabs(x)) * math.fabs(x))))))

Julia

function code(x)
	t_0 = Float64(Float64(abs(x) * abs(x)) * abs(x))
	t_1 = Float64(Float64(t_0 * abs(x)) * abs(x))
	return abs(Float64(Float64(1.0 / sqrt(pi)) * Float64(Float64(Float64(Float64(2.0 * abs(x)) + Float64(Float64(2.0 / 3.0) * t_0)) + Float64(Float64(1.0 / 5.0) * t_1)) + Float64(Float64(1.0 / 21.0) * Float64(Float64(t_1 * abs(x)) * abs(x))))))
end

MATLAB

function tmp = code(x)
	t_0 = (abs(x) * abs(x)) * abs(x);
	t_1 = (t_0 * abs(x)) * abs(x);
	tmp = abs(((1.0 / sqrt(pi)) * ((((2.0 * abs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_1)) + ((1.0 / 21.0) * ((t_1 * abs(x)) * abs(x))))));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, N[Abs[N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(2.0 * N[Abs[x], $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 / 3.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / 5.0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / 21.0), $MachinePrecision] * N[(N[(t$95$1 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Alternative 1: 99.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({\left(\left|x\right|\right)}^{7}, 0.047619047619047616, \mathsf{fma}\left(0.2 \cdot \left|x\right|, \left(\left(x \cdot x\right) \cdot x\right) \cdot x, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right)\right)\right)\right| \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fabs
  (*
   (/ 1.0 (sqrt PI))
   (fma
    (pow (fabs x) 7.0)
    0.047619047619047616
    (fma
     (* 0.2 (fabs x))
     (* (* (* x x) x) x)
     (* (fabs x) (fma (* x x) 0.6666666666666666 2.0)))))))

C

double code(double x) {
	return fabs(((1.0 / sqrt(((double) M_PI))) * fma(pow(fabs(x), 7.0), 0.047619047619047616, fma((0.2 * fabs(x)), (((x * x) * x) * x), (fabs(x) * fma((x * x), 0.6666666666666666, 2.0))))));
}

Julia

function code(x)
	return abs(Float64(Float64(1.0 / sqrt(pi)) * fma((abs(x) ^ 7.0), 0.047619047619047616, fma(Float64(0.2 * abs(x)), Float64(Float64(Float64(x * x) * x) * x), Float64(abs(x) * fma(Float64(x * x), 0.6666666666666666, 2.0))))))
end

Wolfram

code[x_] := N[Abs[N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Abs[x], $MachinePrecision], 7.0], $MachinePrecision] * 0.047619047619047616 + N[(N[(0.2 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] + N[(N[Abs[x], $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.6666666666666666 + 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

2. Taylor expanded in x around 0

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \color{blue}{\left(\frac{1}{21} \cdot {\left(\left|x\right|\right)}^{7} + \left(\frac{1}{5} \cdot {\left(\left|x\right|\right)}^{5} + \left(\frac{2}{3} \cdot {\left(\left|x\right|\right)}^{3} + 2 \cdot \left|x\right|\right)\right)\right)}\right| \]

4. Applied rewrites 99.8%

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \color{blue}{\mathsf{fma}\left({\left(\left|x\right|\right)}^{7}, 0.047619047619047616, \mathsf{fma}\left(0.2 \cdot \left|x\right|, \left(\left(x \cdot x\right) \cdot x\right) \cdot x, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right)\right)\right)}\right| \]
5. Add Preprocessing

Alternative 2: 99.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right), \left|x\right| \cdot \left(\mathsf{fma}\left(x \cdot x, 0.047619047619047616, 0.2\right) \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right)\right)\right)\right| \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fabs
  (*
   (/ 1.0 (sqrt PI))
   (fma
    (fabs x)
    (fma (* x x) 0.6666666666666666 2.0)
    (*
     (fabs x)
     (* (fma (* x x) 0.047619047619047616 0.2) (* (* (* x x) x) x)))))))

C

double code(double x) {
	return fabs(((1.0 / sqrt(((double) M_PI))) * fma(fabs(x), fma((x * x), 0.6666666666666666, 2.0), (fabs(x) * (fma((x * x), 0.047619047619047616, 0.2) * (((x * x) * x) * x))))));
}

Julia

function code(x)
	return abs(Float64(Float64(1.0 / sqrt(pi)) * fma(abs(x), fma(Float64(x * x), 0.6666666666666666, 2.0), Float64(abs(x) * Float64(fma(Float64(x * x), 0.047619047619047616, 0.2) * Float64(Float64(Float64(x * x) * x) * x))))))
end

Wolfram

code[x_] := N[Abs[N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[Abs[x], $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.6666666666666666 + 2.0), $MachinePrecision] + N[(N[Abs[x], $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * 0.047619047619047616 + 0.2), $MachinePrecision] * N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

3. Applied rewrites 99.8%

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \color{blue}{\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right), \left|x\right| \cdot \mathsf{fma}\left(0.2 \cdot \left(x \cdot x\right), x \cdot x, \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot 0.047619047619047616\right)\right)}\right| \]

4. Taylor expanded in x around 0

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right), \left|x\right| \cdot \color{blue}{\left({x}^{4} \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right)\right)}\right)\right| \]
5. Step-by-step derivation

   1. metadata-eval N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right), \left|x\right| \cdot \left({x}^{4} \cdot \left(\frac{1}{5} + \color{blue}{\frac{1}{21}} \cdot {x}^{2}\right)\right)\right)\right| \]

   2. metadata-eval N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right), \left|x\right| \cdot \left({x}^{4} \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot {\color{blue}{x}}^{2}\right)\right)\right)\right| \]

   3. *-commutative N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right), \left|x\right| \cdot \left(\left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right) \cdot \color{blue}{{x}^{4}}\right)\right)\right| \]

   4. lower-*.f64 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right), \left|x\right| \cdot \left(\left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right) \cdot \color{blue}{{x}^{4}}\right)\right)\right| \]

   5. +-commutative N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right), \left|x\right| \cdot \left(\left(\frac{1}{21} \cdot {x}^{2} + \frac{1}{5}\right) \cdot {\color{blue}{x}}^{4}\right)\right)\right| \]

   6. *-commutative N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right), \left|x\right| \cdot \left(\left({x}^{2} \cdot \frac{1}{21} + \frac{1}{5}\right) \cdot {x}^{4}\right)\right)\right| \]

   7. lower-fma.f64 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right), \left|x\right| \cdot \left(\mathsf{fma}\left({x}^{2}, \frac{1}{21}, \frac{1}{5}\right) \cdot {\color{blue}{x}}^{4}\right)\right)\right| \]

   8. pow2 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right), \left|x\right| \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{1}{21}, \frac{1}{5}\right) \cdot {x}^{4}\right)\right)\right| \]

   9. lift-*.f64 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right), \left|x\right| \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{1}{21}, \frac{1}{5}\right) \cdot {x}^{4}\right)\right)\right| \]

   10. metadata-eval N/A

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right), \left|x\right| \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{1}{21}, \frac{1}{5}\right) \cdot {x}^{4}\right)\right)\right| \]

   11. metadata-eval N/A

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right), \left|x\right| \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{1}{21}, \frac{1}{5}\right) \cdot {x}^{4}\right)\right)\right| \]

   12. metadata-eval N/A

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right), \left|x\right| \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{1}{21}, \frac{1}{5}\right) \cdot {x}^{\left(3 + \color{blue}{1}\right)}\right)\right)\right| \]

   13. pow-plus N/A

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right), \left|x\right| \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{1}{21}, \frac{1}{5}\right) \cdot \left({x}^{3} \cdot \color{blue}{x}\right)\right)\right)\right| \]

   14. pow3 N/A

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right), \left|x\right| \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{1}{21}, \frac{1}{5}\right) \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right)\right)\right)\right| \]

   15. lift-*.f64 N/A

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right), \left|x\right| \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{1}{21}, \frac{1}{5}\right) \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right)\right)\right)\right| \]

   16. lift-*.f64 N/A

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right), \left|x\right| \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{1}{21}, \frac{1}{5}\right) \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right)\right)\right)\right| \]

   17. lift-*.f64 99.8

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right), \left|x\right| \cdot \left(\mathsf{fma}\left(x \cdot x, 0.047619047619047616, 0.2\right) \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \color{blue}{x}\right)\right)\right)\right| \]

6. Applied rewrites 99.8%

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right), \left|x\right| \cdot \color{blue}{\left(\mathsf{fma}\left(x \cdot x, 0.047619047619047616, 0.2\right) \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right)\right)}\right)\right| \]
7. Add Preprocessing

Alternative 3: 99.8% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\sqrt{\pi}} \cdot \left|x \cdot \left(\mathsf{fma}\left(x \cdot 0.6666666666666666, x, 2\right) + \left(\mathsf{fma}\left(x \cdot x, 0.047619047619047616, 0.2\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot x\right)\right| \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (/ 1.0 (sqrt PI))
  (fabs
   (*
    x
    (+
     (fma (* x 0.6666666666666666) x 2.0)
     (* (* (fma (* x x) 0.047619047619047616 0.2) (* (* x x) x)) x))))))

C

double code(double x) {
	return (1.0 / sqrt(((double) M_PI))) * fabs((x * (fma((x * 0.6666666666666666), x, 2.0) + ((fma((x * x), 0.047619047619047616, 0.2) * ((x * x) * x)) * x))));
}

Julia

function code(x)
	return Float64(Float64(1.0 / sqrt(pi)) * abs(Float64(x * Float64(fma(Float64(x * 0.6666666666666666), x, 2.0) + Float64(Float64(fma(Float64(x * x), 0.047619047619047616, 0.2) * Float64(Float64(x * x) * x)) * x)))))
end

Wolfram

code[x_] := N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[Abs[N[(x * N[(N[(N[(x * 0.6666666666666666), $MachinePrecision] * x + 2.0), $MachinePrecision] + N[(N[(N[(N[(x * x), $MachinePrecision] * 0.047619047619047616 + 0.2), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

3. Applied rewrites 99.8%

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \color{blue}{\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right), \left|x\right| \cdot \mathsf{fma}\left(0.2 \cdot \left(x \cdot x\right), x \cdot x, \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot 0.047619047619047616\right)\right)}\right| \]

4. Taylor expanded in x around 0

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right), \left|x\right| \cdot \color{blue}{\left({x}^{4} \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right)\right)}\right)\right| \]
5. Step-by-step derivation

   1. metadata-eval N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right), \left|x\right| \cdot \left({x}^{4} \cdot \left(\frac{1}{5} + \color{blue}{\frac{1}{21}} \cdot {x}^{2}\right)\right)\right)\right| \]

   2. metadata-eval N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right), \left|x\right| \cdot \left({x}^{4} \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot {\color{blue}{x}}^{2}\right)\right)\right)\right| \]

   3. *-commutative N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right), \left|x\right| \cdot \left(\left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right) \cdot \color{blue}{{x}^{4}}\right)\right)\right| \]

   4. lower-*.f64 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right), \left|x\right| \cdot \left(\left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right) \cdot \color{blue}{{x}^{4}}\right)\right)\right| \]

   5. +-commutative N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right), \left|x\right| \cdot \left(\left(\frac{1}{21} \cdot {x}^{2} + \frac{1}{5}\right) \cdot {\color{blue}{x}}^{4}\right)\right)\right| \]

   6. *-commutative N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right), \left|x\right| \cdot \left(\left({x}^{2} \cdot \frac{1}{21} + \frac{1}{5}\right) \cdot {x}^{4}\right)\right)\right| \]

   7. lower-fma.f64 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right), \left|x\right| \cdot \left(\mathsf{fma}\left({x}^{2}, \frac{1}{21}, \frac{1}{5}\right) \cdot {\color{blue}{x}}^{4}\right)\right)\right| \]

   8. pow2 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right), \left|x\right| \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{1}{21}, \frac{1}{5}\right) \cdot {x}^{4}\right)\right)\right| \]

   9. lift-*.f64 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right), \left|x\right| \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{1}{21}, \frac{1}{5}\right) \cdot {x}^{4}\right)\right)\right| \]

   10. metadata-eval N/A

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right), \left|x\right| \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{1}{21}, \frac{1}{5}\right) \cdot {x}^{4}\right)\right)\right| \]

   11. metadata-eval N/A

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right), \left|x\right| \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{1}{21}, \frac{1}{5}\right) \cdot {x}^{4}\right)\right)\right| \]

   12. metadata-eval N/A

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right), \left|x\right| \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{1}{21}, \frac{1}{5}\right) \cdot {x}^{\left(3 + \color{blue}{1}\right)}\right)\right)\right| \]

   13. pow-plus N/A

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right), \left|x\right| \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{1}{21}, \frac{1}{5}\right) \cdot \left({x}^{3} \cdot \color{blue}{x}\right)\right)\right)\right| \]

   14. pow3 N/A

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right), \left|x\right| \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{1}{21}, \frac{1}{5}\right) \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right)\right)\right)\right| \]

   15. lift-*.f64 N/A

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right), \left|x\right| \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{1}{21}, \frac{1}{5}\right) \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right)\right)\right)\right| \]

   16. lift-*.f64 N/A

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right), \left|x\right| \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{1}{21}, \frac{1}{5}\right) \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right)\right)\right)\right| \]

   17. lift-*.f64 99.8

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right), \left|x\right| \cdot \left(\mathsf{fma}\left(x \cdot x, 0.047619047619047616, 0.2\right) \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \color{blue}{x}\right)\right)\right)\right| \]

6. Applied rewrites 99.8%

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right), \left|x\right| \cdot \color{blue}{\left(\mathsf{fma}\left(x \cdot x, 0.047619047619047616, 0.2\right) \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right)\right)}\right)\right| \]

8. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\frac{1}{\sqrt{\pi}} \cdot \left|x \cdot \left(\mathsf{fma}\left(x \cdot 0.6666666666666666, x, 2\right) + \left(\mathsf{fma}\left(x \cdot x, 0.047619047619047616, 0.2\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot x\right)\right|} \]
9. Add Preprocessing

Alternative 4: 99.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\sqrt{\pi}} \cdot \left|\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.047619047619047616, x \cdot x, 0.2\right), x \cdot x, 0.6666666666666666\right), x \cdot x, 2\right) \cdot x\right| \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (/ 1.0 (sqrt PI))
  (fabs
   (*
    (fma
     (fma (fma 0.047619047619047616 (* x x) 0.2) (* x x) 0.6666666666666666)
     (* x x)
     2.0)
    x))))

C

double code(double x) {
	return (1.0 / sqrt(((double) M_PI))) * fabs((fma(fma(fma(0.047619047619047616, (x * x), 0.2), (x * x), 0.6666666666666666), (x * x), 2.0) * x));
}

Julia

function code(x)
	return Float64(Float64(1.0 / sqrt(pi)) * abs(Float64(fma(fma(fma(0.047619047619047616, Float64(x * x), 0.2), Float64(x * x), 0.6666666666666666), Float64(x * x), 2.0) * x)))
end

Wolfram

code[x_] := N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[Abs[N[(N[(N[(N[(0.047619047619047616 * N[(x * x), $MachinePrecision] + 0.2), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

3. Applied rewrites 99.8%

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \color{blue}{\mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right), \left|x\right| \cdot \mathsf{fma}\left(0.2 \cdot \left(x \cdot x\right), x \cdot x, \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot 0.047619047619047616\right)\right)}\right| \]

4. Taylor expanded in x around 0

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right), \left|x\right| \cdot \color{blue}{\left({x}^{4} \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right)\right)}\right)\right| \]
5. Step-by-step derivation

   1. metadata-eval N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right), \left|x\right| \cdot \left({x}^{4} \cdot \left(\frac{1}{5} + \color{blue}{\frac{1}{21}} \cdot {x}^{2}\right)\right)\right)\right| \]

   2. metadata-eval N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right), \left|x\right| \cdot \left({x}^{4} \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot {\color{blue}{x}}^{2}\right)\right)\right)\right| \]

   3. *-commutative N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right), \left|x\right| \cdot \left(\left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right) \cdot \color{blue}{{x}^{4}}\right)\right)\right| \]

   4. lower-*.f64 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right), \left|x\right| \cdot \left(\left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right) \cdot \color{blue}{{x}^{4}}\right)\right)\right| \]

   5. +-commutative N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right), \left|x\right| \cdot \left(\left(\frac{1}{21} \cdot {x}^{2} + \frac{1}{5}\right) \cdot {\color{blue}{x}}^{4}\right)\right)\right| \]

   6. *-commutative N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right), \left|x\right| \cdot \left(\left({x}^{2} \cdot \frac{1}{21} + \frac{1}{5}\right) \cdot {x}^{4}\right)\right)\right| \]

   7. lower-fma.f64 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right), \left|x\right| \cdot \left(\mathsf{fma}\left({x}^{2}, \frac{1}{21}, \frac{1}{5}\right) \cdot {\color{blue}{x}}^{4}\right)\right)\right| \]

   8. pow2 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right), \left|x\right| \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{1}{21}, \frac{1}{5}\right) \cdot {x}^{4}\right)\right)\right| \]

   9. lift-*.f64 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right), \left|x\right| \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{1}{21}, \frac{1}{5}\right) \cdot {x}^{4}\right)\right)\right| \]

   10. metadata-eval N/A

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right), \left|x\right| \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{1}{21}, \frac{1}{5}\right) \cdot {x}^{4}\right)\right)\right| \]

   11. metadata-eval N/A

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right), \left|x\right| \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{1}{21}, \frac{1}{5}\right) \cdot {x}^{4}\right)\right)\right| \]

   12. metadata-eval N/A

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right), \left|x\right| \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{1}{21}, \frac{1}{5}\right) \cdot {x}^{\left(3 + \color{blue}{1}\right)}\right)\right)\right| \]

   13. pow-plus N/A

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right), \left|x\right| \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{1}{21}, \frac{1}{5}\right) \cdot \left({x}^{3} \cdot \color{blue}{x}\right)\right)\right)\right| \]

   14. pow3 N/A

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right), \left|x\right| \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{1}{21}, \frac{1}{5}\right) \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right)\right)\right)\right| \]

   15. lift-*.f64 N/A

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right), \left|x\right| \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{1}{21}, \frac{1}{5}\right) \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right)\right)\right)\right| \]

   16. lift-*.f64 N/A

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, \frac{2}{3}, 2\right), \left|x\right| \cdot \left(\mathsf{fma}\left(x \cdot x, \frac{1}{21}, \frac{1}{5}\right) \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right)\right)\right)\right| \]

   17. lift-*.f64 99.8

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right), \left|x\right| \cdot \left(\mathsf{fma}\left(x \cdot x, 0.047619047619047616, 0.2\right) \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \color{blue}{x}\right)\right)\right)\right| \]

6. Applied rewrites 99.8%

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right), \left|x\right| \cdot \color{blue}{\left(\mathsf{fma}\left(x \cdot x, 0.047619047619047616, 0.2\right) \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right)\right)}\right)\right| \]

8. Applied rewrites 99.8%

   \[\leadsto \color{blue}{\frac{1}{\sqrt{\pi}} \cdot \left|x \cdot \left(\mathsf{fma}\left(x \cdot 0.6666666666666666, x, 2\right) + \left(\mathsf{fma}\left(x \cdot x, 0.047619047619047616, 0.2\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot x\right)\right|} \]

9. Taylor expanded in x around 0

   \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \left|\color{blue}{x \cdot \left(2 + {x}^{2} \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right)\right)\right)}\right| \]
10. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \left|\left(2 + {x}^{2} \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right)\right)\right) \cdot \color{blue}{x}\right| \]

    2. lower-*.f64 N/A

       \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \left|\left(2 + {x}^{2} \cdot \left(\frac{2}{3} + {x}^{2} \cdot \left(\frac{1}{5} + \frac{1}{21} \cdot {x}^{2}\right)\right)\right) \cdot \color{blue}{x}\right| \]

11. Applied rewrites 99.8%

    \[\leadsto \frac{1}{\sqrt{\pi}} \cdot \left|\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.047619047619047616, x \cdot x, 0.2\right), x \cdot x, 0.6666666666666666\right), x \cdot x, 2\right) \cdot x}\right| \]
12. Add Preprocessing

Alternative 5: 99.3% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\sqrt{\pi}}\\ \mathbf{if}\;x \leq 2.2:\\ \;\;\;\;\left|\mathsf{fma}\left(2 \cdot t\_0, x, \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot 0.6666666666666666\right) \cdot t\_0\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|t\_0 \cdot \left({x}^{7} \cdot 0.047619047619047616\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (sqrt PI))))
   (if (<= x 2.2)
     (fabs (fma (* 2.0 t_0) x (* (* (* (* x x) x) 0.6666666666666666) t_0)))
     (fabs (* t_0 (* (pow x 7.0) 0.047619047619047616))))))

C

double code(double x) {
	double t_0 = 1.0 / sqrt(((double) M_PI));
	double tmp;
	if (x <= 2.2) {
		tmp = fabs(fma((2.0 * t_0), x, ((((x * x) * x) * 0.6666666666666666) * t_0)));
	} else {
		tmp = fabs((t_0 * (pow(x, 7.0) * 0.047619047619047616)));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(1.0 / sqrt(pi))
	tmp = 0.0
	if (x <= 2.2)
		tmp = abs(fma(Float64(2.0 * t_0), x, Float64(Float64(Float64(Float64(x * x) * x) * 0.6666666666666666) * t_0)));
	else
		tmp = abs(Float64(t_0 * Float64((x ^ 7.0) * 0.047619047619047616)));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 2.2], N[Abs[N[(N[(2.0 * t$95$0), $MachinePrecision] * x + N[(N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * 0.6666666666666666), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(t$95$0 * N[(N[Power[x, 7.0], $MachinePrecision] * 0.047619047619047616), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 2.2000000000000002

   1. Initial program 99.8%

      \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   3. Applied rewrites 99.8%

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \color{blue}{\mathsf{fma}\left(\left|x\right|, 2, \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(0.6666666666666666 \cdot x, x, \left(0.2 \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right), \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot \left|x\right|\right) \cdot x\right) \cdot x\right) \cdot 0.047619047619047616\right)\right)}\right| \]

   4. Taylor expanded in x around 0

      \[\leadsto \left|\color{blue}{\frac{2}{3} \cdot \left(\left({x}^{2} \cdot \left|x\right|\right) \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right) + 2 \cdot \left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left|x\right|\right)}\right| \]

   6. Applied rewrites 89.5%

      \[\leadsto \left|\color{blue}{\mathsf{fma}\left(2 \cdot \frac{1}{\sqrt{\pi}}, x, \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot 0.6666666666666666\right) \cdot \frac{1}{\sqrt{\pi}}\right)}\right| \]

   if 2.2000000000000002 < x

   1. Initial program 99.8%

      \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   2. Taylor expanded in x around 0

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \color{blue}{\left(\frac{1}{21} \cdot {\left(\left|x\right|\right)}^{7} + \left(\frac{1}{5} \cdot {\left(\left|x\right|\right)}^{5} + \left(\frac{2}{3} \cdot {\left(\left|x\right|\right)}^{3} + 2 \cdot \left|x\right|\right)\right)\right)}\right| \]

   4. Applied rewrites 99.8%

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \color{blue}{\mathsf{fma}\left({\left(\left|x\right|\right)}^{7}, 0.047619047619047616, \mathsf{fma}\left(0.2 \cdot \left|x\right|, \left(\left(x \cdot x\right) \cdot x\right) \cdot x, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right)\right)\right)}\right| \]

   5. Taylor expanded in x around 0

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\frac{1}{21} \cdot {\left(\left|x\right|\right)}^{7} + \color{blue}{2 \cdot \left|x\right|}\right)\right| \]
   6. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\frac{1}{21} \cdot {\left(\left|x\right|\right)}^{7} + 2 \cdot \left|x\right|\right)\right| \]

      2. *-commutative N/A

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left({\left(\left|x\right|\right)}^{7} \cdot \frac{1}{21} + 2 \cdot \left|\color{blue}{x}\right|\right)\right| \]

      3. lower-fma.f64 N/A

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({\left(\left|x\right|\right)}^{7}, \frac{1}{\color{blue}{21}}, 2 \cdot \left|x\right|\right)\right| \]

      4. lift-pow.f64 N/A

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({\left(\left|x\right|\right)}^{7}, \frac{1}{21}, 2 \cdot \left|x\right|\right)\right| \]

      5. rem-sqrt-square-rev N/A

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({\left(\sqrt{x \cdot x}\right)}^{7}, \frac{1}{21}, 2 \cdot \left|x\right|\right)\right| \]

      6. pow2 N/A

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({\left(\sqrt{{x}^{2}}\right)}^{7}, \frac{1}{21}, 2 \cdot \left|x\right|\right)\right| \]

      7. sqrt-pow1 N/A

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({\left({x}^{\left(\frac{2}{2}\right)}\right)}^{7}, \frac{1}{21}, 2 \cdot \left|x\right|\right)\right| \]

      8. metadata-eval N/A

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({\left({x}^{1}\right)}^{7}, \frac{1}{21}, 2 \cdot \left|x\right|\right)\right| \]

      9. unpow1 N/A

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({x}^{7}, \frac{1}{21}, 2 \cdot \left|x\right|\right)\right| \]

      10. metadata-eval N/A

          \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({x}^{7}, \frac{1}{21}, 2 \cdot \left|x\right|\right)\right| \]

      11. count-2-rev N/A

          \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({x}^{7}, \frac{1}{21}, \left|x\right| + \left|x\right|\right)\right| \]

      12. lower-+.f64 N/A

          \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({x}^{7}, \frac{1}{21}, \left|x\right| + \left|x\right|\right)\right| \]

      13. rem-sqrt-square-rev N/A

          \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({x}^{7}, \frac{1}{21}, \sqrt{x \cdot x} + \left|x\right|\right)\right| \]

      14. pow2 N/A

          \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({x}^{7}, \frac{1}{21}, \sqrt{{x}^{2}} + \left|x\right|\right)\right| \]

      15. sqrt-pow1 N/A

          \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({x}^{7}, \frac{1}{21}, {x}^{\left(\frac{2}{2}\right)} + \left|x\right|\right)\right| \]

      16. metadata-eval N/A

          \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({x}^{7}, \frac{1}{21}, {x}^{1} + \left|x\right|\right)\right| \]

      17. unpow1 N/A

          \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({x}^{7}, \frac{1}{21}, x + \left|x\right|\right)\right| \]

      18. rem-sqrt-square-rev N/A

          \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({x}^{7}, \frac{1}{21}, x + \sqrt{x \cdot x}\right)\right| \]

      19. pow2 N/A

          \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({x}^{7}, \frac{1}{21}, x + \sqrt{{x}^{2}}\right)\right| \]

      20. sqrt-pow1 N/A

          \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({x}^{7}, \frac{1}{21}, x + {x}^{\left(\frac{2}{2}\right)}\right)\right| \]

      21. metadata-eval N/A

          \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({x}^{7}, \frac{1}{21}, x + {x}^{1}\right)\right| \]

      22. unpow1 99.0

          \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({x}^{7}, 0.047619047619047616, x + x\right)\right| \]

   7. Applied rewrites 99.0%

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({x}^{7}, \color{blue}{0.047619047619047616}, x + x\right)\right| \]

   8. Taylor expanded in x around inf

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\frac{1}{21} \cdot {x}^{\color{blue}{7}}\right)\right| \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left({x}^{7} \cdot \frac{1}{21}\right)\right| \]

      2. lift-pow.f64 N/A

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left({x}^{7} \cdot \frac{1}{21}\right)\right| \]

      3. lift-*.f64 36.9

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left({x}^{7} \cdot 0.047619047619047616\right)\right| \]

   10. Applied rewrites 36.9%

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left({x}^{7} \cdot 0.047619047619047616\right)\right| \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 99.0% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\sqrt{\pi}}\\ \mathbf{if}\;x \leq 2.2:\\ \;\;\;\;\left|t\_0 \cdot \mathsf{fma}\left(x, 2, \left(\left(x \cdot x\right) \cdot x\right) \cdot 0.6666666666666666\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|t\_0 \cdot \left({x}^{7} \cdot 0.047619047619047616\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (sqrt PI))))
   (if (<= x 2.2)
     (fabs (* t_0 (fma x 2.0 (* (* (* x x) x) 0.6666666666666666))))
     (fabs (* t_0 (* (pow x 7.0) 0.047619047619047616))))))

C

double code(double x) {
	double t_0 = 1.0 / sqrt(((double) M_PI));
	double tmp;
	if (x <= 2.2) {
		tmp = fabs((t_0 * fma(x, 2.0, (((x * x) * x) * 0.6666666666666666))));
	} else {
		tmp = fabs((t_0 * (pow(x, 7.0) * 0.047619047619047616)));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(1.0 / sqrt(pi))
	tmp = 0.0
	if (x <= 2.2)
		tmp = abs(Float64(t_0 * fma(x, 2.0, Float64(Float64(Float64(x * x) * x) * 0.6666666666666666))));
	else
		tmp = abs(Float64(t_0 * Float64((x ^ 7.0) * 0.047619047619047616)));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 2.2], N[Abs[N[(t$95$0 * N[(x * 2.0 + N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(t$95$0 * N[(N[Power[x, 7.0], $MachinePrecision] * 0.047619047619047616), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 2.2000000000000002

   1. Initial program 99.8%

      \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   3. Applied rewrites 99.8%

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \color{blue}{\mathsf{fma}\left(\left|x\right|, 2, \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(0.6666666666666666 \cdot x, x, \left(0.2 \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right), \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot \left|x\right|\right) \cdot x\right) \cdot x\right) \cdot 0.047619047619047616\right)\right)}\right| \]

   4. Taylor expanded in x around 0

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, 2, \color{blue}{\frac{2}{3} \cdot \left({x}^{2} \cdot \left|x\right|\right)}\right)\right| \]
   5. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, 2, \frac{2}{3} \cdot \left(\color{blue}{{x}^{2}} \cdot \left|x\right|\right)\right)\right| \]

      2. *-commutative N/A

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, 2, \left({x}^{2} \cdot \left|x\right|\right) \cdot \color{blue}{\frac{2}{3}}\right)\right| \]

      3. lower-*.f64 N/A

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, 2, \left({x}^{2} \cdot \left|x\right|\right) \cdot \color{blue}{\frac{2}{3}}\right)\right| \]

      4. lower-*.f64 N/A

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, 2, \left({x}^{2} \cdot \left|x\right|\right) \cdot \frac{\color{blue}{2}}{3}\right)\right| \]

      5. pow2 N/A

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, 2, \left(\left(x \cdot x\right) \cdot \left|x\right|\right) \cdot \frac{2}{3}\right)\right| \]

      6. lift-*.f64 N/A

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, 2, \left(\left(x \cdot x\right) \cdot \left|x\right|\right) \cdot \frac{2}{3}\right)\right| \]

      7. rem-sqrt-square-rev N/A

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, 2, \left(\left(x \cdot x\right) \cdot \sqrt{x \cdot x}\right) \cdot \frac{2}{3}\right)\right| \]

      8. pow2 N/A

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, 2, \left(\left(x \cdot x\right) \cdot \sqrt{{x}^{2}}\right) \cdot \frac{2}{3}\right)\right| \]

      9. sqrt-pow1 N/A

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, 2, \left(\left(x \cdot x\right) \cdot {x}^{\left(\frac{2}{2}\right)}\right) \cdot \frac{2}{3}\right)\right| \]

      10. metadata-eval N/A

          \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, 2, \left(\left(x \cdot x\right) \cdot {x}^{1}\right) \cdot \frac{2}{3}\right)\right| \]

      11. unpow1 N/A

          \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, 2, \left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{2}{3}\right)\right| \]

      12. metadata-eval 89.4

          \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, 2, \left(\left(x \cdot x\right) \cdot x\right) \cdot 0.6666666666666666\right)\right| \]

   6. Applied rewrites 89.4%

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, 2, \color{blue}{\left(\left(x \cdot x\right) \cdot x\right) \cdot 0.6666666666666666}\right)\right| \]

   7. Taylor expanded in x around 0

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\color{blue}{\left|x\right|}, 2, \left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{2}{3}\right)\right| \]
   8. Step-by-step derivation

      1. rem-sqrt-square-rev N/A

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\sqrt{x \cdot x}, 2, \left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{2}{3}\right)\right| \]

      2. pow2 N/A

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\sqrt{{x}^{2}}, 2, \left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{2}{3}\right)\right| \]

      3. sqrt-pow1 N/A

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({x}^{\color{blue}{\left(\frac{2}{2}\right)}}, 2, \left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{2}{3}\right)\right| \]

      4. metadata-eval N/A

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({x}^{1}, 2, \left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{2}{3}\right)\right| \]

      5. unpow1 89.5

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(x, 2, \left(\left(x \cdot x\right) \cdot x\right) \cdot 0.6666666666666666\right)\right| \]

   9. Applied rewrites 89.5%

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\color{blue}{x}, 2, \left(\left(x \cdot x\right) \cdot x\right) \cdot 0.6666666666666666\right)\right| \]

   if 2.2000000000000002 < x

   1. Initial program 99.8%

      \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   2. Taylor expanded in x around 0

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \color{blue}{\left(\frac{1}{21} \cdot {\left(\left|x\right|\right)}^{7} + \left(\frac{1}{5} \cdot {\left(\left|x\right|\right)}^{5} + \left(\frac{2}{3} \cdot {\left(\left|x\right|\right)}^{3} + 2 \cdot \left|x\right|\right)\right)\right)}\right| \]

   4. Applied rewrites 99.8%

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \color{blue}{\mathsf{fma}\left({\left(\left|x\right|\right)}^{7}, 0.047619047619047616, \mathsf{fma}\left(0.2 \cdot \left|x\right|, \left(\left(x \cdot x\right) \cdot x\right) \cdot x, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right)\right)\right)}\right| \]

   5. Taylor expanded in x around 0

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\frac{1}{21} \cdot {\left(\left|x\right|\right)}^{7} + \color{blue}{2 \cdot \left|x\right|}\right)\right| \]
   6. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\frac{1}{21} \cdot {\left(\left|x\right|\right)}^{7} + 2 \cdot \left|x\right|\right)\right| \]

      2. *-commutative N/A

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left({\left(\left|x\right|\right)}^{7} \cdot \frac{1}{21} + 2 \cdot \left|\color{blue}{x}\right|\right)\right| \]

      3. lower-fma.f64 N/A

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({\left(\left|x\right|\right)}^{7}, \frac{1}{\color{blue}{21}}, 2 \cdot \left|x\right|\right)\right| \]

      4. lift-pow.f64 N/A

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({\left(\left|x\right|\right)}^{7}, \frac{1}{21}, 2 \cdot \left|x\right|\right)\right| \]

      5. rem-sqrt-square-rev N/A

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({\left(\sqrt{x \cdot x}\right)}^{7}, \frac{1}{21}, 2 \cdot \left|x\right|\right)\right| \]

      6. pow2 N/A

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({\left(\sqrt{{x}^{2}}\right)}^{7}, \frac{1}{21}, 2 \cdot \left|x\right|\right)\right| \]

      7. sqrt-pow1 N/A

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({\left({x}^{\left(\frac{2}{2}\right)}\right)}^{7}, \frac{1}{21}, 2 \cdot \left|x\right|\right)\right| \]

      8. metadata-eval N/A

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({\left({x}^{1}\right)}^{7}, \frac{1}{21}, 2 \cdot \left|x\right|\right)\right| \]

      9. unpow1 N/A

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({x}^{7}, \frac{1}{21}, 2 \cdot \left|x\right|\right)\right| \]

      10. metadata-eval N/A

          \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({x}^{7}, \frac{1}{21}, 2 \cdot \left|x\right|\right)\right| \]

      11. count-2-rev N/A

          \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({x}^{7}, \frac{1}{21}, \left|x\right| + \left|x\right|\right)\right| \]

      12. lower-+.f64 N/A

          \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({x}^{7}, \frac{1}{21}, \left|x\right| + \left|x\right|\right)\right| \]

      13. rem-sqrt-square-rev N/A

          \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({x}^{7}, \frac{1}{21}, \sqrt{x \cdot x} + \left|x\right|\right)\right| \]

      14. pow2 N/A

          \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({x}^{7}, \frac{1}{21}, \sqrt{{x}^{2}} + \left|x\right|\right)\right| \]

      15. sqrt-pow1 N/A

          \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({x}^{7}, \frac{1}{21}, {x}^{\left(\frac{2}{2}\right)} + \left|x\right|\right)\right| \]

      16. metadata-eval N/A

          \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({x}^{7}, \frac{1}{21}, {x}^{1} + \left|x\right|\right)\right| \]

      17. unpow1 N/A

          \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({x}^{7}, \frac{1}{21}, x + \left|x\right|\right)\right| \]

      18. rem-sqrt-square-rev N/A

          \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({x}^{7}, \frac{1}{21}, x + \sqrt{x \cdot x}\right)\right| \]

      19. pow2 N/A

          \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({x}^{7}, \frac{1}{21}, x + \sqrt{{x}^{2}}\right)\right| \]

      20. sqrt-pow1 N/A

          \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({x}^{7}, \frac{1}{21}, x + {x}^{\left(\frac{2}{2}\right)}\right)\right| \]

      21. metadata-eval N/A

          \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({x}^{7}, \frac{1}{21}, x + {x}^{1}\right)\right| \]

      22. unpow1 99.0

          \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({x}^{7}, 0.047619047619047616, x + x\right)\right| \]

   7. Applied rewrites 99.0%

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({x}^{7}, \color{blue}{0.047619047619047616}, x + x\right)\right| \]

   8. Taylor expanded in x around inf

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\frac{1}{21} \cdot {x}^{\color{blue}{7}}\right)\right| \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left({x}^{7} \cdot \frac{1}{21}\right)\right| \]

      2. lift-pow.f64 N/A

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left({x}^{7} \cdot \frac{1}{21}\right)\right| \]

      3. lift-*.f64 36.9

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left({x}^{7} \cdot 0.047619047619047616\right)\right| \]

   10. Applied rewrites 36.9%

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left({x}^{7} \cdot 0.047619047619047616\right)\right| \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 99.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \left|\frac{1}{\sqrt{\pi}} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right), 0.047619047619047616, 0.6666666666666666\right), x \cdot x, 2\right) \cdot x\right)\right| \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fabs
  (*
   (/ 1.0 (sqrt PI))
   (*
    (fma
     (fma (* (* x x) (* x x)) 0.047619047619047616 0.6666666666666666)
     (* x x)
     2.0)
    x))))

C

double code(double x) {
	return fabs(((1.0 / sqrt(((double) M_PI))) * (fma(fma(((x * x) * (x * x)), 0.047619047619047616, 0.6666666666666666), (x * x), 2.0) * x)));
}

Julia

function code(x)
	return abs(Float64(Float64(1.0 / sqrt(pi)) * Float64(fma(fma(Float64(Float64(x * x) * Float64(x * x)), 0.047619047619047616, 0.6666666666666666), Float64(x * x), 2.0) * x)))
end

Wolfram

code[x_] := N[Abs[N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * 0.047619047619047616 + 0.6666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

2. Taylor expanded in x around 0

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \color{blue}{\left(\frac{1}{21} \cdot {\left(\left|x\right|\right)}^{7} + \left(\frac{1}{5} \cdot {\left(\left|x\right|\right)}^{5} + \left(\frac{2}{3} \cdot {\left(\left|x\right|\right)}^{3} + 2 \cdot \left|x\right|\right)\right)\right)}\right| \]

4. Applied rewrites 99.8%

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \color{blue}{\mathsf{fma}\left({\left(\left|x\right|\right)}^{7}, 0.047619047619047616, \mathsf{fma}\left(0.2 \cdot \left|x\right|, \left(\left(x \cdot x\right) \cdot x\right) \cdot x, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right)\right)\right)}\right| \]

5. Taylor expanded in x around 0

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\frac{1}{21} \cdot {\left(\left|x\right|\right)}^{7} + \color{blue}{\left(\frac{2}{3} \cdot \left({x}^{2} \cdot \left|x\right|\right) + 2 \cdot \left|x\right|\right)}\right)\right| \]

7. Applied rewrites 99.3%

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.6666666666666666 \cdot x, x, 2\right), \color{blue}{x}, {x}^{7} \cdot 0.047619047619047616\right)\right| \]

8. Taylor expanded in x around 0

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(x \cdot \left(2 + \color{blue}{{x}^{2} \cdot \left(\frac{2}{3} + \frac{1}{21} \cdot {x}^{4}\right)}\right)\right)\right| \]
9. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(2 + {x}^{2} \cdot \left(\frac{2}{3} + \frac{1}{21} \cdot {x}^{4}\right)\right) \cdot x\right)\right| \]

   2. unpow1 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(2 + {x}^{2} \cdot \left(\frac{2}{3} + \frac{1}{21} \cdot {x}^{4}\right)\right) \cdot {x}^{1}\right)\right| \]

   3. metadata-eval N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(2 + {x}^{2} \cdot \left(\frac{2}{3} + \frac{1}{21} \cdot {x}^{4}\right)\right) \cdot {x}^{\left(\frac{2}{2}\right)}\right)\right| \]

   4. sqrt-pow1 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(2 + {x}^{2} \cdot \left(\frac{2}{3} + \frac{1}{21} \cdot {x}^{4}\right)\right) \cdot \sqrt{{x}^{2}}\right)\right| \]

   5. pow2 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(2 + {x}^{2} \cdot \left(\frac{2}{3} + \frac{1}{21} \cdot {x}^{4}\right)\right) \cdot \sqrt{x \cdot x}\right)\right| \]

   6. rem-sqrt-square-rev N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(2 + {x}^{2} \cdot \left(\frac{2}{3} + \frac{1}{21} \cdot {x}^{4}\right)\right) \cdot \left|x\right|\right)\right| \]

   7. lower-*.f64 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(2 + {x}^{2} \cdot \left(\frac{2}{3} + \frac{1}{21} \cdot {x}^{4}\right)\right) \cdot \left|x\right|\right)\right| \]

10. Applied rewrites 99.3%

    \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right), 0.047619047619047616, 0.6666666666666666\right), x \cdot x, 2\right) \cdot x\right)\right| \]
11. Add Preprocessing

Alternative 8: 93.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({x}^{7}, 0.047619047619047616, x + x\right)\right| \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fabs (* (/ 1.0 (sqrt PI)) (fma (pow x 7.0) 0.047619047619047616 (+ x x)))))

C

double code(double x) {
	return fabs(((1.0 / sqrt(((double) M_PI))) * fma(pow(x, 7.0), 0.047619047619047616, (x + x))));
}

Julia

function code(x)
	return abs(Float64(Float64(1.0 / sqrt(pi)) * fma((x ^ 7.0), 0.047619047619047616, Float64(x + x))))
end

Wolfram

code[x_] := N[Abs[N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[Power[x, 7.0], $MachinePrecision] * 0.047619047619047616 + N[(x + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

2. Taylor expanded in x around 0

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \color{blue}{\left(\frac{1}{21} \cdot {\left(\left|x\right|\right)}^{7} + \left(\frac{1}{5} \cdot {\left(\left|x\right|\right)}^{5} + \left(\frac{2}{3} \cdot {\left(\left|x\right|\right)}^{3} + 2 \cdot \left|x\right|\right)\right)\right)}\right| \]

4. Applied rewrites 99.8%

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \color{blue}{\mathsf{fma}\left({\left(\left|x\right|\right)}^{7}, 0.047619047619047616, \mathsf{fma}\left(0.2 \cdot \left|x\right|, \left(\left(x \cdot x\right) \cdot x\right) \cdot x, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right)\right)\right)}\right| \]

5. Taylor expanded in x around 0

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\frac{1}{21} \cdot {\left(\left|x\right|\right)}^{7} + \color{blue}{2 \cdot \left|x\right|}\right)\right| \]
6. Step-by-step derivation

   1. metadata-eval N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\frac{1}{21} \cdot {\left(\left|x\right|\right)}^{7} + 2 \cdot \left|x\right|\right)\right| \]

   2. *-commutative N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left({\left(\left|x\right|\right)}^{7} \cdot \frac{1}{21} + 2 \cdot \left|\color{blue}{x}\right|\right)\right| \]

   3. lower-fma.f64 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({\left(\left|x\right|\right)}^{7}, \frac{1}{\color{blue}{21}}, 2 \cdot \left|x\right|\right)\right| \]

   4. lift-pow.f64 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({\left(\left|x\right|\right)}^{7}, \frac{1}{21}, 2 \cdot \left|x\right|\right)\right| \]

   5. rem-sqrt-square-rev N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({\left(\sqrt{x \cdot x}\right)}^{7}, \frac{1}{21}, 2 \cdot \left|x\right|\right)\right| \]

   6. pow2 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({\left(\sqrt{{x}^{2}}\right)}^{7}, \frac{1}{21}, 2 \cdot \left|x\right|\right)\right| \]

   7. sqrt-pow1 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({\left({x}^{\left(\frac{2}{2}\right)}\right)}^{7}, \frac{1}{21}, 2 \cdot \left|x\right|\right)\right| \]

   8. metadata-eval N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({\left({x}^{1}\right)}^{7}, \frac{1}{21}, 2 \cdot \left|x\right|\right)\right| \]

   9. unpow1 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({x}^{7}, \frac{1}{21}, 2 \cdot \left|x\right|\right)\right| \]

   10. metadata-eval N/A

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({x}^{7}, \frac{1}{21}, 2 \cdot \left|x\right|\right)\right| \]

   11. count-2-rev N/A

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({x}^{7}, \frac{1}{21}, \left|x\right| + \left|x\right|\right)\right| \]

   12. lower-+.f64 N/A

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({x}^{7}, \frac{1}{21}, \left|x\right| + \left|x\right|\right)\right| \]

   13. rem-sqrt-square-rev N/A

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({x}^{7}, \frac{1}{21}, \sqrt{x \cdot x} + \left|x\right|\right)\right| \]

   14. pow2 N/A

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({x}^{7}, \frac{1}{21}, \sqrt{{x}^{2}} + \left|x\right|\right)\right| \]

   15. sqrt-pow1 N/A

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({x}^{7}, \frac{1}{21}, {x}^{\left(\frac{2}{2}\right)} + \left|x\right|\right)\right| \]

   16. metadata-eval N/A

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({x}^{7}, \frac{1}{21}, {x}^{1} + \left|x\right|\right)\right| \]

   17. unpow1 N/A

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({x}^{7}, \frac{1}{21}, x + \left|x\right|\right)\right| \]

   18. rem-sqrt-square-rev N/A

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({x}^{7}, \frac{1}{21}, x + \sqrt{x \cdot x}\right)\right| \]

   19. pow2 N/A

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({x}^{7}, \frac{1}{21}, x + \sqrt{{x}^{2}}\right)\right| \]

   20. sqrt-pow1 N/A

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({x}^{7}, \frac{1}{21}, x + {x}^{\left(\frac{2}{2}\right)}\right)\right| \]

   21. metadata-eval N/A

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({x}^{7}, \frac{1}{21}, x + {x}^{1}\right)\right| \]

   22. unpow1 99.0

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({x}^{7}, 0.047619047619047616, x + x\right)\right| \]

7. Applied rewrites 99.0%

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({x}^{7}, \color{blue}{0.047619047619047616}, x + x\right)\right| \]
8. Add Preprocessing

Alternative 9: 89.5% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \left|\frac{1}{\sqrt{\pi}} \cdot \left(\mathsf{fma}\left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot x, 0.047619047619047616, 2\right) \cdot x\right)\right| \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fabs
  (*
   (/ 1.0 (sqrt PI))
   (* (fma (* (* (* (* x x) x) (* x x)) x) 0.047619047619047616 2.0) x))))

C

double code(double x) {
	return fabs(((1.0 / sqrt(((double) M_PI))) * (fma(((((x * x) * x) * (x * x)) * x), 0.047619047619047616, 2.0) * x)));
}

Julia

function code(x)
	return abs(Float64(Float64(1.0 / sqrt(pi)) * Float64(fma(Float64(Float64(Float64(Float64(x * x) * x) * Float64(x * x)) * x), 0.047619047619047616, 2.0) * x)))
end

Wolfram

code[x_] := N[Abs[N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] * 0.047619047619047616 + 2.0), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

2. Taylor expanded in x around 0

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \color{blue}{\left(\frac{1}{21} \cdot {\left(\left|x\right|\right)}^{7} + \left(\frac{1}{5} \cdot {\left(\left|x\right|\right)}^{5} + \left(\frac{2}{3} \cdot {\left(\left|x\right|\right)}^{3} + 2 \cdot \left|x\right|\right)\right)\right)}\right| \]

4. Applied rewrites 99.8%

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \color{blue}{\mathsf{fma}\left({\left(\left|x\right|\right)}^{7}, 0.047619047619047616, \mathsf{fma}\left(0.2 \cdot \left|x\right|, \left(\left(x \cdot x\right) \cdot x\right) \cdot x, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right)\right)\right)}\right| \]

5. Taylor expanded in x around 0

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\frac{1}{21} \cdot {\left(\left|x\right|\right)}^{7} + \color{blue}{2 \cdot \left|x\right|}\right)\right| \]
6. Step-by-step derivation

   1. metadata-eval N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\frac{1}{21} \cdot {\left(\left|x\right|\right)}^{7} + 2 \cdot \left|x\right|\right)\right| \]

   2. *-commutative N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left({\left(\left|x\right|\right)}^{7} \cdot \frac{1}{21} + 2 \cdot \left|\color{blue}{x}\right|\right)\right| \]

   3. lower-fma.f64 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({\left(\left|x\right|\right)}^{7}, \frac{1}{\color{blue}{21}}, 2 \cdot \left|x\right|\right)\right| \]

   4. lift-pow.f64 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({\left(\left|x\right|\right)}^{7}, \frac{1}{21}, 2 \cdot \left|x\right|\right)\right| \]

   5. rem-sqrt-square-rev N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({\left(\sqrt{x \cdot x}\right)}^{7}, \frac{1}{21}, 2 \cdot \left|x\right|\right)\right| \]

   6. pow2 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({\left(\sqrt{{x}^{2}}\right)}^{7}, \frac{1}{21}, 2 \cdot \left|x\right|\right)\right| \]

   7. sqrt-pow1 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({\left({x}^{\left(\frac{2}{2}\right)}\right)}^{7}, \frac{1}{21}, 2 \cdot \left|x\right|\right)\right| \]

   8. metadata-eval N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({\left({x}^{1}\right)}^{7}, \frac{1}{21}, 2 \cdot \left|x\right|\right)\right| \]

   9. unpow1 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({x}^{7}, \frac{1}{21}, 2 \cdot \left|x\right|\right)\right| \]

   10. metadata-eval N/A

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({x}^{7}, \frac{1}{21}, 2 \cdot \left|x\right|\right)\right| \]

   11. count-2-rev N/A

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({x}^{7}, \frac{1}{21}, \left|x\right| + \left|x\right|\right)\right| \]

   12. lower-+.f64 N/A

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({x}^{7}, \frac{1}{21}, \left|x\right| + \left|x\right|\right)\right| \]

   13. rem-sqrt-square-rev N/A

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({x}^{7}, \frac{1}{21}, \sqrt{x \cdot x} + \left|x\right|\right)\right| \]

   14. pow2 N/A

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({x}^{7}, \frac{1}{21}, \sqrt{{x}^{2}} + \left|x\right|\right)\right| \]

   15. sqrt-pow1 N/A

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({x}^{7}, \frac{1}{21}, {x}^{\left(\frac{2}{2}\right)} + \left|x\right|\right)\right| \]

   16. metadata-eval N/A

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({x}^{7}, \frac{1}{21}, {x}^{1} + \left|x\right|\right)\right| \]

   17. unpow1 N/A

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({x}^{7}, \frac{1}{21}, x + \left|x\right|\right)\right| \]

   18. rem-sqrt-square-rev N/A

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({x}^{7}, \frac{1}{21}, x + \sqrt{x \cdot x}\right)\right| \]

   19. pow2 N/A

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({x}^{7}, \frac{1}{21}, x + \sqrt{{x}^{2}}\right)\right| \]

   20. sqrt-pow1 N/A

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({x}^{7}, \frac{1}{21}, x + {x}^{\left(\frac{2}{2}\right)}\right)\right| \]

   21. metadata-eval N/A

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({x}^{7}, \frac{1}{21}, x + {x}^{1}\right)\right| \]

   22. unpow1 99.0

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({x}^{7}, 0.047619047619047616, x + x\right)\right| \]

7. Applied rewrites 99.0%

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({x}^{7}, \color{blue}{0.047619047619047616}, x + x\right)\right| \]

8. Taylor expanded in x around 0

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(x \cdot \left(2 + \color{blue}{\frac{1}{21} \cdot {x}^{6}}\right)\right)\right| \]
9. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(2 + \frac{1}{21} \cdot {x}^{6}\right) \cdot x\right)\right| \]

   2. unpow1 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(2 + \frac{1}{21} \cdot {x}^{6}\right) \cdot {x}^{1}\right)\right| \]

   3. metadata-eval N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(2 + \frac{1}{21} \cdot {x}^{6}\right) \cdot {x}^{\left(\frac{2}{2}\right)}\right)\right| \]

   4. sqrt-pow1 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(2 + \frac{1}{21} \cdot {x}^{6}\right) \cdot \sqrt{{x}^{2}}\right)\right| \]

   5. pow2 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(2 + \frac{1}{21} \cdot {x}^{6}\right) \cdot \sqrt{x \cdot x}\right)\right| \]

   6. rem-sqrt-square-rev N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(2 + \frac{1}{21} \cdot {x}^{6}\right) \cdot \left|x\right|\right)\right| \]

   7. lower-*.f64 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(2 + \frac{1}{21} \cdot {x}^{6}\right) \cdot \left|x\right|\right)\right| \]

10. Applied rewrites 99.0%

    \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\mathsf{fma}\left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot x, 0.047619047619047616, 2\right) \cdot x\right)\right| \]
11. Add Preprocessing

Alternative 10: 89.5% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\sqrt{\pi}}\\ t_1 := \left(x \cdot x\right) \cdot x\\ \mathbf{if}\;x \leq 2.3:\\ \;\;\;\;\left|t\_0 \cdot \mathsf{fma}\left(x, 2, t\_1 \cdot 0.6666666666666666\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|t\_0 \cdot \left(\left(t\_1 \cdot \left(x \cdot x\right)\right) \cdot 0.2\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (sqrt PI))) (t_1 (* (* x x) x)))
   (if (<= x 2.3)
     (fabs (* t_0 (fma x 2.0 (* t_1 0.6666666666666666))))
     (fabs (* t_0 (* (* t_1 (* x x)) 0.2))))))

C

double code(double x) {
	double t_0 = 1.0 / sqrt(((double) M_PI));
	double t_1 = (x * x) * x;
	double tmp;
	if (x <= 2.3) {
		tmp = fabs((t_0 * fma(x, 2.0, (t_1 * 0.6666666666666666))));
	} else {
		tmp = fabs((t_0 * ((t_1 * (x * x)) * 0.2)));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(1.0 / sqrt(pi))
	t_1 = Float64(Float64(x * x) * x)
	tmp = 0.0
	if (x <= 2.3)
		tmp = abs(Float64(t_0 * fma(x, 2.0, Float64(t_1 * 0.6666666666666666))));
	else
		tmp = abs(Float64(t_0 * Float64(Float64(t_1 * Float64(x * x)) * 0.2)));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, 2.3], N[Abs[N[(t$95$0 * N[(x * 2.0 + N[(t$95$1 * 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(t$95$0 * N[(N[(t$95$1 * N[(x * x), $MachinePrecision]), $MachinePrecision] * 0.2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 2.2999999999999998

   1. Initial program 99.8%

      \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   3. Applied rewrites 99.8%

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \color{blue}{\mathsf{fma}\left(\left|x\right|, 2, \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(0.6666666666666666 \cdot x, x, \left(0.2 \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right), \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot \left|x\right|\right) \cdot x\right) \cdot x\right) \cdot 0.047619047619047616\right)\right)}\right| \]

   4. Taylor expanded in x around 0

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, 2, \color{blue}{\frac{2}{3} \cdot \left({x}^{2} \cdot \left|x\right|\right)}\right)\right| \]
   5. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, 2, \frac{2}{3} \cdot \left(\color{blue}{{x}^{2}} \cdot \left|x\right|\right)\right)\right| \]

      2. *-commutative N/A

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, 2, \left({x}^{2} \cdot \left|x\right|\right) \cdot \color{blue}{\frac{2}{3}}\right)\right| \]

      3. lower-*.f64 N/A

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, 2, \left({x}^{2} \cdot \left|x\right|\right) \cdot \color{blue}{\frac{2}{3}}\right)\right| \]

      4. lower-*.f64 N/A

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, 2, \left({x}^{2} \cdot \left|x\right|\right) \cdot \frac{\color{blue}{2}}{3}\right)\right| \]

      5. pow2 N/A

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, 2, \left(\left(x \cdot x\right) \cdot \left|x\right|\right) \cdot \frac{2}{3}\right)\right| \]

      6. lift-*.f64 N/A

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, 2, \left(\left(x \cdot x\right) \cdot \left|x\right|\right) \cdot \frac{2}{3}\right)\right| \]

      7. rem-sqrt-square-rev N/A

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, 2, \left(\left(x \cdot x\right) \cdot \sqrt{x \cdot x}\right) \cdot \frac{2}{3}\right)\right| \]

      8. pow2 N/A

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, 2, \left(\left(x \cdot x\right) \cdot \sqrt{{x}^{2}}\right) \cdot \frac{2}{3}\right)\right| \]

      9. sqrt-pow1 N/A

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, 2, \left(\left(x \cdot x\right) \cdot {x}^{\left(\frac{2}{2}\right)}\right) \cdot \frac{2}{3}\right)\right| \]

      10. metadata-eval N/A

          \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, 2, \left(\left(x \cdot x\right) \cdot {x}^{1}\right) \cdot \frac{2}{3}\right)\right| \]

      11. unpow1 N/A

          \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, 2, \left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{2}{3}\right)\right| \]

      12. metadata-eval 89.4

          \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, 2, \left(\left(x \cdot x\right) \cdot x\right) \cdot 0.6666666666666666\right)\right| \]

   6. Applied rewrites 89.4%

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, 2, \color{blue}{\left(\left(x \cdot x\right) \cdot x\right) \cdot 0.6666666666666666}\right)\right| \]

   7. Taylor expanded in x around 0

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\color{blue}{\left|x\right|}, 2, \left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{2}{3}\right)\right| \]
   8. Step-by-step derivation

      1. rem-sqrt-square-rev N/A

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\sqrt{x \cdot x}, 2, \left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{2}{3}\right)\right| \]

      2. pow2 N/A

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\sqrt{{x}^{2}}, 2, \left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{2}{3}\right)\right| \]

      3. sqrt-pow1 N/A

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({x}^{\color{blue}{\left(\frac{2}{2}\right)}}, 2, \left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{2}{3}\right)\right| \]

      4. metadata-eval N/A

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({x}^{1}, 2, \left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{2}{3}\right)\right| \]

      5. unpow1 89.5

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(x, 2, \left(\left(x \cdot x\right) \cdot x\right) \cdot 0.6666666666666666\right)\right| \]

   9. Applied rewrites 89.5%

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\color{blue}{x}, 2, \left(\left(x \cdot x\right) \cdot x\right) \cdot 0.6666666666666666\right)\right| \]

   if 2.2999999999999998 < x

   1. Initial program 99.8%

      \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   2. Taylor expanded in x around 0

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \color{blue}{\left(\frac{1}{21} \cdot {\left(\left|x\right|\right)}^{7} + \left(\frac{1}{5} \cdot {\left(\left|x\right|\right)}^{5} + \left(\frac{2}{3} \cdot {\left(\left|x\right|\right)}^{3} + 2 \cdot \left|x\right|\right)\right)\right)}\right| \]

   4. Applied rewrites 99.8%

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \color{blue}{\mathsf{fma}\left({\left(\left|x\right|\right)}^{7}, 0.047619047619047616, \mathsf{fma}\left(0.2 \cdot \left|x\right|, \left(\left(x \cdot x\right) \cdot x\right) \cdot x, \left|x\right| \cdot \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right)\right)\right)}\right| \]

   5. Taylor expanded in x around 0

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\frac{1}{21} \cdot {\left(\left|x\right|\right)}^{7} + \color{blue}{2 \cdot \left|x\right|}\right)\right| \]
   6. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\frac{1}{21} \cdot {\left(\left|x\right|\right)}^{7} + 2 \cdot \left|x\right|\right)\right| \]

      2. *-commutative N/A

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left({\left(\left|x\right|\right)}^{7} \cdot \frac{1}{21} + 2 \cdot \left|\color{blue}{x}\right|\right)\right| \]

      3. lower-fma.f64 N/A

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({\left(\left|x\right|\right)}^{7}, \frac{1}{\color{blue}{21}}, 2 \cdot \left|x\right|\right)\right| \]

      4. lift-pow.f64 N/A

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({\left(\left|x\right|\right)}^{7}, \frac{1}{21}, 2 \cdot \left|x\right|\right)\right| \]

      5. rem-sqrt-square-rev N/A

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({\left(\sqrt{x \cdot x}\right)}^{7}, \frac{1}{21}, 2 \cdot \left|x\right|\right)\right| \]

      6. pow2 N/A

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({\left(\sqrt{{x}^{2}}\right)}^{7}, \frac{1}{21}, 2 \cdot \left|x\right|\right)\right| \]

      7. sqrt-pow1 N/A

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({\left({x}^{\left(\frac{2}{2}\right)}\right)}^{7}, \frac{1}{21}, 2 \cdot \left|x\right|\right)\right| \]

      8. metadata-eval N/A

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({\left({x}^{1}\right)}^{7}, \frac{1}{21}, 2 \cdot \left|x\right|\right)\right| \]

      9. unpow1 N/A

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({x}^{7}, \frac{1}{21}, 2 \cdot \left|x\right|\right)\right| \]

      10. metadata-eval N/A

          \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({x}^{7}, \frac{1}{21}, 2 \cdot \left|x\right|\right)\right| \]

      11. count-2-rev N/A

          \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({x}^{7}, \frac{1}{21}, \left|x\right| + \left|x\right|\right)\right| \]

      12. lower-+.f64 N/A

          \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({x}^{7}, \frac{1}{21}, \left|x\right| + \left|x\right|\right)\right| \]

      13. rem-sqrt-square-rev N/A

          \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({x}^{7}, \frac{1}{21}, \sqrt{x \cdot x} + \left|x\right|\right)\right| \]

      14. pow2 N/A

          \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({x}^{7}, \frac{1}{21}, \sqrt{{x}^{2}} + \left|x\right|\right)\right| \]

      15. sqrt-pow1 N/A

          \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({x}^{7}, \frac{1}{21}, {x}^{\left(\frac{2}{2}\right)} + \left|x\right|\right)\right| \]

      16. metadata-eval N/A

          \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({x}^{7}, \frac{1}{21}, {x}^{1} + \left|x\right|\right)\right| \]

      17. unpow1 N/A

          \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({x}^{7}, \frac{1}{21}, x + \left|x\right|\right)\right| \]

      18. rem-sqrt-square-rev N/A

          \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({x}^{7}, \frac{1}{21}, x + \sqrt{x \cdot x}\right)\right| \]

      19. pow2 N/A

          \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({x}^{7}, \frac{1}{21}, x + \sqrt{{x}^{2}}\right)\right| \]

      20. sqrt-pow1 N/A

          \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({x}^{7}, \frac{1}{21}, x + {x}^{\left(\frac{2}{2}\right)}\right)\right| \]

      21. metadata-eval N/A

          \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({x}^{7}, \frac{1}{21}, x + {x}^{1}\right)\right| \]

      22. unpow1 99.0

          \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({x}^{7}, 0.047619047619047616, x + x\right)\right| \]

   7. Applied rewrites 99.0%

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({x}^{7}, \color{blue}{0.047619047619047616}, x + x\right)\right| \]

   8. Taylor expanded in x around inf

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\frac{1}{5} \cdot \color{blue}{\left({x}^{4} \cdot \left|x\right|\right)}\right)\right| \]
   9. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left({x}^{4} \cdot \left|x\right|\right) \cdot \frac{1}{5}\right)\right| \]

      2. metadata-eval N/A

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left({x}^{\left(3 + 1\right)} \cdot \left|x\right|\right) \cdot \frac{1}{5}\right)\right| \]

      3. unpow1 N/A

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left({\left({x}^{1}\right)}^{\left(3 + 1\right)} \cdot \left|x\right|\right) \cdot \frac{1}{5}\right)\right| \]

      4. metadata-eval N/A

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left({\left({x}^{\left(\frac{2}{2}\right)}\right)}^{\left(3 + 1\right)} \cdot \left|x\right|\right) \cdot \frac{1}{5}\right)\right| \]

      5. sqrt-pow1 N/A

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left({\left(\sqrt{{x}^{2}}\right)}^{\left(3 + 1\right)} \cdot \left|x\right|\right) \cdot \frac{1}{5}\right)\right| \]

      6. pow2 N/A

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left({\left(\sqrt{x \cdot x}\right)}^{\left(3 + 1\right)} \cdot \left|x\right|\right) \cdot \frac{1}{5}\right)\right| \]

      7. rem-sqrt-square-rev N/A

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left({\left(\left|x\right|\right)}^{\left(3 + 1\right)} \cdot \left|x\right|\right) \cdot \frac{1}{5}\right)\right| \]

      8. pow-plus N/A

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left({\left(\left|x\right|\right)}^{3} \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \frac{1}{5}\right)\right| \]

      9. pow3 N/A

         \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \frac{1}{5}\right)\right| \]

      10. lower-*.f64 N/A

          \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \frac{1}{5}\right)\right| \]

   10. Applied rewrites 31.4%

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{0.2}\right)\right| \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 89.5% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \frac{\left|\mathsf{fma}\left(\mathsf{fma}\left(0.2 \cdot x, x, 0.6666666666666666\right), x \cdot x, 2\right) \cdot x\right|}{\sqrt{\pi}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  (fabs (* (fma (fma (* 0.2 x) x 0.6666666666666666) (* x x) 2.0) x))
  (sqrt PI)))

C

double code(double x) {
	return fabs((fma(fma((0.2 * x), x, 0.6666666666666666), (x * x), 2.0) * x)) / sqrt(((double) M_PI));
}

Julia

function code(x)
	return Float64(abs(Float64(fma(fma(Float64(0.2 * x), x, 0.6666666666666666), Float64(x * x), 2.0) * x)) / sqrt(pi))
end

Wolfram

code[x_] := N[(N[Abs[N[(N[(N[(N[(0.2 * x), $MachinePrecision] * x + 0.6666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

3. Applied rewrites 99.8%

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \color{blue}{\mathsf{fma}\left(\left|x\right|, 2, \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(0.6666666666666666 \cdot x, x, \left(0.2 \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right), \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot \left|x\right|\right) \cdot x\right) \cdot x\right) \cdot 0.047619047619047616\right)\right)}\right| \]

4. Taylor expanded in x around 0

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, 2, \color{blue}{\frac{2}{3} \cdot \left({x}^{2} \cdot \left|x\right|\right)}\right)\right| \]
5. Step-by-step derivation

   1. metadata-eval N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, 2, \frac{2}{3} \cdot \left(\color{blue}{{x}^{2}} \cdot \left|x\right|\right)\right)\right| \]

   2. *-commutative N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, 2, \left({x}^{2} \cdot \left|x\right|\right) \cdot \color{blue}{\frac{2}{3}}\right)\right| \]

   3. lower-*.f64 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, 2, \left({x}^{2} \cdot \left|x\right|\right) \cdot \color{blue}{\frac{2}{3}}\right)\right| \]

   4. lower-*.f64 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, 2, \left({x}^{2} \cdot \left|x\right|\right) \cdot \frac{\color{blue}{2}}{3}\right)\right| \]

   5. pow2 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, 2, \left(\left(x \cdot x\right) \cdot \left|x\right|\right) \cdot \frac{2}{3}\right)\right| \]

   6. lift-*.f64 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, 2, \left(\left(x \cdot x\right) \cdot \left|x\right|\right) \cdot \frac{2}{3}\right)\right| \]

   7. rem-sqrt-square-rev N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, 2, \left(\left(x \cdot x\right) \cdot \sqrt{x \cdot x}\right) \cdot \frac{2}{3}\right)\right| \]

   8. pow2 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, 2, \left(\left(x \cdot x\right) \cdot \sqrt{{x}^{2}}\right) \cdot \frac{2}{3}\right)\right| \]

   9. sqrt-pow1 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, 2, \left(\left(x \cdot x\right) \cdot {x}^{\left(\frac{2}{2}\right)}\right) \cdot \frac{2}{3}\right)\right| \]

   10. metadata-eval N/A

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, 2, \left(\left(x \cdot x\right) \cdot {x}^{1}\right) \cdot \frac{2}{3}\right)\right| \]

   11. unpow1 N/A

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, 2, \left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{2}{3}\right)\right| \]

   12. metadata-eval 89.4

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, 2, \left(\left(x \cdot x\right) \cdot x\right) \cdot 0.6666666666666666\right)\right| \]

6. Applied rewrites 89.4%

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, 2, \color{blue}{\left(\left(x \cdot x\right) \cdot x\right) \cdot 0.6666666666666666}\right)\right| \]

8. Applied rewrites 89.1%

   \[\leadsto \color{blue}{\frac{\left|1 \cdot \mathsf{fma}\left(x, 2, \left(\left(x \cdot x\right) \cdot 0.6666666666666666\right) \cdot x\right)\right|}{\sqrt{\pi}}} \]

9. Taylor expanded in x around 0

   \[\leadsto \frac{\left|\color{blue}{x \cdot \left(2 + x \cdot \left(\frac{1}{5} \cdot \left({x}^{2} \cdot \left|x\right|\right) + \frac{2}{3} \cdot \left|x\right|\right)\right)}\right|}{\sqrt{\pi}} \]

11. Applied rewrites 93.3%

    \[\leadsto \frac{\left|\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.2 \cdot x, x, 0.6666666666666666\right), x \cdot x, 2\right) \cdot x}\right|}{\sqrt{\pi}} \]
12. Add Preprocessing

Alternative 12: 89.5% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(x, 2, \left(\left(x \cdot x\right) \cdot x\right) \cdot 0.6666666666666666\right)\right| \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fabs (* (/ 1.0 (sqrt PI)) (fma x 2.0 (* (* (* x x) x) 0.6666666666666666)))))

C

double code(double x) {
	return fabs(((1.0 / sqrt(((double) M_PI))) * fma(x, 2.0, (((x * x) * x) * 0.6666666666666666))));
}

Julia

function code(x)
	return abs(Float64(Float64(1.0 / sqrt(pi)) * fma(x, 2.0, Float64(Float64(Float64(x * x) * x) * 0.6666666666666666))))
end

Wolfram

code[x_] := N[Abs[N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(x * 2.0 + N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * 0.6666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

3. Applied rewrites 99.8%

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \color{blue}{\mathsf{fma}\left(\left|x\right|, 2, \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(0.6666666666666666 \cdot x, x, \left(0.2 \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right), \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot \left|x\right|\right) \cdot x\right) \cdot x\right) \cdot 0.047619047619047616\right)\right)}\right| \]

4. Taylor expanded in x around 0

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, 2, \color{blue}{\frac{2}{3} \cdot \left({x}^{2} \cdot \left|x\right|\right)}\right)\right| \]
5. Step-by-step derivation

   1. metadata-eval N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, 2, \frac{2}{3} \cdot \left(\color{blue}{{x}^{2}} \cdot \left|x\right|\right)\right)\right| \]

   2. *-commutative N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, 2, \left({x}^{2} \cdot \left|x\right|\right) \cdot \color{blue}{\frac{2}{3}}\right)\right| \]

   3. lower-*.f64 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, 2, \left({x}^{2} \cdot \left|x\right|\right) \cdot \color{blue}{\frac{2}{3}}\right)\right| \]

   4. lower-*.f64 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, 2, \left({x}^{2} \cdot \left|x\right|\right) \cdot \frac{\color{blue}{2}}{3}\right)\right| \]

   5. pow2 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, 2, \left(\left(x \cdot x\right) \cdot \left|x\right|\right) \cdot \frac{2}{3}\right)\right| \]

   6. lift-*.f64 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, 2, \left(\left(x \cdot x\right) \cdot \left|x\right|\right) \cdot \frac{2}{3}\right)\right| \]

   7. rem-sqrt-square-rev N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, 2, \left(\left(x \cdot x\right) \cdot \sqrt{x \cdot x}\right) \cdot \frac{2}{3}\right)\right| \]

   8. pow2 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, 2, \left(\left(x \cdot x\right) \cdot \sqrt{{x}^{2}}\right) \cdot \frac{2}{3}\right)\right| \]

   9. sqrt-pow1 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, 2, \left(\left(x \cdot x\right) \cdot {x}^{\left(\frac{2}{2}\right)}\right) \cdot \frac{2}{3}\right)\right| \]

   10. metadata-eval N/A

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, 2, \left(\left(x \cdot x\right) \cdot {x}^{1}\right) \cdot \frac{2}{3}\right)\right| \]

   11. unpow1 N/A

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, 2, \left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{2}{3}\right)\right| \]

   12. metadata-eval 89.4

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, 2, \left(\left(x \cdot x\right) \cdot x\right) \cdot 0.6666666666666666\right)\right| \]

6. Applied rewrites 89.4%

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, 2, \color{blue}{\left(\left(x \cdot x\right) \cdot x\right) \cdot 0.6666666666666666}\right)\right| \]

7. Taylor expanded in x around 0

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\color{blue}{\left|x\right|}, 2, \left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{2}{3}\right)\right| \]
8. Step-by-step derivation

   1. rem-sqrt-square-rev N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\sqrt{x \cdot x}, 2, \left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{2}{3}\right)\right| \]

   2. pow2 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\sqrt{{x}^{2}}, 2, \left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{2}{3}\right)\right| \]

   3. sqrt-pow1 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({x}^{\color{blue}{\left(\frac{2}{2}\right)}}, 2, \left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{2}{3}\right)\right| \]

   4. metadata-eval N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({x}^{1}, 2, \left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{2}{3}\right)\right| \]

   5. unpow1 89.5

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(x, 2, \left(\left(x \cdot x\right) \cdot x\right) \cdot 0.6666666666666666\right)\right| \]

9. Applied rewrites 89.5%

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\color{blue}{x}, 2, \left(\left(x \cdot x\right) \cdot x\right) \cdot 0.6666666666666666\right)\right| \]
10. Add Preprocessing

Alternative 13: 89.5% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \left|\frac{1}{\sqrt{\pi}} \cdot \left(\mathsf{fma}\left(0.6666666666666666 \cdot x, x, 2\right) \cdot x\right)\right| \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fabs (* (/ 1.0 (sqrt PI)) (* (fma (* 0.6666666666666666 x) x 2.0) x))))

C

double code(double x) {
	return fabs(((1.0 / sqrt(((double) M_PI))) * (fma((0.6666666666666666 * x), x, 2.0) * x)));
}

Julia

function code(x)
	return abs(Float64(Float64(1.0 / sqrt(pi)) * Float64(fma(Float64(0.6666666666666666 * x), x, 2.0) * x)))
end

Wolfram

code[x_] := N[Abs[N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(0.6666666666666666 * x), $MachinePrecision] * x + 2.0), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

3. Applied rewrites 99.8%

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \color{blue}{\mathsf{fma}\left(\left|x\right|, 2, \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(0.6666666666666666 \cdot x, x, \left(0.2 \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right), \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot \left|x\right|\right) \cdot x\right) \cdot x\right) \cdot 0.047619047619047616\right)\right)}\right| \]

4. Taylor expanded in x around 0

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \color{blue}{\left(\frac{2}{3} \cdot \left({x}^{2} \cdot \left|x\right|\right) + 2 \cdot \left|x\right|\right)}\right| \]

6. Applied rewrites 89.5%

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \color{blue}{\left(\mathsf{fma}\left(0.6666666666666666 \cdot x, x, 2\right) \cdot x\right)}\right| \]
7. Add Preprocessing

Alternative 14: 89.1% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \frac{\left|\mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right) \cdot x\right|}{\sqrt{\pi}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (fabs (* (fma (* x x) 0.6666666666666666 2.0) x)) (sqrt PI)))

C

double code(double x) {
	return fabs((fma((x * x), 0.6666666666666666, 2.0) * x)) / sqrt(((double) M_PI));
}

Julia

function code(x)
	return Float64(abs(Float64(fma(Float64(x * x), 0.6666666666666666, 2.0) * x)) / sqrt(pi))
end

Wolfram

code[x_] := N[(N[Abs[N[(N[(N[(x * x), $MachinePrecision] * 0.6666666666666666 + 2.0), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

3. Applied rewrites 99.8%

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \color{blue}{\mathsf{fma}\left(\left|x\right|, 2, \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(0.6666666666666666 \cdot x, x, \left(0.2 \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right), \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot \left|x\right|\right) \cdot x\right) \cdot x\right) \cdot 0.047619047619047616\right)\right)}\right| \]

4. Taylor expanded in x around 0

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, 2, \color{blue}{\frac{2}{3} \cdot \left({x}^{2} \cdot \left|x\right|\right)}\right)\right| \]
5. Step-by-step derivation

   1. metadata-eval N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, 2, \frac{2}{3} \cdot \left(\color{blue}{{x}^{2}} \cdot \left|x\right|\right)\right)\right| \]

   2. *-commutative N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, 2, \left({x}^{2} \cdot \left|x\right|\right) \cdot \color{blue}{\frac{2}{3}}\right)\right| \]

   3. lower-*.f64 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, 2, \left({x}^{2} \cdot \left|x\right|\right) \cdot \color{blue}{\frac{2}{3}}\right)\right| \]

   4. lower-*.f64 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, 2, \left({x}^{2} \cdot \left|x\right|\right) \cdot \frac{\color{blue}{2}}{3}\right)\right| \]

   5. pow2 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, 2, \left(\left(x \cdot x\right) \cdot \left|x\right|\right) \cdot \frac{2}{3}\right)\right| \]

   6. lift-*.f64 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, 2, \left(\left(x \cdot x\right) \cdot \left|x\right|\right) \cdot \frac{2}{3}\right)\right| \]

   7. rem-sqrt-square-rev N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, 2, \left(\left(x \cdot x\right) \cdot \sqrt{x \cdot x}\right) \cdot \frac{2}{3}\right)\right| \]

   8. pow2 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, 2, \left(\left(x \cdot x\right) \cdot \sqrt{{x}^{2}}\right) \cdot \frac{2}{3}\right)\right| \]

   9. sqrt-pow1 N/A

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, 2, \left(\left(x \cdot x\right) \cdot {x}^{\left(\frac{2}{2}\right)}\right) \cdot \frac{2}{3}\right)\right| \]

   10. metadata-eval N/A

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, 2, \left(\left(x \cdot x\right) \cdot {x}^{1}\right) \cdot \frac{2}{3}\right)\right| \]

   11. unpow1 N/A

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, 2, \left(\left(x \cdot x\right) \cdot x\right) \cdot \frac{2}{3}\right)\right| \]

   12. metadata-eval 89.4

       \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, 2, \left(\left(x \cdot x\right) \cdot x\right) \cdot 0.6666666666666666\right)\right| \]

6. Applied rewrites 89.4%

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(\left|x\right|, 2, \color{blue}{\left(\left(x \cdot x\right) \cdot x\right) \cdot 0.6666666666666666}\right)\right| \]

8. Applied rewrites 89.1%

   \[\leadsto \color{blue}{\frac{\left|1 \cdot \mathsf{fma}\left(x, 2, \left(\left(x \cdot x\right) \cdot 0.6666666666666666\right) \cdot x\right)\right|}{\sqrt{\pi}}} \]

9. Taylor expanded in x around 0

   \[\leadsto \frac{\left|\color{blue}{x \cdot \left(2 + \frac{2}{3} \cdot \left(x \cdot \left|x\right|\right)\right)}\right|}{\sqrt{\pi}} \]

11. Applied rewrites 89.1%

    \[\leadsto \frac{\left|\color{blue}{\mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right) \cdot x}\right|}{\sqrt{\pi}} \]
12. Add Preprocessing

Alternative 15: 83.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\\ t_1 := \frac{1}{\sqrt{\pi}}\\ t_2 := \left(t\_0 \cdot \left|x\right|\right) \cdot \left|x\right|\\ \mathbf{if}\;\left|t\_1 \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot t\_0\right) + \frac{1}{5} \cdot t\_2\right) + \frac{1}{21} \cdot \left(\left(t\_2 \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \leq 2 \cdot 10^{-16}:\\ \;\;\;\;\left|t\_1 \cdot \left(x + x\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\frac{\left(x + x\right) \cdot \left(x + x\right)}{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (* (fabs x) (fabs x)) (fabs x)))
        (t_1 (/ 1.0 (sqrt PI)))
        (t_2 (* (* t_0 (fabs x)) (fabs x))))
   (if (<=
        (fabs
         (*
          t_1
          (+
           (+ (+ (* 2.0 (fabs x)) (* (/ 2.0 3.0) t_0)) (* (/ 1.0 5.0) t_2))
           (* (/ 1.0 21.0) (* (* t_2 (fabs x)) (fabs x))))))
        2e-16)
     (fabs (* t_1 (+ x x)))
     (sqrt (/ (* (+ x x) (+ x x)) PI)))))

C

double code(double x) {
	double t_0 = (fabs(x) * fabs(x)) * fabs(x);
	double t_1 = 1.0 / sqrt(((double) M_PI));
	double t_2 = (t_0 * fabs(x)) * fabs(x);
	double tmp;
	if (fabs((t_1 * ((((2.0 * fabs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_2)) + ((1.0 / 21.0) * ((t_2 * fabs(x)) * fabs(x)))))) <= 2e-16) {
		tmp = fabs((t_1 * (x + x)));
	} else {
		tmp = sqrt((((x + x) * (x + x)) / ((double) M_PI)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = (Math.abs(x) * Math.abs(x)) * Math.abs(x);
	double t_1 = 1.0 / Math.sqrt(Math.PI);
	double t_2 = (t_0 * Math.abs(x)) * Math.abs(x);
	double tmp;
	if (Math.abs((t_1 * ((((2.0 * Math.abs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_2)) + ((1.0 / 21.0) * ((t_2 * Math.abs(x)) * Math.abs(x)))))) <= 2e-16) {
		tmp = Math.abs((t_1 * (x + x)));
	} else {
		tmp = Math.sqrt((((x + x) * (x + x)) / Math.PI));
	}
	return tmp;
}

Python

def code(x):
	t_0 = (math.fabs(x) * math.fabs(x)) * math.fabs(x)
	t_1 = 1.0 / math.sqrt(math.pi)
	t_2 = (t_0 * math.fabs(x)) * math.fabs(x)
	tmp = 0
	if math.fabs((t_1 * ((((2.0 * math.fabs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_2)) + ((1.0 / 21.0) * ((t_2 * math.fabs(x)) * math.fabs(x)))))) <= 2e-16:
		tmp = math.fabs((t_1 * (x + x)))
	else:
		tmp = math.sqrt((((x + x) * (x + x)) / math.pi))
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(abs(x) * abs(x)) * abs(x))
	t_1 = Float64(1.0 / sqrt(pi))
	t_2 = Float64(Float64(t_0 * abs(x)) * abs(x))
	tmp = 0.0
	if (abs(Float64(t_1 * Float64(Float64(Float64(Float64(2.0 * abs(x)) + Float64(Float64(2.0 / 3.0) * t_0)) + Float64(Float64(1.0 / 5.0) * t_2)) + Float64(Float64(1.0 / 21.0) * Float64(Float64(t_2 * abs(x)) * abs(x)))))) <= 2e-16)
		tmp = abs(Float64(t_1 * Float64(x + x)));
	else
		tmp = sqrt(Float64(Float64(Float64(x + x) * Float64(x + x)) / pi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (abs(x) * abs(x)) * abs(x);
	t_1 = 1.0 / sqrt(pi);
	t_2 = (t_0 * abs(x)) * abs(x);
	tmp = 0.0;
	if (abs((t_1 * ((((2.0 * abs(x)) + ((2.0 / 3.0) * t_0)) + ((1.0 / 5.0) * t_2)) + ((1.0 / 21.0) * ((t_2 * abs(x)) * abs(x)))))) <= 2e-16)
		tmp = abs((t_1 * (x + x)));
	else
		tmp = sqrt((((x + x) * (x + x)) / pi));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$0 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[N[(t$95$1 * N[(N[(N[(N[(2.0 * N[Abs[x], $MachinePrecision]), $MachinePrecision] + N[(N[(2.0 / 3.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / 5.0), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / 21.0), $MachinePrecision] * N[(N[(t$95$2 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 2e-16], N[Abs[N[(t$95$1 * N[(x + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Sqrt[N[(N[(N[(x + x), $MachinePrecision] * N[(x + x), $MachinePrecision]), $MachinePrecision] / Pi), $MachinePrecision]], $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 (*.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 (PI.f64))) (+.f64 (+.f64 (+.f64 (*.f64 #s(literal 2 binary64) (fabs.f64 x)) (*.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)))) (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 5 binary64)) (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)))) (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 21 binary64)) (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)))))) < 2e-16

   1. Initial program 99.9%

      \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   3. Applied rewrites 99.9%

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \color{blue}{\mathsf{fma}\left(\left|x\right|, 2, \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(0.6666666666666666 \cdot x, x, \left(0.2 \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right), \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot \left|x\right|\right) \cdot x\right) \cdot x\right) \cdot 0.047619047619047616\right)\right)}\right| \]

   4. Taylor expanded in x around 0

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \color{blue}{\left(2 \cdot \left|x\right|\right)}\right| \]

   6. Applied rewrites 99.9%

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \color{blue}{\left(x + x\right)}\right| \]

   if 2e-16 < (fabs.f64 (*.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 (PI.f64))) (+.f64 (+.f64 (+.f64 (*.f64 #s(literal 2 binary64) (fabs.f64 x)) (*.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)))) (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 5 binary64)) (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)))) (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 21 binary64)) (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x))))))

   1. Initial program 99.8%

      \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

   3. Applied rewrites 99.8%

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \color{blue}{\mathsf{fma}\left(\left|x\right|, 2, \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(0.6666666666666666 \cdot x, x, \left(0.2 \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right), \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot \left|x\right|\right) \cdot x\right) \cdot x\right) \cdot 0.047619047619047616\right)\right)}\right| \]

   4. Taylor expanded in x around 0

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \color{blue}{\left(2 \cdot \left|x\right|\right)}\right| \]

   6. Applied rewrites 12.7%

      \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \color{blue}{\left(x + x\right)}\right| \]

   8. Applied rewrites 12.7%

      \[\leadsto \color{blue}{\frac{\left|1 \cdot \left(x + x\right)\right|}{\sqrt{\pi}}} \]

   10. Applied rewrites 56.1%

       \[\leadsto \color{blue}{\sqrt{\frac{\left(x + x\right) \cdot \left(x + x\right)}{\pi}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 67.8% accurate, 7.3× speedup

Math

\[\begin{array}{l} \\ \left|\frac{1}{\sqrt{\pi}} \cdot \left(x + x\right)\right| \end{array} \]

FPCore

(FPCore (x) :precision binary64 (fabs (* (/ 1.0 (sqrt PI)) (+ x x))))

C

double code(double x) {
	return fabs(((1.0 / sqrt(((double) M_PI))) * (x + x)));
}

Java

public static double code(double x) {
	return Math.abs(((1.0 / Math.sqrt(Math.PI)) * (x + x)));
}

Python

def code(x):
	return math.fabs(((1.0 / math.sqrt(math.pi)) * (x + x)))

Julia

function code(x)
	return abs(Float64(Float64(1.0 / sqrt(pi)) * Float64(x + x)))
end

MATLAB

function tmp = code(x)
	tmp = abs(((1.0 / sqrt(pi)) * (x + x)));
end

Wolfram

code[x_] := N[Abs[N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(x + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

3. Applied rewrites 99.8%

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \color{blue}{\mathsf{fma}\left(\left|x\right|, 2, \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(0.6666666666666666 \cdot x, x, \left(0.2 \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right), \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot \left|x\right|\right) \cdot x\right) \cdot x\right) \cdot 0.047619047619047616\right)\right)}\right| \]

4. Taylor expanded in x around 0

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \color{blue}{\left(2 \cdot \left|x\right|\right)}\right| \]

6. Applied rewrites 67.8%

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \color{blue}{\left(x + x\right)}\right| \]
7. Add Preprocessing

Alternative 17: 67.4% accurate, 9.4× speedup

Math

\[\begin{array}{l} \\ \frac{\left|x + x\right|}{\sqrt{\pi}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (fabs (+ x x)) (sqrt PI)))

C

double code(double x) {
	return fabs((x + x)) / sqrt(((double) M_PI));
}

Java

public static double code(double x) {
	return Math.abs((x + x)) / Math.sqrt(Math.PI);
}

Python

def code(x):
	return math.fabs((x + x)) / math.sqrt(math.pi)

Julia

function code(x)
	return Float64(abs(Float64(x + x)) / sqrt(pi))
end

MATLAB

function tmp = code(x)
	tmp = abs((x + x)) / sqrt(pi);
end

Wolfram

code[x_] := N[(N[Abs[N[(x + x), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot \left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{5} \cdot \left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right) + \frac{1}{21} \cdot \left(\left(\left(\left(\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \]

3. Applied rewrites 99.8%

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \color{blue}{\mathsf{fma}\left(\left|x\right|, 2, \mathsf{fma}\left(\left|x\right|, \mathsf{fma}\left(0.6666666666666666 \cdot x, x, \left(0.2 \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right)\right), \left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot \left|x\right|\right) \cdot x\right) \cdot x\right) \cdot 0.047619047619047616\right)\right)}\right| \]

4. Taylor expanded in x around 0

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \color{blue}{\left(2 \cdot \left|x\right|\right)}\right| \]

6. Applied rewrites 67.8%

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \color{blue}{\left(x + x\right)}\right| \]

8. Applied rewrites 67.4%

   \[\leadsto \color{blue}{\frac{\left|1 \cdot \left(x + x\right)\right|}{\sqrt{\pi}}} \]

10. Applied rewrites 67.4%

    \[\leadsto \color{blue}{\frac{\left|x + x\right|}{\sqrt{\pi}}} \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025111 
(FPCore (x)
  :name "Jmat.Real.erfi, branch x less than or equal to 0.5"
  :precision binary64
  :pre (<= x 0.5)
  (fabs (* (/ 1.0 (sqrt PI)) (+ (+ (+ (* 2.0 (fabs x)) (* (/ 2.0 3.0) (* (* (fabs x) (fabs x)) (fabs x)))) (* (/ 1.0 5.0) (* (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x)) (fabs x)))) (* (/ 1.0 21.0) (* (* (* (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x)) (fabs x)) (fabs x)) (fabs x)))))))