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

Percentage Accurate: 99.8% → 99.8%

Time: 11.9s

Alternatives: 15

Speedup: 2.5×

• 28.2% 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, 2.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

4. Applied rewrites 99.9%

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

5. Final simplification 99.9%

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

Alternative 2: 99.8% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

4. Applied rewrites 99.9%

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \color{blue}{\left(\left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.2, x \cdot x, 0.047619047619047616 \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right)\right)\right)}\right| \]
5. Step-by-step derivation

   1. lift-fma.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-+r+ N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*l* N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. associate-*l* N/A

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

   10. distribute-lft-out N/A

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

   11. lower-fma.f64 N/A

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

6. Applied rewrites 99.9%

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

7. Final simplification 99.9%

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

Alternative 3: 99.8% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

4. Applied rewrites 99.9%

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

6. Applied rewrites 99.9%

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

7. Final simplification 99.9%

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

Alternative 4: 99.2% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.1:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.2 \cdot x, x, 0.6666666666666666\right), x \cdot x, 2\right)}{\sqrt{\pi}} \cdot \left|x\right|\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(0.047619047619047616 \cdot \left|x\right|\right)\right) \cdot x}{\sqrt{\pi}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 0.1)
   (fabs
    (*
     (/ (fma (fma (* 0.2 x) x 0.6666666666666666) (* x x) 2.0) (sqrt PI))
     (fabs x)))
   (fabs
    (/
     (* (* (* (* (* x x) x) (* x x)) (* 0.047619047619047616 (fabs x))) x)
     (sqrt PI)))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 0.1) {
		tmp = fabs(((fma(fma((0.2 * x), x, 0.6666666666666666), (x * x), 2.0) / sqrt(((double) M_PI))) * fabs(x)));
	} else {
		tmp = fabs(((((((x * x) * x) * (x * x)) * (0.047619047619047616 * fabs(x))) * x) / sqrt(((double) M_PI))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 0.1)
		tmp = abs(Float64(Float64(fma(fma(Float64(0.2 * x), x, 0.6666666666666666), Float64(x * x), 2.0) / sqrt(pi)) * abs(x)));
	else
		tmp = abs(Float64(Float64(Float64(Float64(Float64(Float64(x * x) * x) * Float64(x * x)) * Float64(0.047619047619047616 * abs(x))) * x) / sqrt(pi)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 0.1], N[Abs[N[(N[(N[(N[(N[(0.2 * x), $MachinePrecision] * x + 0.6666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(0.047619047619047616 * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.10000000000000001

   1. Initial program 99.9%

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 99.6%

      \[\leadsto \left|\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.2, x \cdot x, 0.6666666666666666\right), x \cdot x, 2\right) \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left|x\right|\right)}\right| \]
   8. Step-by-step derivation

   9. Applied rewrites 99.6%

      \[\leadsto \left|\mathsf{fma}\left(\mathsf{fma}\left(0.2 \cdot x, x, 0.6666666666666666\right), x \cdot x, 2\right) \cdot \left(\sqrt{\color{blue}{\frac{1}{\pi}}} \cdot \left|x\right|\right)\right| \]
   10. Step-by-step derivation

   11. Applied rewrites 99.6%

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

   if 0.10000000000000001 < (fabs.f64 x)

   1. Initial program 99.8%

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around inf

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

      1. associate-*l* N/A

         \[\leadsto \left|\frac{1}{21} \cdot \color{blue}{\left({x}^{6} \cdot \left(\left|x\right| \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)\right)}\right| \]

      2. *-commutative N/A

         \[\leadsto \left|\frac{1}{21} \cdot \left({x}^{6} \cdot \color{blue}{\left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left|x\right|\right)}\right)\right| \]

      3. associate-*r* N/A

         \[\leadsto \left|\color{blue}{\left(\frac{1}{21} \cdot {x}^{6}\right) \cdot \left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left|x\right|\right)}\right| \]

      4. lower-*.f64 N/A

         \[\leadsto \left|\color{blue}{\left(\frac{1}{21} \cdot {x}^{6}\right) \cdot \left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left|x\right|\right)}\right| \]

   7. Applied rewrites 97.6%

      \[\leadsto \left|\color{blue}{\left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot 0.047619047619047616\right) \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left|x\right|\right)}\right| \]
   8. Step-by-step derivation

   9. Applied rewrites 97.7%

      \[\leadsto \left|\frac{\left(\left(0.047619047619047616 \cdot \left|x\right|\right) \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(x \cdot x\right)\right)\right) \cdot x}{\color{blue}{\sqrt{\pi}}}\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.1:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.2 \cdot x, x, 0.6666666666666666\right), x \cdot x, 2\right)}{\sqrt{\pi}} \cdot \left|x\right|\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(0.047619047619047616 \cdot \left|x\right|\right)\right) \cdot x}{\sqrt{\pi}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.2% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.1:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.2 \cdot x, x, 0.6666666666666666\right), x \cdot x, 2\right)}{\sqrt{\pi}} \cdot \left|x\right|\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(\left(\left(\left(0.047619047619047616 \cdot \left(x \cdot x\right)\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x}{\sqrt{\pi}} \cdot \left|x\right|\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 0.1)
   (fabs
    (*
     (/ (fma (fma (* 0.2 x) x 0.6666666666666666) (* x x) 2.0) (sqrt PI))
     (fabs x)))
   (fabs
    (*
     (/ (* (* (* (* (* 0.047619047619047616 (* x x)) x) x) x) x) (sqrt PI))
     (fabs x)))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 0.1) {
		tmp = fabs(((fma(fma((0.2 * x), x, 0.6666666666666666), (x * x), 2.0) / sqrt(((double) M_PI))) * fabs(x)));
	} else {
		tmp = fabs((((((((0.047619047619047616 * (x * x)) * x) * x) * x) * x) / sqrt(((double) M_PI))) * fabs(x)));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 0.1)
		tmp = abs(Float64(Float64(fma(fma(Float64(0.2 * x), x, 0.6666666666666666), Float64(x * x), 2.0) / sqrt(pi)) * abs(x)));
	else
		tmp = abs(Float64(Float64(Float64(Float64(Float64(Float64(Float64(0.047619047619047616 * Float64(x * x)) * x) * x) * x) * x) / sqrt(pi)) * abs(x)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 0.1], N[Abs[N[(N[(N[(N[(N[(0.2 * x), $MachinePrecision] * x + 0.6666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(N[(N[(N[(N[(0.047619047619047616 * N[(x * x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.10000000000000001

   1. Initial program 99.9%

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 99.6%

      \[\leadsto \left|\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.2, x \cdot x, 0.6666666666666666\right), x \cdot x, 2\right) \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left|x\right|\right)}\right| \]
   8. Step-by-step derivation

   9. Applied rewrites 99.6%

      \[\leadsto \left|\mathsf{fma}\left(\mathsf{fma}\left(0.2 \cdot x, x, 0.6666666666666666\right), x \cdot x, 2\right) \cdot \left(\sqrt{\color{blue}{\frac{1}{\pi}}} \cdot \left|x\right|\right)\right| \]
   10. Step-by-step derivation

   11. Applied rewrites 99.6%

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

   if 0.10000000000000001 < (fabs.f64 x)

   1. Initial program 99.8%

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around inf

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

      1. associate-*l* N/A

         \[\leadsto \left|\frac{1}{21} \cdot \color{blue}{\left({x}^{6} \cdot \left(\left|x\right| \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)\right)}\right| \]

      2. *-commutative N/A

         \[\leadsto \left|\frac{1}{21} \cdot \left({x}^{6} \cdot \color{blue}{\left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left|x\right|\right)}\right)\right| \]

      3. associate-*r* N/A

         \[\leadsto \left|\color{blue}{\left(\frac{1}{21} \cdot {x}^{6}\right) \cdot \left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left|x\right|\right)}\right| \]

      4. lower-*.f64 N/A

         \[\leadsto \left|\color{blue}{\left(\frac{1}{21} \cdot {x}^{6}\right) \cdot \left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left|x\right|\right)}\right| \]

   7. Applied rewrites 97.6%

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

   9. Applied rewrites 97.6%

      \[\leadsto \left|\frac{\left(\left(\left(\left(\left(x \cdot x\right) \cdot 0.047619047619047616\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x}{\sqrt{\pi}} \cdot \color{blue}{\left|x\right|}\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.1:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.2 \cdot x, x, 0.6666666666666666\right), x \cdot x, 2\right)}{\sqrt{\pi}} \cdot \left|x\right|\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(\left(\left(\left(0.047619047619047616 \cdot \left(x \cdot x\right)\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x}{\sqrt{\pi}} \cdot \left|x\right|\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 99.8% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

4. Applied rewrites 99.9%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. +-commutative N/A

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

   5. *-commutative N/A

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

   6. lower-fma.f64 N/A

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

   7. +-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 N/A

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

   11. unpow2 N/A

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

   12. lower-*.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 99.8

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

7. Applied rewrites 99.8%

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

8. Final simplification 99.8%

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

Alternative 7: 99.2% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.1:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.2 \cdot x, x, 0.6666666666666666\right), x \cdot x, 2\right)}{\sqrt{\pi}} \cdot \left|x\right|\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot 0.047619047619047616\right) \cdot x\right) \cdot x}{\sqrt{\pi}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 0.1)
   (fabs
    (*
     (/ (fma (fma (* 0.2 x) x 0.6666666666666666) (* x x) 2.0) (sqrt PI))
     (fabs x)))
   (fabs
    (/
     (* (* (* (* (* (* (* x x) x) x) x) 0.047619047619047616) x) x)
     (sqrt PI)))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 0.1) {
		tmp = fabs(((fma(fma((0.2 * x), x, 0.6666666666666666), (x * x), 2.0) / sqrt(((double) M_PI))) * fabs(x)));
	} else {
		tmp = fabs(((((((((x * x) * x) * x) * x) * 0.047619047619047616) * x) * x) / sqrt(((double) M_PI))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 0.1)
		tmp = abs(Float64(Float64(fma(fma(Float64(0.2 * x), x, 0.6666666666666666), Float64(x * x), 2.0) / sqrt(pi)) * abs(x)));
	else
		tmp = abs(Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(x * x) * x) * x) * x) * 0.047619047619047616) * x) * x) / sqrt(pi)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 0.1], N[Abs[N[(N[(N[(N[(N[(0.2 * x), $MachinePrecision] * x + 0.6666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(N[(N[(N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * 0.047619047619047616), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.10000000000000001

   1. Initial program 99.9%

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 99.6%

      \[\leadsto \left|\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.2, x \cdot x, 0.6666666666666666\right), x \cdot x, 2\right) \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left|x\right|\right)}\right| \]
   8. Step-by-step derivation

   9. Applied rewrites 99.6%

      \[\leadsto \left|\mathsf{fma}\left(\mathsf{fma}\left(0.2 \cdot x, x, 0.6666666666666666\right), x \cdot x, 2\right) \cdot \left(\sqrt{\color{blue}{\frac{1}{\pi}}} \cdot \left|x\right|\right)\right| \]
   10. Step-by-step derivation

   11. Applied rewrites 99.6%

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

   if 0.10000000000000001 < (fabs.f64 x)

   1. Initial program 99.8%

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around inf

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

      1. associate-*l* N/A

         \[\leadsto \left|\frac{1}{21} \cdot \color{blue}{\left({x}^{6} \cdot \left(\left|x\right| \cdot \sqrt{\frac{1}{\mathsf{PI}\left(\right)}}\right)\right)}\right| \]

      2. *-commutative N/A

         \[\leadsto \left|\frac{1}{21} \cdot \left({x}^{6} \cdot \color{blue}{\left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left|x\right|\right)}\right)\right| \]

      3. associate-*r* N/A

         \[\leadsto \left|\color{blue}{\left(\frac{1}{21} \cdot {x}^{6}\right) \cdot \left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left|x\right|\right)}\right| \]

      4. lower-*.f64 N/A

         \[\leadsto \left|\color{blue}{\left(\frac{1}{21} \cdot {x}^{6}\right) \cdot \left(\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left|x\right|\right)}\right| \]

   7. Applied rewrites 97.6%

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

   9. Applied rewrites 97.6%

      \[\leadsto \left|\frac{\left(\left(\left(\left(\left(x \cdot x\right) \cdot 0.047619047619047616\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x}{\sqrt{\pi}} \cdot \color{blue}{\left|x\right|}\right| \]

   11. Applied rewrites 97.6%

       \[\leadsto \color{blue}{\left|\frac{\left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot 0.047619047619047616\right) \cdot x\right) \cdot x}{\sqrt{\pi}}\right|} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.1:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\mathsf{fma}\left(0.2 \cdot x, x, 0.6666666666666666\right), x \cdot x, 2\right)}{\sqrt{\pi}} \cdot \left|x\right|\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left(\left(\left(\left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot 0.047619047619047616\right) \cdot x\right) \cdot x}{\sqrt{\pi}}\right|\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.4% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

4. Applied rewrites 99.9%

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \color{blue}{\left(\left|x\right| \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.2, x \cdot x, 0.047619047619047616 \cdot \left(\left(\left(x \cdot x\right) \cdot x\right) \cdot x\right)\right), \mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right)\right)\right)}\right| \]
5. Step-by-step derivation

   1. lift-fma.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-+r+ N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*l* N/A

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

   7. lift-*.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. associate-*l* N/A

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

   10. distribute-lft-out N/A

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

   11. lower-fma.f64 N/A

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

6. Applied rewrites 99.9%

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

8. Applied rewrites 99.3%

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

9. Final simplification 99.3%

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

Alternative 9: 99.4% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

4. Applied rewrites 99.9%

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

6. Applied rewrites 99.3%

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

8. Applied rewrites 99.3%

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

Alternative 10: 98.7% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

4. Applied rewrites 99.9%

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

6. Applied rewrites 99.3%

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

7. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. unpow3 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 98.2

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

9. Applied rewrites 98.2%

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

10. Final simplification 98.2%

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

Alternative 11: 93.9% accurate, 3.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

4. Applied rewrites 99.9%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 94.3%

   \[\leadsto \left|\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.2, x \cdot x, 0.6666666666666666\right), x \cdot x, 2\right) \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left|x\right|\right)}\right| \]
8. Step-by-step derivation

9. Applied rewrites 94.3%

   \[\leadsto \left|\mathsf{fma}\left(\mathsf{fma}\left(0.2 \cdot x, x, 0.6666666666666666\right), x \cdot x, 2\right) \cdot \left(\sqrt{\color{blue}{\frac{1}{\pi}}} \cdot \left|x\right|\right)\right| \]
10. Step-by-step derivation

11. Applied rewrites 94.3%

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

12. Final simplification 94.3%

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

Alternative 12: 88.9% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.1:\\ \;\;\;\;\frac{\left|2\right|}{\sqrt{\pi}} \cdot \left|x\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(\left(x \cdot x\right) \cdot x\right) \cdot 0.6666666666666666\right|}{\sqrt{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 0.1)
   (* (/ (fabs 2.0) (sqrt PI)) (fabs x))
   (/ (fabs (* (* (* x x) x) 0.6666666666666666)) (sqrt PI))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 0.1) {
		tmp = (fabs(2.0) / sqrt(((double) M_PI))) * fabs(x);
	} else {
		tmp = fabs((((x * x) * x) * 0.6666666666666666)) / sqrt(((double) M_PI));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) <= 0.1) {
		tmp = (Math.abs(2.0) / Math.sqrt(Math.PI)) * Math.abs(x);
	} else {
		tmp = Math.abs((((x * x) * x) * 0.6666666666666666)) / Math.sqrt(Math.PI);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) <= 0.1:
		tmp = (math.fabs(2.0) / math.sqrt(math.pi)) * math.fabs(x)
	else:
		tmp = math.fabs((((x * x) * x) * 0.6666666666666666)) / math.sqrt(math.pi)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 0.1)
		tmp = Float64(Float64(abs(2.0) / sqrt(pi)) * abs(x));
	else
		tmp = Float64(abs(Float64(Float64(Float64(x * x) * x) * 0.6666666666666666)) / sqrt(pi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) <= 0.1)
		tmp = (abs(2.0) / sqrt(pi)) * abs(x);
	else
		tmp = abs((((x * x) * x) * 0.6666666666666666)) / sqrt(pi);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 0.1], N[(N[(N[Abs[2.0], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision], N[(N[Abs[N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * 0.6666666666666666), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.10000000000000001

   1. Initial program 99.9%

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

      \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left|x\right| \cdot \color{blue}{2}\right)\right| \]
   6. Step-by-step derivation

   7. Applied rewrites 98.7%

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

      1. lift-fabs.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. fabs-div N/A

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

   9. Applied rewrites 98.0%

      \[\leadsto \color{blue}{\frac{\left|2\right| \cdot \left|x\right|}{\sqrt{\pi}}} \]
   10. Step-by-step derivation

       1. lift-/.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift-fabs.f64 N/A

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

       4. *-commutative N/A

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

       5. associate-/l* N/A

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

       6. lower-*.f64 N/A

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

       7. lift-fabs.f64 N/A

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

   11. Applied rewrites 98.7%

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

   if 0.10000000000000001 < (fabs.f64 x)

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0

      \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \color{blue}{\left(\frac{1}{21} \cdot {\left(\left|x\right|\right)}^{7} + \left(\frac{1}{5} \cdot {\left(\left|x\right|\right)}^{5} + \left(\frac{2}{3} \cdot {\left(\left|x\right|\right)}^{3} + 2 \cdot \left|x\right|\right)\right)\right)}\right| \]
   4. Step-by-step derivation

      1. associate-+r+ N/A

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

      2. +-commutative N/A

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

      3. +-commutative N/A

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

      4. unpow3 N/A

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

      5. associate-*r* N/A

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

      6. distribute-rgt-out N/A

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

      7. *-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 65.1%

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

   10. Applied rewrites 65.1%

       \[\leadsto \color{blue}{\frac{\left|\left(\left(x \cdot x\right) \cdot x\right) \cdot 0.6666666666666666\right|}{\sqrt{\pi}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 88.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.1:\\ \;\;\;\;\frac{\left|2\right|}{\sqrt{\pi}} \cdot \left|x\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(\left(x \cdot x\right) \cdot x\right) \cdot 0.6666666666666666\right|}{\sqrt{\pi}}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 89.2% accurate, 4.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

4. Applied rewrites 99.9%

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

5. 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. Step-by-step derivation

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*r* N/A

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

   4. associate-*r* N/A

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

   5. distribute-rgt-in N/A

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

   6. associate-*r* N/A

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

   7. distribute-rgt-in N/A

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

   8. *-commutative N/A

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

   9. *-commutative N/A

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

   10. associate-*l* N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 N/A

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

7. Applied rewrites 88.6%

   \[\leadsto \left|\color{blue}{\mathsf{fma}\left(x \cdot x, 0.6666666666666666, 2\right) \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left|x\right|\right)}\right| \]
8. Step-by-step derivation

9. Applied rewrites 88.6%

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

Alternative 14: 68.2% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \frac{\left|2\right|}{\sqrt{\pi}} \cdot \left|x\right| \end{array} \]

FPCore

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

C

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

Java

public static double code(double x) {
	return (Math.abs(2.0) / Math.sqrt(Math.PI)) * Math.abs(x);
}

Python

def code(x):
	return (math.fabs(2.0) / math.sqrt(math.pi)) * math.fabs(x)

Julia

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

MATLAB

function tmp = code(x)
	tmp = (abs(2.0) / sqrt(pi)) * abs(x);
end

Wolfram

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

Derivation

1. Initial program 99.9%

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

4. Applied rewrites 99.9%

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

5. Taylor expanded in x around 0

   \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left|x\right| \cdot \color{blue}{2}\right)\right| \]
6. Step-by-step derivation

7. Applied rewrites 69.7%

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

   1. lift-fabs.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-*l/ N/A

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

   5. fabs-div N/A

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

9. Applied rewrites 69.2%

   \[\leadsto \color{blue}{\frac{\left|2\right| \cdot \left|x\right|}{\sqrt{\pi}}} \]
10. Step-by-step derivation

    1. lift-/.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. lift-fabs.f64 N/A

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

    4. *-commutative N/A

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

    5. associate-/l* N/A

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

    6. lower-*.f64 N/A

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

    7. lift-fabs.f64 N/A

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

11. Applied rewrites 69.7%

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

12. Final simplification 69.7%

    \[\leadsto \frac{\left|2\right|}{\sqrt{\pi}} \cdot \left|x\right| \]
13. Add Preprocessing

Alternative 15: 67.8% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \frac{\left|2 \cdot x\right|}{\sqrt{\pi}} \end{array} \]

FPCore

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

C

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

Java

public static double code(double x) {
	return Math.abs((2.0 * x)) / Math.sqrt(Math.PI);
}

Python

def code(x):
	return math.fabs((2.0 * x)) / math.sqrt(math.pi)

Julia

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

MATLAB

function tmp = code(x)
	tmp = abs((2.0 * x)) / sqrt(pi);
end

Wolfram

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

Derivation

1. Initial program 99.9%

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

4. Applied rewrites 99.9%

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

5. Taylor expanded in x around 0

   \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left|x\right| \cdot \color{blue}{2}\right)\right| \]
6. Step-by-step derivation

7. Applied rewrites 69.7%

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

   1. lift-fabs.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. associate-*l/ N/A

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

   5. fabs-div N/A

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

9. Applied rewrites 69.2%

   \[\leadsto \color{blue}{\frac{\left|2\right| \cdot \left|x\right|}{\sqrt{\pi}}} \]
10. Step-by-step derivation

    1. lift-*.f64 N/A

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

    2. lift-fabs.f64 N/A

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

    3. lift-fabs.f64 N/A

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

    4. mul-fabs N/A

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

    5. lower-fabs.f64 N/A

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

    6. lower-*.f64 69.2

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

11. Applied rewrites 69.2%

    \[\leadsto \color{blue}{\frac{\left|2 \cdot x\right|}{\sqrt{\pi}}} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024236 
(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)))))))