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

Percentage Accurate: 99.8% → 99.8%

Time: 4.9s

Alternatives: 13

Speedup: 2.2×

• 28.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[x \leq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.8%

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

5. Applied rewrites 99.8%

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

Alternative 2: 99.8% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.8%

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

5. Applied rewrites 99.8%

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

7. Applied rewrites 99.8%

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

9. Applied rewrites 99.8%

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

Alternative 3: 99.8% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.8%

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

5. Applied rewrites 99.8%

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

7. Applied rewrites 99.8%

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

9. Applied rewrites 99.8%

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

Alternative 4: 99.4% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.86:\\ \;\;\;\;\frac{2}{\sqrt{\pi}} \cdot \left|x\right|\\ \mathbf{else}:\\ \;\;\;\;\left|x\right| \cdot \left|\frac{0.047619047619047616 \cdot {x}^{6}}{\sqrt{\pi}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.86)
   (* (/ 2.0 (sqrt PI)) (fabs x))
   (* (fabs x) (fabs (/ (* 0.047619047619047616 (pow x 6.0)) (sqrt PI))))))

C

double code(double x) {
	double tmp;
	if (x <= 1.86) {
		tmp = (2.0 / sqrt(((double) M_PI))) * fabs(x);
	} else {
		tmp = fabs(x) * fabs(((0.047619047619047616 * pow(x, 6.0)) / sqrt(((double) M_PI))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.86) {
		tmp = (2.0 / Math.sqrt(Math.PI)) * Math.abs(x);
	} else {
		tmp = Math.abs(x) * Math.abs(((0.047619047619047616 * Math.pow(x, 6.0)) / Math.sqrt(Math.PI)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.86:
		tmp = (2.0 / math.sqrt(math.pi)) * math.fabs(x)
	else:
		tmp = math.fabs(x) * math.fabs(((0.047619047619047616 * math.pow(x, 6.0)) / math.sqrt(math.pi)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.86)
		tmp = Float64(Float64(2.0 / sqrt(pi)) * abs(x));
	else
		tmp = Float64(abs(x) * abs(Float64(Float64(0.047619047619047616 * (x ^ 6.0)) / sqrt(pi))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.86)
		tmp = (2.0 / sqrt(pi)) * abs(x);
	else
		tmp = abs(x) * abs(((0.047619047619047616 * (x ^ 6.0)) / sqrt(pi)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.86], N[(N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision], N[(N[Abs[x], $MachinePrecision] * N[Abs[N[(N[(0.047619047619047616 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.8600000000000001

   1. Initial program 99.8%

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

   3. Applied rewrites 99.4%

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

   4. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-fabs.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-PI.f64 67.8

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

   6. Applied rewrites 67.8%

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

      1. lift-fabs.f64 N/A

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

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

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

      3. sqrt-prod N/A

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

   8. Applied rewrites 67.8%

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

      1. lift-*.f64 N/A

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

   10. Applied rewrites 68.3%

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

   if 1.8600000000000001 < x

   1. Initial program 99.8%

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

   3. Applied rewrites 99.8%

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

         \[\leadsto \left|x\right| \cdot \left|\frac{\frac{1}{21} \cdot \color{blue}{{x}^{6}}}{\sqrt{\pi}}\right| \]

      2. lower-pow.f64 36.4

         \[\leadsto \left|x\right| \cdot \left|\frac{0.047619047619047616 \cdot {x}^{\color{blue}{6}}}{\sqrt{\pi}}\right| \]

   8. Applied rewrites 36.4%

      \[\leadsto \left|x\right| \cdot \left|\frac{\color{blue}{0.047619047619047616 \cdot {x}^{6}}}{\sqrt{\pi}}\right| \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 98.9% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.8%

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

5. Applied rewrites 99.8%

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

7. Applied rewrites 99.8%

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

9. Applied rewrites 99.4%

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

Alternative 6: 98.5% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.86:\\ \;\;\;\;\frac{2}{\sqrt{\pi}} \cdot \left|x\right|\\ \mathbf{else}:\\ \;\;\;\;\left|0.047619047619047616 \cdot \frac{{x}^{6} \cdot \left|x\right|}{\sqrt{\pi}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.86)
   (* (/ 2.0 (sqrt PI)) (fabs x))
   (fabs (* 0.047619047619047616 (/ (* (pow x 6.0) (fabs x)) (sqrt PI))))))

C

double code(double x) {
	double tmp;
	if (x <= 1.86) {
		tmp = (2.0 / sqrt(((double) M_PI))) * fabs(x);
	} else {
		tmp = fabs((0.047619047619047616 * ((pow(x, 6.0) * fabs(x)) / sqrt(((double) M_PI)))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.86) {
		tmp = (2.0 / Math.sqrt(Math.PI)) * Math.abs(x);
	} else {
		tmp = Math.abs((0.047619047619047616 * ((Math.pow(x, 6.0) * Math.abs(x)) / Math.sqrt(Math.PI))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.86:
		tmp = (2.0 / math.sqrt(math.pi)) * math.fabs(x)
	else:
		tmp = math.fabs((0.047619047619047616 * ((math.pow(x, 6.0) * math.fabs(x)) / math.sqrt(math.pi))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.86)
		tmp = Float64(Float64(2.0 / sqrt(pi)) * abs(x));
	else
		tmp = abs(Float64(0.047619047619047616 * Float64(Float64((x ^ 6.0) * abs(x)) / sqrt(pi))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.86)
		tmp = (2.0 / sqrt(pi)) * abs(x);
	else
		tmp = abs((0.047619047619047616 * (((x ^ 6.0) * abs(x)) / sqrt(pi))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.86], N[(N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision], N[Abs[N[(0.047619047619047616 * N[(N[(N[Power[x, 6.0], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.8600000000000001

   1. Initial program 99.8%

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

   3. Applied rewrites 99.4%

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

   4. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-fabs.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-PI.f64 67.8

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

   6. Applied rewrites 67.8%

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

      1. lift-fabs.f64 N/A

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

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

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

      3. sqrt-prod N/A

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

   8. Applied rewrites 67.8%

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

      1. lift-*.f64 N/A

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

   10. Applied rewrites 68.3%

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

   if 1.8600000000000001 < x

   1. Initial program 99.8%

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

   3. Applied rewrites 99.4%

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

   4. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-fabs.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-PI.f64 36.4

         \[\leadsto \left|0.047619047619047616 \cdot \frac{{x}^{6} \cdot \left|x\right|}{\sqrt{\pi}}\right| \]

   6. Applied rewrites 36.4%

      \[\leadsto \left|\color{blue}{0.047619047619047616 \cdot \frac{{x}^{6} \cdot \left|x\right|}{\sqrt{\pi}}}\right| \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 98.5% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.8%

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

4. Taylor expanded in x around 0

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

   1. lower-*.f64 N/A

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

   2. lower-fabs.f64 98.9

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

6. Applied rewrites 98.9%

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

Alternative 8: 98.4% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.86:\\ \;\;\;\;\frac{2}{\sqrt{\pi}} \cdot \left|x\right|\\ \mathbf{else}:\\ \;\;\;\;\left|0.047619047619047616 \cdot {x}^{7}\right| \cdot \frac{1}{\sqrt{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.86)
   (* (/ 2.0 (sqrt PI)) (fabs x))
   (* (fabs (* 0.047619047619047616 (pow x 7.0))) (/ 1.0 (sqrt PI)))))

C

double code(double x) {
	double tmp;
	if (x <= 1.86) {
		tmp = (2.0 / sqrt(((double) M_PI))) * fabs(x);
	} else {
		tmp = fabs((0.047619047619047616 * pow(x, 7.0))) * (1.0 / sqrt(((double) M_PI)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.86) {
		tmp = (2.0 / Math.sqrt(Math.PI)) * Math.abs(x);
	} else {
		tmp = Math.abs((0.047619047619047616 * Math.pow(x, 7.0))) * (1.0 / Math.sqrt(Math.PI));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.86:
		tmp = (2.0 / math.sqrt(math.pi)) * math.fabs(x)
	else:
		tmp = math.fabs((0.047619047619047616 * math.pow(x, 7.0))) * (1.0 / math.sqrt(math.pi))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.86)
		tmp = Float64(Float64(2.0 / sqrt(pi)) * abs(x));
	else
		tmp = Float64(abs(Float64(0.047619047619047616 * (x ^ 7.0))) * Float64(1.0 / sqrt(pi)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.86)
		tmp = (2.0 / sqrt(pi)) * abs(x);
	else
		tmp = abs((0.047619047619047616 * (x ^ 7.0))) * (1.0 / sqrt(pi));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.86], N[(N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision], N[(N[Abs[N[(0.047619047619047616 * N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.8600000000000001

   1. Initial program 99.8%

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

   3. Applied rewrites 99.4%

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

   4. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-fabs.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-PI.f64 67.8

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

   6. Applied rewrites 67.8%

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

      1. lift-fabs.f64 N/A

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

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

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

      3. sqrt-prod N/A

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

   8. Applied rewrites 67.8%

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

      1. lift-*.f64 N/A

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

   10. Applied rewrites 68.3%

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

   if 1.8600000000000001 < x

   1. Initial program 99.8%

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

   3. Applied rewrites 99.8%

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

   5. Applied rewrites 99.8%

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

   7. Applied rewrites 99.8%

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

   8. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-pow.f64 36.4

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

   10. Applied rewrites 36.4%

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

Alternative 9: 68.3% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.8%

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

4. Taylor expanded in x around 0

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

   1. lower-/.f64 N/A

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

   2. lower-fma.f64 N/A

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

   3. lower-pow.f64 N/A

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

   4. lower-fabs.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-fabs.f64 N/A

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

   7. lower-sqrt.f64 N/A

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

   8. lower-PI.f64 98.5

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

6. Applied rewrites 98.5%

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

8. Applied rewrites 98.5%

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

Alternative 10: 68.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \left|\frac{\mathsf{fma}\left(0.047619047619047616, {\left(x \cdot x\right)}^{3.5}, 2 \cdot \left|x\right|\right)}{\sqrt{\pi}}\right| \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fabs
  (/ (fma 0.047619047619047616 (pow (* x x) 3.5) (* 2.0 (fabs x))) (sqrt PI))))

C

double code(double x) {
	return fabs((fma(0.047619047619047616, pow((x * x), 3.5), (2.0 * fabs(x))) / sqrt(((double) M_PI))));
}

Julia

function code(x)
	return abs(Float64(fma(0.047619047619047616, (Float64(x * x) ^ 3.5), Float64(2.0 * abs(x))) / sqrt(pi)))
end

Wolfram

code[x_] := N[Abs[N[(N[(0.047619047619047616 * N[Power[N[(x * x), $MachinePrecision], 3.5], $MachinePrecision] + N[(2.0 * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.8%

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

4. Taylor expanded in x around 0

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

   1. lower-/.f64 N/A

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

   2. lower-fma.f64 N/A

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

   3. lower-pow.f64 N/A

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

   4. lower-fabs.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-fabs.f64 N/A

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

   7. lower-sqrt.f64 N/A

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

   8. lower-PI.f64 98.5

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

6. Applied rewrites 98.5%

   \[\leadsto \left|\color{blue}{\frac{\mathsf{fma}\left(0.047619047619047616, {\left(\left|x\right|\right)}^{7}, 2 \cdot \left|x\right|\right)}{\sqrt{\pi}}}\right| \]
7. Step-by-step derivation

   1. lift-pow.f64 N/A

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

   2. lift-fabs.f64 N/A

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

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

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

   4. lift-*.f64 N/A

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

   5. sqrt-pow2 N/A

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

   6. lower-pow.f64 N/A

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

   7. metadata-eval 98.4

      \[\leadsto \left|\frac{\mathsf{fma}\left(0.047619047619047616, {\left(x \cdot x\right)}^{3.5}, 2 \cdot \left|x\right|\right)}{\sqrt{\pi}}\right| \]

8. Applied rewrites 98.4%

   \[\leadsto \left|\frac{\mathsf{fma}\left(0.047619047619047616, {\left(x \cdot x\right)}^{3.5}, 2 \cdot \left|x\right|\right)}{\sqrt{\pi}}\right| \]
9. Add Preprocessing

Alternative 11: 68.3% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.8%

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

4. Taylor expanded in x around 0

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

   1. lower-/.f64 N/A

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

   2. lower-fma.f64 N/A

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

   3. lower-pow.f64 N/A

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

   4. lower-fabs.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. lower-fabs.f64 N/A

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

   7. lower-sqrt.f64 N/A

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

   8. lower-PI.f64 98.5

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

6. Applied rewrites 98.5%

   \[\leadsto \left|\color{blue}{\frac{\mathsf{fma}\left(0.047619047619047616, {\left(\left|x\right|\right)}^{7}, 2 \cdot \left|x\right|\right)}{\sqrt{\pi}}}\right| \]
7. Step-by-step derivation

   1. lift-fma.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. lift-fabs.f64 N/A

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

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

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

   5. lift-*.f64 N/A

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

   6. sqrt-pow2 N/A

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

   7. metadata-eval N/A

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

   8. metadata-eval N/A

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

   9. pow-prod-up N/A

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

   10. pow1/2 N/A

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

   11. lift-*.f64 N/A

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

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

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

   13. lift-fabs.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. pow-prod-down N/A

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

   16. pow-prod-up N/A

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

   17. metadata-eval N/A

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

   18. lift-pow.f64 N/A

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

   19. associate-*l* N/A

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

   20. *-commutative N/A

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

   21. lower-fma.f64 N/A

       \[\leadsto \left|\frac{\mathsf{fma}\left({x}^{6}, \frac{1}{21} \cdot \left|x\right|, 2 \cdot \left|x\right|\right)}{\sqrt{\color{blue}{\pi}}}\right| \]

8. Applied rewrites 98.5%

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

10. Applied rewrites 98.5%

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

Alternative 12: 68.3% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1000000000:\\ \;\;\;\;\frac{2}{\sqrt{\pi}} \cdot \left|x\right|\\ \mathbf{else}:\\ \;\;\;\;\left|2 \cdot \sqrt{\frac{x \cdot x}{\pi}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1000000000.0)
   (* (/ 2.0 (sqrt PI)) (fabs x))
   (fabs (* 2.0 (sqrt (/ (* x x) PI))))))

C

double code(double x) {
	double tmp;
	if (x <= 1000000000.0) {
		tmp = (2.0 / sqrt(((double) M_PI))) * fabs(x);
	} else {
		tmp = fabs((2.0 * sqrt(((x * x) / ((double) M_PI)))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1000000000.0) {
		tmp = (2.0 / Math.sqrt(Math.PI)) * Math.abs(x);
	} else {
		tmp = Math.abs((2.0 * Math.sqrt(((x * x) / Math.PI))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1000000000.0:
		tmp = (2.0 / math.sqrt(math.pi)) * math.fabs(x)
	else:
		tmp = math.fabs((2.0 * math.sqrt(((x * x) / math.pi))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1000000000.0)
		tmp = Float64(Float64(2.0 / sqrt(pi)) * abs(x));
	else
		tmp = abs(Float64(2.0 * sqrt(Float64(Float64(x * x) / pi))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1000000000.0)
		tmp = (2.0 / sqrt(pi)) * abs(x);
	else
		tmp = abs((2.0 * sqrt(((x * x) / pi))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1000000000.0], N[(N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision], N[Abs[N[(2.0 * N[Sqrt[N[(N[(x * x), $MachinePrecision] / Pi), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1e9

   1. Initial program 99.8%

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

   3. Applied rewrites 99.4%

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

   4. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-fabs.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-PI.f64 67.8

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

   6. Applied rewrites 67.8%

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

      1. lift-fabs.f64 N/A

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

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

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

      3. sqrt-prod N/A

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

   8. Applied rewrites 67.8%

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

      1. lift-*.f64 N/A

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

   10. Applied rewrites 68.3%

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

   if 1e9 < x

   1. Initial program 99.8%

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

   3. Applied rewrites 99.4%

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

   4. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-fabs.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-PI.f64 67.8

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

   6. Applied rewrites 67.8%

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

      1. lift-/.f64 N/A

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

      2. lift-fabs.f64 N/A

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

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

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

      4. lift-*.f64 N/A

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

      5. lift-sqrt.f64 N/A

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

      6. sqrt-undiv N/A

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

      7. lower-sqrt.f64 N/A

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

      8. lower-/.f64 53.6

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

   8. Applied rewrites 53.6%

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

Alternative 13: 68.3% accurate, 9.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.4%

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

4. Taylor expanded in x around 0

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-fabs.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-PI.f64 67.8

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

6. Applied rewrites 67.8%

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

   1. lift-fabs.f64 N/A

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

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

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

   3. sqrt-prod N/A

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

8. Applied rewrites 67.8%

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

   1. lift-*.f64 N/A

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

10. Applied rewrites 68.3%

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

Reproduce

herbie shell --seed 2025164 
(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)))))))