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

Percentage Accurate: 99.8% → 99.8%

Time: 11.3s

Alternatives: 10

Speedup: 1.5×

• 29.2% 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, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|x\right| \cdot \left(x \cdot x\right)\\ t_1 := \left|x\right| \cdot \left(\left|x\right| \cdot t_0\right)\\ \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + 0.6666666666666666 \cdot t_0\right) + 0.2 \cdot t_1\right) + 0.047619047619047616 \cdot \left(\left|x\right| \cdot \left(\left|x\right| \cdot t_1\right)\right)\right)\right| \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (fabs x) (* x x))) (t_1 (* (fabs x) (* (fabs x) t_0))))
   (fabs
    (*
     (/ 1.0 (sqrt PI))
     (+
      (+ (+ (* 2.0 (fabs x)) (* 0.6666666666666666 t_0)) (* 0.2 t_1))
      (* 0.047619047619047616 (* (fabs x) (* (fabs x) t_1))))))))

C

double code(double x) {
	double t_0 = fabs(x) * (x * x);
	double t_1 = fabs(x) * (fabs(x) * t_0);
	return fabs(((1.0 / sqrt(((double) M_PI))) * ((((2.0 * fabs(x)) + (0.6666666666666666 * t_0)) + (0.2 * t_1)) + (0.047619047619047616 * (fabs(x) * (fabs(x) * t_1))))));
}

Java

public static double code(double x) {
	double t_0 = Math.abs(x) * (x * x);
	double t_1 = Math.abs(x) * (Math.abs(x) * t_0);
	return Math.abs(((1.0 / Math.sqrt(Math.PI)) * ((((2.0 * Math.abs(x)) + (0.6666666666666666 * t_0)) + (0.2 * t_1)) + (0.047619047619047616 * (Math.abs(x) * (Math.abs(x) * t_1))))));
}

Python

def code(x):
	t_0 = math.fabs(x) * (x * x)
	t_1 = math.fabs(x) * (math.fabs(x) * t_0)
	return math.fabs(((1.0 / math.sqrt(math.pi)) * ((((2.0 * math.fabs(x)) + (0.6666666666666666 * t_0)) + (0.2 * t_1)) + (0.047619047619047616 * (math.fabs(x) * (math.fabs(x) * t_1))))))

Julia

function code(x)
	t_0 = Float64(abs(x) * Float64(x * x))
	t_1 = Float64(abs(x) * Float64(abs(x) * t_0))
	return abs(Float64(Float64(1.0 / sqrt(pi)) * Float64(Float64(Float64(Float64(2.0 * abs(x)) + Float64(0.6666666666666666 * t_0)) + Float64(0.2 * t_1)) + Float64(0.047619047619047616 * Float64(abs(x) * Float64(abs(x) * t_1))))))
end

MATLAB

function tmp = code(x)
	t_0 = abs(x) * (x * x);
	t_1 = abs(x) * (abs(x) * t_0);
	tmp = abs(((1.0 / sqrt(pi)) * ((((2.0 * abs(x)) + (0.6666666666666666 * t_0)) + (0.2 * t_1)) + (0.047619047619047616 * (abs(x) * (abs(x) * t_1))))));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Abs[x], $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[x], $MachinePrecision] * N[(N[Abs[x], $MachinePrecision] * t$95$0), $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[(0.6666666666666666 * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(0.2 * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(0.047619047619047616 * N[(N[Abs[x], $MachinePrecision] * N[(N[Abs[x], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Initial program 99.8%

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

2. Final simplification 99.8%

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

Alternative 2: 99.8% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x \cdot x\right) \cdot \left(x \cdot x\right)\\ \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\mathsf{fma}\left(2, \left|x\right|, 0.6666666666666666 \cdot \left(\left|x\right| \cdot \left(x \cdot x\right)\right)\right) + 0.2 \cdot \left(\left|x\right| \cdot t_0\right)\right) + 0.047619047619047616 \cdot \left(\left|x\right| \cdot \left(\left(x \cdot x\right) \cdot t_0\right)\right)\right)\right| \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (* x x) (* x x))))
   (fabs
    (*
     (/ 1.0 (sqrt PI))
     (+
      (+
       (fma 2.0 (fabs x) (* 0.6666666666666666 (* (fabs x) (* x x))))
       (* 0.2 (* (fabs x) t_0)))
      (* 0.047619047619047616 (* (fabs x) (* (* x x) t_0))))))))

C

double code(double x) {
	double t_0 = (x * x) * (x * x);
	return fabs(((1.0 / sqrt(((double) M_PI))) * ((fma(2.0, fabs(x), (0.6666666666666666 * (fabs(x) * (x * x)))) + (0.2 * (fabs(x) * t_0))) + (0.047619047619047616 * (fabs(x) * ((x * x) * t_0))))));
}

Julia

function code(x)
	t_0 = Float64(Float64(x * x) * Float64(x * x))
	return abs(Float64(Float64(1.0 / sqrt(pi)) * Float64(Float64(fma(2.0, abs(x), Float64(0.6666666666666666 * Float64(abs(x) * Float64(x * x)))) + Float64(0.2 * Float64(abs(x) * t_0))) + Float64(0.047619047619047616 * Float64(abs(x) * Float64(Float64(x * x) * t_0))))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]}, N[Abs[N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(2.0 * N[Abs[x], $MachinePrecision] + N[(0.6666666666666666 * N[(N[Abs[x], $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.2 * N[(N[Abs[x], $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.047619047619047616 * N[(N[Abs[x], $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

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

4. Final simplification 99.8%

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

Alternative 3: 99.4% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \left|\frac{\mathsf{fma}\left(0.047619047619047616, {x}^{7}, x \cdot \left(\left(2 + 0.6666666666666666 \cdot \left(x \cdot x\right)\right) + 0.2 \cdot {x}^{4}\right)\right)}{\sqrt{\pi}}\right| \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fabs
  (/
   (fma
    0.047619047619047616
    (pow x 7.0)
    (* x (+ (+ 2.0 (* 0.6666666666666666 (* x x))) (* 0.2 (pow x 4.0)))))
   (sqrt PI))))

C

double code(double x) {
	return fabs((fma(0.047619047619047616, pow(x, 7.0), (x * ((2.0 + (0.6666666666666666 * (x * x))) + (0.2 * pow(x, 4.0))))) / sqrt(((double) M_PI))));
}

Julia

function code(x)
	return abs(Float64(fma(0.047619047619047616, (x ^ 7.0), Float64(x * Float64(Float64(2.0 + Float64(0.6666666666666666 * Float64(x * x))) + Float64(0.2 * (x ^ 4.0))))) / sqrt(pi)))
end

Wolfram

code[x_] := N[Abs[N[(N[(0.047619047619047616 * N[Power[x, 7.0], $MachinePrecision] + N[(x * N[(N[(2.0 + N[(0.6666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.2 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $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 egg-rr 99.8%

   \[\leadsto \left|\color{blue}{\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.2, \left|x\right| \cdot {\left(x \cdot x\right)}^{2}, \left|x\right| \cdot \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)\right) + \frac{1}{\sqrt{\pi}} \cdot \left(0.047619047619047616 \cdot \left({\left(x \cdot x\right)}^{2} \cdot {\left(\left|x\right|\right)}^{3}\right)\right)}\right| \]

5. Simplified 99.4%

   \[\leadsto \left|\color{blue}{\frac{\mathsf{fma}\left(0.047619047619047616, {x}^{7}, x \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + 0.2 \cdot {x}^{4}\right)\right)}{\sqrt{\pi}}}\right| \]
6. Step-by-step derivation

   1. metadata-eval 90.4%

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

   2. fma-udef 90.4%

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

   3. metadata-eval 90.4%

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

7. Applied egg-rr 99.4%

   \[\leadsto \left|\frac{\mathsf{fma}\left(0.047619047619047616, {x}^{7}, x \cdot \left(\color{blue}{\left(0.6666666666666666 \cdot \left(x \cdot x\right) + 2\right)} + 0.2 \cdot {x}^{4}\right)\right)}{\sqrt{\pi}}\right| \]

8. Final simplification 99.4%

   \[\leadsto \left|\frac{\mathsf{fma}\left(0.047619047619047616, {x}^{7}, x \cdot \left(\left(2 + 0.6666666666666666 \cdot \left(x \cdot x\right)\right) + 0.2 \cdot {x}^{4}\right)\right)}{\sqrt{\pi}}\right| \]

Alternative 4: 98.6% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \left|\frac{\mathsf{fma}\left(2, x, 0.2 \cdot {x}^{5} + 0.047619047619047616 \cdot {x}^{7}\right)}{\sqrt{\pi}}\right| \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fabs
  (/
   (fma 2.0 x (+ (* 0.2 (pow x 5.0)) (* 0.047619047619047616 (pow x 7.0))))
   (sqrt PI))))

C

double code(double x) {
	return fabs((fma(2.0, x, ((0.2 * pow(x, 5.0)) + (0.047619047619047616 * pow(x, 7.0)))) / sqrt(((double) M_PI))));
}

Julia

function code(x)
	return abs(Float64(fma(2.0, x, Float64(Float64(0.2 * (x ^ 5.0)) + Float64(0.047619047619047616 * (x ^ 7.0)))) / sqrt(pi)))
end

Wolfram

code[x_] := N[Abs[N[(N[(2.0 * x + N[(N[(0.2 * N[Power[x, 5.0], $MachinePrecision]), $MachinePrecision] + N[(0.047619047619047616 * N[Power[x, 7.0], $MachinePrecision]), $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. Simplified 99.4%

   \[\leadsto \color{blue}{\left|\frac{\mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(0.047619047619047616, {x}^{7}, 0.2 \cdot {x}^{5}\right)\right)\right)}{\sqrt{\pi}}\right|} \]

4. Taylor expanded in x around inf 98.6%

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

5. Final simplification 98.6%

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

Alternative 5: 98.5% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[Abs[N[(N[(2.0 * x + N[(0.047619047619047616 * N[Power[x, 7.0], $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. Simplified 99.4%

   \[\leadsto \color{blue}{\left|\frac{\mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(0.047619047619047616, {x}^{7}, 0.2 \cdot {x}^{5}\right)\right)\right)}{\sqrt{\pi}}\right|} \]

4. Taylor expanded in x around inf 98.4%

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

5. Final simplification 98.4%

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

Alternative 6: 99.2% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.2:\\ \;\;\;\;\left|\left(0.047619047619047616 \cdot {x}^{7}\right) \cdot {\pi}^{-0.5}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(2 \cdot x + 0.6666666666666666 \cdot {x}^{3}\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -2.2)
   (fabs (* (* 0.047619047619047616 (pow x 7.0)) (pow PI -0.5)))
   (fabs
    (* (sqrt (/ 1.0 PI)) (+ (* 2.0 x) (* 0.6666666666666666 (pow x 3.0)))))))

C

double code(double x) {
	double tmp;
	if (x <= -2.2) {
		tmp = fabs(((0.047619047619047616 * pow(x, 7.0)) * pow(((double) M_PI), -0.5)));
	} else {
		tmp = fabs((sqrt((1.0 / ((double) M_PI))) * ((2.0 * x) + (0.6666666666666666 * pow(x, 3.0)))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -2.2) {
		tmp = Math.abs(((0.047619047619047616 * Math.pow(x, 7.0)) * Math.pow(Math.PI, -0.5)));
	} else {
		tmp = Math.abs((Math.sqrt((1.0 / Math.PI)) * ((2.0 * x) + (0.6666666666666666 * Math.pow(x, 3.0)))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -2.2:
		tmp = math.fabs(((0.047619047619047616 * math.pow(x, 7.0)) * math.pow(math.pi, -0.5)))
	else:
		tmp = math.fabs((math.sqrt((1.0 / math.pi)) * ((2.0 * x) + (0.6666666666666666 * math.pow(x, 3.0)))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -2.2)
		tmp = abs(Float64(Float64(0.047619047619047616 * (x ^ 7.0)) * (pi ^ -0.5)));
	else
		tmp = abs(Float64(sqrt(Float64(1.0 / pi)) * Float64(Float64(2.0 * x) + Float64(0.6666666666666666 * (x ^ 3.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -2.2)
		tmp = abs(((0.047619047619047616 * (x ^ 7.0)) * (pi ^ -0.5)));
	else
		tmp = abs((sqrt((1.0 / pi)) * ((2.0 * x) + (0.6666666666666666 * (x ^ 3.0)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -2.2], N[Abs[N[(N[(0.047619047619047616 * N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision] * N[Power[Pi, -0.5], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(N[(2.0 * x), $MachinePrecision] + N[(0.6666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.2000000000000002

   1. Initial program 99.8%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\left|\frac{\mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(0.047619047619047616, {x}^{7}, 0.2 \cdot {x}^{5}\right)\right)\right)}{\sqrt{\pi}}\right|} \]

   4. Taylor expanded in x around inf 99.5%

      \[\leadsto \left|\color{blue}{0.047619047619047616 \cdot \left({x}^{7} \cdot \sqrt{\frac{1}{\pi}}\right)}\right| \]
   5. Step-by-step derivation

      1. associate-*r* 99.5%

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

      2. *-commutative 99.5%

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

   6. Simplified 99.5%

      \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot {x}^{7}\right)}\right| \]
   7. Step-by-step derivation

      1. expm1-log1p-u 0.0%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot {x}^{7}\right)\right)\right)}\right| \]

      2. expm1-udef 0.0%

         \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot {x}^{7}\right)\right)} - 1}\right| \]

      3. *-commutative 0.0%

         \[\leadsto \left|e^{\mathsf{log1p}\left(\color{blue}{\left(0.047619047619047616 \cdot {x}^{7}\right) \cdot \sqrt{\frac{1}{\pi}}}\right)} - 1\right| \]

      4. pow1/2 0.0%

         \[\leadsto \left|e^{\mathsf{log1p}\left(\left(0.047619047619047616 \cdot {x}^{7}\right) \cdot \color{blue}{{\left(\frac{1}{\pi}\right)}^{0.5}}\right)} - 1\right| \]

      5. inv-pow 0.0%

         \[\leadsto \left|e^{\mathsf{log1p}\left(\left(0.047619047619047616 \cdot {x}^{7}\right) \cdot {\color{blue}{\left({\pi}^{-1}\right)}}^{0.5}\right)} - 1\right| \]

      6. pow-pow 0.0%

         \[\leadsto \left|e^{\mathsf{log1p}\left(\left(0.047619047619047616 \cdot {x}^{7}\right) \cdot \color{blue}{{\pi}^{\left(-1 \cdot 0.5\right)}}\right)} - 1\right| \]

      7. metadata-eval 0.0%

         \[\leadsto \left|e^{\mathsf{log1p}\left(\left(0.047619047619047616 \cdot {x}^{7}\right) \cdot {\pi}^{\color{blue}{-0.5}}\right)} - 1\right| \]

   8. Applied egg-rr 0.0%

      \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(\left(0.047619047619047616 \cdot {x}^{7}\right) \cdot {\pi}^{-0.5}\right)} - 1}\right| \]
   9. Step-by-step derivation

      1. expm1-def 0.0%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(0.047619047619047616 \cdot {x}^{7}\right) \cdot {\pi}^{-0.5}\right)\right)}\right| \]

      2. expm1-log1p 99.5%

         \[\leadsto \left|\color{blue}{\left(0.047619047619047616 \cdot {x}^{7}\right) \cdot {\pi}^{-0.5}}\right| \]

   10. Simplified 99.5%

       \[\leadsto \left|\color{blue}{\left(0.047619047619047616 \cdot {x}^{7}\right) \cdot {\pi}^{-0.5}}\right| \]

   if -2.2000000000000002 < x

   1. Initial program 99.8%

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

   3. Simplified 99.2%

      \[\leadsto \color{blue}{\left|\frac{\mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(0.047619047619047616, {x}^{7}, 0.2 \cdot {x}^{5}\right)\right)\right)}{\sqrt{\pi}}\right|} \]

   4. Taylor expanded in x around 0 99.4%

      \[\leadsto \left|\color{blue}{0.6666666666666666 \cdot \left({x}^{3} \cdot \sqrt{\frac{1}{\pi}}\right) + 2 \cdot \left(x \cdot \sqrt{\frac{1}{\pi}}\right)}\right| \]
   5. Step-by-step derivation

      1. +-commutative 99.4%

         \[\leadsto \left|\color{blue}{2 \cdot \left(x \cdot \sqrt{\frac{1}{\pi}}\right) + 0.6666666666666666 \cdot \left({x}^{3} \cdot \sqrt{\frac{1}{\pi}}\right)}\right| \]

      2. associate-*r* 99.4%

         \[\leadsto \left|\color{blue}{\left(2 \cdot x\right) \cdot \sqrt{\frac{1}{\pi}}} + 0.6666666666666666 \cdot \left({x}^{3} \cdot \sqrt{\frac{1}{\pi}}\right)\right| \]

      3. associate-*r* 99.4%

         \[\leadsto \left|\left(2 \cdot x\right) \cdot \sqrt{\frac{1}{\pi}} + \color{blue}{\left(0.6666666666666666 \cdot {x}^{3}\right) \cdot \sqrt{\frac{1}{\pi}}}\right| \]

      4. distribute-rgt-out 99.4%

         \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(2 \cdot x + 0.6666666666666666 \cdot {x}^{3}\right)}\right| \]

      5. *-commutative 99.4%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(\color{blue}{x \cdot 2} + 0.6666666666666666 \cdot {x}^{3}\right)\right| \]

   6. Simplified 99.4%

      \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot 2 + 0.6666666666666666 \cdot {x}^{3}\right)}\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.2:\\ \;\;\;\;\left|\left(0.047619047619047616 \cdot {x}^{7}\right) \cdot {\pi}^{-0.5}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(2 \cdot x + 0.6666666666666666 \cdot {x}^{3}\right)\right|\\ \end{array} \]

Alternative 7: 99.2% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.2:\\ \;\;\;\;\left|\left(0.047619047619047616 \cdot {x}^{7}\right) \cdot {\pi}^{-0.5}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot \left(2 + 0.6666666666666666 \cdot \left(x \cdot x\right)\right)\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -2.2)
   (fabs (* (* 0.047619047619047616 (pow x 7.0)) (pow PI -0.5)))
   (fabs (* (sqrt (/ 1.0 PI)) (* x (+ 2.0 (* 0.6666666666666666 (* x x))))))))

C

double code(double x) {
	double tmp;
	if (x <= -2.2) {
		tmp = fabs(((0.047619047619047616 * pow(x, 7.0)) * pow(((double) M_PI), -0.5)));
	} else {
		tmp = fabs((sqrt((1.0 / ((double) M_PI))) * (x * (2.0 + (0.6666666666666666 * (x * x))))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -2.2) {
		tmp = Math.abs(((0.047619047619047616 * Math.pow(x, 7.0)) * Math.pow(Math.PI, -0.5)));
	} else {
		tmp = Math.abs((Math.sqrt((1.0 / Math.PI)) * (x * (2.0 + (0.6666666666666666 * (x * x))))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -2.2:
		tmp = math.fabs(((0.047619047619047616 * math.pow(x, 7.0)) * math.pow(math.pi, -0.5)))
	else:
		tmp = math.fabs((math.sqrt((1.0 / math.pi)) * (x * (2.0 + (0.6666666666666666 * (x * x))))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -2.2)
		tmp = abs(Float64(Float64(0.047619047619047616 * (x ^ 7.0)) * (pi ^ -0.5)));
	else
		tmp = abs(Float64(sqrt(Float64(1.0 / pi)) * Float64(x * Float64(2.0 + Float64(0.6666666666666666 * Float64(x * x))))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -2.2)
		tmp = abs(((0.047619047619047616 * (x ^ 7.0)) * (pi ^ -0.5)));
	else
		tmp = abs((sqrt((1.0 / pi)) * (x * (2.0 + (0.6666666666666666 * (x * x))))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -2.2], N[Abs[N[(N[(0.047619047619047616 * N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision] * N[Power[Pi, -0.5], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(x * N[(2.0 + N[(0.6666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.2000000000000002

   1. Initial program 99.8%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\left|\frac{\mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(0.047619047619047616, {x}^{7}, 0.2 \cdot {x}^{5}\right)\right)\right)}{\sqrt{\pi}}\right|} \]

   4. Taylor expanded in x around inf 99.5%

      \[\leadsto \left|\color{blue}{0.047619047619047616 \cdot \left({x}^{7} \cdot \sqrt{\frac{1}{\pi}}\right)}\right| \]
   5. Step-by-step derivation

      1. associate-*r* 99.5%

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

      2. *-commutative 99.5%

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

   6. Simplified 99.5%

      \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot {x}^{7}\right)}\right| \]
   7. Step-by-step derivation

      1. expm1-log1p-u 0.0%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot {x}^{7}\right)\right)\right)}\right| \]

      2. expm1-udef 0.0%

         \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{1}{\pi}} \cdot \left(0.047619047619047616 \cdot {x}^{7}\right)\right)} - 1}\right| \]

      3. *-commutative 0.0%

         \[\leadsto \left|e^{\mathsf{log1p}\left(\color{blue}{\left(0.047619047619047616 \cdot {x}^{7}\right) \cdot \sqrt{\frac{1}{\pi}}}\right)} - 1\right| \]

      4. pow1/2 0.0%

         \[\leadsto \left|e^{\mathsf{log1p}\left(\left(0.047619047619047616 \cdot {x}^{7}\right) \cdot \color{blue}{{\left(\frac{1}{\pi}\right)}^{0.5}}\right)} - 1\right| \]

      5. inv-pow 0.0%

         \[\leadsto \left|e^{\mathsf{log1p}\left(\left(0.047619047619047616 \cdot {x}^{7}\right) \cdot {\color{blue}{\left({\pi}^{-1}\right)}}^{0.5}\right)} - 1\right| \]

      6. pow-pow 0.0%

         \[\leadsto \left|e^{\mathsf{log1p}\left(\left(0.047619047619047616 \cdot {x}^{7}\right) \cdot \color{blue}{{\pi}^{\left(-1 \cdot 0.5\right)}}\right)} - 1\right| \]

      7. metadata-eval 0.0%

         \[\leadsto \left|e^{\mathsf{log1p}\left(\left(0.047619047619047616 \cdot {x}^{7}\right) \cdot {\pi}^{\color{blue}{-0.5}}\right)} - 1\right| \]

   8. Applied egg-rr 0.0%

      \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(\left(0.047619047619047616 \cdot {x}^{7}\right) \cdot {\pi}^{-0.5}\right)} - 1}\right| \]
   9. Step-by-step derivation

      1. expm1-def 0.0%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(0.047619047619047616 \cdot {x}^{7}\right) \cdot {\pi}^{-0.5}\right)\right)}\right| \]

      2. expm1-log1p 99.5%

         \[\leadsto \left|\color{blue}{\left(0.047619047619047616 \cdot {x}^{7}\right) \cdot {\pi}^{-0.5}}\right| \]

   10. Simplified 99.5%

       \[\leadsto \left|\color{blue}{\left(0.047619047619047616 \cdot {x}^{7}\right) \cdot {\pi}^{-0.5}}\right| \]

   if -2.2000000000000002 < x

   1. Initial program 99.8%

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

   3. Simplified 99.2%

      \[\leadsto \color{blue}{\left|\frac{\mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(0.047619047619047616, {x}^{7}, 0.2 \cdot {x}^{5}\right)\right)\right)}{\sqrt{\pi}}\right|} \]

   4. Taylor expanded in x around 0 99.4%

      \[\leadsto \left|\color{blue}{0.6666666666666666 \cdot \left({x}^{3} \cdot \sqrt{\frac{1}{\pi}}\right) + 2 \cdot \left(x \cdot \sqrt{\frac{1}{\pi}}\right)}\right| \]
   5. Step-by-step derivation

      1. associate-*r* 99.4%

         \[\leadsto \left|\color{blue}{\left(0.6666666666666666 \cdot {x}^{3}\right) \cdot \sqrt{\frac{1}{\pi}}} + 2 \cdot \left(x \cdot \sqrt{\frac{1}{\pi}}\right)\right| \]

      2. associate-*r* 99.4%

         \[\leadsto \left|\left(0.6666666666666666 \cdot {x}^{3}\right) \cdot \sqrt{\frac{1}{\pi}} + \color{blue}{\left(2 \cdot x\right) \cdot \sqrt{\frac{1}{\pi}}}\right| \]

      3. distribute-rgt-out 99.4%

         \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(0.6666666666666666 \cdot {x}^{3} + 2 \cdot x\right)}\right| \]

      4. *-commutative 99.4%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(\color{blue}{{x}^{3} \cdot 0.6666666666666666} + 2 \cdot x\right)\right| \]

      5. cube-mult 99.4%

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

      6. unpow2 99.4%

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

      7. associate-*l* 99.4%

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

      8. *-commutative 99.4%

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

      9. unpow2 99.4%

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

      10. unpow1 99.4%

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

      11. sqr-pow 43.0%

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

      12. fabs-sqr 43.0%

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

      13. sqr-pow 98.9%

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

      14. unpow1 98.9%

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

      15. *-commutative 98.9%

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

      16. unpow1 98.9%

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

      17. sqr-pow 42.8%

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

      18. fabs-sqr 42.8%

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

      19. sqr-pow 99.4%

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

      20. unpow1 99.4%

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

   6. Simplified 99.4%

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

      1. metadata-eval 99.4%

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

      2. fma-udef 99.4%

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

      3. metadata-eval 99.4%

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

   8. Applied egg-rr 99.4%

      \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot \color{blue}{\left(0.6666666666666666 \cdot \left(x \cdot x\right) + 2\right)}\right)\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.2:\\ \;\;\;\;\left|\left(0.047619047619047616 \cdot {x}^{7}\right) \cdot {\pi}^{-0.5}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot \left(2 + 0.6666666666666666 \cdot \left(x \cdot x\right)\right)\right)\right|\\ \end{array} \]

Alternative 8: 88.8% accurate, 6.2× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double x) {
	return Math.abs((Math.sqrt((1.0 / Math.PI)) * (x * (2.0 + (0.6666666666666666 * (x * x))))));
}

Python

def code(x):
	return math.fabs((math.sqrt((1.0 / math.pi)) * (x * (2.0 + (0.6666666666666666 * (x * x))))))

Julia

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

MATLAB

function tmp = code(x)
	tmp = abs((sqrt((1.0 / pi)) * (x * (2.0 + (0.6666666666666666 * (x * x))))));
end

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Simplified 99.4%

   \[\leadsto \color{blue}{\left|\frac{\mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(0.047619047619047616, {x}^{7}, 0.2 \cdot {x}^{5}\right)\right)\right)}{\sqrt{\pi}}\right|} \]

4. Taylor expanded in x around 0 90.4%

   \[\leadsto \left|\color{blue}{0.6666666666666666 \cdot \left({x}^{3} \cdot \sqrt{\frac{1}{\pi}}\right) + 2 \cdot \left(x \cdot \sqrt{\frac{1}{\pi}}\right)}\right| \]
5. Step-by-step derivation

   1. associate-*r* 90.4%

      \[\leadsto \left|\color{blue}{\left(0.6666666666666666 \cdot {x}^{3}\right) \cdot \sqrt{\frac{1}{\pi}}} + 2 \cdot \left(x \cdot \sqrt{\frac{1}{\pi}}\right)\right| \]

   2. associate-*r* 90.4%

      \[\leadsto \left|\left(0.6666666666666666 \cdot {x}^{3}\right) \cdot \sqrt{\frac{1}{\pi}} + \color{blue}{\left(2 \cdot x\right) \cdot \sqrt{\frac{1}{\pi}}}\right| \]

   3. distribute-rgt-out 90.4%

      \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(0.6666666666666666 \cdot {x}^{3} + 2 \cdot x\right)}\right| \]

   4. *-commutative 90.4%

      \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(\color{blue}{{x}^{3} \cdot 0.6666666666666666} + 2 \cdot x\right)\right| \]

   5. cube-mult 90.4%

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

   6. unpow2 90.4%

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

   7. associate-*l* 90.4%

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

   8. *-commutative 90.4%

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

   9. unpow2 90.4%

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

   10. unpow1 90.4%

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

   11. sqr-pow 29.2%

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

   12. fabs-sqr 29.2%

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

   13. sqr-pow 90.1%

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

   14. unpow1 90.1%

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

   15. *-commutative 90.1%

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

   16. unpow1 90.1%

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

   17. sqr-pow 29.1%

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

   18. fabs-sqr 29.1%

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

   19. sqr-pow 90.4%

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

   20. unpow1 90.4%

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

6. Simplified 90.4%

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

   1. metadata-eval 90.4%

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

   2. fma-udef 90.4%

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

   3. metadata-eval 90.4%

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

8. Applied egg-rr 90.4%

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

9. Final simplification 90.4%

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

Alternative 9: 88.4% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.75:\\ \;\;\;\;\left|{\pi}^{-0.5} \cdot \left(x \cdot \left(0.6666666666666666 \cdot \left(x \cdot x\right)\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|{\pi}^{-0.5} \cdot \left(2 \cdot x\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.75)
   (fabs (* (pow PI -0.5) (* x (* 0.6666666666666666 (* x x)))))
   (fabs (* (pow PI -0.5) (* 2.0 x)))))

C

double code(double x) {
	double tmp;
	if (x <= -1.75) {
		tmp = fabs((pow(((double) M_PI), -0.5) * (x * (0.6666666666666666 * (x * x)))));
	} else {
		tmp = fabs((pow(((double) M_PI), -0.5) * (2.0 * x)));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= -1.75) {
		tmp = Math.abs((Math.pow(Math.PI, -0.5) * (x * (0.6666666666666666 * (x * x)))));
	} else {
		tmp = Math.abs((Math.pow(Math.PI, -0.5) * (2.0 * x)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= -1.75:
		tmp = math.fabs((math.pow(math.pi, -0.5) * (x * (0.6666666666666666 * (x * x)))))
	else:
		tmp = math.fabs((math.pow(math.pi, -0.5) * (2.0 * x)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.75)
		tmp = abs(Float64((pi ^ -0.5) * Float64(x * Float64(0.6666666666666666 * Float64(x * x)))));
	else
		tmp = abs(Float64((pi ^ -0.5) * Float64(2.0 * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= -1.75)
		tmp = abs(((pi ^ -0.5) * (x * (0.6666666666666666 * (x * x)))));
	else
		tmp = abs(((pi ^ -0.5) * (2.0 * x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, -1.75], N[Abs[N[(N[Power[Pi, -0.5], $MachinePrecision] * N[(x * N[(0.6666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[Power[Pi, -0.5], $MachinePrecision] * N[(2.0 * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.75

   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. Simplified 99.9%

      \[\leadsto \color{blue}{\left|\frac{\mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(0.047619047619047616, {x}^{7}, 0.2 \cdot {x}^{5}\right)\right)\right)}{\sqrt{\pi}}\right|} \]

   4. Taylor expanded in x around 0 71.4%

      \[\leadsto \left|\color{blue}{0.6666666666666666 \cdot \left({x}^{3} \cdot \sqrt{\frac{1}{\pi}}\right) + 2 \cdot \left(x \cdot \sqrt{\frac{1}{\pi}}\right)}\right| \]
   5. Step-by-step derivation

      1. associate-*r* 71.4%

         \[\leadsto \left|\color{blue}{\left(0.6666666666666666 \cdot {x}^{3}\right) \cdot \sqrt{\frac{1}{\pi}}} + 2 \cdot \left(x \cdot \sqrt{\frac{1}{\pi}}\right)\right| \]

      2. associate-*r* 71.4%

         \[\leadsto \left|\left(0.6666666666666666 \cdot {x}^{3}\right) \cdot \sqrt{\frac{1}{\pi}} + \color{blue}{\left(2 \cdot x\right) \cdot \sqrt{\frac{1}{\pi}}}\right| \]

      3. distribute-rgt-out 71.4%

         \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(0.6666666666666666 \cdot {x}^{3} + 2 \cdot x\right)}\right| \]

      4. *-commutative 71.4%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(\color{blue}{{x}^{3} \cdot 0.6666666666666666} + 2 \cdot x\right)\right| \]

      5. cube-mult 71.4%

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

      6. unpow2 71.4%

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

      7. associate-*l* 71.4%

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

      8. *-commutative 71.4%

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

      9. unpow2 71.4%

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

      10. unpow1 71.4%

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

      11. sqr-pow 0.0%

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

      12. fabs-sqr 0.0%

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

      13. sqr-pow 71.4%

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

      14. unpow1 71.4%

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

      15. *-commutative 71.4%

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

      16. unpow1 71.4%

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

      17. sqr-pow 0.0%

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

      18. fabs-sqr 0.0%

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

      19. sqr-pow 71.4%

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

      20. unpow1 71.4%

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

   6. Simplified 71.4%

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

   7. Taylor expanded in x around inf 71.4%

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

      1. unpow2 71.4%

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

   9. Simplified 71.4%

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

       1. expm1-log1p-u 0.0%

          \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot \left(0.6666666666666666 \cdot \left(x \cdot x\right)\right)\right)\right)\right)}\right| \]

       2. expm1-udef 0.0%

          \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot \left(0.6666666666666666 \cdot \left(x \cdot x\right)\right)\right)\right)} - 1}\right| \]

       3. pow1/2 0.0%

          \[\leadsto \left|e^{\mathsf{log1p}\left(\color{blue}{{\left(\frac{1}{\pi}\right)}^{0.5}} \cdot \left(x \cdot \left(0.6666666666666666 \cdot \left(x \cdot x\right)\right)\right)\right)} - 1\right| \]

       4. inv-pow 0.0%

          \[\leadsto \left|e^{\mathsf{log1p}\left({\color{blue}{\left({\pi}^{-1}\right)}}^{0.5} \cdot \left(x \cdot \left(0.6666666666666666 \cdot \left(x \cdot x\right)\right)\right)\right)} - 1\right| \]

       5. pow-pow 0.0%

          \[\leadsto \left|e^{\mathsf{log1p}\left(\color{blue}{{\pi}^{\left(-1 \cdot 0.5\right)}} \cdot \left(x \cdot \left(0.6666666666666666 \cdot \left(x \cdot x\right)\right)\right)\right)} - 1\right| \]

       6. metadata-eval 0.0%

          \[\leadsto \left|e^{\mathsf{log1p}\left({\pi}^{\color{blue}{-0.5}} \cdot \left(x \cdot \left(0.6666666666666666 \cdot \left(x \cdot x\right)\right)\right)\right)} - 1\right| \]

   11. Applied egg-rr 0.0%

       \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left({\pi}^{-0.5} \cdot \left(x \cdot \left(0.6666666666666666 \cdot \left(x \cdot x\right)\right)\right)\right)} - 1}\right| \]
   12. Step-by-step derivation

       1. expm1-def 0.0%

          \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left({\pi}^{-0.5} \cdot \left(x \cdot \left(0.6666666666666666 \cdot \left(x \cdot x\right)\right)\right)\right)\right)}\right| \]

       2. expm1-log1p 71.4%

          \[\leadsto \left|\color{blue}{{\pi}^{-0.5} \cdot \left(x \cdot \left(0.6666666666666666 \cdot \left(x \cdot x\right)\right)\right)}\right| \]

   13. Simplified 71.4%

       \[\leadsto \left|\color{blue}{{\pi}^{-0.5} \cdot \left(x \cdot \left(0.6666666666666666 \cdot \left(x \cdot x\right)\right)\right)}\right| \]

   if -1.75 < 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. Simplified 99.2%

      \[\leadsto \color{blue}{\left|\frac{\mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(0.047619047619047616, {x}^{7}, 0.2 \cdot {x}^{5}\right)\right)\right)}{\sqrt{\pi}}\right|} \]

   4. Taylor expanded in x around 0 98.6%

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

      1. associate-*r* 98.6%

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

   6. Simplified 98.6%

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

      1. expm1-log1p-u 98.6%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(2 \cdot x\right) \cdot \sqrt{\frac{1}{\pi}}\right)\right)}\right| \]

      2. expm1-udef 7.3%

         \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(\left(2 \cdot x\right) \cdot \sqrt{\frac{1}{\pi}}\right)} - 1}\right| \]

      3. associate-*l* 7.3%

         \[\leadsto \left|e^{\mathsf{log1p}\left(\color{blue}{2 \cdot \left(x \cdot \sqrt{\frac{1}{\pi}}\right)}\right)} - 1\right| \]

      4. pow1/2 7.3%

         \[\leadsto \left|e^{\mathsf{log1p}\left(2 \cdot \left(x \cdot \color{blue}{{\left(\frac{1}{\pi}\right)}^{0.5}}\right)\right)} - 1\right| \]

      5. inv-pow 7.3%

         \[\leadsto \left|e^{\mathsf{log1p}\left(2 \cdot \left(x \cdot {\color{blue}{\left({\pi}^{-1}\right)}}^{0.5}\right)\right)} - 1\right| \]

      6. pow-pow 7.3%

         \[\leadsto \left|e^{\mathsf{log1p}\left(2 \cdot \left(x \cdot \color{blue}{{\pi}^{\left(-1 \cdot 0.5\right)}}\right)\right)} - 1\right| \]

      7. metadata-eval 7.3%

         \[\leadsto \left|e^{\mathsf{log1p}\left(2 \cdot \left(x \cdot {\pi}^{\color{blue}{-0.5}}\right)\right)} - 1\right| \]

   8. Applied egg-rr 7.3%

      \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(2 \cdot \left(x \cdot {\pi}^{-0.5}\right)\right)} - 1}\right| \]
   9. Step-by-step derivation

      1. expm1-def 98.6%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(2 \cdot \left(x \cdot {\pi}^{-0.5}\right)\right)\right)}\right| \]

      2. expm1-log1p 98.6%

         \[\leadsto \left|\color{blue}{2 \cdot \left(x \cdot {\pi}^{-0.5}\right)}\right| \]

      3. associate-*r* 98.6%

         \[\leadsto \left|\color{blue}{\left(2 \cdot x\right) \cdot {\pi}^{-0.5}}\right| \]

   10. Simplified 98.6%

       \[\leadsto \left|\color{blue}{\left(2 \cdot x\right) \cdot {\pi}^{-0.5}}\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.75:\\ \;\;\;\;\left|{\pi}^{-0.5} \cdot \left(x \cdot \left(0.6666666666666666 \cdot \left(x \cdot x\right)\right)\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|{\pi}^{-0.5} \cdot \left(2 \cdot x\right)\right|\\ \end{array} \]

Alternative 10: 67.1% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \left|{\pi}^{-0.5} \cdot \left(2 \cdot x\right)\right| \end{array} \]

FPCore

(FPCore (x) :precision binary64 (fabs (* (pow PI -0.5) (* 2.0 x))))

C

double code(double x) {
	return fabs((pow(((double) M_PI), -0.5) * (2.0 * x)));
}

Java

public static double code(double x) {
	return Math.abs((Math.pow(Math.PI, -0.5) * (2.0 * x)));
}

Python

def code(x):
	return math.fabs((math.pow(math.pi, -0.5) * (2.0 * x)))

Julia

function code(x)
	return abs(Float64((pi ^ -0.5) * Float64(2.0 * x)))
end

MATLAB

function tmp = code(x)
	tmp = abs(((pi ^ -0.5) * (2.0 * x)));
end

Wolfram

code[x_] := N[Abs[N[(N[Power[Pi, -0.5], $MachinePrecision] * N[(2.0 * 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. Simplified 99.4%

   \[\leadsto \color{blue}{\left|\frac{\mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, \mathsf{fma}\left(0.047619047619047616, {x}^{7}, 0.2 \cdot {x}^{5}\right)\right)\right)}{\sqrt{\pi}}\right|} \]

4. Taylor expanded in x around 0 68.8%

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

   1. associate-*r* 68.8%

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

6. Simplified 68.8%

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

   1. expm1-log1p-u 67.0%

      \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(2 \cdot x\right) \cdot \sqrt{\frac{1}{\pi}}\right)\right)}\right| \]

   2. expm1-udef 5.0%

      \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(\left(2 \cdot x\right) \cdot \sqrt{\frac{1}{\pi}}\right)} - 1}\right| \]

   3. associate-*l* 5.0%

      \[\leadsto \left|e^{\mathsf{log1p}\left(\color{blue}{2 \cdot \left(x \cdot \sqrt{\frac{1}{\pi}}\right)}\right)} - 1\right| \]

   4. pow1/2 5.0%

      \[\leadsto \left|e^{\mathsf{log1p}\left(2 \cdot \left(x \cdot \color{blue}{{\left(\frac{1}{\pi}\right)}^{0.5}}\right)\right)} - 1\right| \]

   5. inv-pow 5.0%

      \[\leadsto \left|e^{\mathsf{log1p}\left(2 \cdot \left(x \cdot {\color{blue}{\left({\pi}^{-1}\right)}}^{0.5}\right)\right)} - 1\right| \]

   6. pow-pow 5.0%

      \[\leadsto \left|e^{\mathsf{log1p}\left(2 \cdot \left(x \cdot \color{blue}{{\pi}^{\left(-1 \cdot 0.5\right)}}\right)\right)} - 1\right| \]

   7. metadata-eval 5.0%

      \[\leadsto \left|e^{\mathsf{log1p}\left(2 \cdot \left(x \cdot {\pi}^{\color{blue}{-0.5}}\right)\right)} - 1\right| \]

8. Applied egg-rr 5.0%

   \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(2 \cdot \left(x \cdot {\pi}^{-0.5}\right)\right)} - 1}\right| \]
9. Step-by-step derivation

   1. expm1-def 67.0%

      \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(2 \cdot \left(x \cdot {\pi}^{-0.5}\right)\right)\right)}\right| \]

   2. expm1-log1p 68.8%

      \[\leadsto \left|\color{blue}{2 \cdot \left(x \cdot {\pi}^{-0.5}\right)}\right| \]

   3. associate-*r* 68.8%

      \[\leadsto \left|\color{blue}{\left(2 \cdot x\right) \cdot {\pi}^{-0.5}}\right| \]

10. Simplified 68.8%

    \[\leadsto \left|\color{blue}{\left(2 \cdot x\right) \cdot {\pi}^{-0.5}}\right| \]

11. Final simplification 68.8%

    \[\leadsto \left|{\pi}^{-0.5} \cdot \left(2 \cdot x\right)\right| \]

Reproduce

herbie shell --seed 2023182 
(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)))))))