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

Percentage Accurate: 99.8% → 99.8%

Time: 11.3s

Alternatives: 13

Speedup: N/A×

• 27.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[x \leq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

4. Applied rewrites 99.6%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 99.9%

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

9. Applied rewrites 99.9%

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

10. Final simplification 99.9%

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

Alternative 2: 98.2% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 10^{-6}:\\ \;\;\;\;\left|\left(\mathsf{fma}\left(0.6666666666666666 \cdot x, x, 2\right) \cdot \left|x\right|\right) \cdot \frac{1}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(\left(\left(\left(\left(x \cdot x\right) \cdot 0.047619047619047616\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right| \cdot \frac{\left|x\right|}{\sqrt{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 1e-6)
   (fabs
    (* (* (fma (* 0.6666666666666666 x) x 2.0) (fabs x)) (/ 1.0 (sqrt PI))))
   (*
    (fabs (* (* (* (* (* (* x x) 0.047619047619047616) x) x) x) x))
    (/ (fabs x) (sqrt PI)))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 1e-6) {
		tmp = fabs(((fma((0.6666666666666666 * x), x, 2.0) * fabs(x)) * (1.0 / sqrt(((double) M_PI)))));
	} else {
		tmp = fabs(((((((x * x) * 0.047619047619047616) * x) * x) * x) * x)) * (fabs(x) / sqrt(((double) M_PI)));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 1e-6)
		tmp = abs(Float64(Float64(fma(Float64(0.6666666666666666 * x), x, 2.0) * abs(x)) * Float64(1.0 / sqrt(pi))));
	else
		tmp = Float64(abs(Float64(Float64(Float64(Float64(Float64(Float64(x * x) * 0.047619047619047616) * x) * x) * x) * x)) * Float64(abs(x) / sqrt(pi)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 1e-6], N[Abs[N[(N[(N[(N[(0.6666666666666666 * x), $MachinePrecision] * x + 2.0), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Abs[N[(N[(N[(N[(N[(N[(x * x), $MachinePrecision] * 0.047619047619047616), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision] * N[(N[Abs[x], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 9.99999999999999955e-7

   1. Initial program 99.9%

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. unpow2 N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-*.f64 99.9

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

   7. Applied rewrites 99.9%

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

   if 9.99999999999999955e-7 < (fabs.f64 x)

   1. Initial program 99.8%

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in x around inf

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

      1. metadata-eval N/A

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

      2. pow-plus N/A

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

      3. metadata-eval N/A

         \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left|x\right| \cdot \left(\frac{1}{21} \cdot \left({x}^{\color{blue}{\left(4 + 1\right)}} \cdot x\right)\right)\right)\right| \]

      4. pow-plus N/A

         \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left|x\right| \cdot \left(\frac{1}{21} \cdot \left(\color{blue}{\left({x}^{4} \cdot x\right)} \cdot x\right)\right)\right)\right| \]

      5. associate-*r* N/A

         \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left|x\right| \cdot \left(\frac{1}{21} \cdot \color{blue}{\left({x}^{4} \cdot \left(x \cdot x\right)\right)}\right)\right)\right| \]

      6. unpow2 N/A

         \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left|x\right| \cdot \left(\frac{1}{21} \cdot \left({x}^{4} \cdot \color{blue}{{x}^{2}}\right)\right)\right)\right| \]

      7. *-commutative N/A

         \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left|x\right| \cdot \color{blue}{\left(\left({x}^{4} \cdot {x}^{2}\right) \cdot \frac{1}{21}\right)}\right)\right| \]

      8. associate-*r* N/A

         \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left|x\right| \cdot \color{blue}{\left({x}^{4} \cdot \left({x}^{2} \cdot \frac{1}{21}\right)\right)}\right)\right| \]

      9. unpow2 N/A

         \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left|x\right| \cdot \left({x}^{4} \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{21}\right)\right)\right)\right| \]

      10. associate-*l* N/A

          \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left|x\right| \cdot \left({x}^{4} \cdot \color{blue}{\left(x \cdot \left(x \cdot \frac{1}{21}\right)\right)}\right)\right)\right| \]

      11. associate-*r* N/A

          \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left|x\right| \cdot \color{blue}{\left(\left({x}^{4} \cdot x\right) \cdot \left(x \cdot \frac{1}{21}\right)\right)}\right)\right| \]

      12. *-commutative N/A

          \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left|x\right| \cdot \left(\left({x}^{4} \cdot x\right) \cdot \color{blue}{\left(\frac{1}{21} \cdot x\right)}\right)\right)\right| \]

      13. lower-*.f64 N/A

          \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left|x\right| \cdot \color{blue}{\left(\left({x}^{4} \cdot x\right) \cdot \left(\frac{1}{21} \cdot x\right)\right)}\right)\right| \]

      14. lower-*.f64 N/A

          \[\leadsto \left|\frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \left(\left|x\right| \cdot \left(\color{blue}{\left({x}^{4} \cdot x\right)} \cdot \left(\frac{1}{21} \cdot x\right)\right)\right)\right| \]

      15. metadata-eval N/A

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

      16. pow-plus N/A

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

      17. lower-*.f64 N/A

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

      18. unpow3 N/A

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

      19. unpow2 N/A

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

      20. lower-*.f64 N/A

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

      21. unpow2 N/A

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

      22. lower-*.f64 N/A

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

      23. lower-*.f64 95.0

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

   7. Applied rewrites 95.0%

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

   9. Applied rewrites 95.0%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 10^{-6}:\\ \;\;\;\;\left|\left(\mathsf{fma}\left(0.6666666666666666 \cdot x, x, 2\right) \cdot \left|x\right|\right) \cdot \frac{1}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(\left(\left(\left(\left(x \cdot x\right) \cdot 0.047619047619047616\right) \cdot x\right) \cdot x\right) \cdot x\right) \cdot x\right| \cdot \frac{\left|x\right|}{\sqrt{\pi}}\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024234 
(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)))))))