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

Percentage Accurate: 99.8% → 99.8%

Time: 2.8s

Alternatives: 10

Speedup: 2.0×

• 28.2% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[x \leq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left({\left(\left|x\right|\right)}^{7}, 0.047619047619047616, \mathsf{fma}\left({\left(\left|x\right|\right)}^{5}, 0.2, \mathsf{fma}\left(0.6666666666666666 \cdot \left(x \cdot x\right), \left|x\right|, \left|x\right| \cdot 2\right)\right)\right)\right| \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fabs
  (*
   (/ 1.0 (sqrt PI))
   (fma
    (pow (fabs x) 7.0)
    0.047619047619047616
    (fma
     (pow (fabs x) 5.0)
     0.2
     (fma (* 0.6666666666666666 (* x x)) (fabs x) (* (fabs x) 2.0)))))))

C

double code(double x) {
	return fabs(((1.0 / sqrt(((double) M_PI))) * fma(pow(fabs(x), 7.0), 0.047619047619047616, fma(pow(fabs(x), 5.0), 0.2, fma((0.6666666666666666 * (x * x)), fabs(x), (fabs(x) * 2.0))))));
}

Julia

function code(x)
	return abs(Float64(Float64(1.0 / sqrt(pi)) * fma((abs(x) ^ 7.0), 0.047619047619047616, fma((abs(x) ^ 5.0), 0.2, fma(Float64(0.6666666666666666 * Float64(x * x)), abs(x), Float64(abs(x) * 2.0))))))
end

Wolfram

code[x_] := N[Abs[N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Abs[x], $MachinePrecision], 7.0], $MachinePrecision] * 0.047619047619047616 + N[(N[Power[N[Abs[x], $MachinePrecision], 5.0], $MachinePrecision] * 0.2 + N[(N[(0.6666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision] + N[(N[Abs[x], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 99.8%

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

Alternative 2: 99.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.8%

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

4. Taylor expanded in x around 0

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

6. Applied rewrites 99.8%

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

   1. metadata-eval N/A

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

   2. lift-fabs.f64 N/A

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

   3. metadata-eval N/A

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

   4. lift-fma.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-fabs.f64 N/A

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

   7. distribute-rgt-out N/A

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

   8. lower-*.f64 N/A

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

   9. lift-fabs.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f64 N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval 99.8

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

8. Applied rewrites 99.8%

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

Alternative 3: 99.4% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.85:\\ \;\;\;\;\left|\mathsf{fma}\left(0.6666666666666666 \cdot \left(x \cdot x\right), \left|x\right|, \left|x\right| \cdot 2\right) \cdot \frac{1}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\mathsf{fma}\left(\frac{\left|x\right|}{x \cdot x}, 0.2, 0.047619047619047616 \cdot \left|x\right|\right) \cdot {x}^{6}}{\sqrt{\pi}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.85)
   (fabs
    (*
     (fma (* 0.6666666666666666 (* x x)) (fabs x) (* (fabs x) 2.0))
     (/ 1.0 (sqrt PI))))
   (fabs
    (/
     (*
      (fma (/ (fabs x) (* x x)) 0.2 (* 0.047619047619047616 (fabs x)))
      (pow x 6.0))
     (sqrt PI)))))

C

double code(double x) {
	double tmp;
	if (x <= 1.85) {
		tmp = fabs((fma((0.6666666666666666 * (x * x)), fabs(x), (fabs(x) * 2.0)) * (1.0 / sqrt(((double) M_PI)))));
	} else {
		tmp = fabs(((fma((fabs(x) / (x * x)), 0.2, (0.047619047619047616 * fabs(x))) * pow(x, 6.0)) / sqrt(((double) M_PI))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.85)
		tmp = abs(Float64(fma(Float64(0.6666666666666666 * Float64(x * x)), abs(x), Float64(abs(x) * 2.0)) * Float64(1.0 / sqrt(pi))));
	else
		tmp = abs(Float64(Float64(fma(Float64(abs(x) / Float64(x * x)), 0.2, Float64(0.047619047619047616 * abs(x))) * (x ^ 6.0)) / sqrt(pi)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.85], N[Abs[N[(N[(N[(0.6666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision] + N[(N[Abs[x], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(N[(N[Abs[x], $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision] * 0.2 + N[(0.047619047619047616 * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.8500000000000001

   1. Initial program 99.8%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in x around 0

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

      1. metadata-eval N/A

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

      2. pow2-fabs-rev N/A

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

      3. pow2 N/A

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

      4. pow3 N/A

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

      5. pow3 N/A

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

      6. sqr-abs-rev N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval N/A

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

      12. lift-fabs.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-fabs.f64 N/A

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

      15. lift-*.f64 89.3

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

   9. Applied rewrites 89.3%

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

   if 1.8500000000000001 < x

   1. Initial program 99.8%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around inf

      \[\leadsto \left|\color{blue}{\left({x}^{6} \cdot \left(\frac{1}{21} \cdot \left|x\right| + \frac{1}{5} \cdot \frac{\left|x\right|}{{x}^{2}}\right)\right)} \cdot \frac{1}{\sqrt{\pi}}\right| \]

   6. Applied rewrites 34.3%

      \[\leadsto \left|\color{blue}{\left(\mathsf{fma}\left(\frac{\left|x\right|}{x \cdot x}, 0.2, 0.047619047619047616 \cdot \left|x\right|\right) \cdot {x}^{6}\right)} \cdot \frac{1}{\sqrt{\pi}}\right| \]
   7. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \left|\left(\mathsf{fma}\left(\frac{\left|x\right|}{x \cdot x}, \frac{1}{5}, \frac{1}{21} \cdot \left|x\right|\right) \cdot {x}^{6}\right) \cdot \frac{1}{\sqrt{\pi}}\right| \]

      2. metadata-eval N/A

         \[\leadsto \left|\left(\mathsf{fma}\left(\frac{\left|x\right|}{x \cdot x}, \frac{1}{5}, \frac{1}{21} \cdot \left|x\right|\right) \cdot {x}^{6}\right) \cdot \frac{1}{\sqrt{\pi}}\right| \]

   8. Applied rewrites 34.4%

      \[\leadsto \left|\color{blue}{\frac{\mathsf{fma}\left(\frac{\left|x\right|}{x \cdot x}, 0.2, 0.047619047619047616 \cdot \left|x\right|\right) \cdot {x}^{6}}{\sqrt{\pi}}}\right| \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 89.3% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\sqrt{\pi}}\\ \mathbf{if}\;x \leq 1.85:\\ \;\;\;\;\left|\mathsf{fma}\left(0.6666666666666666 \cdot \left(x \cdot x\right), \left|x\right|, \left|x\right| \cdot 2\right) \cdot t\_0\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left({x}^{4} \cdot \left(\left|x\right| \cdot \mathsf{fma}\left(x \cdot x, 0.047619047619047616, 0.2\right)\right)\right) \cdot t\_0\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (sqrt PI))))
   (if (<= x 1.85)
     (fabs
      (* (fma (* 0.6666666666666666 (* x x)) (fabs x) (* (fabs x) 2.0)) t_0))
     (fabs
      (*
       (* (pow x 4.0) (* (fabs x) (fma (* x x) 0.047619047619047616 0.2)))
       t_0)))))

C

double code(double x) {
	double t_0 = 1.0 / sqrt(((double) M_PI));
	double tmp;
	if (x <= 1.85) {
		tmp = fabs((fma((0.6666666666666666 * (x * x)), fabs(x), (fabs(x) * 2.0)) * t_0));
	} else {
		tmp = fabs(((pow(x, 4.0) * (fabs(x) * fma((x * x), 0.047619047619047616, 0.2))) * t_0));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(1.0 / sqrt(pi))
	tmp = 0.0
	if (x <= 1.85)
		tmp = abs(Float64(fma(Float64(0.6666666666666666 * Float64(x * x)), abs(x), Float64(abs(x) * 2.0)) * t_0));
	else
		tmp = abs(Float64(Float64((x ^ 4.0) * Float64(abs(x) * fma(Float64(x * x), 0.047619047619047616, 0.2))) * t_0));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 1.85], N[Abs[N[(N[(N[(0.6666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision] + N[(N[Abs[x], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[Power[x, 4.0], $MachinePrecision] * N[(N[Abs[x], $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.047619047619047616 + 0.2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1.8500000000000001

   1. Initial program 99.8%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in x around 0

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

      1. metadata-eval N/A

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

      2. pow2-fabs-rev N/A

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

      3. pow2 N/A

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

      4. pow3 N/A

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

      5. pow3 N/A

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

      6. sqr-abs-rev N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval N/A

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

      12. lift-fabs.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-fabs.f64 N/A

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

      15. lift-*.f64 89.3

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

   9. Applied rewrites 89.3%

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

   if 1.8500000000000001 < x

   1. Initial program 99.8%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around inf

      \[\leadsto \left|\color{blue}{\left({x}^{6} \cdot \left(\frac{1}{21} \cdot \left|x\right| + \frac{1}{5} \cdot \frac{\left|x\right|}{{x}^{2}}\right)\right)} \cdot \frac{1}{\sqrt{\pi}}\right| \]

   6. Applied rewrites 34.3%

      \[\leadsto \left|\color{blue}{\left(\mathsf{fma}\left(\frac{\left|x\right|}{x \cdot x}, 0.2, 0.047619047619047616 \cdot \left|x\right|\right) \cdot {x}^{6}\right)} \cdot \frac{1}{\sqrt{\pi}}\right| \]

   7. Taylor expanded in x around 0

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

      1. metadata-eval N/A

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

      2. metadata-eval N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. pow2-fabs-rev N/A

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

      6. pow2 N/A

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

      7. pow2 N/A

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

      8. pow2-fabs-rev N/A

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

      9. pow2 N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. lift-*.f64 N/A

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

   9. Applied rewrites 36.6%

      \[\leadsto \left|\left({x}^{4} \cdot \color{blue}{\left(\left|x\right| \cdot \mathsf{fma}\left(x \cdot x, 0.047619047619047616, 0.2\right)\right)}\right) \cdot \frac{1}{\sqrt{\pi}}\right| \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 89.3% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.2:\\ \;\;\;\;\left|\mathsf{fma}\left(0.6666666666666666 \cdot \left(x \cdot x\right), \left|x\right|, \left|x\right| \cdot 2\right) \cdot \frac{1}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left({x}^{6} \cdot 0.047619047619047616\right) \cdot \left|x\right|}{\sqrt{\pi}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.2)
   (fabs
    (*
     (fma (* 0.6666666666666666 (* x x)) (fabs x) (* (fabs x) 2.0))
     (/ 1.0 (sqrt PI))))
   (fabs (/ (* (* (pow x 6.0) 0.047619047619047616) (fabs x)) (sqrt PI)))))

C

double code(double x) {
	double tmp;
	if (x <= 2.2) {
		tmp = fabs((fma((0.6666666666666666 * (x * x)), fabs(x), (fabs(x) * 2.0)) * (1.0 / sqrt(((double) M_PI)))));
	} else {
		tmp = fabs((((pow(x, 6.0) * 0.047619047619047616) * fabs(x)) / sqrt(((double) M_PI))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.2)
		tmp = abs(Float64(fma(Float64(0.6666666666666666 * Float64(x * x)), abs(x), Float64(abs(x) * 2.0)) * Float64(1.0 / sqrt(pi))));
	else
		tmp = abs(Float64(Float64(Float64((x ^ 6.0) * 0.047619047619047616) * abs(x)) / sqrt(pi)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 2.2], N[Abs[N[(N[(N[(0.6666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision] + N[(N[Abs[x], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[(N[Power[x, 6.0], $MachinePrecision] * 0.047619047619047616), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.2000000000000002

   1. Initial program 99.8%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 99.8%

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

   7. Taylor expanded in x around 0

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

      1. metadata-eval N/A

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

      2. pow2-fabs-rev N/A

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

      3. pow2 N/A

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

      4. pow3 N/A

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

      5. pow3 N/A

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

      6. sqr-abs-rev N/A

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

      7. lift-*.f64 N/A

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

      8. associate-*r* N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. metadata-eval N/A

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

      12. lift-fabs.f64 N/A

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

      13. *-commutative N/A

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

      14. lift-fabs.f64 N/A

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

      15. lift-*.f64 89.3

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

   9. Applied rewrites 89.3%

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

   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. Applied rewrites 99.8%

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

   4. Taylor expanded in x around inf

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

   6. Applied rewrites 36.3%

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

Alternative 6: 89.3% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.8%

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

4. Taylor expanded in x around 0

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

6. Applied rewrites 99.8%

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

8. Applied rewrites 99.4%

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

Alternative 7: 89.3% accurate, 3.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.8%

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

4. Taylor expanded in x around 0

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

6. Applied rewrites 99.8%

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

7. Taylor expanded in x around 0

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

   1. metadata-eval N/A

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

   2. pow2-fabs-rev N/A

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

   3. pow2 N/A

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

   4. pow3 N/A

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

   5. pow3 N/A

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

   6. sqr-abs-rev N/A

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

   7. lift-*.f64 N/A

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

   8. associate-*r* N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. metadata-eval N/A

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

   12. lift-fabs.f64 N/A

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

   13. *-commutative N/A

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

   14. lift-fabs.f64 N/A

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

   15. lift-*.f64 89.3

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

9. Applied rewrites 89.3%

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

Alternative 8: 68.2% accurate, 4.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.7:\\ \;\;\;\;\left|\left|x\right| \cdot \frac{2}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\left(\left(x \cdot x\right) \cdot \left|x\right|\right) \cdot 0.6666666666666666\right|}{\sqrt{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.7)
   (fabs (* (fabs x) (/ 2.0 (sqrt PI))))
   (/ (fabs (* (* (* x x) (fabs x)) 0.6666666666666666)) (sqrt PI))))

C

double code(double x) {
	double tmp;
	if (x <= 1.7) {
		tmp = fabs((fabs(x) * (2.0 / sqrt(((double) M_PI)))));
	} else {
		tmp = fabs((((x * x) * fabs(x)) * 0.6666666666666666)) / sqrt(((double) M_PI));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.7) {
		tmp = Math.abs((Math.abs(x) * (2.0 / Math.sqrt(Math.PI))));
	} else {
		tmp = Math.abs((((x * x) * Math.abs(x)) * 0.6666666666666666)) / Math.sqrt(Math.PI);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.7:
		tmp = math.fabs((math.fabs(x) * (2.0 / math.sqrt(math.pi))))
	else:
		tmp = math.fabs((((x * x) * math.fabs(x)) * 0.6666666666666666)) / math.sqrt(math.pi)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.7)
		tmp = abs(Float64(abs(x) * Float64(2.0 / sqrt(pi))));
	else
		tmp = Float64(abs(Float64(Float64(Float64(x * x) * abs(x)) * 0.6666666666666666)) / sqrt(pi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.7)
		tmp = abs((abs(x) * (2.0 / sqrt(pi))));
	else
		tmp = abs((((x * x) * abs(x)) * 0.6666666666666666)) / sqrt(pi);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.7], N[Abs[N[(N[Abs[x], $MachinePrecision] * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(N[Abs[N[(N[(N[(x * x), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * 0.6666666666666666), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.69999999999999996

   1. Initial program 99.8%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 67.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fabs.f64 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-fabs.f64 N/A

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

      7. lower-/.f64 68.2

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

   8. Applied rewrites 68.2%

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

   if 1.69999999999999996 < x

   1. Initial program 99.8%

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

   2. Taylor expanded in x around 0

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in x around inf

      \[\leadsto \frac{\left|\frac{2}{3} \cdot \left({x}^{2} \cdot \left|x\right|\right)\right|}{\sqrt{\pi}} \]
   6. Step-by-step derivation

      1. metadata-eval N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. pow2-fabs-rev N/A

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

      5. pow2 N/A

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

      6. lift-fabs.f64 N/A

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

      7. lift-fabs.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. lift-fabs.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lift-fabs.f64 N/A

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

      13. lift-fabs.f64 N/A

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

      14. sqr-abs-rev N/A

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

      15. lift-*.f64 N/A

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

      16. metadata-eval 26.9

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

   7. Applied rewrites 26.9%

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

Alternative 9: 68.2% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 10^{-8}:\\ \;\;\;\;\left|\left|x\right| \cdot \frac{2}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\sqrt{x \cdot x} \cdot 2}{\sqrt{\pi}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1e-8)
   (fabs (* (fabs x) (/ 2.0 (sqrt PI))))
   (fabs (/ (* (sqrt (* x x)) 2.0) (sqrt PI)))))

C

double code(double x) {
	double tmp;
	if (x <= 1e-8) {
		tmp = fabs((fabs(x) * (2.0 / sqrt(((double) M_PI)))));
	} else {
		tmp = fabs(((sqrt((x * x)) * 2.0) / sqrt(((double) M_PI))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1e-8) {
		tmp = Math.abs((Math.abs(x) * (2.0 / Math.sqrt(Math.PI))));
	} else {
		tmp = Math.abs(((Math.sqrt((x * x)) * 2.0) / Math.sqrt(Math.PI)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1e-8:
		tmp = math.fabs((math.fabs(x) * (2.0 / math.sqrt(math.pi))))
	else:
		tmp = math.fabs(((math.sqrt((x * x)) * 2.0) / math.sqrt(math.pi)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1e-8)
		tmp = abs(Float64(abs(x) * Float64(2.0 / sqrt(pi))));
	else
		tmp = abs(Float64(Float64(sqrt(Float64(x * x)) * 2.0) / sqrt(pi)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1e-8)
		tmp = abs((abs(x) * (2.0 / sqrt(pi))));
	else
		tmp = abs(((sqrt((x * x)) * 2.0) / sqrt(pi)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1e-8], N[Abs[N[(N[Abs[x], $MachinePrecision] * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[Sqrt[N[(x * x), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1e-8

   1. Initial program 99.8%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 67.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-fabs.f64 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-fabs.f64 N/A

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

      7. lower-/.f64 68.2

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

   8. Applied rewrites 68.2%

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

   if 1e-8 < x

   1. Initial program 99.8%

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

   3. Applied rewrites 99.8%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 67.7%

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

      1. lift-fabs.f64 N/A

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

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

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

      3. lift-*.f64 N/A

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

      4. lower-sqrt.f64 53.2

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

   8. Applied rewrites 53.2%

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

Alternative 10: 68.2% accurate, 8.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.8%

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

4. Taylor expanded in x around 0

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

6. Applied rewrites 67.7%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-fabs.f64 N/A

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

   4. associate-/l* N/A

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

   5. lower-*.f64 N/A

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

   6. lift-fabs.f64 N/A

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

   7. lower-/.f64 68.2

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

8. Applied rewrites 68.2%

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

Reproduce

herbie shell --seed 2025135 
(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)))))))