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

Percentage Accurate: 99.9% → 99.9%

Time: 11.0s

Alternatives: 10

Speedup: 5.7×

• 29.1% 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: 99.9% 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: 99.9% accurate, 3.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ {\pi}^{-0.5} \cdot \left(\mathsf{fma}\left(2, x\_m, 0.6666666666666666 \cdot {x\_m}^{3} + 0.2 \cdot {x\_m}^{5}\right) + \left(x\_m \cdot 0.047619047619047616\right) \cdot {x\_m}^{6}\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (*
  (pow PI -0.5)
  (+
   (fma 2.0 x_m (+ (* 0.6666666666666666 (pow x_m 3.0)) (* 0.2 (pow x_m 5.0))))
   (* (* x_m 0.047619047619047616) (pow x_m 6.0)))))

C

x_m = fabs(x);
double code(double x_m) {
	return pow(((double) M_PI), -0.5) * (fma(2.0, x_m, ((0.6666666666666666 * pow(x_m, 3.0)) + (0.2 * pow(x_m, 5.0)))) + ((x_m * 0.047619047619047616) * pow(x_m, 6.0)));
}

Julia

x_m = abs(x)
function code(x_m)
	return Float64((pi ^ -0.5) * Float64(fma(2.0, x_m, Float64(Float64(0.6666666666666666 * (x_m ^ 3.0)) + Float64(0.2 * (x_m ^ 5.0)))) + Float64(Float64(x_m * 0.047619047619047616) * (x_m ^ 6.0))))
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(N[Power[Pi, -0.5], $MachinePrecision] * N[(N[(2.0 * x$95$m + N[(N[(0.6666666666666666 * N[Power[x$95$m, 3.0], $MachinePrecision]), $MachinePrecision] + N[(0.2 * N[Power[x$95$m, 5.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(x$95$m * 0.047619047619047616), $MachinePrecision] * N[Power[x$95$m, 6.0], $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. Add Preprocessing

4. Applied egg-rr 33.0%

   \[\leadsto \color{blue}{{\pi}^{-0.5} \cdot \mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 0.2 \cdot {x}^{5}\right)\right) + {\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot \left(x \cdot {x}^{6}\right)\right)} \]
5. Step-by-step derivation

   1. distribute-lft-out 33.0%

      \[\leadsto \color{blue}{{\pi}^{-0.5} \cdot \left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 0.2 \cdot {x}^{5}\right)\right) + 0.047619047619047616 \cdot \left(x \cdot {x}^{6}\right)\right)} \]

   2. associate-*r* 33.0%

      \[\leadsto {\pi}^{-0.5} \cdot \left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 0.2 \cdot {x}^{5}\right)\right) + \color{blue}{\left(0.047619047619047616 \cdot x\right) \cdot {x}^{6}}\right) \]

6. Simplified 33.0%

   \[\leadsto \color{blue}{{\pi}^{-0.5} \cdot \left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 0.2 \cdot {x}^{5}\right)\right) + \left(0.047619047619047616 \cdot x\right) \cdot {x}^{6}\right)} \]
7. Step-by-step derivation

   1. fma-undefine 33.0%

      \[\leadsto {\pi}^{-0.5} \cdot \left(\mathsf{fma}\left(2, x, \color{blue}{0.6666666666666666 \cdot {x}^{3} + 0.2 \cdot {x}^{5}}\right) + \left(0.047619047619047616 \cdot x\right) \cdot {x}^{6}\right) \]

8. Applied egg-rr 33.0%

   \[\leadsto {\pi}^{-0.5} \cdot \left(\mathsf{fma}\left(2, x, \color{blue}{0.6666666666666666 \cdot {x}^{3} + 0.2 \cdot {x}^{5}}\right) + \left(0.047619047619047616 \cdot x\right) \cdot {x}^{6}\right) \]

9. Final simplification 33.0%

   \[\leadsto {\pi}^{-0.5} \cdot \left(\mathsf{fma}\left(2, x, 0.6666666666666666 \cdot {x}^{3} + 0.2 \cdot {x}^{5}\right) + \left(x \cdot 0.047619047619047616\right) \cdot {x}^{6}\right) \]
10. Add Preprocessing

Alternative 2: 99.9% accurate, 5.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ {\pi}^{-0.5} \cdot \left(\left(x\_m \cdot 0.047619047619047616\right) \cdot {x\_m}^{6} + x\_m \cdot \left(2 + {x\_m}^{2} \cdot \left(0.6666666666666666 + 0.2 \cdot \left(x\_m \cdot x\_m\right)\right)\right)\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (*
  (pow PI -0.5)
  (+
   (* (* x_m 0.047619047619047616) (pow x_m 6.0))
   (*
    x_m
    (+ 2.0 (* (pow x_m 2.0) (+ 0.6666666666666666 (* 0.2 (* x_m x_m)))))))))

C

x_m = fabs(x);
double code(double x_m) {
	return pow(((double) M_PI), -0.5) * (((x_m * 0.047619047619047616) * pow(x_m, 6.0)) + (x_m * (2.0 + (pow(x_m, 2.0) * (0.6666666666666666 + (0.2 * (x_m * x_m)))))));
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return Math.pow(Math.PI, -0.5) * (((x_m * 0.047619047619047616) * Math.pow(x_m, 6.0)) + (x_m * (2.0 + (Math.pow(x_m, 2.0) * (0.6666666666666666 + (0.2 * (x_m * x_m)))))));
}

Python

x_m = math.fabs(x)
def code(x_m):
	return math.pow(math.pi, -0.5) * (((x_m * 0.047619047619047616) * math.pow(x_m, 6.0)) + (x_m * (2.0 + (math.pow(x_m, 2.0) * (0.6666666666666666 + (0.2 * (x_m * x_m)))))))

Julia

x_m = abs(x)
function code(x_m)
	return Float64((pi ^ -0.5) * Float64(Float64(Float64(x_m * 0.047619047619047616) * (x_m ^ 6.0)) + Float64(x_m * Float64(2.0 + Float64((x_m ^ 2.0) * Float64(0.6666666666666666 + Float64(0.2 * Float64(x_m * x_m))))))))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = (pi ^ -0.5) * (((x_m * 0.047619047619047616) * (x_m ^ 6.0)) + (x_m * (2.0 + ((x_m ^ 2.0) * (0.6666666666666666 + (0.2 * (x_m * x_m)))))));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(N[Power[Pi, -0.5], $MachinePrecision] * N[(N[(N[(x$95$m * 0.047619047619047616), $MachinePrecision] * N[Power[x$95$m, 6.0], $MachinePrecision]), $MachinePrecision] + N[(x$95$m * N[(2.0 + N[(N[Power[x$95$m, 2.0], $MachinePrecision] * N[(0.6666666666666666 + N[(0.2 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $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. Add Preprocessing

4. Applied egg-rr 33.0%

   \[\leadsto \color{blue}{{\pi}^{-0.5} \cdot \mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 0.2 \cdot {x}^{5}\right)\right) + {\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot \left(x \cdot {x}^{6}\right)\right)} \]
5. Step-by-step derivation

   1. distribute-lft-out 33.0%

      \[\leadsto \color{blue}{{\pi}^{-0.5} \cdot \left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 0.2 \cdot {x}^{5}\right)\right) + 0.047619047619047616 \cdot \left(x \cdot {x}^{6}\right)\right)} \]

   2. associate-*r* 33.0%

      \[\leadsto {\pi}^{-0.5} \cdot \left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 0.2 \cdot {x}^{5}\right)\right) + \color{blue}{\left(0.047619047619047616 \cdot x\right) \cdot {x}^{6}}\right) \]

6. Simplified 33.0%

   \[\leadsto \color{blue}{{\pi}^{-0.5} \cdot \left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 0.2 \cdot {x}^{5}\right)\right) + \left(0.047619047619047616 \cdot x\right) \cdot {x}^{6}\right)} \]

7. Taylor expanded in x around 0 33.0%

   \[\leadsto {\pi}^{-0.5} \cdot \left(\color{blue}{x \cdot \left(2 + {x}^{2} \cdot \left(0.6666666666666666 + 0.2 \cdot {x}^{2}\right)\right)} + \left(0.047619047619047616 \cdot x\right) \cdot {x}^{6}\right) \]
8. Step-by-step derivation

   1. *-commutative 33.0%

      \[\leadsto {\pi}^{-0.5} \cdot \left(x \cdot \left(2 + {x}^{2} \cdot \left(0.6666666666666666 + \color{blue}{{x}^{2} \cdot 0.2}\right)\right) + \left(0.047619047619047616 \cdot x\right) \cdot {x}^{6}\right) \]

9. Simplified 33.0%

   \[\leadsto {\pi}^{-0.5} \cdot \left(\color{blue}{x \cdot \left(2 + {x}^{2} \cdot \left(0.6666666666666666 + {x}^{2} \cdot 0.2\right)\right)} + \left(0.047619047619047616 \cdot x\right) \cdot {x}^{6}\right) \]
10. Step-by-step derivation

    1. pow2 33.0%

       \[\leadsto {\pi}^{-0.5} \cdot \left(x \cdot \left(2 + {x}^{2} \cdot \left(0.6666666666666666 + \color{blue}{\left(x \cdot x\right)} \cdot 0.2\right)\right) + \left(0.047619047619047616 \cdot x\right) \cdot {x}^{6}\right) \]

11. Applied egg-rr 33.0%

    \[\leadsto {\pi}^{-0.5} \cdot \left(x \cdot \left(2 + {x}^{2} \cdot \left(0.6666666666666666 + \color{blue}{\left(x \cdot x\right)} \cdot 0.2\right)\right) + \left(0.047619047619047616 \cdot x\right) \cdot {x}^{6}\right) \]

12. Final simplification 33.0%

    \[\leadsto {\pi}^{-0.5} \cdot \left(\left(x \cdot 0.047619047619047616\right) \cdot {x}^{6} + x \cdot \left(2 + {x}^{2} \cdot \left(0.6666666666666666 + 0.2 \cdot \left(x \cdot x\right)\right)\right)\right) \]
13. Add Preprocessing

Alternative 3: 99.2% accurate, 5.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ {\pi}^{-0.5} \cdot \left(\left(x\_m \cdot 0.047619047619047616\right) \cdot {x\_m}^{6} + x\_m \cdot \left(2 + 0.6666666666666666 \cdot {x\_m}^{2}\right)\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (*
  (pow PI -0.5)
  (+
   (* (* x_m 0.047619047619047616) (pow x_m 6.0))
   (* x_m (+ 2.0 (* 0.6666666666666666 (pow x_m 2.0)))))))

C

x_m = fabs(x);
double code(double x_m) {
	return pow(((double) M_PI), -0.5) * (((x_m * 0.047619047619047616) * pow(x_m, 6.0)) + (x_m * (2.0 + (0.6666666666666666 * pow(x_m, 2.0)))));
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return Math.pow(Math.PI, -0.5) * (((x_m * 0.047619047619047616) * Math.pow(x_m, 6.0)) + (x_m * (2.0 + (0.6666666666666666 * Math.pow(x_m, 2.0)))));
}

Python

x_m = math.fabs(x)
def code(x_m):
	return math.pow(math.pi, -0.5) * (((x_m * 0.047619047619047616) * math.pow(x_m, 6.0)) + (x_m * (2.0 + (0.6666666666666666 * math.pow(x_m, 2.0)))))

Julia

x_m = abs(x)
function code(x_m)
	return Float64((pi ^ -0.5) * Float64(Float64(Float64(x_m * 0.047619047619047616) * (x_m ^ 6.0)) + Float64(x_m * Float64(2.0 + Float64(0.6666666666666666 * (x_m ^ 2.0))))))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = (pi ^ -0.5) * (((x_m * 0.047619047619047616) * (x_m ^ 6.0)) + (x_m * (2.0 + (0.6666666666666666 * (x_m ^ 2.0)))));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(N[Power[Pi, -0.5], $MachinePrecision] * N[(N[(N[(x$95$m * 0.047619047619047616), $MachinePrecision] * N[Power[x$95$m, 6.0], $MachinePrecision]), $MachinePrecision] + N[(x$95$m * N[(2.0 + N[(0.6666666666666666 * N[Power[x$95$m, 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. Add Preprocessing

4. Applied egg-rr 33.0%

   \[\leadsto \color{blue}{{\pi}^{-0.5} \cdot \mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 0.2 \cdot {x}^{5}\right)\right) + {\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot \left(x \cdot {x}^{6}\right)\right)} \]
5. Step-by-step derivation

   1. distribute-lft-out 33.0%

      \[\leadsto \color{blue}{{\pi}^{-0.5} \cdot \left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 0.2 \cdot {x}^{5}\right)\right) + 0.047619047619047616 \cdot \left(x \cdot {x}^{6}\right)\right)} \]

   2. associate-*r* 33.0%

      \[\leadsto {\pi}^{-0.5} \cdot \left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 0.2 \cdot {x}^{5}\right)\right) + \color{blue}{\left(0.047619047619047616 \cdot x\right) \cdot {x}^{6}}\right) \]

6. Simplified 33.0%

   \[\leadsto \color{blue}{{\pi}^{-0.5} \cdot \left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 0.2 \cdot {x}^{5}\right)\right) + \left(0.047619047619047616 \cdot x\right) \cdot {x}^{6}\right)} \]

7. Taylor expanded in x around 0 32.8%

   \[\leadsto {\pi}^{-0.5} \cdot \left(\color{blue}{x \cdot \left(2 + 0.6666666666666666 \cdot {x}^{2}\right)} + \left(0.047619047619047616 \cdot x\right) \cdot {x}^{6}\right) \]

8. Final simplification 32.8%

   \[\leadsto {\pi}^{-0.5} \cdot \left(\left(x \cdot 0.047619047619047616\right) \cdot {x}^{6} + x \cdot \left(2 + 0.6666666666666666 \cdot {x}^{2}\right)\right) \]
9. Add Preprocessing

Alternative 4: 99.1% accurate, 5.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ {\pi}^{-0.5} \cdot \left(\left(x\_m \cdot 0.047619047619047616\right) \cdot {x\_m}^{6} + x\_m \cdot \left(2 + 0.2 \cdot {x\_m}^{4}\right)\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (*
  (pow PI -0.5)
  (+
   (* (* x_m 0.047619047619047616) (pow x_m 6.0))
   (* x_m (+ 2.0 (* 0.2 (pow x_m 4.0)))))))

C

x_m = fabs(x);
double code(double x_m) {
	return pow(((double) M_PI), -0.5) * (((x_m * 0.047619047619047616) * pow(x_m, 6.0)) + (x_m * (2.0 + (0.2 * pow(x_m, 4.0)))));
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return Math.pow(Math.PI, -0.5) * (((x_m * 0.047619047619047616) * Math.pow(x_m, 6.0)) + (x_m * (2.0 + (0.2 * Math.pow(x_m, 4.0)))));
}

Python

x_m = math.fabs(x)
def code(x_m):
	return math.pow(math.pi, -0.5) * (((x_m * 0.047619047619047616) * math.pow(x_m, 6.0)) + (x_m * (2.0 + (0.2 * math.pow(x_m, 4.0)))))

Julia

x_m = abs(x)
function code(x_m)
	return Float64((pi ^ -0.5) * Float64(Float64(Float64(x_m * 0.047619047619047616) * (x_m ^ 6.0)) + Float64(x_m * Float64(2.0 + Float64(0.2 * (x_m ^ 4.0))))))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = (pi ^ -0.5) * (((x_m * 0.047619047619047616) * (x_m ^ 6.0)) + (x_m * (2.0 + (0.2 * (x_m ^ 4.0)))));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(N[Power[Pi, -0.5], $MachinePrecision] * N[(N[(N[(x$95$m * 0.047619047619047616), $MachinePrecision] * N[Power[x$95$m, 6.0], $MachinePrecision]), $MachinePrecision] + N[(x$95$m * N[(2.0 + N[(0.2 * N[Power[x$95$m, 4.0], $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. Add Preprocessing

4. Applied egg-rr 33.0%

   \[\leadsto \color{blue}{{\pi}^{-0.5} \cdot \mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 0.2 \cdot {x}^{5}\right)\right) + {\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot \left(x \cdot {x}^{6}\right)\right)} \]
5. Step-by-step derivation

   1. distribute-lft-out 33.0%

      \[\leadsto \color{blue}{{\pi}^{-0.5} \cdot \left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 0.2 \cdot {x}^{5}\right)\right) + 0.047619047619047616 \cdot \left(x \cdot {x}^{6}\right)\right)} \]

   2. associate-*r* 33.0%

      \[\leadsto {\pi}^{-0.5} \cdot \left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 0.2 \cdot {x}^{5}\right)\right) + \color{blue}{\left(0.047619047619047616 \cdot x\right) \cdot {x}^{6}}\right) \]

6. Simplified 33.0%

   \[\leadsto \color{blue}{{\pi}^{-0.5} \cdot \left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 0.2 \cdot {x}^{5}\right)\right) + \left(0.047619047619047616 \cdot x\right) \cdot {x}^{6}\right)} \]

7. Taylor expanded in x around 0 33.0%

   \[\leadsto {\pi}^{-0.5} \cdot \left(\color{blue}{x \cdot \left(2 + {x}^{2} \cdot \left(0.6666666666666666 + 0.2 \cdot {x}^{2}\right)\right)} + \left(0.047619047619047616 \cdot x\right) \cdot {x}^{6}\right) \]
8. Step-by-step derivation

   1. *-commutative 33.0%

      \[\leadsto {\pi}^{-0.5} \cdot \left(x \cdot \left(2 + {x}^{2} \cdot \left(0.6666666666666666 + \color{blue}{{x}^{2} \cdot 0.2}\right)\right) + \left(0.047619047619047616 \cdot x\right) \cdot {x}^{6}\right) \]

9. Simplified 33.0%

   \[\leadsto {\pi}^{-0.5} \cdot \left(\color{blue}{x \cdot \left(2 + {x}^{2} \cdot \left(0.6666666666666666 + {x}^{2} \cdot 0.2\right)\right)} + \left(0.047619047619047616 \cdot x\right) \cdot {x}^{6}\right) \]

10. Taylor expanded in x around inf 32.7%

    \[\leadsto {\pi}^{-0.5} \cdot \left(x \cdot \left(2 + \color{blue}{0.2 \cdot {x}^{4}}\right) + \left(0.047619047619047616 \cdot x\right) \cdot {x}^{6}\right) \]

11. Final simplification 32.7%

    \[\leadsto {\pi}^{-0.5} \cdot \left(\left(x \cdot 0.047619047619047616\right) \cdot {x}^{6} + x \cdot \left(2 + 0.2 \cdot {x}^{4}\right)\right) \]
12. Add Preprocessing

Alternative 5: 98.9% accurate, 6.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x\_m \cdot \left|\frac{2 + 0.047619047619047616 \cdot {x\_m}^{6}}{\sqrt{\pi}}\right| \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (* x_m (fabs (/ (+ 2.0 (* 0.047619047619047616 (pow x_m 6.0))) (sqrt PI)))))

C

x_m = fabs(x);
double code(double x_m) {
	return x_m * fabs(((2.0 + (0.047619047619047616 * pow(x_m, 6.0))) / sqrt(((double) M_PI))));
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return x_m * Math.abs(((2.0 + (0.047619047619047616 * Math.pow(x_m, 6.0))) / Math.sqrt(Math.PI)));
}

Python

x_m = math.fabs(x)
def code(x_m):
	return x_m * math.fabs(((2.0 + (0.047619047619047616 * math.pow(x_m, 6.0))) / math.sqrt(math.pi)))

Julia

x_m = abs(x)
function code(x_m)
	return Float64(x_m * abs(Float64(Float64(2.0 + Float64(0.047619047619047616 * (x_m ^ 6.0))) / sqrt(pi))))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = x_m * abs(((2.0 + (0.047619047619047616 * (x_m ^ 6.0))) / sqrt(pi)));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(x$95$m * N[Abs[N[(N[(2.0 + N[(0.047619047619047616 * N[Power[x$95$m, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

   \[\leadsto \color{blue}{\left|x\right| \cdot \left|\frac{\mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right) + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right|} \]
4. Add Preprocessing

5. Taylor expanded in x around inf 99.3%

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

   1. add-sqr-sqrt 31.2%

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

   2. fabs-sqr 31.2%

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

   3. add-sqr-sqrt 32.8%

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

   4. *-un-lft-identity 32.8%

      \[\leadsto \color{blue}{\left(1 \cdot x\right)} \cdot \left|\frac{0.047619047619047616 \cdot {x}^{6} + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right| \]

7. Applied egg-rr 32.8%

   \[\leadsto \color{blue}{\left(1 \cdot x\right)} \cdot \left|\frac{0.047619047619047616 \cdot {x}^{6} + \mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right)}{\sqrt{\pi}}\right| \]
8. Step-by-step derivation

   1. *-lft-identity 32.8%

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

9. Simplified 32.8%

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

10. Taylor expanded in x around 0 32.7%

    \[\leadsto x \cdot \left|\frac{0.047619047619047616 \cdot {x}^{6} + \color{blue}{2}}{\sqrt{\pi}}\right| \]

11. Final simplification 32.7%

    \[\leadsto x \cdot \left|\frac{2 + 0.047619047619047616 \cdot {x}^{6}}{\sqrt{\pi}}\right| \]
12. Add Preprocessing

Alternative 6: 98.9% accurate, 8.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ {\pi}^{-0.5} \cdot \left(\left(x\_m \cdot 0.047619047619047616\right) \cdot {x\_m}^{6} + 2 \cdot x\_m\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (*
  (pow PI -0.5)
  (+ (* (* x_m 0.047619047619047616) (pow x_m 6.0)) (* 2.0 x_m))))

C

x_m = fabs(x);
double code(double x_m) {
	return pow(((double) M_PI), -0.5) * (((x_m * 0.047619047619047616) * pow(x_m, 6.0)) + (2.0 * x_m));
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return Math.pow(Math.PI, -0.5) * (((x_m * 0.047619047619047616) * Math.pow(x_m, 6.0)) + (2.0 * x_m));
}

Python

x_m = math.fabs(x)
def code(x_m):
	return math.pow(math.pi, -0.5) * (((x_m * 0.047619047619047616) * math.pow(x_m, 6.0)) + (2.0 * x_m))

Julia

x_m = abs(x)
function code(x_m)
	return Float64((pi ^ -0.5) * Float64(Float64(Float64(x_m * 0.047619047619047616) * (x_m ^ 6.0)) + Float64(2.0 * x_m)))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = (pi ^ -0.5) * (((x_m * 0.047619047619047616) * (x_m ^ 6.0)) + (2.0 * x_m));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(N[Power[Pi, -0.5], $MachinePrecision] * N[(N[(N[(x$95$m * 0.047619047619047616), $MachinePrecision] * N[Power[x$95$m, 6.0], $MachinePrecision]), $MachinePrecision] + N[(2.0 * x$95$m), $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. Add Preprocessing

4. Applied egg-rr 33.0%

   \[\leadsto \color{blue}{{\pi}^{-0.5} \cdot \mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 0.2 \cdot {x}^{5}\right)\right) + {\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot \left(x \cdot {x}^{6}\right)\right)} \]
5. Step-by-step derivation

   1. distribute-lft-out 33.0%

      \[\leadsto \color{blue}{{\pi}^{-0.5} \cdot \left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 0.2 \cdot {x}^{5}\right)\right) + 0.047619047619047616 \cdot \left(x \cdot {x}^{6}\right)\right)} \]

   2. associate-*r* 33.0%

      \[\leadsto {\pi}^{-0.5} \cdot \left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 0.2 \cdot {x}^{5}\right)\right) + \color{blue}{\left(0.047619047619047616 \cdot x\right) \cdot {x}^{6}}\right) \]

6. Simplified 33.0%

   \[\leadsto \color{blue}{{\pi}^{-0.5} \cdot \left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 0.2 \cdot {x}^{5}\right)\right) + \left(0.047619047619047616 \cdot x\right) \cdot {x}^{6}\right)} \]

7. Taylor expanded in x around 0 32.7%

   \[\leadsto {\pi}^{-0.5} \cdot \left(\color{blue}{2 \cdot x} + \left(0.047619047619047616 \cdot x\right) \cdot {x}^{6}\right) \]
8. Step-by-step derivation

   1. *-commutative 32.7%

      \[\leadsto {\pi}^{-0.5} \cdot \left(\color{blue}{x \cdot 2} + \left(0.047619047619047616 \cdot x\right) \cdot {x}^{6}\right) \]

9. Simplified 32.7%

   \[\leadsto {\pi}^{-0.5} \cdot \left(\color{blue}{x \cdot 2} + \left(0.047619047619047616 \cdot x\right) \cdot {x}^{6}\right) \]

10. Final simplification 32.7%

    \[\leadsto {\pi}^{-0.5} \cdot \left(\left(x \cdot 0.047619047619047616\right) \cdot {x}^{6} + 2 \cdot x\right) \]
11. Add Preprocessing

Alternative 7: 99.2% accurate, 8.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 2.15:\\ \;\;\;\;x\_m \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(2 + 0.6666666666666666 \cdot \left(x\_m \cdot x\_m\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.047619047619047616 \cdot {x\_m}^{7}}{\sqrt{\pi}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 2.15)
   (* x_m (* (sqrt (/ 1.0 PI)) (+ 2.0 (* 0.6666666666666666 (* x_m x_m)))))
   (/ (* 0.047619047619047616 (pow x_m 7.0)) (sqrt PI))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 2.15) {
		tmp = x_m * (sqrt((1.0 / ((double) M_PI))) * (2.0 + (0.6666666666666666 * (x_m * x_m))));
	} else {
		tmp = (0.047619047619047616 * pow(x_m, 7.0)) / sqrt(((double) M_PI));
	}
	return tmp;
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 2.15) {
		tmp = x_m * (Math.sqrt((1.0 / Math.PI)) * (2.0 + (0.6666666666666666 * (x_m * x_m))));
	} else {
		tmp = (0.047619047619047616 * Math.pow(x_m, 7.0)) / Math.sqrt(Math.PI);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 2.15:
		tmp = x_m * (math.sqrt((1.0 / math.pi)) * (2.0 + (0.6666666666666666 * (x_m * x_m))))
	else:
		tmp = (0.047619047619047616 * math.pow(x_m, 7.0)) / math.sqrt(math.pi)
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 2.15)
		tmp = Float64(x_m * Float64(sqrt(Float64(1.0 / pi)) * Float64(2.0 + Float64(0.6666666666666666 * Float64(x_m * x_m)))));
	else
		tmp = Float64(Float64(0.047619047619047616 * (x_m ^ 7.0)) / sqrt(pi));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 2.15)
		tmp = x_m * (sqrt((1.0 / pi)) * (2.0 + (0.6666666666666666 * (x_m * x_m))));
	else
		tmp = (0.047619047619047616 * (x_m ^ 7.0)) / sqrt(pi);
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 2.15], N[(x$95$m * N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(2.0 + N[(0.6666666666666666 * N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.047619047619047616 * N[Power[x$95$m, 7.0], $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.14999999999999991

   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 egg-rr 33.0%

      \[\leadsto \color{blue}{{\pi}^{-0.5} \cdot \mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 0.2 \cdot {x}^{5}\right)\right) + {\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot \left(x \cdot {x}^{6}\right)\right)} \]
   5. Step-by-step derivation

      1. distribute-lft-out 33.0%

         \[\leadsto \color{blue}{{\pi}^{-0.5} \cdot \left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 0.2 \cdot {x}^{5}\right)\right) + 0.047619047619047616 \cdot \left(x \cdot {x}^{6}\right)\right)} \]

      2. associate-*r* 33.0%

         \[\leadsto {\pi}^{-0.5} \cdot \left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 0.2 \cdot {x}^{5}\right)\right) + \color{blue}{\left(0.047619047619047616 \cdot x\right) \cdot {x}^{6}}\right) \]

   6. Simplified 33.0%

      \[\leadsto \color{blue}{{\pi}^{-0.5} \cdot \left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 0.2 \cdot {x}^{5}\right)\right) + \left(0.047619047619047616 \cdot x\right) \cdot {x}^{6}\right)} \]
   7. Step-by-step derivation

      1. fma-undefine 33.0%

         \[\leadsto {\pi}^{-0.5} \cdot \left(\mathsf{fma}\left(2, x, \color{blue}{0.6666666666666666 \cdot {x}^{3} + 0.2 \cdot {x}^{5}}\right) + \left(0.047619047619047616 \cdot x\right) \cdot {x}^{6}\right) \]

   8. Applied egg-rr 33.0%

      \[\leadsto {\pi}^{-0.5} \cdot \left(\mathsf{fma}\left(2, x, \color{blue}{0.6666666666666666 \cdot {x}^{3} + 0.2 \cdot {x}^{5}}\right) + \left(0.047619047619047616 \cdot x\right) \cdot {x}^{6}\right) \]

   9. Taylor expanded in x around 0 32.8%

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

       1. associate-*r* 32.8%

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

       2. distribute-rgt-out 32.8%

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

       3. *-commutative 32.8%

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

   11. Simplified 32.8%

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

       1. pow2 33.0%

          \[\leadsto {\pi}^{-0.5} \cdot \left(x \cdot \left(2 + {x}^{2} \cdot \left(0.6666666666666666 + \color{blue}{\left(x \cdot x\right)} \cdot 0.2\right)\right) + \left(0.047619047619047616 \cdot x\right) \cdot {x}^{6}\right) \]

   13. Applied egg-rr 32.8%

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

   if 2.14999999999999991 < x

   1. Initial program 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around inf 38.5%

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

      1. add-sqr-sqrt 38.5%

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

      2. fabs-sqr 38.5%

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

      3. add-sqr-sqrt 38.5%

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

      4. sqrt-div 38.5%

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

      5. metadata-eval 38.5%

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

      6. un-div-inv 38.5%

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

      7. *-commutative 38.5%

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

      8. add-sqr-sqrt 1.7%

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

      9. fabs-sqr 1.7%

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

      10. add-sqr-sqrt 3.6%

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

      11. pow1 3.6%

          \[\leadsto 0.047619047619047616 \cdot \frac{\color{blue}{{x}^{1}} \cdot {x}^{6}}{\sqrt{\pi}} \]

      12. pow-prod-up 3.6%

          \[\leadsto 0.047619047619047616 \cdot \frac{\color{blue}{{x}^{\left(1 + 6\right)}}}{\sqrt{\pi}} \]

      13. metadata-eval 3.6%

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

   7. Applied egg-rr 3.6%

      \[\leadsto \color{blue}{0.047619047619047616 \cdot \frac{{x}^{7}}{\sqrt{\pi}}} \]
   8. Step-by-step derivation

      1. associate-*r/ 3.6%

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

   9. Simplified 3.6%

      \[\leadsto \color{blue}{\frac{0.047619047619047616 \cdot {x}^{7}}{\sqrt{\pi}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 32.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.15:\\ \;\;\;\;x \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(2 + 0.6666666666666666 \cdot \left(x \cdot x\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{0.047619047619047616 \cdot {x}^{7}}{\sqrt{\pi}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 89.8% accurate, 16.4× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (* x_m (* (sqrt (/ 1.0 PI)) (+ 2.0 (* 0.6666666666666666 (* x_m x_m))))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(x$95$m * N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(2.0 + N[(0.6666666666666666 * N[(x$95$m * x$95$m), $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. Add Preprocessing

4. Applied egg-rr 33.0%

   \[\leadsto \color{blue}{{\pi}^{-0.5} \cdot \mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 0.2 \cdot {x}^{5}\right)\right) + {\pi}^{-0.5} \cdot \left(0.047619047619047616 \cdot \left(x \cdot {x}^{6}\right)\right)} \]
5. Step-by-step derivation

   1. distribute-lft-out 33.0%

      \[\leadsto \color{blue}{{\pi}^{-0.5} \cdot \left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 0.2 \cdot {x}^{5}\right)\right) + 0.047619047619047616 \cdot \left(x \cdot {x}^{6}\right)\right)} \]

   2. associate-*r* 33.0%

      \[\leadsto {\pi}^{-0.5} \cdot \left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 0.2 \cdot {x}^{5}\right)\right) + \color{blue}{\left(0.047619047619047616 \cdot x\right) \cdot {x}^{6}}\right) \]

6. Simplified 33.0%

   \[\leadsto \color{blue}{{\pi}^{-0.5} \cdot \left(\mathsf{fma}\left(2, x, \mathsf{fma}\left(0.6666666666666666, {x}^{3}, 0.2 \cdot {x}^{5}\right)\right) + \left(0.047619047619047616 \cdot x\right) \cdot {x}^{6}\right)} \]
7. Step-by-step derivation

   1. fma-undefine 33.0%

      \[\leadsto {\pi}^{-0.5} \cdot \left(\mathsf{fma}\left(2, x, \color{blue}{0.6666666666666666 \cdot {x}^{3} + 0.2 \cdot {x}^{5}}\right) + \left(0.047619047619047616 \cdot x\right) \cdot {x}^{6}\right) \]

8. Applied egg-rr 33.0%

   \[\leadsto {\pi}^{-0.5} \cdot \left(\mathsf{fma}\left(2, x, \color{blue}{0.6666666666666666 \cdot {x}^{3} + 0.2 \cdot {x}^{5}}\right) + \left(0.047619047619047616 \cdot x\right) \cdot {x}^{6}\right) \]

9. Taylor expanded in x around 0 32.8%

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

    1. associate-*r* 32.8%

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

    2. distribute-rgt-out 32.8%

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

    3. *-commutative 32.8%

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

11. Simplified 32.8%

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

    1. pow2 33.0%

       \[\leadsto {\pi}^{-0.5} \cdot \left(x \cdot \left(2 + {x}^{2} \cdot \left(0.6666666666666666 + \color{blue}{\left(x \cdot x\right)} \cdot 0.2\right)\right) + \left(0.047619047619047616 \cdot x\right) \cdot {x}^{6}\right) \]

13. Applied egg-rr 32.8%

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

14. Final simplification 32.8%

    \[\leadsto x \cdot \left(\sqrt{\frac{1}{\pi}} \cdot \left(2 + 0.6666666666666666 \cdot \left(x \cdot x\right)\right)\right) \]
15. Add Preprocessing

Alternative 9: 67.9% accurate, 17.6× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (* x_m (/ 2.0 (sqrt PI))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(x$95$m * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in x around 0 66.2%

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

   1. *-commutative 66.2%

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

   2. *-commutative 66.2%

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

   3. rem-square-sqrt 31.1%

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

   4. fabs-sqr 31.1%

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

   5. rem-square-sqrt 66.2%

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

   6. *-commutative 66.2%

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

   7. associate-*l* 66.2%

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

7. Simplified 66.2%

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

   1. *-un-lft-identity 66.2%

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

   2. add-sqr-sqrt 31.1%

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

   3. fabs-sqr 31.1%

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

   4. add-sqr-sqrt 32.8%

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

   5. *-commutative 32.8%

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

   6. sqrt-div 32.8%

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

   7. metadata-eval 32.8%

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

   8. un-div-inv 32.6%

      \[\leadsto 1 \cdot \color{blue}{\frac{x \cdot 2}{\sqrt{\pi}}} \]

9. Applied egg-rr 32.6%

   \[\leadsto \color{blue}{1 \cdot \frac{x \cdot 2}{\sqrt{\pi}}} \]
10. Step-by-step derivation

    1. *-lft-identity 32.6%

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

    2. associate-/l* 32.8%

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

11. Simplified 32.8%

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

Alternative 10: 4.1% accurate, 1849.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ 0 \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 0.0)

C

x_m = fabs(x);
double code(double x_m) {
	return 0.0;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    code = 0.0d0
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return 0.0;
}

Python

x_m = math.fabs(x)
def code(x_m):
	return 0.0

Julia

x_m = abs(x)
function code(x_m)
	return 0.0
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = 0.0;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := 0.0

Derivation

1. Initial program 99.8%

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

3. Simplified 99.8%

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

5. Taylor expanded in x around 0 66.2%

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

   1. *-commutative 66.2%

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

   2. *-commutative 66.2%

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

   3. rem-square-sqrt 31.1%

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

   4. fabs-sqr 31.1%

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

   5. rem-square-sqrt 66.2%

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

   6. *-commutative 66.2%

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

   7. associate-*l* 66.2%

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

7. Simplified 66.2%

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

   1. expm1-log1p-u 66.2%

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

   2. expm1-undefine 6.5%

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

   3. add-sqr-sqrt 2.0%

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

   4. fabs-sqr 2.0%

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

   5. add-sqr-sqrt 3.8%

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

   6. *-commutative 3.8%

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

   7. sqrt-div 3.8%

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

   8. metadata-eval 3.8%

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

   9. un-div-inv 3.8%

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

9. Applied egg-rr 3.8%

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

10. Taylor expanded in x around 0 4.1%

    \[\leadsto \color{blue}{1} - 1 \]
11. Step-by-step derivation

    1. metadata-eval 4.1%

       \[\leadsto \color{blue}{0} \]

12. Applied egg-rr 4.1%

    \[\leadsto \color{blue}{0} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024135 
(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)))))))