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

Percentage Accurate: 99.8% → 99.8%

Time: 8.8s

Alternatives: 12

Speedup: 2.1×

• 27.5% 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, 1.2× speedup

Math

\[\begin{array}{l} \\ \left|\frac{1}{\sqrt{\pi}} \cdot \mathsf{fma}\left(0.047619047619047616, {\left(\left|x\right|\right)}^{7}, \mathsf{fma}\left(0.2, {\left(\left|x\right|\right)}^{5}, \mathsf{fma}\left(0.6666666666666666, {\left(\left|x\right|\right)}^{3}, 2 \cdot \left|x\right|\right)\right)\right)\right| \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fabs
  (*
   (/ 1.0 (sqrt PI))
   (fma
    0.047619047619047616
    (pow (fabs x) 7.0)
    (fma
     0.2
     (pow (fabs x) 5.0)
     (fma 0.6666666666666666 (pow (fabs x) 3.0) (* 2.0 (fabs x))))))))

C

double code(double x) {
	return fabs(((1.0 / sqrt(((double) M_PI))) * fma(0.047619047619047616, pow(fabs(x), 7.0), fma(0.2, pow(fabs(x), 5.0), fma(0.6666666666666666, pow(fabs(x), 3.0), (2.0 * fabs(x)))))));
}

Julia

function code(x)
	return abs(Float64(Float64(1.0 / sqrt(pi)) * fma(0.047619047619047616, (abs(x) ^ 7.0), fma(0.2, (abs(x) ^ 5.0), fma(0.6666666666666666, (abs(x) ^ 3.0), Float64(2.0 * abs(x)))))))
end

Wolfram

code[x_] := N[Abs[N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[(0.047619047619047616 * N[Power[N[Abs[x], $MachinePrecision], 7.0], $MachinePrecision] + N[(0.2 * N[Power[N[Abs[x], $MachinePrecision], 5.0], $MachinePrecision] + N[(0.6666666666666666 * N[Power[N[Abs[x], $MachinePrecision], 3.0], $MachinePrecision] + N[(2.0 * N[Abs[x], $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. 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%

   \[\leadsto \left|\frac{1}{\sqrt{\pi}} \cdot \color{blue}{\mathsf{fma}\left(0.047619047619047616, {\left(\left|x\right|\right)}^{7}, \mathsf{fma}\left(0.2, {\left(\left|x\right|\right)}^{5}, \mathsf{fma}\left(0.6666666666666666, {\left(\left|x\right|\right)}^{3}, 2 \cdot \left|x\right|\right)\right)\right)}\right| \]
5. Add Preprocessing

Alternative 2: 99.8% accurate, 2.1× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (*
  (/ 1.0 (sqrt PI))
  (fabs
   (*
    (fma (* x x) (fma (* x x) (fma (/ x 21.0) x (/ 1.0 5.0)) (/ 2.0 3.0)) 2.0)
    (fabs x)))))

C

double code(double x) {
	return (1.0 / sqrt(((double) M_PI))) * fabs((fma((x * x), fma((x * x), fma((x / 21.0), x, (1.0 / 5.0)), (2.0 / 3.0)), 2.0) * fabs(x)));
}

Julia

function code(x)
	return Float64(Float64(1.0 / sqrt(pi)) * abs(Float64(fma(Float64(x * x), fma(Float64(x * x), fma(Float64(x / 21.0), x, Float64(1.0 / 5.0)), Float64(2.0 / 3.0)), 2.0) * abs(x))))
end

Wolfram

code[x_] := N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[Abs[N[(N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x / 21.0), $MachinePrecision] * x + N[(1.0 / 5.0), $MachinePrecision]), $MachinePrecision] + N[(2.0 / 3.0), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] * N[Abs[x], $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 99.4%

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

5. Applied rewrites 99.8%

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

Alternative 3: 99.4% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \frac{\left|\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.047619047619047616 \cdot x, x, \frac{1}{5}\right), \frac{2}{3}\right) \cdot x, x, 2\right)\right| \cdot \left|x\right|}{\sqrt{\pi}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  (*
   (fabs
    (fma
     (*
      (fma (* x x) (fma (* 0.047619047619047616 x) x (/ 1.0 5.0)) (/ 2.0 3.0))
      x)
     x
     2.0))
   (fabs x))
  (sqrt PI)))

C

double code(double x) {
	return (fabs(fma((fma((x * x), fma((0.047619047619047616 * x), x, (1.0 / 5.0)), (2.0 / 3.0)) * x), x, 2.0)) * fabs(x)) / sqrt(((double) M_PI));
}

Julia

function code(x)
	return Float64(Float64(abs(fma(Float64(fma(Float64(x * x), fma(Float64(0.047619047619047616 * x), x, Float64(1.0 / 5.0)), Float64(2.0 / 3.0)) * x), x, 2.0)) * abs(x)) / sqrt(pi))
end

Wolfram

code[x_] := N[(N[(N[Abs[N[(N[(N[(N[(x * x), $MachinePrecision] * N[(N[(0.047619047619047616 * x), $MachinePrecision] * x + N[(1.0 / 5.0), $MachinePrecision]), $MachinePrecision] + N[(2.0 / 3.0), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] * x + 2.0), $MachinePrecision]], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.4%

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

5. Applied rewrites 99.4%

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

7. Applied rewrites 99.4%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 99.4%

    \[\leadsto \frac{\left|\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(\color{blue}{0.047619047619047616 \cdot x}, x, \frac{1}{5}\right), \frac{2}{3}\right) \cdot x, x, 2\right)\right| \cdot \left|x\right|}{\sqrt{\pi}} \]
11. Add Preprocessing

Alternative 4: 99.4% accurate, 2.3× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (/
  (fabs
   (*
    (fma (* x x) (fma (* x x) (fma (/ x 21.0) x (/ 1.0 5.0)) (/ 2.0 3.0)) 2.0)
    x))
  (sqrt PI)))

C

double code(double x) {
	return fabs((fma((x * x), fma((x * x), fma((x / 21.0), x, (1.0 / 5.0)), (2.0 / 3.0)), 2.0) * x)) / sqrt(((double) M_PI));
}

Julia

function code(x)
	return Float64(abs(Float64(fma(Float64(x * x), fma(Float64(x * x), fma(Float64(x / 21.0), x, Float64(1.0 / 5.0)), Float64(2.0 / 3.0)), 2.0) * x)) / sqrt(pi))
end

Wolfram

code[x_] := N[(N[Abs[N[(N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x / 21.0), $MachinePrecision] * x + N[(1.0 / 5.0), $MachinePrecision]), $MachinePrecision] + N[(2.0 / 3.0), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.4%

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

5. Applied rewrites 99.8%

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

7. Applied rewrites 99.4%

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

Alternative 5: 98.8% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\sqrt{\pi}} \cdot \left|\mathsf{fma}\left(x \cdot x, 0.047619047619047616 \cdot {x}^{4}, 2\right) \cdot \left|x\right|\right| \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (/ 1.0 (sqrt PI))
  (fabs (* (fma (* x x) (* 0.047619047619047616 (pow x 4.0)) 2.0) (fabs x)))))

C

double code(double x) {
	return (1.0 / sqrt(((double) M_PI))) * fabs((fma((x * x), (0.047619047619047616 * pow(x, 4.0)), 2.0) * fabs(x)));
}

Julia

function code(x)
	return Float64(Float64(1.0 / sqrt(pi)) * abs(Float64(fma(Float64(x * x), Float64(0.047619047619047616 * (x ^ 4.0)), 2.0) * abs(x))))
end

Wolfram

code[x_] := N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[Abs[N[(N[(N[(x * x), $MachinePrecision] * N[(0.047619047619047616 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] * N[Abs[x], $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 99.4%

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

5. Applied rewrites 99.8%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 98.8%

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

Alternative 6: 98.3% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.85) {
		tmp = Math.sqrt((1.0 / Math.PI)) * Math.abs((Math.abs(x) * 2.0));
	} else {
		tmp = Math.abs((0.047619047619047616 * (Math.pow(x, 6.0) * Math.abs(x)))) / Math.sqrt(Math.PI);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.85:
		tmp = math.sqrt((1.0 / math.pi)) * math.fabs((math.fabs(x) * 2.0))
	else:
		tmp = math.fabs((0.047619047619047616 * (math.pow(x, 6.0) * math.fabs(x)))) / math.sqrt(math.pi)
	return tmp

Julia

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

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.85)
		tmp = sqrt((1.0 / pi)) * abs((abs(x) * 2.0));
	else
		tmp = abs((0.047619047619047616 * ((x ^ 6.0) * abs(x)))) / sqrt(pi);
	end
	tmp_2 = tmp;
end

Wolfram

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

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 68.8%

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

   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 99.4%

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

   4. Taylor expanded in x around inf

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

   6. Applied rewrites 35.7%

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

Alternative 7: 98.3% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(N[(N[Abs[N[(N[(x * x), $MachinePrecision] * N[(0.047619047619047616 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision]], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.4%

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

5. Applied rewrites 99.4%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 98.3%

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

Alternative 8: 88.9% accurate, 2.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.4%

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

5. Applied rewrites 99.4%

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

7. Applied rewrites 99.4%

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

8. Taylor expanded in x around inf

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

10. Applied rewrites 98.3%

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

Alternative 9: 83.8% accurate, 4.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.4%

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

4. Taylor expanded in x around 0

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

6. Applied rewrites 88.9%

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

8. Applied rewrites 88.9%

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

Alternative 10: 68.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\\ t_1 := \left(t\_0 \cdot \left|x\right|\right) \cdot \left|x\right|\\ \mathbf{if}\;\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot t\_0\right) + \frac{1}{5} \cdot t\_1\right) + \frac{1}{21} \cdot \left(\left(t\_1 \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \leq 5 \cdot 10^{-10}:\\ \;\;\;\;\sqrt{\frac{1}{\pi}} \cdot \left|\left|x\right| \cdot 2\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|\sqrt{x \cdot x} \cdot 2\right|}{\sqrt{\pi}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (* (fabs x) (fabs x)) (fabs x)))
        (t_1 (* (* t_0 (fabs x)) (fabs x))))
   (if (<=
        (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))))))
        5e-10)
     (* (sqrt (/ 1.0 PI)) (fabs (* (fabs x) 2.0)))
     (/ (fabs (* (sqrt (* x x)) 2.0)) (sqrt PI)))))

C

double code(double x) {
	double t_0 = (fabs(x) * fabs(x)) * fabs(x);
	double t_1 = (t_0 * fabs(x)) * fabs(x);
	double tmp;
	if (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)))))) <= 5e-10) {
		tmp = sqrt((1.0 / ((double) M_PI))) * fabs((fabs(x) * 2.0));
	} else {
		tmp = fabs((sqrt((x * x)) * 2.0)) / sqrt(((double) M_PI));
	}
	return tmp;
}

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);
	double tmp;
	if (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)))))) <= 5e-10) {
		tmp = Math.sqrt((1.0 / Math.PI)) * Math.abs((Math.abs(x) * 2.0));
	} else {
		tmp = Math.abs((Math.sqrt((x * x)) * 2.0)) / Math.sqrt(Math.PI);
	}
	return tmp;
}

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)
	tmp = 0
	if 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)))))) <= 5e-10:
		tmp = math.sqrt((1.0 / math.pi)) * math.fabs((math.fabs(x) * 2.0))
	else:
		tmp = math.fabs((math.sqrt((x * x)) * 2.0)) / math.sqrt(math.pi)
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(abs(x) * abs(x)) * abs(x))
	t_1 = Float64(Float64(t_0 * abs(x)) * abs(x))
	tmp = 0.0
	if (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)))))) <= 5e-10)
		tmp = Float64(sqrt(Float64(1.0 / pi)) * abs(Float64(abs(x) * 2.0)));
	else
		tmp = Float64(abs(Float64(sqrt(Float64(x * x)) * 2.0)) / sqrt(pi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (abs(x) * abs(x)) * abs(x);
	t_1 = (t_0 * abs(x)) * abs(x);
	tmp = 0.0;
	if (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)))))) <= 5e-10)
		tmp = sqrt((1.0 / pi)) * abs((abs(x) * 2.0));
	else
		tmp = abs((sqrt((x * x)) * 2.0)) / sqrt(pi);
	end
	tmp_2 = tmp;
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]}, If[LessEqual[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], 5e-10], N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[Abs[N[(N[Abs[x], $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Abs[N[(N[Sqrt[N[(x * x), $MachinePrecision]], $MachinePrecision] * 2.0), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 (*.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 (PI.f64))) (+.f64 (+.f64 (+.f64 (*.f64 #s(literal 2 binary64) (fabs.f64 x)) (*.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)))) (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 5 binary64)) (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)))) (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 21 binary64)) (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)))))) < 5.00000000000000031e-10

   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%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 68.8%

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

   if 5.00000000000000031e-10 < (fabs.f64 (*.f64 (/.f64 #s(literal 1 binary64) (sqrt.f64 (PI.f64))) (+.f64 (+.f64 (+.f64 (*.f64 #s(literal 2 binary64) (fabs.f64 x)) (*.f64 (/.f64 #s(literal 2 binary64) #s(literal 3 binary64)) (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)))) (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 5 binary64)) (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)))) (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 21 binary64)) (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (*.f64 (fabs.f64 x) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x)) (fabs.f64 x))))))

   1. Initial program 99.8%

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

   3. Applied rewrites 99.4%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 68.3%

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

   8. Applied rewrites 53.5%

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

Alternative 11: 68.8% accurate, 6.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

4. Taylor expanded in x around 0

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

6. Applied rewrites 68.8%

   \[\leadsto \sqrt{\frac{1}{\pi}} \cdot \left|\left|x\right| \cdot \color{blue}{2}\right| \]
7. Add Preprocessing

Alternative 12: 68.3% accurate, 8.3× speedup

Math

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

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 Float64(abs(Float64(abs(x) * 2.0)) / sqrt(pi))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.4%

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

4. Taylor expanded in x around 0

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

6. Applied rewrites 68.3%

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

Reproduce

herbie shell --seed 2025153 
(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)))))))