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

Percentage Accurate: 99.8% → 99.8%

Time: 5.0s

Alternatives: 14

Speedup: 2.1×

• 72.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[x \leq 0.5\]

Math

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

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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.8%

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

Alternative 2: 99.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[Abs[N[(N[(N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * N[(N[(x * 0.047619047619047616), $MachinePrecision] * x + 0.2), $MachinePrecision] + N[(0.6666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * x + N[(2.0 * 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 99.8%

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

5. Applied rewrites 99.8%

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

7. Applied rewrites 99.8%

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

9. Applied rewrites 99.8%

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

Alternative 3: 99.8% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.8%

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

5. Applied rewrites 99.8%

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

7. Applied rewrites 99.8%

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

9. Applied rewrites 99.8%

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

Alternative 4: 99.4% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.8%

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

5. Applied rewrites 99.8%

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

7. Applied rewrites 99.8%

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

9. Applied rewrites 99.4%

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

Alternative 5: 99.1% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.8%

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

5. Applied rewrites 99.4%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 98.6%

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

10. Applied rewrites 99.1%

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

Alternative 6: 98.8% accurate, 2.3× speedup

Math

\[\begin{array}{l} t_0 := \left|x\right| \cdot \left|x\right|\\ t_1 := \left|\left|x\right|\right|\\ \mathbf{if}\;\left|x\right| \leq 520:\\ \;\;\;\;\left|t\_1 \cdot 1.1283791670955126\right|\\ \mathbf{else}:\\ \;\;\;\;\left|0.047619047619047616 \cdot \frac{\left(\left(\left(t\_0 \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot t\_0\right) \cdot t\_1}{\sqrt{\pi}}\right|\\ \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  (let* ((t_0 (* (fabs x) (fabs x))) (t_1 (fabs (fabs x))))
  (if (<= (fabs x) 520.0)
    (fabs (* t_1 1.1283791670955126))
    (fabs
     (*
      0.047619047619047616
      (/ (* (* (* (* t_0 (fabs x)) (fabs x)) t_0) t_1) (sqrt PI)))))))

C

double code(double x) {
	double t_0 = fabs(x) * fabs(x);
	double t_1 = fabs(fabs(x));
	double tmp;
	if (fabs(x) <= 520.0) {
		tmp = fabs((t_1 * 1.1283791670955126));
	} else {
		tmp = fabs((0.047619047619047616 * (((((t_0 * fabs(x)) * fabs(x)) * t_0) * t_1) / sqrt(((double) M_PI)))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = Math.abs(x) * Math.abs(x);
	double t_1 = Math.abs(Math.abs(x));
	double tmp;
	if (Math.abs(x) <= 520.0) {
		tmp = Math.abs((t_1 * 1.1283791670955126));
	} else {
		tmp = Math.abs((0.047619047619047616 * (((((t_0 * Math.abs(x)) * Math.abs(x)) * t_0) * t_1) / Math.sqrt(Math.PI))));
	}
	return tmp;
}

Python

def code(x):
	t_0 = math.fabs(x) * math.fabs(x)
	t_1 = math.fabs(math.fabs(x))
	tmp = 0
	if math.fabs(x) <= 520.0:
		tmp = math.fabs((t_1 * 1.1283791670955126))
	else:
		tmp = math.fabs((0.047619047619047616 * (((((t_0 * math.fabs(x)) * math.fabs(x)) * t_0) * t_1) / math.sqrt(math.pi))))
	return tmp

Julia

function code(x)
	t_0 = Float64(abs(x) * abs(x))
	t_1 = abs(abs(x))
	tmp = 0.0
	if (abs(x) <= 520.0)
		tmp = abs(Float64(t_1 * 1.1283791670955126));
	else
		tmp = abs(Float64(0.047619047619047616 * Float64(Float64(Float64(Float64(Float64(t_0 * abs(x)) * abs(x)) * t_0) * t_1) / sqrt(pi))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = abs(x) * abs(x);
	t_1 = abs(abs(x));
	tmp = 0.0;
	if (abs(x) <= 520.0)
		tmp = abs((t_1 * 1.1283791670955126));
	else
		tmp = abs((0.047619047619047616 * (((((t_0 * abs(x)) * abs(x)) * t_0) * t_1) / sqrt(pi))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Abs[N[Abs[x], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 520.0], N[Abs[N[(t$95$1 * 1.1283791670955126), $MachinePrecision]], $MachinePrecision], N[Abs[N[(0.047619047619047616 * N[(N[(N[(N[(N[(t$95$0 * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * t$95$1), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 520

   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%

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

   4. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-fabs.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-PI.f64 67.8%

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

   6. Applied rewrites 67.8%

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

   7. Evaluated real constant 68.0%

      \[\leadsto \left|2 \cdot \frac{\left|x\right|}{1.772453850905516}\right| \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left|2 \cdot \color{blue}{\frac{\left|x\right|}{\frac{7982422502469483}{4503599627370496}}}\right| \]

      2. count-2-rev N/A

         \[\leadsto \left|\frac{\left|x\right|}{\frac{7982422502469483}{4503599627370496}} + \color{blue}{\frac{\left|x\right|}{\frac{7982422502469483}{4503599627370496}}}\right| \]

      3. lift-/.f64 N/A

         \[\leadsto \left|\frac{\left|x\right|}{\frac{7982422502469483}{4503599627370496}} + \frac{\color{blue}{\left|x\right|}}{\frac{7982422502469483}{4503599627370496}}\right| \]

      4. mult-flip N/A

         \[\leadsto \left|\left|x\right| \cdot \frac{1}{\frac{7982422502469483}{4503599627370496}} + \frac{\color{blue}{\left|x\right|}}{\frac{7982422502469483}{4503599627370496}}\right| \]

      5. lift-/.f64 N/A

         \[\leadsto \left|\left|x\right| \cdot \frac{1}{\frac{7982422502469483}{4503599627370496}} + \frac{\left|x\right|}{\color{blue}{\frac{7982422502469483}{4503599627370496}}}\right| \]

      6. mult-flip N/A

         \[\leadsto \left|\left|x\right| \cdot \frac{1}{\frac{7982422502469483}{4503599627370496}} + \left|x\right| \cdot \color{blue}{\frac{1}{\frac{7982422502469483}{4503599627370496}}}\right| \]

      7. distribute-lft-out N/A

         \[\leadsto \left|\left|x\right| \cdot \color{blue}{\left(\frac{1}{\frac{7982422502469483}{4503599627370496}} + \frac{1}{\frac{7982422502469483}{4503599627370496}}\right)}\right| \]

      8. lower-*.f64 N/A

         \[\leadsto \left|\left|x\right| \cdot \color{blue}{\left(\frac{1}{\frac{7982422502469483}{4503599627370496}} + \frac{1}{\frac{7982422502469483}{4503599627370496}}\right)}\right| \]

      9. metadata-eval N/A

         \[\leadsto \left|\left|x\right| \cdot \left(\frac{4503599627370496}{7982422502469483} + \frac{\color{blue}{1}}{\frac{7982422502469483}{4503599627370496}}\right)\right| \]

      10. metadata-eval N/A

          \[\leadsto \left|\left|x\right| \cdot \left(\frac{4503599627370496}{7982422502469483} + \frac{4503599627370496}{7982422502469483}\right)\right| \]

      11. metadata-eval 68.2%

          \[\leadsto \left|\left|x\right| \cdot 1.1283791670955126\right| \]

   9. Applied rewrites 68.2%

      \[\leadsto \left|\left|x\right| \cdot \color{blue}{1.1283791670955126}\right| \]

   if 520 < 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%

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

   4. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-fabs.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-PI.f64 67.8%

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

   6. Applied rewrites 67.8%

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

   7. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-fabs.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-PI.f64 36.4%

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

   9. Applied rewrites 36.4%

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

       1. lift-pow.f64 N/A

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

       2. metadata-eval N/A

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

       3. pow-add N/A

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

       4. unpow-prod-down N/A

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

       5. lift-*.f64 N/A

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

       6. pow3 N/A

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

       7. lift-*.f64 N/A

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

       8. associate-*l* N/A

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

       9. lift-*.f64 N/A

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

       10. lift-*.f64 N/A

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

       11. lower-*.f64 36.3%

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

   11. Applied rewrites 36.3%

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

Alternative 7: 98.8% accurate, 2.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|x\right| \leq 520:\\ \;\;\;\;\left|\left|\left|x\right|\right| \cdot 1.1283791670955126\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{\left|{\left(\left|x\right|\right)}^{7}\right|}{\sqrt{\pi}} \cdot 0.047619047619047616\right|\\ \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  (if (<= (fabs x) 520.0)
  (fabs (* (fabs (fabs x)) 1.1283791670955126))
  (fabs
   (* (/ (fabs (pow (fabs x) 7.0)) (sqrt PI)) 0.047619047619047616))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 520.0) {
		tmp = fabs((fabs(fabs(x)) * 1.1283791670955126));
	} else {
		tmp = fabs(((fabs(pow(fabs(x), 7.0)) / sqrt(((double) M_PI))) * 0.047619047619047616));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) <= 520.0) {
		tmp = Math.abs((Math.abs(Math.abs(x)) * 1.1283791670955126));
	} else {
		tmp = Math.abs(((Math.abs(Math.pow(Math.abs(x), 7.0)) / Math.sqrt(Math.PI)) * 0.047619047619047616));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) <= 520.0:
		tmp = math.fabs((math.fabs(math.fabs(x)) * 1.1283791670955126))
	else:
		tmp = math.fabs(((math.fabs(math.pow(math.fabs(x), 7.0)) / math.sqrt(math.pi)) * 0.047619047619047616))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 520.0)
		tmp = abs(Float64(abs(abs(x)) * 1.1283791670955126));
	else
		tmp = abs(Float64(Float64(abs((abs(x) ^ 7.0)) / sqrt(pi)) * 0.047619047619047616));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) <= 520.0)
		tmp = abs((abs(abs(x)) * 1.1283791670955126));
	else
		tmp = abs(((abs((abs(x) ^ 7.0)) / sqrt(pi)) * 0.047619047619047616));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 520.0], N[Abs[N[(N[Abs[N[Abs[x], $MachinePrecision]], $MachinePrecision] * 1.1283791670955126), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(N[Abs[N[Power[N[Abs[x], $MachinePrecision], 7.0], $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * 0.047619047619047616), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 520

   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%

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

   4. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-fabs.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-PI.f64 67.8%

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

   6. Applied rewrites 67.8%

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

   7. Evaluated real constant 68.0%

      \[\leadsto \left|2 \cdot \frac{\left|x\right|}{1.772453850905516}\right| \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left|2 \cdot \color{blue}{\frac{\left|x\right|}{\frac{7982422502469483}{4503599627370496}}}\right| \]

      2. count-2-rev N/A

         \[\leadsto \left|\frac{\left|x\right|}{\frac{7982422502469483}{4503599627370496}} + \color{blue}{\frac{\left|x\right|}{\frac{7982422502469483}{4503599627370496}}}\right| \]

      3. lift-/.f64 N/A

         \[\leadsto \left|\frac{\left|x\right|}{\frac{7982422502469483}{4503599627370496}} + \frac{\color{blue}{\left|x\right|}}{\frac{7982422502469483}{4503599627370496}}\right| \]

      4. mult-flip N/A

         \[\leadsto \left|\left|x\right| \cdot \frac{1}{\frac{7982422502469483}{4503599627370496}} + \frac{\color{blue}{\left|x\right|}}{\frac{7982422502469483}{4503599627370496}}\right| \]

      5. lift-/.f64 N/A

         \[\leadsto \left|\left|x\right| \cdot \frac{1}{\frac{7982422502469483}{4503599627370496}} + \frac{\left|x\right|}{\color{blue}{\frac{7982422502469483}{4503599627370496}}}\right| \]

      6. mult-flip N/A

         \[\leadsto \left|\left|x\right| \cdot \frac{1}{\frac{7982422502469483}{4503599627370496}} + \left|x\right| \cdot \color{blue}{\frac{1}{\frac{7982422502469483}{4503599627370496}}}\right| \]

      7. distribute-lft-out N/A

         \[\leadsto \left|\left|x\right| \cdot \color{blue}{\left(\frac{1}{\frac{7982422502469483}{4503599627370496}} + \frac{1}{\frac{7982422502469483}{4503599627370496}}\right)}\right| \]

      8. lower-*.f64 N/A

         \[\leadsto \left|\left|x\right| \cdot \color{blue}{\left(\frac{1}{\frac{7982422502469483}{4503599627370496}} + \frac{1}{\frac{7982422502469483}{4503599627370496}}\right)}\right| \]

      9. metadata-eval N/A

         \[\leadsto \left|\left|x\right| \cdot \left(\frac{4503599627370496}{7982422502469483} + \frac{\color{blue}{1}}{\frac{7982422502469483}{4503599627370496}}\right)\right| \]

      10. metadata-eval N/A

          \[\leadsto \left|\left|x\right| \cdot \left(\frac{4503599627370496}{7982422502469483} + \frac{4503599627370496}{7982422502469483}\right)\right| \]

      11. metadata-eval 68.2%

          \[\leadsto \left|\left|x\right| \cdot 1.1283791670955126\right| \]

   9. Applied rewrites 68.2%

      \[\leadsto \left|\left|x\right| \cdot \color{blue}{1.1283791670955126}\right| \]

   if 520 < 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%

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

   4. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-fabs.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-PI.f64 67.8%

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

   6. Applied rewrites 67.8%

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

   7. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower-pow.f64 N/A

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

      5. lower-fabs.f64 N/A

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

      6. lower-sqrt.f64 N/A

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

      7. lower-PI.f64 36.4%

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

   9. Applied rewrites 36.4%

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

       1. lift-*.f64 N/A

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

       2. *-commutative N/A

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

       3. lower-*.f64 36.4%

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

   11. Applied rewrites 36.4%

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

Alternative 8: 98.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|x\right| \leq 520:\\ \;\;\;\;\left|\left|\left|x\right|\right| \cdot 1.1283791670955126\right|\\ \mathbf{else}:\\ \;\;\;\;\frac{\left|0.047619047619047616 \cdot {\left(\left|x\right|\right)}^{7}\right|}{\sqrt{\pi}}\\ \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  (if (<= (fabs x) 520.0)
  (fabs (* (fabs (fabs x)) 1.1283791670955126))
  (/ (fabs (* 0.047619047619047616 (pow (fabs x) 7.0))) (sqrt PI))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 520.0) {
		tmp = fabs((fabs(fabs(x)) * 1.1283791670955126));
	} else {
		tmp = fabs((0.047619047619047616 * pow(fabs(x), 7.0))) / sqrt(((double) M_PI));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) <= 520.0) {
		tmp = Math.abs((Math.abs(Math.abs(x)) * 1.1283791670955126));
	} else {
		tmp = Math.abs((0.047619047619047616 * Math.pow(Math.abs(x), 7.0))) / Math.sqrt(Math.PI);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) <= 520.0:
		tmp = math.fabs((math.fabs(math.fabs(x)) * 1.1283791670955126))
	else:
		tmp = math.fabs((0.047619047619047616 * math.pow(math.fabs(x), 7.0))) / math.sqrt(math.pi)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 520.0)
		tmp = abs(Float64(abs(abs(x)) * 1.1283791670955126));
	else
		tmp = Float64(abs(Float64(0.047619047619047616 * (abs(x) ^ 7.0))) / sqrt(pi));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) <= 520.0)
		tmp = abs((abs(abs(x)) * 1.1283791670955126));
	else
		tmp = abs((0.047619047619047616 * (abs(x) ^ 7.0))) / sqrt(pi);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 520.0], N[Abs[N[(N[Abs[N[Abs[x], $MachinePrecision]], $MachinePrecision] * 1.1283791670955126), $MachinePrecision]], $MachinePrecision], N[(N[Abs[N[(0.047619047619047616 * N[Power[N[Abs[x], $MachinePrecision], 7.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 520

   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%

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

   4. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-fabs.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-PI.f64 67.8%

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

   6. Applied rewrites 67.8%

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

   7. Evaluated real constant 68.0%

      \[\leadsto \left|2 \cdot \frac{\left|x\right|}{1.772453850905516}\right| \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left|2 \cdot \color{blue}{\frac{\left|x\right|}{\frac{7982422502469483}{4503599627370496}}}\right| \]

      2. count-2-rev N/A

         \[\leadsto \left|\frac{\left|x\right|}{\frac{7982422502469483}{4503599627370496}} + \color{blue}{\frac{\left|x\right|}{\frac{7982422502469483}{4503599627370496}}}\right| \]

      3. lift-/.f64 N/A

         \[\leadsto \left|\frac{\left|x\right|}{\frac{7982422502469483}{4503599627370496}} + \frac{\color{blue}{\left|x\right|}}{\frac{7982422502469483}{4503599627370496}}\right| \]

      4. mult-flip N/A

         \[\leadsto \left|\left|x\right| \cdot \frac{1}{\frac{7982422502469483}{4503599627370496}} + \frac{\color{blue}{\left|x\right|}}{\frac{7982422502469483}{4503599627370496}}\right| \]

      5. lift-/.f64 N/A

         \[\leadsto \left|\left|x\right| \cdot \frac{1}{\frac{7982422502469483}{4503599627370496}} + \frac{\left|x\right|}{\color{blue}{\frac{7982422502469483}{4503599627370496}}}\right| \]

      6. mult-flip N/A

         \[\leadsto \left|\left|x\right| \cdot \frac{1}{\frac{7982422502469483}{4503599627370496}} + \left|x\right| \cdot \color{blue}{\frac{1}{\frac{7982422502469483}{4503599627370496}}}\right| \]

      7. distribute-lft-out N/A

         \[\leadsto \left|\left|x\right| \cdot \color{blue}{\left(\frac{1}{\frac{7982422502469483}{4503599627370496}} + \frac{1}{\frac{7982422502469483}{4503599627370496}}\right)}\right| \]

      8. lower-*.f64 N/A

         \[\leadsto \left|\left|x\right| \cdot \color{blue}{\left(\frac{1}{\frac{7982422502469483}{4503599627370496}} + \frac{1}{\frac{7982422502469483}{4503599627370496}}\right)}\right| \]

      9. metadata-eval N/A

         \[\leadsto \left|\left|x\right| \cdot \left(\frac{4503599627370496}{7982422502469483} + \frac{\color{blue}{1}}{\frac{7982422502469483}{4503599627370496}}\right)\right| \]

      10. metadata-eval N/A

          \[\leadsto \left|\left|x\right| \cdot \left(\frac{4503599627370496}{7982422502469483} + \frac{4503599627370496}{7982422502469483}\right)\right| \]

      11. metadata-eval 68.2%

          \[\leadsto \left|\left|x\right| \cdot 1.1283791670955126\right| \]

   9. Applied rewrites 68.2%

      \[\leadsto \left|\left|x\right| \cdot \color{blue}{1.1283791670955126}\right| \]

   if 520 < x

   1. Initial program 99.8%

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

   3. Applied rewrites 99.8%

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

   5. Applied rewrites 99.4%

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

   6. Taylor expanded in x around inf

      \[\leadsto \frac{\left|\color{blue}{\frac{1}{21} \cdot {x}^{7}}\right|}{\sqrt{\pi}} \]
   7. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\left|\frac{1}{21} \cdot \color{blue}{{x}^{7}}\right|}{\sqrt{\pi}} \]

      2. lower-pow.f64 36.4%

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

   8. Applied rewrites 36.4%

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

Alternative 9: 98.6% accurate, 2.9× speedup

Math

\[\left|\frac{\mathsf{fma}\left(0.047619047619047616, {\left(\left|x\right|\right)}^{7}, 2 \cdot \left|x\right|\right)}{1.772453850905516}\right| \]

FPCore

(FPCore (x)
  :precision binary64
  (fabs
 (/
  (fma 0.047619047619047616 (pow (fabs x) 7.0) (* 2.0 (fabs x)))
  1.772453850905516)))

C

double code(double x) {
	return fabs((fma(0.047619047619047616, pow(fabs(x), 7.0), (2.0 * fabs(x))) / 1.772453850905516));
}

Julia

function code(x)
	return abs(Float64(fma(0.047619047619047616, (abs(x) ^ 7.0), Float64(2.0 * abs(x))) / 1.772453850905516))
end

Wolfram

code[x_] := N[Abs[N[(N[(0.047619047619047616 * N[Power[N[Abs[x], $MachinePrecision], 7.0], $MachinePrecision] + N[(2.0 * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 1.772453850905516), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.8%

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

4. Taylor expanded in x around 0

   \[\leadsto \left|\color{blue}{\frac{\frac{1}{21} \cdot {\left(\left|x\right|\right)}^{7} + 2 \cdot \left|x\right|}{\sqrt{\pi}}}\right| \]
5. Step-by-step derivation

   1. lower-/.f64 N/A

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

   2. lower-fma.f64 N/A

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

   3. lower-pow.f64 N/A

      \[\leadsto \left|\frac{\mathsf{fma}\left(\frac{1}{21}, {\left(\left|x\right|\right)}^{7}, 2 \cdot \left|x\right|\right)}{\sqrt{\mathsf{PI}\left(\right)}}\right| \]

   4. lower-fabs.f64 N/A

      \[\leadsto \left|\frac{\mathsf{fma}\left(\frac{1}{21}, {\left(\left|x\right|\right)}^{7}, 2 \cdot \left|x\right|\right)}{\sqrt{\mathsf{PI}\left(\right)}}\right| \]

   5. lower-*.f64 N/A

      \[\leadsto \left|\frac{\mathsf{fma}\left(\frac{1}{21}, {\left(\left|x\right|\right)}^{7}, 2 \cdot \left|x\right|\right)}{\sqrt{\mathsf{PI}\left(\right)}}\right| \]

   6. lower-fabs.f64 N/A

      \[\leadsto \left|\frac{\mathsf{fma}\left(\frac{1}{21}, {\left(\left|x\right|\right)}^{7}, 2 \cdot \left|x\right|\right)}{\sqrt{\mathsf{PI}\left(\right)}}\right| \]

   7. lower-sqrt.f64 N/A

      \[\leadsto \left|\frac{\mathsf{fma}\left(\frac{1}{21}, {\left(\left|x\right|\right)}^{7}, 2 \cdot \left|x\right|\right)}{\sqrt{\mathsf{PI}\left(\right)}}\right| \]

   8. lower-PI.f64 98.4%

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

6. Applied rewrites 98.4%

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

7. Evaluated real constant 98.6%

   \[\leadsto \left|\frac{\mathsf{fma}\left(0.047619047619047616, {\left(\left|x\right|\right)}^{7}, 2 \cdot \left|x\right|\right)}{1.772453850905516}\right| \]
8. Add Preprocessing

Alternative 10: 91.3% accurate, 2.9× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|x\right| \leq 2 \cdot 10^{-9}:\\ \;\;\;\;\left|\left|\left|x\right|\right| \cdot 1.1283791670955126\right|\\ \mathbf{else}:\\ \;\;\;\;\left|2 \cdot \frac{\sqrt{\sqrt{\left(\left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\right) \cdot \left|x\right|}}}{\sqrt{\pi}}\right|\\ \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  (if (<= (fabs x) 2e-9)
  (fabs (* (fabs (fabs x)) 1.1283791670955126))
  (fabs
   (*
    2.0
    (/
     (sqrt (sqrt (* (* (* (fabs x) (fabs x)) (fabs x)) (fabs x))))
     (sqrt PI))))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 2e-9) {
		tmp = fabs((fabs(fabs(x)) * 1.1283791670955126));
	} else {
		tmp = fabs((2.0 * (sqrt(sqrt((((fabs(x) * fabs(x)) * fabs(x)) * fabs(x)))) / sqrt(((double) M_PI)))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) <= 2e-9) {
		tmp = Math.abs((Math.abs(Math.abs(x)) * 1.1283791670955126));
	} else {
		tmp = Math.abs((2.0 * (Math.sqrt(Math.sqrt((((Math.abs(x) * Math.abs(x)) * Math.abs(x)) * Math.abs(x)))) / Math.sqrt(Math.PI))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) <= 2e-9:
		tmp = math.fabs((math.fabs(math.fabs(x)) * 1.1283791670955126))
	else:
		tmp = math.fabs((2.0 * (math.sqrt(math.sqrt((((math.fabs(x) * math.fabs(x)) * math.fabs(x)) * math.fabs(x)))) / math.sqrt(math.pi))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 2e-9)
		tmp = abs(Float64(abs(abs(x)) * 1.1283791670955126));
	else
		tmp = abs(Float64(2.0 * Float64(sqrt(sqrt(Float64(Float64(Float64(abs(x) * abs(x)) * abs(x)) * abs(x)))) / sqrt(pi))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) <= 2e-9)
		tmp = abs((abs(abs(x)) * 1.1283791670955126));
	else
		tmp = abs((2.0 * (sqrt(sqrt((((abs(x) * abs(x)) * abs(x)) * abs(x)))) / sqrt(pi))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 2e-9], N[Abs[N[(N[Abs[N[Abs[x], $MachinePrecision]], $MachinePrecision] * 1.1283791670955126), $MachinePrecision]], $MachinePrecision], N[Abs[N[(2.0 * N[(N[Sqrt[N[Sqrt[N[(N[(N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.0000000000000001e-9

   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%

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

   4. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-fabs.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-PI.f64 67.8%

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

   6. Applied rewrites 67.8%

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

   7. Evaluated real constant 68.0%

      \[\leadsto \left|2 \cdot \frac{\left|x\right|}{1.772453850905516}\right| \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left|2 \cdot \color{blue}{\frac{\left|x\right|}{\frac{7982422502469483}{4503599627370496}}}\right| \]

      2. count-2-rev N/A

         \[\leadsto \left|\frac{\left|x\right|}{\frac{7982422502469483}{4503599627370496}} + \color{blue}{\frac{\left|x\right|}{\frac{7982422502469483}{4503599627370496}}}\right| \]

      3. lift-/.f64 N/A

         \[\leadsto \left|\frac{\left|x\right|}{\frac{7982422502469483}{4503599627370496}} + \frac{\color{blue}{\left|x\right|}}{\frac{7982422502469483}{4503599627370496}}\right| \]

      4. mult-flip N/A

         \[\leadsto \left|\left|x\right| \cdot \frac{1}{\frac{7982422502469483}{4503599627370496}} + \frac{\color{blue}{\left|x\right|}}{\frac{7982422502469483}{4503599627370496}}\right| \]

      5. lift-/.f64 N/A

         \[\leadsto \left|\left|x\right| \cdot \frac{1}{\frac{7982422502469483}{4503599627370496}} + \frac{\left|x\right|}{\color{blue}{\frac{7982422502469483}{4503599627370496}}}\right| \]

      6. mult-flip N/A

         \[\leadsto \left|\left|x\right| \cdot \frac{1}{\frac{7982422502469483}{4503599627370496}} + \left|x\right| \cdot \color{blue}{\frac{1}{\frac{7982422502469483}{4503599627370496}}}\right| \]

      7. distribute-lft-out N/A

         \[\leadsto \left|\left|x\right| \cdot \color{blue}{\left(\frac{1}{\frac{7982422502469483}{4503599627370496}} + \frac{1}{\frac{7982422502469483}{4503599627370496}}\right)}\right| \]

      8. lower-*.f64 N/A

         \[\leadsto \left|\left|x\right| \cdot \color{blue}{\left(\frac{1}{\frac{7982422502469483}{4503599627370496}} + \frac{1}{\frac{7982422502469483}{4503599627370496}}\right)}\right| \]

      9. metadata-eval N/A

         \[\leadsto \left|\left|x\right| \cdot \left(\frac{4503599627370496}{7982422502469483} + \frac{\color{blue}{1}}{\frac{7982422502469483}{4503599627370496}}\right)\right| \]

      10. metadata-eval N/A

          \[\leadsto \left|\left|x\right| \cdot \left(\frac{4503599627370496}{7982422502469483} + \frac{4503599627370496}{7982422502469483}\right)\right| \]

      11. metadata-eval 68.2%

          \[\leadsto \left|\left|x\right| \cdot 1.1283791670955126\right| \]

   9. Applied rewrites 68.2%

      \[\leadsto \left|\left|x\right| \cdot \color{blue}{1.1283791670955126}\right| \]

   if 2.0000000000000001e-9 < 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%

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

   4. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-fabs.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-PI.f64 67.8%

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

   6. Applied rewrites 67.8%

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

      1. lift-fabs.f64 N/A

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

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

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

      3. lift-*.f64 N/A

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

      4. lower-sqrt.f64 54.2%

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

   8. Applied rewrites 54.2%

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

      1. rem-square-sqrt N/A

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

      2. sqrt-unprod N/A

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

      3. lift-*.f64 N/A

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

      4. associate-*l* N/A

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

      5. lift-*.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. lower-sqrt.f64 45.1%

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

   10. Applied rewrites 45.1%

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

Alternative 11: 89.0% accurate, 4.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.8%

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

3. Applied rewrites 99.4%

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

4. Taylor expanded in x around 0

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

   1. lower-fma.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-*.f64 N/A

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

   4. lower-pow.f64 N/A

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

   5. lower-fabs.f64 N/A

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

   6. lower-sqrt.f64 N/A

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

   7. lower-PI.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 N/A

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

   10. lower-fabs.f64 N/A

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

   11. lower-sqrt.f64 N/A

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

   12. lower-PI.f64 89.0%

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

6. Applied rewrites 89.0%

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

   1. lift-fma.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. lift-*.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. associate-*r/ N/A

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

   7. lift-*.f64 N/A

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

   8. div-add-rev N/A

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

8. Applied rewrites 89.0%

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

Alternative 12: 84.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_0 := \left(\left|x\right| \cdot \left|x\right|\right) \cdot \left|x\right|\\ t_1 := \left(t\_0 \cdot \left|x\right|\right) \cdot \left|x\right|\\ \mathbf{if}\;\left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + \frac{2}{3} \cdot t\_0\right) + \frac{1}{5} \cdot t\_1\right) + \frac{1}{21} \cdot \left(\left(t\_1 \cdot \left|x\right|\right) \cdot \left|x\right|\right)\right)\right| \leq 2 \cdot 10^{-9}:\\ \;\;\;\;\left|\left|x\right| \cdot 1.1283791670955126\right|\\ \mathbf{else}:\\ \;\;\;\;\left|2 \cdot \frac{\sqrt{x \cdot x}}{\sqrt{\pi}}\right|\\ \end{array} \]

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))))))
       2e-9)
    (fabs (* (fabs x) 1.1283791670955126))
    (fabs (* 2.0 (/ (sqrt (* x x)) (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)))))) <= 2e-9) {
		tmp = fabs((fabs(x) * 1.1283791670955126));
	} else {
		tmp = fabs((2.0 * (sqrt((x * x)) / 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)))))) <= 2e-9) {
		tmp = Math.abs((Math.abs(x) * 1.1283791670955126));
	} else {
		tmp = Math.abs((2.0 * (Math.sqrt((x * x)) / 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)))))) <= 2e-9:
		tmp = math.fabs((math.fabs(x) * 1.1283791670955126))
	else:
		tmp = math.fabs((2.0 * (math.sqrt((x * x)) / 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)))))) <= 2e-9)
		tmp = abs(Float64(abs(x) * 1.1283791670955126));
	else
		tmp = abs(Float64(2.0 * Float64(sqrt(Float64(x * x)) / 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)))))) <= 2e-9)
		tmp = abs((abs(x) * 1.1283791670955126));
	else
		tmp = abs((2.0 * (sqrt((x * x)) / 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], 2e-9], N[Abs[N[(N[Abs[x], $MachinePrecision] * 1.1283791670955126), $MachinePrecision]], $MachinePrecision], N[Abs[N[(2.0 * N[(N[Sqrt[N[(x * x), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $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)))))) < 2.0000000000000001e-9

   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%

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

   4. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-fabs.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-PI.f64 67.8%

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

   6. Applied rewrites 67.8%

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

   7. Evaluated real constant 68.0%

      \[\leadsto \left|2 \cdot \frac{\left|x\right|}{1.772453850905516}\right| \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left|2 \cdot \color{blue}{\frac{\left|x\right|}{\frac{7982422502469483}{4503599627370496}}}\right| \]

      2. count-2-rev N/A

         \[\leadsto \left|\frac{\left|x\right|}{\frac{7982422502469483}{4503599627370496}} + \color{blue}{\frac{\left|x\right|}{\frac{7982422502469483}{4503599627370496}}}\right| \]

      3. lift-/.f64 N/A

         \[\leadsto \left|\frac{\left|x\right|}{\frac{7982422502469483}{4503599627370496}} + \frac{\color{blue}{\left|x\right|}}{\frac{7982422502469483}{4503599627370496}}\right| \]

      4. mult-flip N/A

         \[\leadsto \left|\left|x\right| \cdot \frac{1}{\frac{7982422502469483}{4503599627370496}} + \frac{\color{blue}{\left|x\right|}}{\frac{7982422502469483}{4503599627370496}}\right| \]

      5. lift-/.f64 N/A

         \[\leadsto \left|\left|x\right| \cdot \frac{1}{\frac{7982422502469483}{4503599627370496}} + \frac{\left|x\right|}{\color{blue}{\frac{7982422502469483}{4503599627370496}}}\right| \]

      6. mult-flip N/A

         \[\leadsto \left|\left|x\right| \cdot \frac{1}{\frac{7982422502469483}{4503599627370496}} + \left|x\right| \cdot \color{blue}{\frac{1}{\frac{7982422502469483}{4503599627370496}}}\right| \]

      7. distribute-lft-out N/A

         \[\leadsto \left|\left|x\right| \cdot \color{blue}{\left(\frac{1}{\frac{7982422502469483}{4503599627370496}} + \frac{1}{\frac{7982422502469483}{4503599627370496}}\right)}\right| \]

      8. lower-*.f64 N/A

         \[\leadsto \left|\left|x\right| \cdot \color{blue}{\left(\frac{1}{\frac{7982422502469483}{4503599627370496}} + \frac{1}{\frac{7982422502469483}{4503599627370496}}\right)}\right| \]

      9. metadata-eval N/A

         \[\leadsto \left|\left|x\right| \cdot \left(\frac{4503599627370496}{7982422502469483} + \frac{\color{blue}{1}}{\frac{7982422502469483}{4503599627370496}}\right)\right| \]

      10. metadata-eval N/A

          \[\leadsto \left|\left|x\right| \cdot \left(\frac{4503599627370496}{7982422502469483} + \frac{4503599627370496}{7982422502469483}\right)\right| \]

      11. metadata-eval 68.2%

          \[\leadsto \left|\left|x\right| \cdot 1.1283791670955126\right| \]

   9. Applied rewrites 68.2%

      \[\leadsto \left|\left|x\right| \cdot \color{blue}{1.1283791670955126}\right| \]

   if 2.0000000000000001e-9 < (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%

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

   4. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-fabs.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-PI.f64 67.8%

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

   6. Applied rewrites 67.8%

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

      1. lift-fabs.f64 N/A

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

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

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

      3. lift-*.f64 N/A

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

      4. lower-sqrt.f64 54.2%

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

   8. Applied rewrites 54.2%

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

Alternative 13: 84.0% accurate, 4.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\left|x\right| \leq 5 \cdot 10^{+125}:\\ \;\;\;\;\left|\left|\left|x\right|\right| \cdot 1.1283791670955126\right|\\ \mathbf{else}:\\ \;\;\;\;\left|2 \cdot \frac{\sqrt{\left|x\right| \cdot \left|x\right|}}{1.772453850905516}\right|\\ \end{array} \]

FPCore

(FPCore (x)
  :precision binary64
  (if (<= (fabs x) 5e+125)
  (fabs (* (fabs (fabs x)) 1.1283791670955126))
  (fabs (* 2.0 (/ (sqrt (* (fabs x) (fabs x))) 1.772453850905516)))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 5e+125) {
		tmp = fabs((fabs(fabs(x)) * 1.1283791670955126));
	} else {
		tmp = fabs((2.0 * (sqrt((fabs(x) * fabs(x))) / 1.772453850905516)));
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8) :: tmp
    if (abs(x) <= 5d+125) then
        tmp = abs((abs(abs(x)) * 1.1283791670955126d0))
    else
        tmp = abs((2.0d0 * (sqrt((abs(x) * abs(x))) / 1.772453850905516d0)))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) <= 5e+125) {
		tmp = Math.abs((Math.abs(Math.abs(x)) * 1.1283791670955126));
	} else {
		tmp = Math.abs((2.0 * (Math.sqrt((Math.abs(x) * Math.abs(x))) / 1.772453850905516)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) <= 5e+125:
		tmp = math.fabs((math.fabs(math.fabs(x)) * 1.1283791670955126))
	else:
		tmp = math.fabs((2.0 * (math.sqrt((math.fabs(x) * math.fabs(x))) / 1.772453850905516)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 5e+125)
		tmp = abs(Float64(abs(abs(x)) * 1.1283791670955126));
	else
		tmp = abs(Float64(2.0 * Float64(sqrt(Float64(abs(x) * abs(x))) / 1.772453850905516)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) <= 5e+125)
		tmp = abs((abs(abs(x)) * 1.1283791670955126));
	else
		tmp = abs((2.0 * (sqrt((abs(x) * abs(x))) / 1.772453850905516)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 5e+125], N[Abs[N[(N[Abs[N[Abs[x], $MachinePrecision]], $MachinePrecision] * 1.1283791670955126), $MachinePrecision]], $MachinePrecision], N[Abs[N[(2.0 * N[(N[Sqrt[N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / 1.772453850905516), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.9999999999999996e125

   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%

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

   4. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-fabs.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-PI.f64 67.8%

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

   6. Applied rewrites 67.8%

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

   7. Evaluated real constant 68.0%

      \[\leadsto \left|2 \cdot \frac{\left|x\right|}{1.772453850905516}\right| \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \left|2 \cdot \color{blue}{\frac{\left|x\right|}{\frac{7982422502469483}{4503599627370496}}}\right| \]

      2. count-2-rev N/A

         \[\leadsto \left|\frac{\left|x\right|}{\frac{7982422502469483}{4503599627370496}} + \color{blue}{\frac{\left|x\right|}{\frac{7982422502469483}{4503599627370496}}}\right| \]

      3. lift-/.f64 N/A

         \[\leadsto \left|\frac{\left|x\right|}{\frac{7982422502469483}{4503599627370496}} + \frac{\color{blue}{\left|x\right|}}{\frac{7982422502469483}{4503599627370496}}\right| \]

      4. mult-flip N/A

         \[\leadsto \left|\left|x\right| \cdot \frac{1}{\frac{7982422502469483}{4503599627370496}} + \frac{\color{blue}{\left|x\right|}}{\frac{7982422502469483}{4503599627370496}}\right| \]

      5. lift-/.f64 N/A

         \[\leadsto \left|\left|x\right| \cdot \frac{1}{\frac{7982422502469483}{4503599627370496}} + \frac{\left|x\right|}{\color{blue}{\frac{7982422502469483}{4503599627370496}}}\right| \]

      6. mult-flip N/A

         \[\leadsto \left|\left|x\right| \cdot \frac{1}{\frac{7982422502469483}{4503599627370496}} + \left|x\right| \cdot \color{blue}{\frac{1}{\frac{7982422502469483}{4503599627370496}}}\right| \]

      7. distribute-lft-out N/A

         \[\leadsto \left|\left|x\right| \cdot \color{blue}{\left(\frac{1}{\frac{7982422502469483}{4503599627370496}} + \frac{1}{\frac{7982422502469483}{4503599627370496}}\right)}\right| \]

      8. lower-*.f64 N/A

         \[\leadsto \left|\left|x\right| \cdot \color{blue}{\left(\frac{1}{\frac{7982422502469483}{4503599627370496}} + \frac{1}{\frac{7982422502469483}{4503599627370496}}\right)}\right| \]

      9. metadata-eval N/A

         \[\leadsto \left|\left|x\right| \cdot \left(\frac{4503599627370496}{7982422502469483} + \frac{\color{blue}{1}}{\frac{7982422502469483}{4503599627370496}}\right)\right| \]

      10. metadata-eval N/A

          \[\leadsto \left|\left|x\right| \cdot \left(\frac{4503599627370496}{7982422502469483} + \frac{4503599627370496}{7982422502469483}\right)\right| \]

      11. metadata-eval 68.2%

          \[\leadsto \left|\left|x\right| \cdot 1.1283791670955126\right| \]

   9. Applied rewrites 68.2%

      \[\leadsto \left|\left|x\right| \cdot \color{blue}{1.1283791670955126}\right| \]

   if 4.9999999999999996e125 < 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%

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

   4. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-fabs.f64 N/A

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

      4. lower-sqrt.f64 N/A

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

      5. lower-PI.f64 67.8%

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

   6. Applied rewrites 67.8%

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

      1. lift-fabs.f64 N/A

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

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

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

      3. lift-*.f64 N/A

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

      4. lower-sqrt.f64 54.2%

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

   8. Applied rewrites 54.2%

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

   9. Evaluated real constant 54.3%

      \[\leadsto \left|2 \cdot \frac{\sqrt{x \cdot x}}{1.772453850905516}\right| \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 68.2% accurate, 15.7× speedup

Math

\[\left|\left|x\right| \cdot 1.1283791670955126\right| \]

FPCore

(FPCore (x)
  :precision binary64
  (fabs (* (fabs x) 1.1283791670955126)))

C

double code(double x) {
	return fabs((fabs(x) * 1.1283791670955126));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    code = abs((abs(x) * 1.1283791670955126d0))
end function

Java

public static double code(double x) {
	return Math.abs((Math.abs(x) * 1.1283791670955126));
}

Python

def code(x):
	return math.fabs((math.fabs(x) * 1.1283791670955126))

Julia

function code(x)
	return abs(Float64(abs(x) * 1.1283791670955126))
end

MATLAB

function tmp = code(x)
	tmp = abs((abs(x) * 1.1283791670955126));
end

Wolfram

code[x_] := N[Abs[N[(N[Abs[x], $MachinePrecision] * 1.1283791670955126), $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%

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

4. Taylor expanded in x around 0

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

   1. lower-*.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-fabs.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-PI.f64 67.8%

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

6. Applied rewrites 67.8%

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

7. Evaluated real constant 68.0%

   \[\leadsto \left|2 \cdot \frac{\left|x\right|}{1.772453850905516}\right| \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \left|2 \cdot \color{blue}{\frac{\left|x\right|}{\frac{7982422502469483}{4503599627370496}}}\right| \]

   2. count-2-rev N/A

      \[\leadsto \left|\frac{\left|x\right|}{\frac{7982422502469483}{4503599627370496}} + \color{blue}{\frac{\left|x\right|}{\frac{7982422502469483}{4503599627370496}}}\right| \]

   3. lift-/.f64 N/A

      \[\leadsto \left|\frac{\left|x\right|}{\frac{7982422502469483}{4503599627370496}} + \frac{\color{blue}{\left|x\right|}}{\frac{7982422502469483}{4503599627370496}}\right| \]

   4. mult-flip N/A

      \[\leadsto \left|\left|x\right| \cdot \frac{1}{\frac{7982422502469483}{4503599627370496}} + \frac{\color{blue}{\left|x\right|}}{\frac{7982422502469483}{4503599627370496}}\right| \]

   5. lift-/.f64 N/A

      \[\leadsto \left|\left|x\right| \cdot \frac{1}{\frac{7982422502469483}{4503599627370496}} + \frac{\left|x\right|}{\color{blue}{\frac{7982422502469483}{4503599627370496}}}\right| \]

   6. mult-flip N/A

      \[\leadsto \left|\left|x\right| \cdot \frac{1}{\frac{7982422502469483}{4503599627370496}} + \left|x\right| \cdot \color{blue}{\frac{1}{\frac{7982422502469483}{4503599627370496}}}\right| \]

   7. distribute-lft-out N/A

      \[\leadsto \left|\left|x\right| \cdot \color{blue}{\left(\frac{1}{\frac{7982422502469483}{4503599627370496}} + \frac{1}{\frac{7982422502469483}{4503599627370496}}\right)}\right| \]

   8. lower-*.f64 N/A

      \[\leadsto \left|\left|x\right| \cdot \color{blue}{\left(\frac{1}{\frac{7982422502469483}{4503599627370496}} + \frac{1}{\frac{7982422502469483}{4503599627370496}}\right)}\right| \]

   9. metadata-eval N/A

      \[\leadsto \left|\left|x\right| \cdot \left(\frac{4503599627370496}{7982422502469483} + \frac{\color{blue}{1}}{\frac{7982422502469483}{4503599627370496}}\right)\right| \]

   10. metadata-eval N/A

       \[\leadsto \left|\left|x\right| \cdot \left(\frac{4503599627370496}{7982422502469483} + \frac{4503599627370496}{7982422502469483}\right)\right| \]

   11. metadata-eval 68.2%

       \[\leadsto \left|\left|x\right| \cdot 1.1283791670955126\right| \]

9. Applied rewrites 68.2%

   \[\leadsto \left|\left|x\right| \cdot \color{blue}{1.1283791670955126}\right| \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025212 
(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)))))))