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

Percentage Accurate: 99.8% → 99.9%

Time: 8.7s

Alternatives: 12

Speedup: 3.7×

• 29.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[x \leq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \left|\left(x \cdot {\pi}^{-0.5}\right) \cdot \left(\left(0.6666666666666666 \cdot \left(x \cdot x\right) + 2\right) + \left(0.2 \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fabs
  (*
   (* x (pow PI -0.5))
   (+
    (+ (* 0.6666666666666666 (* x x)) 2.0)
    (+ (* 0.2 (pow x 4.0)) (* 0.047619047619047616 (pow x 6.0)))))))

C

double code(double x) {
	return fabs(((x * pow(((double) M_PI), -0.5)) * (((0.6666666666666666 * (x * x)) + 2.0) + ((0.2 * pow(x, 4.0)) + (0.047619047619047616 * pow(x, 6.0))))));
}

Java

public static double code(double x) {
	return Math.abs(((x * Math.pow(Math.PI, -0.5)) * (((0.6666666666666666 * (x * x)) + 2.0) + ((0.2 * Math.pow(x, 4.0)) + (0.047619047619047616 * Math.pow(x, 6.0))))));
}

Python

def code(x):
	return math.fabs(((x * math.pow(math.pi, -0.5)) * (((0.6666666666666666 * (x * x)) + 2.0) + ((0.2 * math.pow(x, 4.0)) + (0.047619047619047616 * math.pow(x, 6.0))))))

Julia

function code(x)
	return abs(Float64(Float64(x * (pi ^ -0.5)) * Float64(Float64(Float64(0.6666666666666666 * Float64(x * x)) + 2.0) + Float64(Float64(0.2 * (x ^ 4.0)) + Float64(0.047619047619047616 * (x ^ 6.0))))))
end

MATLAB

function tmp = code(x)
	tmp = abs(((x * (pi ^ -0.5)) * (((0.6666666666666666 * (x * x)) + 2.0) + ((0.2 * (x ^ 4.0)) + (0.047619047619047616 * (x ^ 6.0))))));
end

Wolfram

code[x_] := N[Abs[N[(N[(x * N[Power[Pi, -0.5], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(0.6666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] + N[(N[(0.2 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(0.047619047619047616 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.9%

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

3. Simplified 99.5%

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

   1. div-inv 99.9%

      \[\leadsto \left|\color{blue}{\left(\left|x\right| \cdot \frac{1}{\sqrt{\pi}}\right)} \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

   2. add-sqr-sqrt 24.4%

      \[\leadsto \left|\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| \cdot \frac{1}{\sqrt{\pi}}\right) \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

   3. fabs-sqr 24.4%

      \[\leadsto \left|\left(\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot \frac{1}{\sqrt{\pi}}\right) \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

   4. add-sqr-sqrt 99.9%

      \[\leadsto \left|\left(\color{blue}{x} \cdot \frac{1}{\sqrt{\pi}}\right) \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

   5. pow1/2 99.9%

      \[\leadsto \left|\left(x \cdot \frac{1}{\color{blue}{{\pi}^{0.5}}}\right) \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

   6. pow-flip 99.9%

      \[\leadsto \left|\left(x \cdot \color{blue}{{\pi}^{\left(-0.5\right)}}\right) \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

   7. metadata-eval 99.9%

      \[\leadsto \left|\left(x \cdot {\pi}^{\color{blue}{-0.5}}\right) \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

5. Applied egg-rr 99.9%

   \[\leadsto \left|\color{blue}{\left(x \cdot {\pi}^{-0.5}\right)} \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

6. Taylor expanded in x around 0 99.9%

   \[\leadsto \left|\left(x \cdot {\pi}^{-0.5}\right) \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \color{blue}{\left(0.2 \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right)}\right)\right| \]
7. Step-by-step derivation

   1. fma-udef 99.9%

      \[\leadsto \left|\left(x \cdot {\pi}^{-0.5}\right) \cdot \left(\color{blue}{\left(0.6666666666666666 \cdot \left(x \cdot x\right) + 2\right)} + \left(0.2 \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

8. Applied egg-rr 99.9%

   \[\leadsto \left|\left(x \cdot {\pi}^{-0.5}\right) \cdot \left(\color{blue}{\left(0.6666666666666666 \cdot \left(x \cdot x\right) + 2\right)} + \left(0.2 \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

9. Final simplification 99.9%

   \[\leadsto \left|\left(x \cdot {\pi}^{-0.5}\right) \cdot \left(\left(0.6666666666666666 \cdot \left(x \cdot x\right) + 2\right) + \left(0.2 \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

Alternative 2: 99.4% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \left|\left(\left(0.6666666666666666 \cdot \left(x \cdot x\right) + 2\right) + \left(0.2 \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right)\right) \cdot \frac{x}{\sqrt{\pi}}\right| \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fabs
  (*
   (+
    (+ (* 0.6666666666666666 (* x x)) 2.0)
    (+ (* 0.2 (pow x 4.0)) (* 0.047619047619047616 (pow x 6.0))))
   (/ x (sqrt PI)))))

C

double code(double x) {
	return fabs(((((0.6666666666666666 * (x * x)) + 2.0) + ((0.2 * pow(x, 4.0)) + (0.047619047619047616 * pow(x, 6.0)))) * (x / sqrt(((double) M_PI)))));
}

Java

public static double code(double x) {
	return Math.abs(((((0.6666666666666666 * (x * x)) + 2.0) + ((0.2 * Math.pow(x, 4.0)) + (0.047619047619047616 * Math.pow(x, 6.0)))) * (x / Math.sqrt(Math.PI))));
}

Python

def code(x):
	return math.fabs(((((0.6666666666666666 * (x * x)) + 2.0) + ((0.2 * math.pow(x, 4.0)) + (0.047619047619047616 * math.pow(x, 6.0)))) * (x / math.sqrt(math.pi))))

Julia

function code(x)
	return abs(Float64(Float64(Float64(Float64(0.6666666666666666 * Float64(x * x)) + 2.0) + Float64(Float64(0.2 * (x ^ 4.0)) + Float64(0.047619047619047616 * (x ^ 6.0)))) * Float64(x / sqrt(pi))))
end

MATLAB

function tmp = code(x)
	tmp = abs(((((0.6666666666666666 * (x * x)) + 2.0) + ((0.2 * (x ^ 4.0)) + (0.047619047619047616 * (x ^ 6.0)))) * (x / sqrt(pi))));
end

Wolfram

code[x_] := N[Abs[N[(N[(N[(N[(0.6666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] + N[(N[(0.2 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(0.047619047619047616 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(x / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.9%

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

3. Simplified 99.5%

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

   1. div-inv 99.9%

      \[\leadsto \left|\color{blue}{\left(\left|x\right| \cdot \frac{1}{\sqrt{\pi}}\right)} \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

   2. add-sqr-sqrt 24.4%

      \[\leadsto \left|\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| \cdot \frac{1}{\sqrt{\pi}}\right) \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

   3. fabs-sqr 24.4%

      \[\leadsto \left|\left(\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot \frac{1}{\sqrt{\pi}}\right) \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

   4. add-sqr-sqrt 99.9%

      \[\leadsto \left|\left(\color{blue}{x} \cdot \frac{1}{\sqrt{\pi}}\right) \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

   5. pow1/2 99.9%

      \[\leadsto \left|\left(x \cdot \frac{1}{\color{blue}{{\pi}^{0.5}}}\right) \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

   6. pow-flip 99.9%

      \[\leadsto \left|\left(x \cdot \color{blue}{{\pi}^{\left(-0.5\right)}}\right) \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

   7. metadata-eval 99.9%

      \[\leadsto \left|\left(x \cdot {\pi}^{\color{blue}{-0.5}}\right) \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

5. Applied egg-rr 99.9%

   \[\leadsto \left|\color{blue}{\left(x \cdot {\pi}^{-0.5}\right)} \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

6. Taylor expanded in x around 0 99.9%

   \[\leadsto \left|\left(x \cdot {\pi}^{-0.5}\right) \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \color{blue}{\left(0.2 \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right)}\right)\right| \]
7. Step-by-step derivation

   1. fma-udef 99.9%

      \[\leadsto \left|\left(x \cdot {\pi}^{-0.5}\right) \cdot \left(\color{blue}{\left(0.6666666666666666 \cdot \left(x \cdot x\right) + 2\right)} + \left(0.2 \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

8. Applied egg-rr 99.9%

   \[\leadsto \left|\left(x \cdot {\pi}^{-0.5}\right) \cdot \left(\color{blue}{\left(0.6666666666666666 \cdot \left(x \cdot x\right) + 2\right)} + \left(0.2 \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]
9. Step-by-step derivation

   1. expm1-log1p-u 60.8%

      \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot {\pi}^{-0.5}\right)\right)} \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + 0.047619047619047616 \cdot {x}^{6}\right)\right| \]

   2. expm1-udef 4.2%

      \[\leadsto \left|\color{blue}{\left(e^{\mathsf{log1p}\left(x \cdot {\pi}^{-0.5}\right)} - 1\right)} \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + 0.047619047619047616 \cdot {x}^{6}\right)\right| \]

   3. metadata-eval 4.2%

      \[\leadsto \left|\left(e^{\mathsf{log1p}\left(x \cdot {\pi}^{\color{blue}{\left(\frac{-1}{2}\right)}}\right)} - 1\right) \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + 0.047619047619047616 \cdot {x}^{6}\right)\right| \]

   4. sqrt-pow1 4.2%

      \[\leadsto \left|\left(e^{\mathsf{log1p}\left(x \cdot \color{blue}{\sqrt{{\pi}^{-1}}}\right)} - 1\right) \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + 0.047619047619047616 \cdot {x}^{6}\right)\right| \]

   5. inv-pow 4.2%

      \[\leadsto \left|\left(e^{\mathsf{log1p}\left(x \cdot \sqrt{\color{blue}{\frac{1}{\pi}}}\right)} - 1\right) \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + 0.047619047619047616 \cdot {x}^{6}\right)\right| \]

   6. sqrt-div 4.2%

      \[\leadsto \left|\left(e^{\mathsf{log1p}\left(x \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\pi}}}\right)} - 1\right) \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + 0.047619047619047616 \cdot {x}^{6}\right)\right| \]

   7. metadata-eval 4.2%

      \[\leadsto \left|\left(e^{\mathsf{log1p}\left(x \cdot \frac{\color{blue}{1}}{\sqrt{\pi}}\right)} - 1\right) \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + 0.047619047619047616 \cdot {x}^{6}\right)\right| \]

   8. un-div-inv 4.2%

      \[\leadsto \left|\left(e^{\mathsf{log1p}\left(\color{blue}{\frac{x}{\sqrt{\pi}}}\right)} - 1\right) \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + 0.047619047619047616 \cdot {x}^{6}\right)\right| \]

10. Applied egg-rr 4.6%

    \[\leadsto \left|\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{x}{\sqrt{\pi}}\right)} - 1\right)} \cdot \left(\left(0.6666666666666666 \cdot \left(x \cdot x\right) + 2\right) + \left(0.2 \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]
11. Step-by-step derivation

    1. expm1-def 60.4%

       \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{\sqrt{\pi}}\right)\right)} \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + 0.047619047619047616 \cdot {x}^{6}\right)\right| \]

    2. expm1-log1p 99.0%

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

12. Simplified 99.5%

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

13. Final simplification 99.5%

    \[\leadsto \left|\left(\left(0.6666666666666666 \cdot \left(x \cdot x\right) + 2\right) + \left(0.2 \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right)\right) \cdot \frac{x}{\sqrt{\pi}}\right| \]

Alternative 3: 99.3% accurate, 4.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[Abs[N[(N[(x * N[Power[Pi, -0.5], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(0.6666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision] + 2.0), $MachinePrecision] + N[(0.047619047619047616 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.9%

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

3. Simplified 99.5%

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

   1. div-inv 99.9%

      \[\leadsto \left|\color{blue}{\left(\left|x\right| \cdot \frac{1}{\sqrt{\pi}}\right)} \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

   2. add-sqr-sqrt 24.4%

      \[\leadsto \left|\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| \cdot \frac{1}{\sqrt{\pi}}\right) \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

   3. fabs-sqr 24.4%

      \[\leadsto \left|\left(\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot \frac{1}{\sqrt{\pi}}\right) \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

   4. add-sqr-sqrt 99.9%

      \[\leadsto \left|\left(\color{blue}{x} \cdot \frac{1}{\sqrt{\pi}}\right) \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

   5. pow1/2 99.9%

      \[\leadsto \left|\left(x \cdot \frac{1}{\color{blue}{{\pi}^{0.5}}}\right) \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

   6. pow-flip 99.9%

      \[\leadsto \left|\left(x \cdot \color{blue}{{\pi}^{\left(-0.5\right)}}\right) \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

   7. metadata-eval 99.9%

      \[\leadsto \left|\left(x \cdot {\pi}^{\color{blue}{-0.5}}\right) \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

5. Applied egg-rr 99.9%

   \[\leadsto \left|\color{blue}{\left(x \cdot {\pi}^{-0.5}\right)} \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

6. Taylor expanded in x around inf 99.4%

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

   1. fma-udef 99.9%

      \[\leadsto \left|\left(x \cdot {\pi}^{-0.5}\right) \cdot \left(\color{blue}{\left(0.6666666666666666 \cdot \left(x \cdot x\right) + 2\right)} + \left(0.2 \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

8. Applied egg-rr 99.4%

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

9. Final simplification 99.4%

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

Alternative 4: 98.9% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[Abs[N[(N[(x * N[Power[Pi, -0.5], $MachinePrecision]), $MachinePrecision] * N[(2.0 + N[(0.047619047619047616 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.9%

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

3. Simplified 99.5%

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

   1. div-inv 99.9%

      \[\leadsto \left|\color{blue}{\left(\left|x\right| \cdot \frac{1}{\sqrt{\pi}}\right)} \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

   2. add-sqr-sqrt 24.4%

      \[\leadsto \left|\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| \cdot \frac{1}{\sqrt{\pi}}\right) \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

   3. fabs-sqr 24.4%

      \[\leadsto \left|\left(\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot \frac{1}{\sqrt{\pi}}\right) \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

   4. add-sqr-sqrt 99.9%

      \[\leadsto \left|\left(\color{blue}{x} \cdot \frac{1}{\sqrt{\pi}}\right) \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

   5. pow1/2 99.9%

      \[\leadsto \left|\left(x \cdot \frac{1}{\color{blue}{{\pi}^{0.5}}}\right) \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

   6. pow-flip 99.9%

      \[\leadsto \left|\left(x \cdot \color{blue}{{\pi}^{\left(-0.5\right)}}\right) \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

   7. metadata-eval 99.9%

      \[\leadsto \left|\left(x \cdot {\pi}^{\color{blue}{-0.5}}\right) \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

5. Applied egg-rr 99.9%

   \[\leadsto \left|\color{blue}{\left(x \cdot {\pi}^{-0.5}\right)} \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

6. Taylor expanded in x around inf 99.4%

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

7. Taylor expanded in x around 0 99.2%

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

8. Final simplification 99.2%

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

Alternative 5: 98.4% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 99.5%

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

   1. div-inv 99.9%

      \[\leadsto \left|\color{blue}{\left(\left|x\right| \cdot \frac{1}{\sqrt{\pi}}\right)} \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

   2. add-sqr-sqrt 24.4%

      \[\leadsto \left|\left(\left|\color{blue}{\sqrt{x} \cdot \sqrt{x}}\right| \cdot \frac{1}{\sqrt{\pi}}\right) \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

   3. fabs-sqr 24.4%

      \[\leadsto \left|\left(\color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot \frac{1}{\sqrt{\pi}}\right) \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

   4. add-sqr-sqrt 99.9%

      \[\leadsto \left|\left(\color{blue}{x} \cdot \frac{1}{\sqrt{\pi}}\right) \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

   5. pow1/2 99.9%

      \[\leadsto \left|\left(x \cdot \frac{1}{\color{blue}{{\pi}^{0.5}}}\right) \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

   6. pow-flip 99.9%

      \[\leadsto \left|\left(x \cdot \color{blue}{{\pi}^{\left(-0.5\right)}}\right) \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

   7. metadata-eval 99.9%

      \[\leadsto \left|\left(x \cdot {\pi}^{\color{blue}{-0.5}}\right) \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

5. Applied egg-rr 99.9%

   \[\leadsto \left|\color{blue}{\left(x \cdot {\pi}^{-0.5}\right)} \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + \mathsf{fma}\left(0.2, {x}^{4}, 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

6. Taylor expanded in x around inf 99.4%

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

   1. expm1-log1p-u 60.8%

      \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(x \cdot {\pi}^{-0.5}\right)\right)} \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + 0.047619047619047616 \cdot {x}^{6}\right)\right| \]

   2. expm1-udef 4.2%

      \[\leadsto \left|\color{blue}{\left(e^{\mathsf{log1p}\left(x \cdot {\pi}^{-0.5}\right)} - 1\right)} \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + 0.047619047619047616 \cdot {x}^{6}\right)\right| \]

   3. metadata-eval 4.2%

      \[\leadsto \left|\left(e^{\mathsf{log1p}\left(x \cdot {\pi}^{\color{blue}{\left(\frac{-1}{2}\right)}}\right)} - 1\right) \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + 0.047619047619047616 \cdot {x}^{6}\right)\right| \]

   4. sqrt-pow1 4.2%

      \[\leadsto \left|\left(e^{\mathsf{log1p}\left(x \cdot \color{blue}{\sqrt{{\pi}^{-1}}}\right)} - 1\right) \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + 0.047619047619047616 \cdot {x}^{6}\right)\right| \]

   5. inv-pow 4.2%

      \[\leadsto \left|\left(e^{\mathsf{log1p}\left(x \cdot \sqrt{\color{blue}{\frac{1}{\pi}}}\right)} - 1\right) \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + 0.047619047619047616 \cdot {x}^{6}\right)\right| \]

   6. sqrt-div 4.2%

      \[\leadsto \left|\left(e^{\mathsf{log1p}\left(x \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\pi}}}\right)} - 1\right) \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + 0.047619047619047616 \cdot {x}^{6}\right)\right| \]

   7. metadata-eval 4.2%

      \[\leadsto \left|\left(e^{\mathsf{log1p}\left(x \cdot \frac{\color{blue}{1}}{\sqrt{\pi}}\right)} - 1\right) \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + 0.047619047619047616 \cdot {x}^{6}\right)\right| \]

   8. un-div-inv 4.2%

      \[\leadsto \left|\left(e^{\mathsf{log1p}\left(\color{blue}{\frac{x}{\sqrt{\pi}}}\right)} - 1\right) \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + 0.047619047619047616 \cdot {x}^{6}\right)\right| \]

8. Applied egg-rr 4.2%

   \[\leadsto \left|\color{blue}{\left(e^{\mathsf{log1p}\left(\frac{x}{\sqrt{\pi}}\right)} - 1\right)} \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + 0.047619047619047616 \cdot {x}^{6}\right)\right| \]
9. Step-by-step derivation

   1. expm1-def 60.4%

      \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{x}{\sqrt{\pi}}\right)\right)} \cdot \left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right) + 0.047619047619047616 \cdot {x}^{6}\right)\right| \]

   2. expm1-log1p 99.0%

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

10. Simplified 99.0%

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

11. Taylor expanded in x around 0 98.8%

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

12. Final simplification 98.8%

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

Alternative 6: 67.8% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.85:\\ \;\;\;\;\left|x \cdot \left({\pi}^{-0.5} \cdot 2\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|0.047619047619047616 \cdot \left({\pi}^{-0.5} \cdot {x}^{7}\right)\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.85)
   (fabs (* x (* (pow PI -0.5) 2.0)))
   (fabs (* 0.047619047619047616 (* (pow PI -0.5) (pow x 7.0))))))

C

double code(double x) {
	double tmp;
	if (x <= 1.85) {
		tmp = fabs((x * (pow(((double) M_PI), -0.5) * 2.0)));
	} else {
		tmp = fabs((0.047619047619047616 * (pow(((double) M_PI), -0.5) * pow(x, 7.0))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.85) {
		tmp = Math.abs((x * (Math.pow(Math.PI, -0.5) * 2.0)));
	} else {
		tmp = Math.abs((0.047619047619047616 * (Math.pow(Math.PI, -0.5) * Math.pow(x, 7.0))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.85:
		tmp = math.fabs((x * (math.pow(math.pi, -0.5) * 2.0)))
	else:
		tmp = math.fabs((0.047619047619047616 * (math.pow(math.pi, -0.5) * math.pow(x, 7.0))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.85)
		tmp = abs(Float64(x * Float64((pi ^ -0.5) * 2.0)));
	else
		tmp = abs(Float64(0.047619047619047616 * Float64((pi ^ -0.5) * (x ^ 7.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.85)
		tmp = abs((x * ((pi ^ -0.5) * 2.0)));
	else
		tmp = abs((0.047619047619047616 * ((pi ^ -0.5) * (x ^ 7.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.85], N[Abs[N[(x * N[(N[Power[Pi, -0.5], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(0.047619047619047616 * N[(N[Power[Pi, -0.5], $MachinePrecision] * N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.8500000000000001

   1. Initial program 99.9%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in x around 0 63.0%

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

      1. *-commutative 63.0%

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

      2. associate-*l* 63.0%

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

   6. Simplified 63.0%

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

      1. *-commutative 63.0%

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

      2. associate-*r* 63.4%

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

      3. *-commutative 63.4%

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

      4. add-exp-log 63.4%

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

      5. add-exp-log 22.0%

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

      6. prod-exp 22.0%

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

      7. inv-pow 22.0%

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

      8. sqrt-pow1 22.0%

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

      9. metadata-eval 22.0%

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

      10. log-pow 22.0%

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

   8. Applied egg-rr 22.0%

      \[\leadsto \left|\color{blue}{e^{-0.5 \cdot \log \pi + \log \left(x \cdot 2\right)}}\right| \]
   9. Step-by-step derivation

      1. +-commutative 22.0%

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

      2. prod-exp 22.0%

         \[\leadsto \left|\color{blue}{e^{\log \left(x \cdot 2\right)} \cdot e^{-0.5 \cdot \log \pi}}\right| \]

      3. rem-exp-log 63.4%

         \[\leadsto \left|\color{blue}{\left(x \cdot 2\right)} \cdot e^{-0.5 \cdot \log \pi}\right| \]

      4. *-commutative 63.4%

         \[\leadsto \left|\left(x \cdot 2\right) \cdot e^{\color{blue}{\log \pi \cdot -0.5}}\right| \]

      5. exp-to-pow 63.4%

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

      6. associate-*l* 63.0%

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

   10. Simplified 63.0%

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

   if 1.8500000000000001 < x

   1. Initial program 99.9%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in x around inf 41.9%

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

      1. expm1-log1p-u 3.4%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.047619047619047616 \cdot \left({x}^{7} \cdot \sqrt{\frac{1}{\pi}}\right)\right)\right)}\right| \]

      2. expm1-udef 3.3%

         \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(0.047619047619047616 \cdot \left({x}^{7} \cdot \sqrt{\frac{1}{\pi}}\right)\right)} - 1}\right| \]

      3. pow1/2 3.3%

         \[\leadsto \left|e^{\mathsf{log1p}\left(0.047619047619047616 \cdot \left({x}^{7} \cdot \color{blue}{{\left(\frac{1}{\pi}\right)}^{0.5}}\right)\right)} - 1\right| \]

      4. inv-pow 3.3%

         \[\leadsto \left|e^{\mathsf{log1p}\left(0.047619047619047616 \cdot \left({x}^{7} \cdot {\color{blue}{\left({\pi}^{-1}\right)}}^{0.5}\right)\right)} - 1\right| \]

      5. pow-pow 3.3%

         \[\leadsto \left|e^{\mathsf{log1p}\left(0.047619047619047616 \cdot \left({x}^{7} \cdot \color{blue}{{\pi}^{\left(-1 \cdot 0.5\right)}}\right)\right)} - 1\right| \]

      6. metadata-eval 3.3%

         \[\leadsto \left|e^{\mathsf{log1p}\left(0.047619047619047616 \cdot \left({x}^{7} \cdot {\pi}^{\color{blue}{-0.5}}\right)\right)} - 1\right| \]

   6. Applied egg-rr 3.3%

      \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(0.047619047619047616 \cdot \left({x}^{7} \cdot {\pi}^{-0.5}\right)\right)} - 1}\right| \]
   7. Step-by-step derivation

      1. expm1-def 3.4%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.047619047619047616 \cdot \left({x}^{7} \cdot {\pi}^{-0.5}\right)\right)\right)}\right| \]

      2. expm1-log1p 41.9%

         \[\leadsto \left|\color{blue}{0.047619047619047616 \cdot \left({x}^{7} \cdot {\pi}^{-0.5}\right)}\right| \]

   8. Simplified 41.9%

      \[\leadsto \left|\color{blue}{0.047619047619047616 \cdot \left({x}^{7} \cdot {\pi}^{-0.5}\right)}\right| \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.85:\\ \;\;\;\;\left|x \cdot \left({\pi}^{-0.5} \cdot 2\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|0.047619047619047616 \cdot \left({\pi}^{-0.5} \cdot {x}^{7}\right)\right|\\ \end{array} \]

Alternative 7: 74.3% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.6666666666666666 \cdot \left(x \cdot x\right)\\ \mathbf{if}\;x \leq 10^{+103}:\\ \;\;\;\;\left|\frac{x \cdot \frac{t_0 \cdot t_0 - 4}{t_0 - 2}}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x \cdot t_0}{\sqrt{\pi}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* 0.6666666666666666 (* x x))))
   (if (<= x 1e+103)
     (fabs (/ (* x (/ (- (* t_0 t_0) 4.0) (- t_0 2.0))) (sqrt PI)))
     (fabs (/ (* x t_0) (sqrt PI))))))

C

double code(double x) {
	double t_0 = 0.6666666666666666 * (x * x);
	double tmp;
	if (x <= 1e+103) {
		tmp = fabs(((x * (((t_0 * t_0) - 4.0) / (t_0 - 2.0))) / sqrt(((double) M_PI))));
	} else {
		tmp = fabs(((x * t_0) / sqrt(((double) M_PI))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = 0.6666666666666666 * (x * x);
	double tmp;
	if (x <= 1e+103) {
		tmp = Math.abs(((x * (((t_0 * t_0) - 4.0) / (t_0 - 2.0))) / Math.sqrt(Math.PI)));
	} else {
		tmp = Math.abs(((x * t_0) / Math.sqrt(Math.PI)));
	}
	return tmp;
}

Python

def code(x):
	t_0 = 0.6666666666666666 * (x * x)
	tmp = 0
	if x <= 1e+103:
		tmp = math.fabs(((x * (((t_0 * t_0) - 4.0) / (t_0 - 2.0))) / math.sqrt(math.pi)))
	else:
		tmp = math.fabs(((x * t_0) / math.sqrt(math.pi)))
	return tmp

Julia

function code(x)
	t_0 = Float64(0.6666666666666666 * Float64(x * x))
	tmp = 0.0
	if (x <= 1e+103)
		tmp = abs(Float64(Float64(x * Float64(Float64(Float64(t_0 * t_0) - 4.0) / Float64(t_0 - 2.0))) / sqrt(pi)));
	else
		tmp = abs(Float64(Float64(x * t_0) / sqrt(pi)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = 0.6666666666666666 * (x * x);
	tmp = 0.0;
	if (x <= 1e+103)
		tmp = abs(((x * (((t_0 * t_0) - 4.0) / (t_0 - 2.0))) / sqrt(pi)));
	else
		tmp = abs(((x * t_0) / sqrt(pi)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.6666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 1e+103], N[Abs[N[(N[(x * N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] - 4.0), $MachinePrecision] / N[(t$95$0 - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(x * t$95$0), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 1e103

   1. Initial program 99.9%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in x around 0 87.9%

      \[\leadsto \left|\color{blue}{0.6666666666666666 \cdot \left({x}^{3} \cdot \sqrt{\frac{1}{\pi}}\right) + 2 \cdot \left(x \cdot \sqrt{\frac{1}{\pi}}\right)}\right| \]
   5. Step-by-step derivation

      1. +-commutative 87.9%

         \[\leadsto \left|\color{blue}{2 \cdot \left(x \cdot \sqrt{\frac{1}{\pi}}\right) + 0.6666666666666666 \cdot \left({x}^{3} \cdot \sqrt{\frac{1}{\pi}}\right)}\right| \]

      2. associate-*r* 87.9%

         \[\leadsto \left|\color{blue}{\left(2 \cdot x\right) \cdot \sqrt{\frac{1}{\pi}}} + 0.6666666666666666 \cdot \left({x}^{3} \cdot \sqrt{\frac{1}{\pi}}\right)\right| \]

      3. associate-*r* 87.9%

         \[\leadsto \left|\left(2 \cdot x\right) \cdot \sqrt{\frac{1}{\pi}} + \color{blue}{\left(0.6666666666666666 \cdot {x}^{3}\right) \cdot \sqrt{\frac{1}{\pi}}}\right| \]

      4. distribute-rgt-out 87.9%

         \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(2 \cdot x + 0.6666666666666666 \cdot {x}^{3}\right)}\right| \]

      5. *-commutative 87.9%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(\color{blue}{x \cdot 2} + 0.6666666666666666 \cdot {x}^{3}\right)\right| \]

   6. Simplified 87.9%

      \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot 2 + 0.6666666666666666 \cdot {x}^{3}\right)}\right| \]
   7. Step-by-step derivation

      1. expm1-log1p-u 60.8%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot 2 + 0.6666666666666666 \cdot {x}^{3}\right)\right)\right)}\right| \]

      2. expm1-udef 4.2%

         \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot 2 + 0.6666666666666666 \cdot {x}^{3}\right)\right)} - 1}\right| \]

      3. *-commutative 4.2%

         \[\leadsto \left|e^{\mathsf{log1p}\left(\color{blue}{\left(x \cdot 2 + 0.6666666666666666 \cdot {x}^{3}\right) \cdot \sqrt{\frac{1}{\pi}}}\right)} - 1\right| \]

      4. sqrt-div 4.2%

         \[\leadsto \left|e^{\mathsf{log1p}\left(\left(x \cdot 2 + 0.6666666666666666 \cdot {x}^{3}\right) \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\pi}}}\right)} - 1\right| \]

      5. metadata-eval 4.2%

         \[\leadsto \left|e^{\mathsf{log1p}\left(\left(x \cdot 2 + 0.6666666666666666 \cdot {x}^{3}\right) \cdot \frac{\color{blue}{1}}{\sqrt{\pi}}\right)} - 1\right| \]

      6. un-div-inv 4.2%

         \[\leadsto \left|e^{\mathsf{log1p}\left(\color{blue}{\frac{x \cdot 2 + 0.6666666666666666 \cdot {x}^{3}}{\sqrt{\pi}}}\right)} - 1\right| \]

      7. fma-def 4.2%

         \[\leadsto \left|e^{\mathsf{log1p}\left(\frac{\color{blue}{\mathsf{fma}\left(x, 2, 0.6666666666666666 \cdot {x}^{3}\right)}}{\sqrt{\pi}}\right)} - 1\right| \]

   8. Applied egg-rr 4.2%

      \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(\frac{\mathsf{fma}\left(x, 2, 0.6666666666666666 \cdot {x}^{3}\right)}{\sqrt{\pi}}\right)} - 1}\right| \]
   9. Step-by-step derivation

      1. expm1-def 60.4%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left(x, 2, 0.6666666666666666 \cdot {x}^{3}\right)}{\sqrt{\pi}}\right)\right)}\right| \]

      2. expm1-log1p 87.5%

         \[\leadsto \left|\color{blue}{\frac{\mathsf{fma}\left(x, 2, 0.6666666666666666 \cdot {x}^{3}\right)}{\sqrt{\pi}}}\right| \]

   10. Simplified 87.5%

       \[\leadsto \left|\color{blue}{\frac{\mathsf{fma}\left(x, 2, 0.6666666666666666 \cdot {x}^{3}\right)}{\sqrt{\pi}}}\right| \]
   11. Step-by-step derivation

       1. expm1-log1p-u 60.4%

          \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left(x, 2, 0.6666666666666666 \cdot {x}^{3}\right)}{\sqrt{\pi}}\right)\right)}\right| \]

       2. expm1-udef 4.2%

          \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(\frac{\mathsf{fma}\left(x, 2, 0.6666666666666666 \cdot {x}^{3}\right)}{\sqrt{\pi}}\right)} - 1}\right| \]

   12. Applied egg-rr 4.2%

       \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(\frac{\mathsf{fma}\left(x, 2, 0.6666666666666666 \cdot {x}^{3}\right)}{\sqrt{\pi}}\right)} - 1}\right| \]
   13. Step-by-step derivation

       1. expm1-def 60.4%

          \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left(x, 2, 0.6666666666666666 \cdot {x}^{3}\right)}{\sqrt{\pi}}\right)\right)}\right| \]

       2. expm1-log1p 87.5%

          \[\leadsto \left|\color{blue}{\frac{\mathsf{fma}\left(x, 2, 0.6666666666666666 \cdot {x}^{3}\right)}{\sqrt{\pi}}}\right| \]

       3. fma-udef 87.5%

          \[\leadsto \left|\frac{\color{blue}{x \cdot 2 + 0.6666666666666666 \cdot {x}^{3}}}{\sqrt{\pi}}\right| \]

       4. +-commutative 87.5%

          \[\leadsto \left|\frac{\color{blue}{0.6666666666666666 \cdot {x}^{3} + x \cdot 2}}{\sqrt{\pi}}\right| \]

       5. unpow3 87.5%

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

       6. associate-*r* 87.5%

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

       7. *-commutative 87.5%

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

       8. distribute-rgt-in 87.5%

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

       9. fma-udef 87.5%

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

   14. Simplified 87.5%

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

       1. fma-udef 87.5%

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

       2. flip-+ 70.3%

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

       3. metadata-eval 70.3%

          \[\leadsto \left|\frac{x \cdot \frac{\left(0.6666666666666666 \cdot \left(x \cdot x\right)\right) \cdot \left(0.6666666666666666 \cdot \left(x \cdot x\right)\right) - \color{blue}{4}}{0.6666666666666666 \cdot \left(x \cdot x\right) - 2}}{\sqrt{\pi}}\right| \]

   16. Applied egg-rr 70.3%

       \[\leadsto \left|\frac{x \cdot \color{blue}{\frac{\left(0.6666666666666666 \cdot \left(x \cdot x\right)\right) \cdot \left(0.6666666666666666 \cdot \left(x \cdot x\right)\right) - 4}{0.6666666666666666 \cdot \left(x \cdot x\right) - 2}}}{\sqrt{\pi}}\right| \]

   if 1e103 < x

   1. Initial program 99.9%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in x around 0 87.9%

      \[\leadsto \left|\color{blue}{0.6666666666666666 \cdot \left({x}^{3} \cdot \sqrt{\frac{1}{\pi}}\right) + 2 \cdot \left(x \cdot \sqrt{\frac{1}{\pi}}\right)}\right| \]
   5. Step-by-step derivation

      1. +-commutative 87.9%

         \[\leadsto \left|\color{blue}{2 \cdot \left(x \cdot \sqrt{\frac{1}{\pi}}\right) + 0.6666666666666666 \cdot \left({x}^{3} \cdot \sqrt{\frac{1}{\pi}}\right)}\right| \]

      2. associate-*r* 87.9%

         \[\leadsto \left|\color{blue}{\left(2 \cdot x\right) \cdot \sqrt{\frac{1}{\pi}}} + 0.6666666666666666 \cdot \left({x}^{3} \cdot \sqrt{\frac{1}{\pi}}\right)\right| \]

      3. associate-*r* 87.9%

         \[\leadsto \left|\left(2 \cdot x\right) \cdot \sqrt{\frac{1}{\pi}} + \color{blue}{\left(0.6666666666666666 \cdot {x}^{3}\right) \cdot \sqrt{\frac{1}{\pi}}}\right| \]

      4. distribute-rgt-out 87.9%

         \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(2 \cdot x + 0.6666666666666666 \cdot {x}^{3}\right)}\right| \]

      5. *-commutative 87.9%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(\color{blue}{x \cdot 2} + 0.6666666666666666 \cdot {x}^{3}\right)\right| \]

   6. Simplified 87.9%

      \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot 2 + 0.6666666666666666 \cdot {x}^{3}\right)}\right| \]
   7. Step-by-step derivation

      1. expm1-log1p-u 60.8%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot 2 + 0.6666666666666666 \cdot {x}^{3}\right)\right)\right)}\right| \]

      2. expm1-udef 4.2%

         \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot 2 + 0.6666666666666666 \cdot {x}^{3}\right)\right)} - 1}\right| \]

      3. *-commutative 4.2%

         \[\leadsto \left|e^{\mathsf{log1p}\left(\color{blue}{\left(x \cdot 2 + 0.6666666666666666 \cdot {x}^{3}\right) \cdot \sqrt{\frac{1}{\pi}}}\right)} - 1\right| \]

      4. sqrt-div 4.2%

         \[\leadsto \left|e^{\mathsf{log1p}\left(\left(x \cdot 2 + 0.6666666666666666 \cdot {x}^{3}\right) \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\pi}}}\right)} - 1\right| \]

      5. metadata-eval 4.2%

         \[\leadsto \left|e^{\mathsf{log1p}\left(\left(x \cdot 2 + 0.6666666666666666 \cdot {x}^{3}\right) \cdot \frac{\color{blue}{1}}{\sqrt{\pi}}\right)} - 1\right| \]

      6. un-div-inv 4.2%

         \[\leadsto \left|e^{\mathsf{log1p}\left(\color{blue}{\frac{x \cdot 2 + 0.6666666666666666 \cdot {x}^{3}}{\sqrt{\pi}}}\right)} - 1\right| \]

      7. fma-def 4.2%

         \[\leadsto \left|e^{\mathsf{log1p}\left(\frac{\color{blue}{\mathsf{fma}\left(x, 2, 0.6666666666666666 \cdot {x}^{3}\right)}}{\sqrt{\pi}}\right)} - 1\right| \]

   8. Applied egg-rr 4.2%

      \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(\frac{\mathsf{fma}\left(x, 2, 0.6666666666666666 \cdot {x}^{3}\right)}{\sqrt{\pi}}\right)} - 1}\right| \]
   9. Step-by-step derivation

      1. expm1-def 60.4%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left(x, 2, 0.6666666666666666 \cdot {x}^{3}\right)}{\sqrt{\pi}}\right)\right)}\right| \]

      2. expm1-log1p 87.5%

         \[\leadsto \left|\color{blue}{\frac{\mathsf{fma}\left(x, 2, 0.6666666666666666 \cdot {x}^{3}\right)}{\sqrt{\pi}}}\right| \]

   10. Simplified 87.5%

       \[\leadsto \left|\color{blue}{\frac{\mathsf{fma}\left(x, 2, 0.6666666666666666 \cdot {x}^{3}\right)}{\sqrt{\pi}}}\right| \]
   11. Step-by-step derivation

       1. expm1-log1p-u 60.4%

          \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left(x, 2, 0.6666666666666666 \cdot {x}^{3}\right)}{\sqrt{\pi}}\right)\right)}\right| \]

       2. expm1-udef 4.2%

          \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(\frac{\mathsf{fma}\left(x, 2, 0.6666666666666666 \cdot {x}^{3}\right)}{\sqrt{\pi}}\right)} - 1}\right| \]

   12. Applied egg-rr 4.2%

       \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(\frac{\mathsf{fma}\left(x, 2, 0.6666666666666666 \cdot {x}^{3}\right)}{\sqrt{\pi}}\right)} - 1}\right| \]
   13. Step-by-step derivation

       1. expm1-def 60.4%

          \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left(x, 2, 0.6666666666666666 \cdot {x}^{3}\right)}{\sqrt{\pi}}\right)\right)}\right| \]

       2. expm1-log1p 87.5%

          \[\leadsto \left|\color{blue}{\frac{\mathsf{fma}\left(x, 2, 0.6666666666666666 \cdot {x}^{3}\right)}{\sqrt{\pi}}}\right| \]

       3. fma-udef 87.5%

          \[\leadsto \left|\frac{\color{blue}{x \cdot 2 + 0.6666666666666666 \cdot {x}^{3}}}{\sqrt{\pi}}\right| \]

       4. +-commutative 87.5%

          \[\leadsto \left|\frac{\color{blue}{0.6666666666666666 \cdot {x}^{3} + x \cdot 2}}{\sqrt{\pi}}\right| \]

       5. unpow3 87.5%

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

       6. associate-*r* 87.5%

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

       7. *-commutative 87.5%

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

       8. distribute-rgt-in 87.5%

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

       9. fma-udef 87.5%

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

   14. Simplified 87.5%

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

   15. Taylor expanded in x around inf 30.7%

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

       1. unpow2 30.7%

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

   17. Simplified 30.7%

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

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 10^{+103}:\\ \;\;\;\;\left|\frac{x \cdot \frac{\left(0.6666666666666666 \cdot \left(x \cdot x\right)\right) \cdot \left(0.6666666666666666 \cdot \left(x \cdot x\right)\right) - 4}{0.6666666666666666 \cdot \left(x \cdot x\right) - 2}}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x \cdot \left(0.6666666666666666 \cdot \left(x \cdot x\right)\right)}{\sqrt{\pi}}\right|\\ \end{array} \]

Alternative 8: 67.8% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.75:\\ \;\;\;\;\left|x \cdot \left({\pi}^{-0.5} \cdot 2\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x \cdot \left(0.6666666666666666 \cdot \left(x \cdot x\right)\right)}{\sqrt{\pi}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.75)
   (fabs (* x (* (pow PI -0.5) 2.0)))
   (fabs (/ (* x (* 0.6666666666666666 (* x x))) (sqrt PI)))))

C

double code(double x) {
	double tmp;
	if (x <= 1.75) {
		tmp = fabs((x * (pow(((double) M_PI), -0.5) * 2.0)));
	} else {
		tmp = fabs(((x * (0.6666666666666666 * (x * x))) / sqrt(((double) M_PI))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.75) {
		tmp = Math.abs((x * (Math.pow(Math.PI, -0.5) * 2.0)));
	} else {
		tmp = Math.abs(((x * (0.6666666666666666 * (x * x))) / Math.sqrt(Math.PI)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.75:
		tmp = math.fabs((x * (math.pow(math.pi, -0.5) * 2.0)))
	else:
		tmp = math.fabs(((x * (0.6666666666666666 * (x * x))) / math.sqrt(math.pi)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.75)
		tmp = abs(Float64(x * Float64((pi ^ -0.5) * 2.0)));
	else
		tmp = abs(Float64(Float64(x * Float64(0.6666666666666666 * Float64(x * x))) / sqrt(pi)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.75)
		tmp = abs((x * ((pi ^ -0.5) * 2.0)));
	else
		tmp = abs(((x * (0.6666666666666666 * (x * x))) / sqrt(pi)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.75], N[Abs[N[(x * N[(N[Power[Pi, -0.5], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(x * N[(0.6666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.75

   1. Initial program 99.9%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in x around 0 63.0%

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

      1. *-commutative 63.0%

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

      2. associate-*l* 63.0%

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

   6. Simplified 63.0%

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

      1. *-commutative 63.0%

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

      2. associate-*r* 63.4%

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

      3. *-commutative 63.4%

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

      4. add-exp-log 63.4%

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

      5. add-exp-log 22.0%

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

      6. prod-exp 22.0%

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

      7. inv-pow 22.0%

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

      8. sqrt-pow1 22.0%

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

      9. metadata-eval 22.0%

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

      10. log-pow 22.0%

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

   8. Applied egg-rr 22.0%

      \[\leadsto \left|\color{blue}{e^{-0.5 \cdot \log \pi + \log \left(x \cdot 2\right)}}\right| \]
   9. Step-by-step derivation

      1. +-commutative 22.0%

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

      2. prod-exp 22.0%

         \[\leadsto \left|\color{blue}{e^{\log \left(x \cdot 2\right)} \cdot e^{-0.5 \cdot \log \pi}}\right| \]

      3. rem-exp-log 63.4%

         \[\leadsto \left|\color{blue}{\left(x \cdot 2\right)} \cdot e^{-0.5 \cdot \log \pi}\right| \]

      4. *-commutative 63.4%

         \[\leadsto \left|\left(x \cdot 2\right) \cdot e^{\color{blue}{\log \pi \cdot -0.5}}\right| \]

      5. exp-to-pow 63.4%

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

      6. associate-*l* 63.0%

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

   10. Simplified 63.0%

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

   if 1.75 < x

   1. Initial program 99.9%

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

   3. Simplified 99.5%

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

   4. Taylor expanded in x around 0 87.9%

      \[\leadsto \left|\color{blue}{0.6666666666666666 \cdot \left({x}^{3} \cdot \sqrt{\frac{1}{\pi}}\right) + 2 \cdot \left(x \cdot \sqrt{\frac{1}{\pi}}\right)}\right| \]
   5. Step-by-step derivation

      1. +-commutative 87.9%

         \[\leadsto \left|\color{blue}{2 \cdot \left(x \cdot \sqrt{\frac{1}{\pi}}\right) + 0.6666666666666666 \cdot \left({x}^{3} \cdot \sqrt{\frac{1}{\pi}}\right)}\right| \]

      2. associate-*r* 87.9%

         \[\leadsto \left|\color{blue}{\left(2 \cdot x\right) \cdot \sqrt{\frac{1}{\pi}}} + 0.6666666666666666 \cdot \left({x}^{3} \cdot \sqrt{\frac{1}{\pi}}\right)\right| \]

      3. associate-*r* 87.9%

         \[\leadsto \left|\left(2 \cdot x\right) \cdot \sqrt{\frac{1}{\pi}} + \color{blue}{\left(0.6666666666666666 \cdot {x}^{3}\right) \cdot \sqrt{\frac{1}{\pi}}}\right| \]

      4. distribute-rgt-out 87.9%

         \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(2 \cdot x + 0.6666666666666666 \cdot {x}^{3}\right)}\right| \]

      5. *-commutative 87.9%

         \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(\color{blue}{x \cdot 2} + 0.6666666666666666 \cdot {x}^{3}\right)\right| \]

   6. Simplified 87.9%

      \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot 2 + 0.6666666666666666 \cdot {x}^{3}\right)}\right| \]
   7. Step-by-step derivation

      1. expm1-log1p-u 60.8%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot 2 + 0.6666666666666666 \cdot {x}^{3}\right)\right)\right)}\right| \]

      2. expm1-udef 4.2%

         \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot 2 + 0.6666666666666666 \cdot {x}^{3}\right)\right)} - 1}\right| \]

      3. *-commutative 4.2%

         \[\leadsto \left|e^{\mathsf{log1p}\left(\color{blue}{\left(x \cdot 2 + 0.6666666666666666 \cdot {x}^{3}\right) \cdot \sqrt{\frac{1}{\pi}}}\right)} - 1\right| \]

      4. sqrt-div 4.2%

         \[\leadsto \left|e^{\mathsf{log1p}\left(\left(x \cdot 2 + 0.6666666666666666 \cdot {x}^{3}\right) \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\pi}}}\right)} - 1\right| \]

      5. metadata-eval 4.2%

         \[\leadsto \left|e^{\mathsf{log1p}\left(\left(x \cdot 2 + 0.6666666666666666 \cdot {x}^{3}\right) \cdot \frac{\color{blue}{1}}{\sqrt{\pi}}\right)} - 1\right| \]

      6. un-div-inv 4.2%

         \[\leadsto \left|e^{\mathsf{log1p}\left(\color{blue}{\frac{x \cdot 2 + 0.6666666666666666 \cdot {x}^{3}}{\sqrt{\pi}}}\right)} - 1\right| \]

      7. fma-def 4.2%

         \[\leadsto \left|e^{\mathsf{log1p}\left(\frac{\color{blue}{\mathsf{fma}\left(x, 2, 0.6666666666666666 \cdot {x}^{3}\right)}}{\sqrt{\pi}}\right)} - 1\right| \]

   8. Applied egg-rr 4.2%

      \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(\frac{\mathsf{fma}\left(x, 2, 0.6666666666666666 \cdot {x}^{3}\right)}{\sqrt{\pi}}\right)} - 1}\right| \]
   9. Step-by-step derivation

      1. expm1-def 60.4%

         \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left(x, 2, 0.6666666666666666 \cdot {x}^{3}\right)}{\sqrt{\pi}}\right)\right)}\right| \]

      2. expm1-log1p 87.5%

         \[\leadsto \left|\color{blue}{\frac{\mathsf{fma}\left(x, 2, 0.6666666666666666 \cdot {x}^{3}\right)}{\sqrt{\pi}}}\right| \]

   10. Simplified 87.5%

       \[\leadsto \left|\color{blue}{\frac{\mathsf{fma}\left(x, 2, 0.6666666666666666 \cdot {x}^{3}\right)}{\sqrt{\pi}}}\right| \]
   11. Step-by-step derivation

       1. expm1-log1p-u 60.4%

          \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left(x, 2, 0.6666666666666666 \cdot {x}^{3}\right)}{\sqrt{\pi}}\right)\right)}\right| \]

       2. expm1-udef 4.2%

          \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(\frac{\mathsf{fma}\left(x, 2, 0.6666666666666666 \cdot {x}^{3}\right)}{\sqrt{\pi}}\right)} - 1}\right| \]

   12. Applied egg-rr 4.2%

       \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(\frac{\mathsf{fma}\left(x, 2, 0.6666666666666666 \cdot {x}^{3}\right)}{\sqrt{\pi}}\right)} - 1}\right| \]
   13. Step-by-step derivation

       1. expm1-def 60.4%

          \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left(x, 2, 0.6666666666666666 \cdot {x}^{3}\right)}{\sqrt{\pi}}\right)\right)}\right| \]

       2. expm1-log1p 87.5%

          \[\leadsto \left|\color{blue}{\frac{\mathsf{fma}\left(x, 2, 0.6666666666666666 \cdot {x}^{3}\right)}{\sqrt{\pi}}}\right| \]

       3. fma-udef 87.5%

          \[\leadsto \left|\frac{\color{blue}{x \cdot 2 + 0.6666666666666666 \cdot {x}^{3}}}{\sqrt{\pi}}\right| \]

       4. +-commutative 87.5%

          \[\leadsto \left|\frac{\color{blue}{0.6666666666666666 \cdot {x}^{3} + x \cdot 2}}{\sqrt{\pi}}\right| \]

       5. unpow3 87.5%

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

       6. associate-*r* 87.5%

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

       7. *-commutative 87.5%

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

       8. distribute-rgt-in 87.5%

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

       9. fma-udef 87.5%

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

   14. Simplified 87.5%

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

   15. Taylor expanded in x around inf 30.7%

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

       1. unpow2 30.7%

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

   17. Simplified 30.7%

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

4. Final simplification 63.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.75:\\ \;\;\;\;\left|x \cdot \left({\pi}^{-0.5} \cdot 2\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{x \cdot \left(0.6666666666666666 \cdot \left(x \cdot x\right)\right)}{\sqrt{\pi}}\right|\\ \end{array} \]

Alternative 9: 89.2% accurate, 6.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 99.5%

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

4. Taylor expanded in x around 0 87.9%

   \[\leadsto \left|\color{blue}{0.6666666666666666 \cdot \left({x}^{3} \cdot \sqrt{\frac{1}{\pi}}\right) + 2 \cdot \left(x \cdot \sqrt{\frac{1}{\pi}}\right)}\right| \]
5. Step-by-step derivation

   1. +-commutative 87.9%

      \[\leadsto \left|\color{blue}{2 \cdot \left(x \cdot \sqrt{\frac{1}{\pi}}\right) + 0.6666666666666666 \cdot \left({x}^{3} \cdot \sqrt{\frac{1}{\pi}}\right)}\right| \]

   2. associate-*r* 87.9%

      \[\leadsto \left|\color{blue}{\left(2 \cdot x\right) \cdot \sqrt{\frac{1}{\pi}}} + 0.6666666666666666 \cdot \left({x}^{3} \cdot \sqrt{\frac{1}{\pi}}\right)\right| \]

   3. associate-*r* 87.9%

      \[\leadsto \left|\left(2 \cdot x\right) \cdot \sqrt{\frac{1}{\pi}} + \color{blue}{\left(0.6666666666666666 \cdot {x}^{3}\right) \cdot \sqrt{\frac{1}{\pi}}}\right| \]

   4. distribute-rgt-out 87.9%

      \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(2 \cdot x + 0.6666666666666666 \cdot {x}^{3}\right)}\right| \]

   5. *-commutative 87.9%

      \[\leadsto \left|\sqrt{\frac{1}{\pi}} \cdot \left(\color{blue}{x \cdot 2} + 0.6666666666666666 \cdot {x}^{3}\right)\right| \]

6. Simplified 87.9%

   \[\leadsto \left|\color{blue}{\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot 2 + 0.6666666666666666 \cdot {x}^{3}\right)}\right| \]
7. Step-by-step derivation

   1. expm1-log1p-u 60.8%

      \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot 2 + 0.6666666666666666 \cdot {x}^{3}\right)\right)\right)}\right| \]

   2. expm1-udef 4.2%

      \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(\sqrt{\frac{1}{\pi}} \cdot \left(x \cdot 2 + 0.6666666666666666 \cdot {x}^{3}\right)\right)} - 1}\right| \]

   3. *-commutative 4.2%

      \[\leadsto \left|e^{\mathsf{log1p}\left(\color{blue}{\left(x \cdot 2 + 0.6666666666666666 \cdot {x}^{3}\right) \cdot \sqrt{\frac{1}{\pi}}}\right)} - 1\right| \]

   4. sqrt-div 4.2%

      \[\leadsto \left|e^{\mathsf{log1p}\left(\left(x \cdot 2 + 0.6666666666666666 \cdot {x}^{3}\right) \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{\pi}}}\right)} - 1\right| \]

   5. metadata-eval 4.2%

      \[\leadsto \left|e^{\mathsf{log1p}\left(\left(x \cdot 2 + 0.6666666666666666 \cdot {x}^{3}\right) \cdot \frac{\color{blue}{1}}{\sqrt{\pi}}\right)} - 1\right| \]

   6. un-div-inv 4.2%

      \[\leadsto \left|e^{\mathsf{log1p}\left(\color{blue}{\frac{x \cdot 2 + 0.6666666666666666 \cdot {x}^{3}}{\sqrt{\pi}}}\right)} - 1\right| \]

   7. fma-def 4.2%

      \[\leadsto \left|e^{\mathsf{log1p}\left(\frac{\color{blue}{\mathsf{fma}\left(x, 2, 0.6666666666666666 \cdot {x}^{3}\right)}}{\sqrt{\pi}}\right)} - 1\right| \]

8. Applied egg-rr 4.2%

   \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(\frac{\mathsf{fma}\left(x, 2, 0.6666666666666666 \cdot {x}^{3}\right)}{\sqrt{\pi}}\right)} - 1}\right| \]
9. Step-by-step derivation

   1. expm1-def 60.4%

      \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left(x, 2, 0.6666666666666666 \cdot {x}^{3}\right)}{\sqrt{\pi}}\right)\right)}\right| \]

   2. expm1-log1p 87.5%

      \[\leadsto \left|\color{blue}{\frac{\mathsf{fma}\left(x, 2, 0.6666666666666666 \cdot {x}^{3}\right)}{\sqrt{\pi}}}\right| \]

10. Simplified 87.5%

    \[\leadsto \left|\color{blue}{\frac{\mathsf{fma}\left(x, 2, 0.6666666666666666 \cdot {x}^{3}\right)}{\sqrt{\pi}}}\right| \]
11. Step-by-step derivation

    1. expm1-log1p-u 60.4%

       \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left(x, 2, 0.6666666666666666 \cdot {x}^{3}\right)}{\sqrt{\pi}}\right)\right)}\right| \]

    2. expm1-udef 4.2%

       \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(\frac{\mathsf{fma}\left(x, 2, 0.6666666666666666 \cdot {x}^{3}\right)}{\sqrt{\pi}}\right)} - 1}\right| \]

12. Applied egg-rr 4.2%

    \[\leadsto \left|\color{blue}{e^{\mathsf{log1p}\left(\frac{\mathsf{fma}\left(x, 2, 0.6666666666666666 \cdot {x}^{3}\right)}{\sqrt{\pi}}\right)} - 1}\right| \]
13. Step-by-step derivation

    1. expm1-def 60.4%

       \[\leadsto \left|\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\mathsf{fma}\left(x, 2, 0.6666666666666666 \cdot {x}^{3}\right)}{\sqrt{\pi}}\right)\right)}\right| \]

    2. expm1-log1p 87.5%

       \[\leadsto \left|\color{blue}{\frac{\mathsf{fma}\left(x, 2, 0.6666666666666666 \cdot {x}^{3}\right)}{\sqrt{\pi}}}\right| \]

    3. fma-udef 87.5%

       \[\leadsto \left|\frac{\color{blue}{x \cdot 2 + 0.6666666666666666 \cdot {x}^{3}}}{\sqrt{\pi}}\right| \]

    4. +-commutative 87.5%

       \[\leadsto \left|\frac{\color{blue}{0.6666666666666666 \cdot {x}^{3} + x \cdot 2}}{\sqrt{\pi}}\right| \]

    5. unpow3 87.5%

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

    6. associate-*r* 87.5%

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

    7. *-commutative 87.5%

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

    8. distribute-rgt-in 87.5%

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

    9. fma-udef 87.5%

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

14. Simplified 87.5%

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

    1. fma-udef 99.9%

       \[\leadsto \left|\left(x \cdot {\pi}^{-0.5}\right) \cdot \left(\color{blue}{\left(0.6666666666666666 \cdot \left(x \cdot x\right) + 2\right)} + \left(0.2 \cdot {x}^{4} + 0.047619047619047616 \cdot {x}^{6}\right)\right)\right| \]

16. Applied egg-rr 87.5%

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

17. Final simplification 87.5%

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

Alternative 10: 67.8% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \left|x \cdot \left({\pi}^{-0.5} \cdot 2\right)\right| \end{array} \]

FPCore

(FPCore (x) :precision binary64 (fabs (* x (* (pow PI -0.5) 2.0))))

C

double code(double x) {
	return fabs((x * (pow(((double) M_PI), -0.5) * 2.0)));
}

Java

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

Python

def code(x):
	return math.fabs((x * (math.pow(math.pi, -0.5) * 2.0)))

Julia

function code(x)
	return abs(Float64(x * Float64((pi ^ -0.5) * 2.0)))
end

MATLAB

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

Wolfram

code[x_] := N[Abs[N[(x * N[(N[Power[Pi, -0.5], $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.9%

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

3. Simplified 99.5%

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

4. Taylor expanded in x around 0 63.0%

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

   1. *-commutative 63.0%

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

   2. associate-*l* 63.0%

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

6. Simplified 63.0%

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

   1. *-commutative 63.0%

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

   2. associate-*r* 63.4%

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

   3. *-commutative 63.4%

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

   4. add-exp-log 63.4%

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

   5. add-exp-log 22.0%

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

   6. prod-exp 22.0%

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

   7. inv-pow 22.0%

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

   8. sqrt-pow1 22.0%

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

   9. metadata-eval 22.0%

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

   10. log-pow 22.0%

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

8. Applied egg-rr 22.0%

   \[\leadsto \left|\color{blue}{e^{-0.5 \cdot \log \pi + \log \left(x \cdot 2\right)}}\right| \]
9. Step-by-step derivation

   1. +-commutative 22.0%

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

   2. prod-exp 22.0%

      \[\leadsto \left|\color{blue}{e^{\log \left(x \cdot 2\right)} \cdot e^{-0.5 \cdot \log \pi}}\right| \]

   3. rem-exp-log 63.4%

      \[\leadsto \left|\color{blue}{\left(x \cdot 2\right)} \cdot e^{-0.5 \cdot \log \pi}\right| \]

   4. *-commutative 63.4%

      \[\leadsto \left|\left(x \cdot 2\right) \cdot e^{\color{blue}{\log \pi \cdot -0.5}}\right| \]

   5. exp-to-pow 63.4%

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

   6. associate-*l* 63.0%

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

10. Simplified 63.0%

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

11. Final simplification 63.0%

    \[\leadsto \left|x \cdot \left({\pi}^{-0.5} \cdot 2\right)\right| \]

Alternative 11: 67.8% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \left|{\pi}^{-0.5} \cdot \left(x \cdot 2\right)\right| \end{array} \]

FPCore

(FPCore (x) :precision binary64 (fabs (* (pow PI -0.5) (* x 2.0))))

C

double code(double x) {
	return fabs((pow(((double) M_PI), -0.5) * (x * 2.0)));
}

Java

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

Python

def code(x):
	return math.fabs((math.pow(math.pi, -0.5) * (x * 2.0)))

Julia

function code(x)
	return abs(Float64((pi ^ -0.5) * Float64(x * 2.0)))
end

MATLAB

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

Wolfram

code[x_] := N[Abs[N[(N[Power[Pi, -0.5], $MachinePrecision] * N[(x * 2.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 99.9%

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

3. Simplified 99.5%

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

4. Taylor expanded in x around 0 63.0%

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

   1. *-commutative 63.0%

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

   2. associate-*l* 63.0%

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

6. Simplified 63.0%

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

   1. *-commutative 63.0%

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

   2. associate-*r* 63.4%

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

   3. sqrt-div 63.4%

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

   4. metadata-eval 63.4%

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

   5. un-div-inv 63.0%

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

8. Applied egg-rr 63.0%

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

   1. div-inv 63.4%

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

   2. pow1/2 63.4%

      \[\leadsto \left|\left(x \cdot 2\right) \cdot \frac{1}{\color{blue}{{\pi}^{0.5}}}\right| \]

   3. pow-flip 63.4%

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

   4. metadata-eval 63.4%

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

10. Applied egg-rr 63.4%

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

11. Final simplification 63.4%

    \[\leadsto \left|{\pi}^{-0.5} \cdot \left(x \cdot 2\right)\right| \]

Alternative 12: 67.3% accurate, 6.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

3. Simplified 99.5%

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

4. Taylor expanded in x around 0 63.0%

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

   1. *-commutative 63.0%

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

   2. associate-*l* 63.0%

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

6. Simplified 63.0%

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

   1. *-commutative 63.0%

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

   2. associate-*r* 63.4%

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

   3. sqrt-div 63.4%

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

   4. metadata-eval 63.4%

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

   5. un-div-inv 63.0%

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

8. Applied egg-rr 63.0%

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

9. Final simplification 63.0%

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

Reproduce

herbie shell --seed 2023227 
(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)))))))