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

Percentage Accurate: 99.8% → 99.8%

Time: 8.6s

Alternatives: 13

Speedup: 3.7×

• 29.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[x \leq 0.5\]

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left|x\right| \cdot \left(x \cdot x\right)\\ t_1 := \left|x\right| \cdot \left(\left|x\right| \cdot t_0\right)\\ \left|\frac{1}{\sqrt{\pi}} \cdot \left(\left(\left(2 \cdot \left|x\right| + 0.6666666666666666 \cdot t_0\right) + 0.2 \cdot t_1\right) + 0.047619047619047616 \cdot \left(\left|x\right| \cdot \left(\left|x\right| \cdot t_1\right)\right)\right)\right| \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (fabs x) (* x x))) (t_1 (* (fabs x) (* (fabs x) t_0))))
   (fabs
    (*
     (/ 1.0 (sqrt PI))
     (+
      (+ (+ (* 2.0 (fabs x)) (* 0.6666666666666666 t_0)) (* 0.2 t_1))
      (* 0.047619047619047616 (* (fabs x) (* (fabs x) t_1))))))))

C

double code(double x) {
	double t_0 = fabs(x) * (x * x);
	double t_1 = fabs(x) * (fabs(x) * t_0);
	return fabs(((1.0 / sqrt(((double) M_PI))) * ((((2.0 * fabs(x)) + (0.6666666666666666 * t_0)) + (0.2 * t_1)) + (0.047619047619047616 * (fabs(x) * (fabs(x) * t_1))))));
}

Java

public static double code(double x) {
	double t_0 = Math.abs(x) * (x * x);
	double t_1 = Math.abs(x) * (Math.abs(x) * t_0);
	return Math.abs(((1.0 / Math.sqrt(Math.PI)) * ((((2.0 * Math.abs(x)) + (0.6666666666666666 * t_0)) + (0.2 * t_1)) + (0.047619047619047616 * (Math.abs(x) * (Math.abs(x) * t_1))))));
}

Python

def code(x):
	t_0 = math.fabs(x) * (x * x)
	t_1 = math.fabs(x) * (math.fabs(x) * t_0)
	return math.fabs(((1.0 / math.sqrt(math.pi)) * ((((2.0 * math.fabs(x)) + (0.6666666666666666 * t_0)) + (0.2 * t_1)) + (0.047619047619047616 * (math.fabs(x) * (math.fabs(x) * t_1))))))

Julia

function code(x)
	t_0 = Float64(abs(x) * Float64(x * x))
	t_1 = Float64(abs(x) * Float64(abs(x) * t_0))
	return abs(Float64(Float64(1.0 / sqrt(pi)) * Float64(Float64(Float64(Float64(2.0 * abs(x)) + Float64(0.6666666666666666 * t_0)) + Float64(0.2 * t_1)) + Float64(0.047619047619047616 * Float64(abs(x) * Float64(abs(x) * t_1))))))
end

MATLAB

function tmp = code(x)
	t_0 = abs(x) * (x * x);
	t_1 = abs(x) * (abs(x) * t_0);
	tmp = abs(((1.0 / sqrt(pi)) * ((((2.0 * abs(x)) + (0.6666666666666666 * t_0)) + (0.2 * t_1)) + (0.047619047619047616 * (abs(x) * (abs(x) * t_1))))));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Abs[x], $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[x], $MachinePrecision] * N[(N[Abs[x], $MachinePrecision] * t$95$0), $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[(0.6666666666666666 * t$95$0), $MachinePrecision]), $MachinePrecision] + N[(0.2 * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(0.047619047619047616 * N[(N[Abs[x], $MachinePrecision] * N[(N[Abs[x], $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Initial program 99.9%

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

2. Final simplification 99.9%

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

Alternative 2: 99.9% accurate, 3.7× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (fabs
  (*
   (* x (pow PI -0.5))
   (+
    (+ 2.0 (* 0.6666666666666666 (* x x)))
    (+ (* 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)) * ((2.0 + (0.6666666666666666 * (x * x))) + ((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)) * ((2.0 + (0.6666666666666666 * (x * x))) + ((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)) * ((2.0 + (0.6666666666666666 * (x * x))) + ((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(2.0 + Float64(0.6666666666666666 * Float64(x * x))) + 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)) * ((2.0 + (0.6666666666666666 * (x * x))) + ((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[(2.0 + N[(0.6666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $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 99.4%

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

   1. add-sqr-sqrt 31.7%

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

   2. fabs-sqr 31.7%

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

   3. add-sqr-sqrt 99.4%

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

   4. div-inv 99.9%

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

   5. *-commutative 99.9%

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

   6. pow1/2 99.9%

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

   7. pow-flip 99.9%

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

   8. metadata-eval 99.9%

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

5. Applied egg-rr 99.9%

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

6. Taylor expanded in x around 0 99.9%

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

   1. fma-udef 90.1%

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

8. Applied egg-rr 99.9%

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

9. Final simplification 99.9%

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

Alternative 3: 98.4% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[Abs[N[(N[(2.0 * x + N[(0.047619047619047616 * N[Power[x, 7.0], $MachinePrecision]), $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 99.4%

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

4. Taylor expanded in x around inf 98.8%

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

5. Final simplification 98.8%

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

Alternative 4: 93.7% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.65:\\ \;\;\;\;\left|\left(x \cdot {\pi}^{-0.5}\right) \cdot \left(\left(2 + 0.6666666666666666 \cdot \left(x \cdot x\right)\right) + 0.2 \cdot {x}^{4}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{0.047619047619047616 \cdot {x}^{7}}{\sqrt{\pi}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.65)
   (fabs
    (*
     (* x (pow PI -0.5))
     (+ (+ 2.0 (* 0.6666666666666666 (* x x))) (* 0.2 (pow x 4.0)))))
   (fabs (/ (* 0.047619047619047616 (pow x 7.0)) (sqrt PI)))))

C

double code(double x) {
	double tmp;
	if (x <= 2.65) {
		tmp = fabs(((x * pow(((double) M_PI), -0.5)) * ((2.0 + (0.6666666666666666 * (x * x))) + (0.2 * pow(x, 4.0)))));
	} else {
		tmp = fabs(((0.047619047619047616 * pow(x, 7.0)) / sqrt(((double) M_PI))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 2.65) {
		tmp = Math.abs(((x * Math.pow(Math.PI, -0.5)) * ((2.0 + (0.6666666666666666 * (x * x))) + (0.2 * Math.pow(x, 4.0)))));
	} else {
		tmp = Math.abs(((0.047619047619047616 * Math.pow(x, 7.0)) / Math.sqrt(Math.PI)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2.65:
		tmp = math.fabs(((x * math.pow(math.pi, -0.5)) * ((2.0 + (0.6666666666666666 * (x * x))) + (0.2 * math.pow(x, 4.0)))))
	else:
		tmp = math.fabs(((0.047619047619047616 * math.pow(x, 7.0)) / math.sqrt(math.pi)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.65)
		tmp = abs(Float64(Float64(x * (pi ^ -0.5)) * Float64(Float64(2.0 + Float64(0.6666666666666666 * Float64(x * x))) + Float64(0.2 * (x ^ 4.0)))));
	else
		tmp = abs(Float64(Float64(0.047619047619047616 * (x ^ 7.0)) / sqrt(pi)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2.65)
		tmp = abs(((x * (pi ^ -0.5)) * ((2.0 + (0.6666666666666666 * (x * x))) + (0.2 * (x ^ 4.0)))));
	else
		tmp = abs(((0.047619047619047616 * (x ^ 7.0)) / sqrt(pi)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2.65], N[Abs[N[(N[(x * N[Power[Pi, -0.5], $MachinePrecision]), $MachinePrecision] * N[(N[(2.0 + N[(0.6666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(0.2 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(0.047619047619047616 * N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.64999999999999991

   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 99.4%

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

      1. add-sqr-sqrt 31.7%

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

      2. fabs-sqr 31.7%

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

      3. add-sqr-sqrt 99.4%

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

      4. div-inv 99.9%

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

      5. *-commutative 99.9%

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

      6. pow1/2 99.9%

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

      7. pow-flip 99.9%

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

      8. metadata-eval 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in x around 0 95.5%

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

      1. fma-udef 90.1%

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

   8. Applied egg-rr 95.5%

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

   if 2.64999999999999991 < 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 99.4%

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

   4. Taylor expanded in x around inf 38.7%

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

      1. associate-*r* 38.7%

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

   6. Simplified 38.7%

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

      1. sqrt-div 38.7%

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

      2. metadata-eval 38.7%

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

      3. un-div-inv 38.7%

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

   8. Applied egg-rr 38.7%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.65:\\ \;\;\;\;\left|\left(x \cdot {\pi}^{-0.5}\right) \cdot \left(\left(2 + 0.6666666666666666 \cdot \left(x \cdot x\right)\right) + 0.2 \cdot {x}^{4}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{0.047619047619047616 \cdot {x}^{7}}{\sqrt{\pi}}\right|\\ \end{array} \]

Alternative 5: 89.4% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.2:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(2 \cdot x + 0.6666666666666666 \cdot {x}^{3}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{0.047619047619047616 \cdot {x}^{7}}{\sqrt{\pi}}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.2)
   (fabs
    (* (sqrt (/ 1.0 PI)) (+ (* 2.0 x) (* 0.6666666666666666 (pow x 3.0)))))
   (fabs (/ (* 0.047619047619047616 (pow x 7.0)) (sqrt PI)))))

C

double code(double x) {
	double tmp;
	if (x <= 2.2) {
		tmp = fabs((sqrt((1.0 / ((double) M_PI))) * ((2.0 * x) + (0.6666666666666666 * pow(x, 3.0)))));
	} else {
		tmp = fabs(((0.047619047619047616 * pow(x, 7.0)) / sqrt(((double) M_PI))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 2.2) {
		tmp = Math.abs((Math.sqrt((1.0 / Math.PI)) * ((2.0 * x) + (0.6666666666666666 * Math.pow(x, 3.0)))));
	} else {
		tmp = Math.abs(((0.047619047619047616 * Math.pow(x, 7.0)) / Math.sqrt(Math.PI)));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2.2:
		tmp = math.fabs((math.sqrt((1.0 / math.pi)) * ((2.0 * x) + (0.6666666666666666 * math.pow(x, 3.0)))))
	else:
		tmp = math.fabs(((0.047619047619047616 * math.pow(x, 7.0)) / math.sqrt(math.pi)))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.2)
		tmp = abs(Float64(sqrt(Float64(1.0 / pi)) * Float64(Float64(2.0 * x) + Float64(0.6666666666666666 * (x ^ 3.0)))));
	else
		tmp = abs(Float64(Float64(0.047619047619047616 * (x ^ 7.0)) / sqrt(pi)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2.2)
		tmp = abs((sqrt((1.0 / pi)) * ((2.0 * x) + (0.6666666666666666 * (x ^ 3.0)))));
	else
		tmp = abs(((0.047619047619047616 * (x ^ 7.0)) / sqrt(pi)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2.2], N[Abs[N[(N[Sqrt[N[(1.0 / Pi), $MachinePrecision]], $MachinePrecision] * N[(N[(2.0 * x), $MachinePrecision] + N[(0.6666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(0.047619047619047616 * N[Power[x, 7.0], $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.2000000000000002

   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 99.4%

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

   4. Taylor expanded in x around 0 90.6%

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

      1. +-commutative 90.6%

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

      2. associate-*r* 90.6%

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

      3. associate-*r* 90.6%

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

      4. distribute-rgt-out 90.6%

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

      5. *-commutative 90.6%

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

   6. Simplified 90.6%

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

   if 2.2000000000000002 < 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 99.4%

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

   4. Taylor expanded in x around inf 38.7%

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

      1. associate-*r* 38.7%

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

   6. Simplified 38.7%

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

      1. sqrt-div 38.7%

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

      2. metadata-eval 38.7%

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

      3. un-div-inv 38.7%

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

   8. Applied egg-rr 38.7%

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

4. Final simplification 90.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.2:\\ \;\;\;\;\left|\sqrt{\frac{1}{\pi}} \cdot \left(2 \cdot x + 0.6666666666666666 \cdot {x}^{3}\right)\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\frac{0.047619047619047616 \cdot {x}^{7}}{\sqrt{\pi}}\right|\\ \end{array} \]

Alternative 6: 67.5% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := If[LessEqual[x, 1.85], N[Abs[N[(x * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(0.047619047619047616 * N[(N[Power[x, 7.0], $MachinePrecision] / N[Sqrt[Pi], $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 99.4%

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

   4. Taylor expanded in x around 0 66.2%

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

      1. associate-*r* 66.2%

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

   6. Simplified 66.2%

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

      1. expm1-log1p-u 64.2%

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

      2. expm1-udef 5.4%

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

      3. associate-*l* 5.4%

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

      4. sqrt-div 5.4%

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

      5. metadata-eval 5.4%

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

   8. Applied egg-rr 5.4%

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

      1. expm1-def 64.2%

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

      2. expm1-log1p 66.2%

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

      3. associate-*r* 66.2%

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

      4. *-commutative 66.2%

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

      5. associate-*l* 66.2%

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

      6. associate-*r/ 66.2%

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

      7. metadata-eval 66.2%

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

   10. Simplified 66.2%

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

   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 99.4%

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

   4. Taylor expanded in x around inf 38.7%

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

      1. associate-*r* 38.7%

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

   6. Simplified 38.7%

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

      1. expm1-log1p-u 3.7%

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

      2. expm1-udef 3.4%

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

      3. sqrt-div 3.4%

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

      4. metadata-eval 3.4%

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

      5. un-div-inv 3.4%

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

   8. Applied egg-rr 3.4%

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

      1. expm1-def 3.7%

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

      2. expm1-log1p 38.7%

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

      3. *-commutative 38.7%

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

      4. associate-*l/ 38.7%

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

      5. *-commutative 38.7%

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

   10. Simplified 38.7%

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

4. Final simplification 66.2%

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

Alternative 7: 67.5% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1.8500000000000001

   1. Initial program 99.9%

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

   3. Simplified 99.4%

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

   4. Taylor expanded in x around 0 66.2%

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

      1. associate-*r* 66.2%

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

   6. Simplified 66.2%

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

      1. expm1-log1p-u 64.2%

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

      2. expm1-udef 5.4%

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

      3. associate-*l* 5.4%

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

      4. sqrt-div 5.4%

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

      5. metadata-eval 5.4%

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

   8. Applied egg-rr 5.4%

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

      1. expm1-def 64.2%

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

      2. expm1-log1p 66.2%

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

      3. associate-*r* 66.2%

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

      4. *-commutative 66.2%

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

      5. associate-*l* 66.2%

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

      6. associate-*r/ 66.2%

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

      7. metadata-eval 66.2%

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

   10. Simplified 66.2%

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

   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 99.4%

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

   4. Taylor expanded in x around inf 38.7%

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

      1. associate-*r* 38.7%

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

   6. Simplified 38.7%

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

      1. sqrt-div 38.7%

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

      2. metadata-eval 38.7%

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

      3. un-div-inv 38.7%

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

   8. Applied egg-rr 38.7%

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

4. Final simplification 66.2%

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

Alternative 8: 74.3% accurate, 6.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{\sqrt{\pi}}\\ t_1 := 0.6666666666666666 \cdot \left(x \cdot x\right)\\ \mathbf{if}\;x \leq 5 \cdot 10^{+102}:\\ \;\;\;\;\left|t_0 \cdot \frac{t_1 \cdot t_1 - 4}{t_1 - 2}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|t_1 \cdot t_0\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ x (sqrt PI))) (t_1 (* 0.6666666666666666 (* x x))))
   (if (<= x 5e+102)
     (fabs (* t_0 (/ (- (* t_1 t_1) 4.0) (- t_1 2.0))))
     (fabs (* t_1 t_0)))))

C

double code(double x) {
	double t_0 = x / sqrt(((double) M_PI));
	double t_1 = 0.6666666666666666 * (x * x);
	double tmp;
	if (x <= 5e+102) {
		tmp = fabs((t_0 * (((t_1 * t_1) - 4.0) / (t_1 - 2.0))));
	} else {
		tmp = fabs((t_1 * t_0));
	}
	return tmp;
}

Java

public static double code(double x) {
	double t_0 = x / Math.sqrt(Math.PI);
	double t_1 = 0.6666666666666666 * (x * x);
	double tmp;
	if (x <= 5e+102) {
		tmp = Math.abs((t_0 * (((t_1 * t_1) - 4.0) / (t_1 - 2.0))));
	} else {
		tmp = Math.abs((t_1 * t_0));
	}
	return tmp;
}

Python

def code(x):
	t_0 = x / math.sqrt(math.pi)
	t_1 = 0.6666666666666666 * (x * x)
	tmp = 0
	if x <= 5e+102:
		tmp = math.fabs((t_0 * (((t_1 * t_1) - 4.0) / (t_1 - 2.0))))
	else:
		tmp = math.fabs((t_1 * t_0))
	return tmp

Julia

function code(x)
	t_0 = Float64(x / sqrt(pi))
	t_1 = Float64(0.6666666666666666 * Float64(x * x))
	tmp = 0.0
	if (x <= 5e+102)
		tmp = abs(Float64(t_0 * Float64(Float64(Float64(t_1 * t_1) - 4.0) / Float64(t_1 - 2.0))));
	else
		tmp = abs(Float64(t_1 * t_0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = x / sqrt(pi);
	t_1 = 0.6666666666666666 * (x * x);
	tmp = 0.0;
	if (x <= 5e+102)
		tmp = abs((t_0 * (((t_1 * t_1) - 4.0) / (t_1 - 2.0))));
	else
		tmp = abs((t_1 * t_0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.6666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 5e+102], N[Abs[N[(t$95$0 * N[(N[(N[(t$95$1 * t$95$1), $MachinePrecision] - 4.0), $MachinePrecision] / N[(t$95$1 - 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(t$95$1 * t$95$0), $MachinePrecision]], $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if x < 5e102

   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 99.4%

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

   4. Taylor expanded in x around 0 90.6%

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

      1. +-commutative 90.6%

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

      2. associate-*r* 90.6%

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

      3. associate-*r* 90.6%

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

      4. distribute-rgt-out 90.6%

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

      5. *-commutative 90.6%

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

   6. Simplified 90.6%

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

      1. distribute-lft-in 90.6%

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

      2. *-commutative 90.6%

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

      3. *-commutative 90.6%

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

      4. associate-*l* 90.6%

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

      5. sqrt-div 90.6%

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

      6. metadata-eval 90.6%

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

      7. sqrt-div 90.6%

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

      8. metadata-eval 90.6%

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

   8. Applied egg-rr 90.6%

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

      1. *-commutative 90.6%

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

      2. *-commutative 90.6%

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

      3. associate-*r* 90.6%

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

      4. distribute-lft-in 90.6%

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

      5. fma-udef 90.6%

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

      6. associate-*l/ 90.1%

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

      7. *-lft-identity 90.1%

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

   10. Simplified 90.1%

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

       1. expm1-log1p-u 64.0%

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

       2. expm1-udef 5.6%

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

   12. Applied egg-rr 5.6%

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

       1. expm1-def 64.0%

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

       2. expm1-log1p 90.1%

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

       3. fma-udef 90.1%

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

       4. +-commutative 90.1%

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

       5. unpow3 90.1%

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

       6. associate-*r* 90.1%

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

       7. *-commutative 90.1%

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

       8. distribute-rgt-in 90.1%

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

       9. fma-udef 90.1%

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

       10. associate-*l/ 90.1%

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

   14. Simplified 90.1%

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

       1. fma-udef 90.1%

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

       2. flip-+ 73.4%

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

       3. metadata-eval 73.4%

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

   16. Applied egg-rr 73.4%

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

   if 5e102 < 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 99.4%

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

   4. Taylor expanded in x around 0 90.6%

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

      1. +-commutative 90.6%

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

      2. associate-*r* 90.6%

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

      3. associate-*r* 90.6%

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

      4. distribute-rgt-out 90.6%

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

      5. *-commutative 90.6%

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

   6. Simplified 90.6%

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

      1. distribute-lft-in 90.6%

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

      2. *-commutative 90.6%

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

      3. *-commutative 90.6%

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

      4. associate-*l* 90.6%

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

      5. sqrt-div 90.6%

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

      6. metadata-eval 90.6%

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

      7. sqrt-div 90.6%

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

      8. metadata-eval 90.6%

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

   8. Applied egg-rr 90.6%

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

      1. *-commutative 90.6%

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

      2. *-commutative 90.6%

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

      3. associate-*r* 90.6%

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

      4. distribute-lft-in 90.6%

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

      5. fma-udef 90.6%

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

      6. associate-*l/ 90.1%

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

      7. *-lft-identity 90.1%

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

   10. Simplified 90.1%

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

       1. expm1-log1p-u 64.0%

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

       2. expm1-udef 5.6%

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

   12. Applied egg-rr 5.6%

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

       1. expm1-def 64.0%

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

       2. expm1-log1p 90.1%

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

       3. fma-udef 90.1%

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

       4. +-commutative 90.1%

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

       5. unpow3 90.1%

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

       6. associate-*r* 90.1%

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

       7. *-commutative 90.1%

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

       8. distribute-rgt-in 90.1%

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

       9. fma-udef 90.1%

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

       10. associate-*l/ 90.1%

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

   14. Simplified 90.1%

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

   15. Taylor expanded in x around inf 30.1%

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

       1. unpow2 30.1%

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

   17. Simplified 30.1%

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

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{+102}:\\ \;\;\;\;\left|\frac{x}{\sqrt{\pi}} \cdot \frac{\left(0.6666666666666666 \cdot \left(x \cdot x\right)\right) \cdot \left(0.6666666666666666 \cdot \left(x \cdot x\right)\right) - 4}{0.6666666666666666 \cdot \left(x \cdot x\right) - 2}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\left(0.6666666666666666 \cdot \left(x \cdot x\right)\right) \cdot \frac{x}{\sqrt{\pi}}\right|\\ \end{array} \]

Alternative 9: 67.5% accurate, 6.3× speedup

Math

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

FPCore

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

C

double code(double x) {
	double tmp;
	if (x <= 1.75) {
		tmp = fabs((x * (2.0 / sqrt(((double) M_PI)))));
	} else {
		tmp = fabs(((0.6666666666666666 * (x * 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 * (2.0 / Math.sqrt(Math.PI))));
	} else {
		tmp = Math.abs(((0.6666666666666666 * (x * x)) * (x / Math.sqrt(Math.PI))));
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := If[LessEqual[x, 1.75], N[Abs[N[(x * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[(N[(0.6666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x / N[Sqrt[Pi], $MachinePrecision]), $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 99.4%

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

   4. Taylor expanded in x around 0 66.2%

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

      1. associate-*r* 66.2%

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

   6. Simplified 66.2%

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

      1. expm1-log1p-u 64.2%

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

      2. expm1-udef 5.4%

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

      3. associate-*l* 5.4%

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

      4. sqrt-div 5.4%

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

      5. metadata-eval 5.4%

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

   8. Applied egg-rr 5.4%

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

      1. expm1-def 64.2%

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

      2. expm1-log1p 66.2%

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

      3. associate-*r* 66.2%

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

      4. *-commutative 66.2%

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

      5. associate-*l* 66.2%

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

      6. associate-*r/ 66.2%

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

      7. metadata-eval 66.2%

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

   10. Simplified 66.2%

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

   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 99.4%

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

   4. Taylor expanded in x around 0 90.6%

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

      1. +-commutative 90.6%

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

      2. associate-*r* 90.6%

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

      3. associate-*r* 90.6%

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

      4. distribute-rgt-out 90.6%

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

      5. *-commutative 90.6%

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

   6. Simplified 90.6%

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

      1. distribute-lft-in 90.6%

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

      2. *-commutative 90.6%

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

      3. *-commutative 90.6%

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

      4. associate-*l* 90.6%

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

      5. sqrt-div 90.6%

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

      6. metadata-eval 90.6%

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

      7. sqrt-div 90.6%

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

      8. metadata-eval 90.6%

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

   8. Applied egg-rr 90.6%

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

      1. *-commutative 90.6%

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

      2. *-commutative 90.6%

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

      3. associate-*r* 90.6%

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

      4. distribute-lft-in 90.6%

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

      5. fma-udef 90.6%

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

      6. associate-*l/ 90.1%

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

      7. *-lft-identity 90.1%

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

   10. Simplified 90.1%

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

       1. expm1-log1p-u 64.0%

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

       2. expm1-udef 5.6%

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

   12. Applied egg-rr 5.6%

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

       1. expm1-def 64.0%

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

       2. expm1-log1p 90.1%

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

       3. fma-udef 90.1%

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

       4. +-commutative 90.1%

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

       5. unpow3 90.1%

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

       6. associate-*r* 90.1%

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

       7. *-commutative 90.1%

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

       8. distribute-rgt-in 90.1%

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

       9. fma-udef 90.1%

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

       10. associate-*l/ 90.1%

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

   14. Simplified 90.1%

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

   15. Taylor expanded in x around inf 30.1%

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

       1. unpow2 30.1%

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

   17. Simplified 30.1%

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

4. Final simplification 66.2%

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

Alternative 10: 89.0% accurate, 6.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[Abs[N[(N[(2.0 + N[(0.6666666666666666 * N[(x * x), $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 99.4%

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

4. Taylor expanded in x around 0 90.6%

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

   1. +-commutative 90.6%

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

   2. associate-*r* 90.6%

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

   3. associate-*r* 90.6%

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

   4. distribute-rgt-out 90.6%

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

   5. *-commutative 90.6%

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

6. Simplified 90.6%

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

   1. distribute-lft-in 90.6%

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

   2. *-commutative 90.6%

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

   3. *-commutative 90.6%

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

   4. associate-*l* 90.6%

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

   5. sqrt-div 90.6%

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

   6. metadata-eval 90.6%

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

   7. sqrt-div 90.6%

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

   8. metadata-eval 90.6%

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

8. Applied egg-rr 90.6%

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

   1. *-commutative 90.6%

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

   2. *-commutative 90.6%

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

   3. associate-*r* 90.6%

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

   4. distribute-lft-in 90.6%

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

   5. fma-udef 90.6%

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

   6. associate-*l/ 90.1%

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

   7. *-lft-identity 90.1%

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

10. Simplified 90.1%

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

    1. expm1-log1p-u 64.0%

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

    2. expm1-udef 5.6%

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

12. Applied egg-rr 5.6%

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

    1. expm1-def 64.0%

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

    2. expm1-log1p 90.1%

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

    3. fma-udef 90.1%

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

    4. +-commutative 90.1%

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

    5. unpow3 90.1%

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

    6. associate-*r* 90.1%

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

    7. *-commutative 90.1%

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

    8. distribute-rgt-in 90.1%

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

    9. fma-udef 90.1%

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

    10. associate-*l/ 90.1%

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

14. Simplified 90.1%

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

    1. fma-udef 90.1%

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

16. Applied egg-rr 90.1%

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

17. Final simplification 90.1%

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

Alternative 11: 67.5% accurate, 6.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.8 \cdot 10^{-27}:\\ \;\;\;\;\left|x \cdot \frac{2}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt{x \cdot \left(x \cdot \frac{4}{\pi}\right)}\right|\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.8e-27)
   (fabs (* x (/ 2.0 (sqrt PI))))
   (fabs (sqrt (* x (* x (/ 4.0 PI)))))))

C

double code(double x) {
	double tmp;
	if (x <= 2.8e-27) {
		tmp = fabs((x * (2.0 / sqrt(((double) M_PI)))));
	} else {
		tmp = fabs(sqrt((x * (x * (4.0 / ((double) M_PI))))));
	}
	return tmp;
}

Java

public static double code(double x) {
	double tmp;
	if (x <= 2.8e-27) {
		tmp = Math.abs((x * (2.0 / Math.sqrt(Math.PI))));
	} else {
		tmp = Math.abs(Math.sqrt((x * (x * (4.0 / Math.PI)))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2.8e-27:
		tmp = math.fabs((x * (2.0 / math.sqrt(math.pi))))
	else:
		tmp = math.fabs(math.sqrt((x * (x * (4.0 / math.pi)))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.8e-27)
		tmp = abs(Float64(x * Float64(2.0 / sqrt(pi))));
	else
		tmp = abs(sqrt(Float64(x * Float64(x * Float64(4.0 / pi)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2.8e-27)
		tmp = abs((x * (2.0 / sqrt(pi))));
	else
		tmp = abs(sqrt((x * (x * (4.0 / pi)))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2.8e-27], N[Abs[N[(x * N[(2.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Abs[N[Sqrt[N[(x * N[(x * N[(4.0 / Pi), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.8e-27

   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 99.4%

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

   4. Taylor expanded in x around 0 64.7%

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

      1. associate-*r* 64.7%

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

   6. Simplified 64.7%

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

      1. expm1-log1p-u 62.7%

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

      2. expm1-udef 4.3%

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

      3. associate-*l* 4.3%

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

      4. sqrt-div 4.3%

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

      5. metadata-eval 4.3%

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

   8. Applied egg-rr 4.3%

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

      1. expm1-def 62.7%

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

      2. expm1-log1p 64.7%

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

      3. associate-*r* 64.7%

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

      4. *-commutative 64.7%

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

      5. associate-*l* 64.7%

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

      6. associate-*r/ 64.7%

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

      7. metadata-eval 64.7%

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

   10. Simplified 64.7%

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

   if 2.8e-27 < x

   1. Initial program 99.7%

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

   3. Simplified 99.6%

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

   4. Taylor expanded in x around 0 95.8%

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

      1. associate-*r* 95.8%

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

   6. Simplified 95.8%

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

      1. expm1-log1p-u 95.8%

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

      2. expm1-udef 27.5%

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

      3. associate-*l* 27.5%

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

      4. sqrt-div 27.5%

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

      5. metadata-eval 27.5%

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

   8. Applied egg-rr 27.5%

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

      1. expm1-def 95.8%

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

      2. expm1-log1p 95.8%

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

      3. associate-*r* 95.8%

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

      4. *-commutative 95.8%

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

      5. associate-*l* 95.8%

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

      6. associate-*r/ 95.8%

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

      7. metadata-eval 95.8%

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

   10. Simplified 95.8%

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

       1. add-sqr-sqrt 95.2%

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

       2. sqrt-unprod 95.8%

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

       3. swap-sqr 95.3%

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

       4. frac-times 94.8%

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

       5. metadata-eval 94.8%

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

       6. add-sqr-sqrt 95.9%

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

   12. Applied egg-rr 95.9%

       \[\leadsto \left|\color{blue}{\sqrt{\left(x \cdot x\right) \cdot \frac{4}{\pi}}}\right| \]
   13. Step-by-step derivation

       1. associate-*l* 95.9%

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

   14. Simplified 95.9%

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

4. Final simplification 66.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.8 \cdot 10^{-27}:\\ \;\;\;\;\left|x \cdot \frac{2}{\sqrt{\pi}}\right|\\ \mathbf{else}:\\ \;\;\;\;\left|\sqrt{x \cdot \left(x \cdot \frac{4}{\pi}\right)}\right|\\ \end{array} \]

Alternative 12: 67.0% accurate, 6.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[Abs[N[(2.0 * 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 99.4%

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

4. Taylor expanded in x around 0 66.2%

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

   1. associate-*r* 66.2%

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

6. Simplified 66.2%

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

   1. expm1-log1p-u 64.2%

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

   2. expm1-udef 5.4%

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

   3. associate-*l* 5.4%

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

   4. sqrt-div 5.4%

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

   5. metadata-eval 5.4%

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

8. Applied egg-rr 5.4%

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

   1. expm1-def 64.2%

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

   2. expm1-log1p 66.2%

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

   3. associate-*r/ 65.7%

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

   4. *-rgt-identity 65.7%

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

10. Simplified 65.7%

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

11. Final simplification 65.7%

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

Alternative 13: 67.5% accurate, 6.4× speedup

Math

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

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

MATLAB

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

Wolfram

code[x_] := N[Abs[N[(x * N[(2.0 / 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 99.4%

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

4. Taylor expanded in x around 0 66.2%

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

   1. associate-*r* 66.2%

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

6. Simplified 66.2%

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

   1. expm1-log1p-u 64.2%

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

   2. expm1-udef 5.4%

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

   3. associate-*l* 5.4%

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

   4. sqrt-div 5.4%

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

   5. metadata-eval 5.4%

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

8. Applied egg-rr 5.4%

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

   1. expm1-def 64.2%

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

   2. expm1-log1p 66.2%

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

   3. associate-*r* 66.2%

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

   4. *-commutative 66.2%

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

   5. associate-*l* 66.2%

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

   6. associate-*r/ 66.2%

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

   7. metadata-eval 66.2%

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

10. Simplified 66.2%

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

11. Final simplification 66.2%

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

Reproduce

herbie shell --seed 2023252 
(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)))))))