Jmat.Real.erfi, branch x greater than or equal to 5

Percentage Accurate: 100.0% → 100.0%

Time: 4.4s

Alternatives: 8

Speedup: N/A×

• 99.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[x \geq 0.5\]

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\left|x\right|}\\ t_1 := \left(t\_0 \cdot t\_0\right) \cdot t\_0\\ t_2 := \left(t\_1 \cdot t\_0\right) \cdot t\_0\\ \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(t\_0 + \frac{1}{2} \cdot t\_1\right) + \frac{3}{4} \cdot t\_2\right) + \frac{15}{8} \cdot \left(\left(t\_2 \cdot t\_0\right) \cdot t\_0\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (fabs x)))
        (t_1 (* (* t_0 t_0) t_0))
        (t_2 (* (* t_1 t_0) t_0)))
   (*
    (* (/ 1.0 (sqrt PI)) (exp (* (fabs x) (fabs x))))
    (+
     (+ (+ t_0 (* (/ 1.0 2.0) t_1)) (* (/ 3.0 4.0) t_2))
     (* (/ 15.0 8.0) (* (* t_2 t_0) t_0))))))

C

double code(double x) {
	double t_0 = 1.0 / fabs(x);
	double t_1 = (t_0 * t_0) * t_0;
	double t_2 = (t_1 * t_0) * t_0;
	return ((1.0 / sqrt(((double) M_PI))) * exp((fabs(x) * fabs(x)))) * (((t_0 + ((1.0 / 2.0) * t_1)) + ((3.0 / 4.0) * t_2)) + ((15.0 / 8.0) * ((t_2 * t_0) * t_0)));
}

Java

public static double code(double x) {
	double t_0 = 1.0 / Math.abs(x);
	double t_1 = (t_0 * t_0) * t_0;
	double t_2 = (t_1 * t_0) * t_0;
	return ((1.0 / Math.sqrt(Math.PI)) * Math.exp((Math.abs(x) * Math.abs(x)))) * (((t_0 + ((1.0 / 2.0) * t_1)) + ((3.0 / 4.0) * t_2)) + ((15.0 / 8.0) * ((t_2 * t_0) * t_0)));
}

Python

def code(x):
	t_0 = 1.0 / math.fabs(x)
	t_1 = (t_0 * t_0) * t_0
	t_2 = (t_1 * t_0) * t_0
	return ((1.0 / math.sqrt(math.pi)) * math.exp((math.fabs(x) * math.fabs(x)))) * (((t_0 + ((1.0 / 2.0) * t_1)) + ((3.0 / 4.0) * t_2)) + ((15.0 / 8.0) * ((t_2 * t_0) * t_0)))

Julia

function code(x)
	t_0 = Float64(1.0 / abs(x))
	t_1 = Float64(Float64(t_0 * t_0) * t_0)
	t_2 = Float64(Float64(t_1 * t_0) * t_0)
	return Float64(Float64(Float64(1.0 / sqrt(pi)) * exp(Float64(abs(x) * abs(x)))) * Float64(Float64(Float64(t_0 + Float64(Float64(1.0 / 2.0) * t_1)) + Float64(Float64(3.0 / 4.0) * t_2)) + Float64(Float64(15.0 / 8.0) * Float64(Float64(t_2 * t_0) * t_0))))
end

MATLAB

function tmp = code(x)
	t_0 = 1.0 / abs(x);
	t_1 = (t_0 * t_0) * t_0;
	t_2 = (t_1 * t_0) * t_0;
	tmp = ((1.0 / sqrt(pi)) * exp((abs(x) * abs(x)))) * (((t_0 + ((1.0 / 2.0) * t_1)) + ((3.0 / 4.0) * t_2)) + ((15.0 / 8.0) * ((t_2 * t_0) * t_0)));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]}, N[(N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(t$95$0 + N[(N[(1.0 / 2.0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(3.0 / 4.0), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(N[(15.0 / 8.0), $MachinePrecision] * N[(N[(t$95$2 * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{\left|x\right|}\\ t_1 := \left(t\_0 \cdot t\_0\right) \cdot t\_0\\ t_2 := \left(t\_1 \cdot t\_0\right) \cdot t\_0\\ \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(t\_0 + \frac{1}{2} \cdot t\_1\right) + \frac{3}{4} \cdot t\_2\right) + \frac{15}{8} \cdot \left(\left(t\_2 \cdot t\_0\right) \cdot t\_0\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (fabs x)))
        (t_1 (* (* t_0 t_0) t_0))
        (t_2 (* (* t_1 t_0) t_0)))
   (*
    (* (/ 1.0 (sqrt PI)) (exp (* (fabs x) (fabs x))))
    (+
     (+ (+ t_0 (* (/ 1.0 2.0) t_1)) (* (/ 3.0 4.0) t_2))
     (* (/ 15.0 8.0) (* (* t_2 t_0) t_0))))))

C

double code(double x) {
	double t_0 = 1.0 / fabs(x);
	double t_1 = (t_0 * t_0) * t_0;
	double t_2 = (t_1 * t_0) * t_0;
	return ((1.0 / sqrt(((double) M_PI))) * exp((fabs(x) * fabs(x)))) * (((t_0 + ((1.0 / 2.0) * t_1)) + ((3.0 / 4.0) * t_2)) + ((15.0 / 8.0) * ((t_2 * t_0) * t_0)));
}

Java

public static double code(double x) {
	double t_0 = 1.0 / Math.abs(x);
	double t_1 = (t_0 * t_0) * t_0;
	double t_2 = (t_1 * t_0) * t_0;
	return ((1.0 / Math.sqrt(Math.PI)) * Math.exp((Math.abs(x) * Math.abs(x)))) * (((t_0 + ((1.0 / 2.0) * t_1)) + ((3.0 / 4.0) * t_2)) + ((15.0 / 8.0) * ((t_2 * t_0) * t_0)));
}

Python

def code(x):
	t_0 = 1.0 / math.fabs(x)
	t_1 = (t_0 * t_0) * t_0
	t_2 = (t_1 * t_0) * t_0
	return ((1.0 / math.sqrt(math.pi)) * math.exp((math.fabs(x) * math.fabs(x)))) * (((t_0 + ((1.0 / 2.0) * t_1)) + ((3.0 / 4.0) * t_2)) + ((15.0 / 8.0) * ((t_2 * t_0) * t_0)))

Julia

function code(x)
	t_0 = Float64(1.0 / abs(x))
	t_1 = Float64(Float64(t_0 * t_0) * t_0)
	t_2 = Float64(Float64(t_1 * t_0) * t_0)
	return Float64(Float64(Float64(1.0 / sqrt(pi)) * exp(Float64(abs(x) * abs(x)))) * Float64(Float64(Float64(t_0 + Float64(Float64(1.0 / 2.0) * t_1)) + Float64(Float64(3.0 / 4.0) * t_2)) + Float64(Float64(15.0 / 8.0) * Float64(Float64(t_2 * t_0) * t_0))))
end

MATLAB

function tmp = code(x)
	t_0 = 1.0 / abs(x);
	t_1 = (t_0 * t_0) * t_0;
	t_2 = (t_1 * t_0) * t_0;
	tmp = ((1.0 / sqrt(pi)) * exp((abs(x) * abs(x)))) * (((t_0 + ((1.0 / 2.0) * t_1)) + ((3.0 / 4.0) * t_2)) + ((15.0 / 8.0) * ((t_2 * t_0) * t_0)));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(t$95$0 * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t$95$1 * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]}, N[(N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[Exp[N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(t$95$0 + N[(N[(1.0 / 2.0), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision] + N[(N[(3.0 / 4.0), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] + N[(N[(15.0 / 8.0), $MachinePrecision] * N[(N[(t$95$2 * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Alternative 1: 100.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\left(\frac{\frac{1.875}{x \cdot x} + 0.75}{-\left(\left(x \cdot x\right) \cdot x\right) \cdot x} - 1\right) - \frac{0.5}{x \cdot x}}{-x} \cdot e^{x \cdot x}}{\sqrt{\pi}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  (*
   (/
    (-
     (- (/ (+ (/ 1.875 (* x x)) 0.75) (- (* (* (* x x) x) x))) 1.0)
     (/ 0.5 (* x x)))
    (- x))
   (exp (* x x)))
  (sqrt PI)))

C

double code(double x) {
	return (((((((1.875 / (x * x)) + 0.75) / -(((x * x) * x) * x)) - 1.0) - (0.5 / (x * x))) / -x) * exp((x * x))) / sqrt(((double) M_PI));
}

Java

public static double code(double x) {
	return (((((((1.875 / (x * x)) + 0.75) / -(((x * x) * x) * x)) - 1.0) - (0.5 / (x * x))) / -x) * Math.exp((x * x))) / Math.sqrt(Math.PI);
}

Python

def code(x):
	return (((((((1.875 / (x * x)) + 0.75) / -(((x * x) * x) * x)) - 1.0) - (0.5 / (x * x))) / -x) * math.exp((x * x))) / math.sqrt(math.pi)

Julia

function code(x)
	return Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(1.875 / Float64(x * x)) + 0.75) / Float64(-Float64(Float64(Float64(x * x) * x) * x))) - 1.0) - Float64(0.5 / Float64(x * x))) / Float64(-x)) * exp(Float64(x * x))) / sqrt(pi))
end

MATLAB

function tmp = code(x)
	tmp = (((((((1.875 / (x * x)) + 0.75) / -(((x * x) * x) * x)) - 1.0) - (0.5 / (x * x))) / -x) * exp((x * x))) / sqrt(pi);
end

Wolfram

code[x_] := N[(N[(N[(N[(N[(N[(N[(N[(1.875 / N[(x * x), $MachinePrecision]), $MachinePrecision] + 0.75), $MachinePrecision] / (-N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision])), $MachinePrecision] - 1.0), $MachinePrecision] - N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / (-x)), $MachinePrecision] * N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

2. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(e^{{\left(\left|x\right|\right)}^{2}} \cdot \left(\frac{3}{4} \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}} + \left(\frac{15}{8} \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}} + \left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}\right)\right)\right)\right)} \]

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0.5}{x \cdot x} + 1, \frac{1}{\left|x\right|}, \mathsf{fma}\left({\left(\left|x\right|\right)}^{-7}, 1.875, {\left(\left|x\right|\right)}^{-5} \cdot 0.75\right)\right) \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}}} \]

6. Applied rewrites 100.0%

   \[\leadsto \left(\mathsf{fma}\left(\frac{0.5}{x \cdot x} + 1, \frac{1}{x}, {x}^{-7} \cdot 1.875\right) + {x}^{-5} \cdot 0.75\right) \cdot \frac{\color{blue}{e^{x \cdot x}}}{\sqrt{\pi}} \]

7. Taylor expanded in x around -inf

   \[\leadsto \left(-1 \cdot \frac{-1 \cdot \frac{\frac{3}{4} + \frac{15}{8} \cdot \frac{1}{{x}^{2}}}{{x}^{4}} - \left(1 + \frac{1}{2} \cdot \frac{1}{{x}^{2}}\right)}{x}\right) \cdot \frac{\color{blue}{e^{x \cdot x}}}{\sqrt{\pi}} \]

9. Applied rewrites 100.0%

   \[\leadsto \left(-\frac{\left(\left(-\frac{\frac{1.875}{x \cdot x} + 0.75}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}\right) - 1\right) - \frac{0.5}{x \cdot x}}{x}\right) \cdot \frac{\color{blue}{e^{x \cdot x}}}{\sqrt{\pi}} \]

11. Applied rewrites 100.0%

    \[\leadsto \frac{\frac{\left(\frac{\frac{1.875}{x \cdot x} + 0.75}{-\left(\left(x \cdot x\right) \cdot x\right) \cdot x} - 1\right) - \frac{0.5}{x \cdot x}}{-x} \cdot e^{x \cdot x}}{\color{blue}{\sqrt{\pi}}} \]
12. Add Preprocessing

Alternative 2: 99.7% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \left(-\frac{\left(-\frac{\frac{1.875}{x \cdot x} + 0.75}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}\right) - 1}{x}\right) \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (- (/ (- (- (/ (+ (/ 1.875 (* x x)) 0.75) (* (* x x) (* x x)))) 1.0) x))
  (/ (exp (* x x)) (sqrt PI))))

C

double code(double x) {
	return -((-(((1.875 / (x * x)) + 0.75) / ((x * x) * (x * x))) - 1.0) / x) * (exp((x * x)) / sqrt(((double) M_PI)));
}

Java

public static double code(double x) {
	return -((-(((1.875 / (x * x)) + 0.75) / ((x * x) * (x * x))) - 1.0) / x) * (Math.exp((x * x)) / Math.sqrt(Math.PI));
}

Python

def code(x):
	return -((-(((1.875 / (x * x)) + 0.75) / ((x * x) * (x * x))) - 1.0) / x) * (math.exp((x * x)) / math.sqrt(math.pi))

Julia

function code(x)
	return Float64(Float64(-Float64(Float64(Float64(-Float64(Float64(Float64(1.875 / Float64(x * x)) + 0.75) / Float64(Float64(x * x) * Float64(x * x)))) - 1.0) / x)) * Float64(exp(Float64(x * x)) / sqrt(pi)))
end

MATLAB

function tmp = code(x)
	tmp = -((-(((1.875 / (x * x)) + 0.75) / ((x * x) * (x * x))) - 1.0) / x) * (exp((x * x)) / sqrt(pi));
end

Wolfram

code[x_] := N[((-N[(N[((-N[(N[(N[(1.875 / N[(x * x), $MachinePrecision]), $MachinePrecision] + 0.75), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) - 1.0), $MachinePrecision] / x), $MachinePrecision]) * N[(N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

2. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(e^{{\left(\left|x\right|\right)}^{2}} \cdot \left(\frac{3}{4} \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}} + \left(\frac{15}{8} \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}} + \left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}\right)\right)\right)\right)} \]

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0.5}{x \cdot x} + 1, \frac{1}{\left|x\right|}, \mathsf{fma}\left({\left(\left|x\right|\right)}^{-7}, 1.875, {\left(\left|x\right|\right)}^{-5} \cdot 0.75\right)\right) \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}}} \]

5. Taylor expanded in x around inf

   \[\leadsto \left(\frac{3}{4} \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}} + \left(\frac{1}{\left|x\right|} + \frac{15}{8} \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}}\right)\right) \cdot \frac{\color{blue}{e^{x \cdot x}}}{\sqrt{\pi}} \]
6. Step-by-step derivation

   1. metadata-eval N/A

      \[\leadsto \left(\frac{3}{4} \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}} + \left(\frac{1}{\left|x\right|} + \frac{15}{8} \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}}\right)\right) \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}} \]

   2. metadata-eval N/A

      \[\leadsto \left(\frac{3}{4} \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}} + \left(\frac{1}{\left|x\right|} + \frac{15}{8} \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}}\right)\right) \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}} \]

   3. pow-flip N/A

      \[\leadsto \left(\frac{3}{4} \cdot {\left(\left|x\right|\right)}^{\left(\mathsf{neg}\left(5\right)\right)} + \left(\frac{1}{\left|x\right|} + \frac{15}{8} \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}}\right)\right) \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}} \]

   4. metadata-eval N/A

      \[\leadsto \left(\frac{3}{4} \cdot {\left(\left|x\right|\right)}^{-5} + \left(\frac{1}{\left|x\right|} + \frac{15}{8} \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}}\right)\right) \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}} \]

   5. *-commutative N/A

      \[\leadsto \left({\left(\left|x\right|\right)}^{-5} \cdot \frac{3}{4} + \left(\frac{1}{\left|x\right|} + \frac{15}{8} \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}}\right)\right) \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}} \]

   6. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left({\left(\left|x\right|\right)}^{-5}, \frac{3}{4}, \frac{1}{\left|x\right|} + \frac{15}{8} \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}}\right) \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}} \]

7. Applied rewrites 99.7%

   \[\leadsto \mathsf{fma}\left({x}^{-5}, 0.75, \mathsf{fma}\left({x}^{-7}, 1.875, \frac{1}{x}\right)\right) \cdot \frac{\color{blue}{e^{x \cdot x}}}{\sqrt{\pi}} \]

8. Taylor expanded in x around -inf

   \[\leadsto \left(-1 \cdot \frac{-1 \cdot \frac{\frac{3}{4} + \frac{15}{8} \cdot \frac{1}{{x}^{2}}}{{x}^{4}} - 1}{x}\right) \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}} \]
9. Step-by-step derivation

   1. metadata-eval N/A

      \[\leadsto \left(-1 \cdot \frac{-1 \cdot \frac{\frac{3}{4} + \frac{15}{8} \cdot \frac{1}{{x}^{2}}}{{x}^{4}} - 1}{x}\right) \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}} \]

   2. metadata-eval N/A

      \[\leadsto \left(-1 \cdot \frac{-1 \cdot \frac{\frac{3}{4} + \frac{15}{8} \cdot \frac{1}{{x}^{2}}}{{x}^{4}} - 1}{x}\right) \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}} \]

   3. mul-1-neg N/A

      \[\leadsto \left(\mathsf{neg}\left(\frac{-1 \cdot \frac{\frac{3}{4} + \frac{15}{8} \cdot \frac{1}{{x}^{2}}}{{x}^{4}} - 1}{x}\right)\right) \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}} \]

   4. lower-neg.f64 N/A

      \[\leadsto \left(-\frac{-1 \cdot \frac{\frac{3}{4} + \frac{15}{8} \cdot \frac{1}{{x}^{2}}}{{x}^{4}} - 1}{x}\right) \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}} \]

   5. lower-/.f64 N/A

      \[\leadsto \left(-\frac{-1 \cdot \frac{\frac{3}{4} + \frac{15}{8} \cdot \frac{1}{{x}^{2}}}{{x}^{4}} - 1}{x}\right) \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}} \]

10. Applied rewrites 99.7%

    \[\leadsto \left(-\frac{\left(-\frac{\frac{1.875}{x \cdot x} + 0.75}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}\right) - 1}{x}\right) \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}} \]
11. Add Preprocessing

Alternative 3: 99.7% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.75}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)} + 1}{x} \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (/ (+ (/ 0.75 (* (* x x) (* x x))) 1.0) x) (/ (exp (* x x)) (sqrt PI))))

C

double code(double x) {
	return (((0.75 / ((x * x) * (x * x))) + 1.0) / x) * (exp((x * x)) / sqrt(((double) M_PI)));
}

Java

public static double code(double x) {
	return (((0.75 / ((x * x) * (x * x))) + 1.0) / x) * (Math.exp((x * x)) / Math.sqrt(Math.PI));
}

Python

def code(x):
	return (((0.75 / ((x * x) * (x * x))) + 1.0) / x) * (math.exp((x * x)) / math.sqrt(math.pi))

Julia

function code(x)
	return Float64(Float64(Float64(Float64(0.75 / Float64(Float64(x * x) * Float64(x * x))) + 1.0) / x) * Float64(exp(Float64(x * x)) / sqrt(pi)))
end

MATLAB

function tmp = code(x)
	tmp = (((0.75 / ((x * x) * (x * x))) + 1.0) / x) * (exp((x * x)) / sqrt(pi));
end

Wolfram

code[x_] := N[(N[(N[(N[(0.75 / N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision] * N[(N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

2. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(e^{{\left(\left|x\right|\right)}^{2}} \cdot \left(\frac{3}{4} \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}} + \left(\frac{15}{8} \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}} + \left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}\right)\right)\right)\right)} \]

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0.5}{x \cdot x} + 1, \frac{1}{\left|x\right|}, \mathsf{fma}\left({\left(\left|x\right|\right)}^{-7}, 1.875, {\left(\left|x\right|\right)}^{-5} \cdot 0.75\right)\right) \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}}} \]

5. Taylor expanded in x around inf

   \[\leadsto \left(\frac{3}{4} \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}} + \left(\frac{1}{\left|x\right|} + \frac{15}{8} \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}}\right)\right) \cdot \frac{\color{blue}{e^{x \cdot x}}}{\sqrt{\pi}} \]
6. Step-by-step derivation

   1. metadata-eval N/A

      \[\leadsto \left(\frac{3}{4} \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}} + \left(\frac{1}{\left|x\right|} + \frac{15}{8} \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}}\right)\right) \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}} \]

   2. metadata-eval N/A

      \[\leadsto \left(\frac{3}{4} \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}} + \left(\frac{1}{\left|x\right|} + \frac{15}{8} \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}}\right)\right) \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}} \]

   3. pow-flip N/A

      \[\leadsto \left(\frac{3}{4} \cdot {\left(\left|x\right|\right)}^{\left(\mathsf{neg}\left(5\right)\right)} + \left(\frac{1}{\left|x\right|} + \frac{15}{8} \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}}\right)\right) \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}} \]

   4. metadata-eval N/A

      \[\leadsto \left(\frac{3}{4} \cdot {\left(\left|x\right|\right)}^{-5} + \left(\frac{1}{\left|x\right|} + \frac{15}{8} \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}}\right)\right) \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}} \]

   5. *-commutative N/A

      \[\leadsto \left({\left(\left|x\right|\right)}^{-5} \cdot \frac{3}{4} + \left(\frac{1}{\left|x\right|} + \frac{15}{8} \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}}\right)\right) \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}} \]

   6. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left({\left(\left|x\right|\right)}^{-5}, \frac{3}{4}, \frac{1}{\left|x\right|} + \frac{15}{8} \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}}\right) \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}} \]

7. Applied rewrites 99.7%

   \[\leadsto \mathsf{fma}\left({x}^{-5}, 0.75, \mathsf{fma}\left({x}^{-7}, 1.875, \frac{1}{x}\right)\right) \cdot \frac{\color{blue}{e^{x \cdot x}}}{\sqrt{\pi}} \]

8. Taylor expanded in x around inf

   \[\leadsto \frac{1 + \frac{3}{4} \cdot \frac{1}{{x}^{4}}}{x} \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}} \]
9. Step-by-step derivation

   1. metadata-eval N/A

      \[\leadsto \frac{1 + \frac{3}{4} \cdot \frac{1}{{x}^{4}}}{x} \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}} \]

   2. lower-/.f64 N/A

      \[\leadsto \frac{1 + \frac{3}{4} \cdot \frac{1}{{x}^{4}}}{x} \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}} \]

   3. associate-*r/ N/A

      \[\leadsto \frac{1 + \frac{\frac{3}{4} \cdot 1}{{x}^{4}}}{x} \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}} \]

   4. metadata-eval N/A

      \[\leadsto \frac{1 + \frac{\frac{3}{4} \cdot 1}{{x}^{4}}}{x} \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}} \]

   5. metadata-eval N/A

      \[\leadsto \frac{1 + \frac{\frac{3}{4}}{{x}^{4}}}{x} \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}} \]

   6. metadata-eval N/A

      \[\leadsto \frac{1 + \frac{\frac{3}{4}}{{x}^{4}}}{x} \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}} \]

   7. +-commutative N/A

      \[\leadsto \frac{\frac{\frac{3}{4}}{{x}^{4}} + 1}{x} \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}} \]

   8. lower-+.f64 N/A

      \[\leadsto \frac{\frac{\frac{3}{4}}{{x}^{4}} + 1}{x} \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}} \]

   9. lower-/.f64 N/A

      \[\leadsto \frac{\frac{\frac{3}{4}}{{x}^{4}} + 1}{x} \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}} \]

   10. metadata-eval N/A

       \[\leadsto \frac{\frac{\frac{3}{4}}{{x}^{4}} + 1}{x} \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}} \]

   11. metadata-eval N/A

       \[\leadsto \frac{\frac{\frac{3}{4}}{{x}^{\left(2 + 2\right)}} + 1}{x} \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}} \]

   12. pow-prod-up N/A

       \[\leadsto \frac{\frac{\frac{3}{4}}{{x}^{2} \cdot {x}^{2}} + 1}{x} \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}} \]

   13. lower-*.f64 N/A

       \[\leadsto \frac{\frac{\frac{3}{4}}{{x}^{2} \cdot {x}^{2}} + 1}{x} \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}} \]

   14. pow2 N/A

       \[\leadsto \frac{\frac{\frac{3}{4}}{\left(x \cdot x\right) \cdot {x}^{2}} + 1}{x} \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}} \]

   15. lift-*.f64 N/A

       \[\leadsto \frac{\frac{\frac{3}{4}}{\left(x \cdot x\right) \cdot {x}^{2}} + 1}{x} \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}} \]

   16. pow2 N/A

       \[\leadsto \frac{\frac{\frac{3}{4}}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)} + 1}{x} \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}} \]

   17. lift-*.f64 99.7

       \[\leadsto \frac{\frac{0.75}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)} + 1}{x} \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}} \]

10. Applied rewrites 99.7%

    \[\leadsto \frac{\frac{0.75}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)} + 1}{x} \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}} \]
11. Add Preprocessing

Alternative 4: 99.7% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.5}{x \cdot x} + 1}{x} \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (/ (+ (/ 0.5 (* x x)) 1.0) x) (/ (exp (* x x)) (sqrt PI))))

C

double code(double x) {
	return (((0.5 / (x * x)) + 1.0) / x) * (exp((x * x)) / sqrt(((double) M_PI)));
}

Java

public static double code(double x) {
	return (((0.5 / (x * x)) + 1.0) / x) * (Math.exp((x * x)) / Math.sqrt(Math.PI));
}

Python

def code(x):
	return (((0.5 / (x * x)) + 1.0) / x) * (math.exp((x * x)) / math.sqrt(math.pi))

Julia

function code(x)
	return Float64(Float64(Float64(Float64(0.5 / Float64(x * x)) + 1.0) / x) * Float64(exp(Float64(x * x)) / sqrt(pi)))
end

MATLAB

function tmp = code(x)
	tmp = (((0.5 / (x * x)) + 1.0) / x) * (exp((x * x)) / sqrt(pi));
end

Wolfram

code[x_] := N[(N[(N[(N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] / x), $MachinePrecision] * N[(N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

2. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(e^{{\left(\left|x\right|\right)}^{2}} \cdot \left(\frac{3}{4} \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}} + \left(\frac{15}{8} \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}} + \left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}\right)\right)\right)\right)} \]

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0.5}{x \cdot x} + 1, \frac{1}{\left|x\right|}, \mathsf{fma}\left({\left(\left|x\right|\right)}^{-7}, 1.875, {\left(\left|x\right|\right)}^{-5} \cdot 0.75\right)\right) \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}}} \]

6. Applied rewrites 100.0%

   \[\leadsto \left(\mathsf{fma}\left(\frac{0.5}{x \cdot x} + 1, \frac{1}{x}, {x}^{-7} \cdot 1.875\right) + {x}^{-5} \cdot 0.75\right) \cdot \frac{\color{blue}{e^{x \cdot x}}}{\sqrt{\pi}} \]

7. Taylor expanded in x around inf

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

9. Applied rewrites 99.7%

   \[\leadsto \frac{\frac{0.5}{x \cdot x} + 1}{x} \cdot \frac{\color{blue}{e^{x \cdot x}}}{\sqrt{\pi}} \]
10. Add Preprocessing

Alternative 5: 99.7% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \frac{1}{x} \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

2. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(e^{{\left(\left|x\right|\right)}^{2}} \cdot \left(\frac{3}{4} \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}} + \left(\frac{15}{8} \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}} + \left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}\right)\right)\right)\right)} \]

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0.5}{x \cdot x} + 1, \frac{1}{\left|x\right|}, \mathsf{fma}\left({\left(\left|x\right|\right)}^{-7}, 1.875, {\left(\left|x\right|\right)}^{-5} \cdot 0.75\right)\right) \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}}} \]

6. Applied rewrites 100.0%

   \[\leadsto \left(\mathsf{fma}\left(\frac{0.5}{x \cdot x} + 1, \frac{1}{x}, {x}^{-7} \cdot 1.875\right) + {x}^{-5} \cdot 0.75\right) \cdot \frac{\color{blue}{e^{x \cdot x}}}{\sqrt{\pi}} \]

7. Taylor expanded in x around inf

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

9. Applied rewrites 99.7%

   \[\leadsto \frac{1}{x} \cdot \frac{\color{blue}{e^{x \cdot x}}}{\sqrt{\pi}} \]
10. Add Preprocessing

Alternative 6: 99.6% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ \frac{e^{0 + x \cdot x}}{\sqrt{\pi} \cdot x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (exp (+ 0.0 (* x x))) (* (sqrt PI) x)))

C

double code(double x) {
	return exp((0.0 + (x * x))) / (sqrt(((double) M_PI)) * x);
}

Java

public static double code(double x) {
	return Math.exp((0.0 + (x * x))) / (Math.sqrt(Math.PI) * x);
}

Python

def code(x):
	return math.exp((0.0 + (x * x))) / (math.sqrt(math.pi) * x)

Julia

function code(x)
	return Float64(exp(Float64(0.0 + Float64(x * x))) / Float64(sqrt(pi) * x))
end

MATLAB

function tmp = code(x)
	tmp = exp((0.0 + (x * x))) / (sqrt(pi) * x);
end

Wolfram

code[x_] := N[(N[Exp[N[(0.0 + N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[(N[Sqrt[Pi], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

2. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(e^{{\left(\left|x\right|\right)}^{2}} \cdot \left(\frac{3}{4} \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}} + \left(\frac{15}{8} \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}} + \left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}\right)\right)\right)\right)} \]

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0.5}{x \cdot x} + 1, \frac{1}{\left|x\right|}, \mathsf{fma}\left({\left(\left|x\right|\right)}^{-7}, 1.875, {\left(\left|x\right|\right)}^{-5} \cdot 0.75\right)\right) \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}}} \]

6. Applied rewrites 100.0%

   \[\leadsto \left(\mathsf{fma}\left(\frac{0.5}{x \cdot x} + 1, \frac{1}{x}, {x}^{-7} \cdot 1.875\right) + {x}^{-5} \cdot 0.75\right) \cdot \frac{\color{blue}{e^{x \cdot x}}}{\sqrt{\pi}} \]

7. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. sqrt-div N/A

      \[\leadsto \frac{\sqrt{1}}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \frac{e^{{x}^{2}}}{x} \]

   3. metadata-eval N/A

      \[\leadsto \frac{1}{\sqrt{\mathsf{PI}\left(\right)}} \cdot \frac{e^{{x}^{2}}}{x} \]

   4. frac-times N/A

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

   5. pow2 N/A

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

   6. sqr-abs-rev N/A

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

   7. lower-/.f64 N/A

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

9. Applied rewrites 99.6%

   \[\leadsto \frac{e^{0 + x \cdot x}}{\color{blue}{\sqrt{\pi} \cdot x}} \]
10. Add Preprocessing

Alternative 7: 1.8% accurate, 10.6× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{0.5}{x \cdot x}}{\sqrt{\pi} \cdot x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (/ 0.5 (* x x)) (* (sqrt PI) x)))

C

double code(double x) {
	return (0.5 / (x * x)) / (sqrt(((double) M_PI)) * x);
}

Java

public static double code(double x) {
	return (0.5 / (x * x)) / (Math.sqrt(Math.PI) * x);
}

Python

def code(x):
	return (0.5 / (x * x)) / (math.sqrt(math.pi) * x)

Julia

function code(x)
	return Float64(Float64(0.5 / Float64(x * x)) / Float64(sqrt(pi) * x))
end

MATLAB

function tmp = code(x)
	tmp = (0.5 / (x * x)) / (sqrt(pi) * x);
end

Wolfram

code[x_] := N[(N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[Pi], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

2. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(e^{{\left(\left|x\right|\right)}^{2}} \cdot \left(\frac{3}{4} \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}} + \left(\frac{15}{8} \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}} + \left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}\right)\right)\right)\right)} \]

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0.5}{x \cdot x} + 1, \frac{1}{\left|x\right|}, \mathsf{fma}\left({\left(\left|x\right|\right)}^{-7}, 1.875, {\left(\left|x\right|\right)}^{-5} \cdot 0.75\right)\right) \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}}} \]

5. Taylor expanded in x around 0

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

   1. metadata-eval N/A

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

   2. associate-*r* N/A

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

   3. associate-*r/ N/A

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

   4. metadata-eval N/A

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

   5. metadata-eval N/A

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

   6. metadata-eval N/A

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

   7. sqrt-div N/A

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

   8. metadata-eval N/A

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

   9. frac-times N/A

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. lower-/.f64 N/A

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

   14. metadata-eval N/A

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

7. Applied rewrites 1.8%

   \[\leadsto \frac{0.5}{\color{blue}{\left(\left(x \cdot x\right) \cdot x\right) \cdot \sqrt{\pi}}} \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. pow2 N/A

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

   5. lift-PI.f64 N/A

      \[\leadsto \frac{\frac{1}{2}}{\left({x}^{2} \cdot x\right) \cdot \sqrt{\mathsf{PI}\left(\right)}} \]

   6. lift-sqrt.f64 N/A

      \[\leadsto \frac{\frac{1}{2}}{\left({x}^{2} \cdot x\right) \cdot \sqrt{\mathsf{PI}\left(\right)}} \]

   7. associate-*l* N/A

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

   8. lower-*.f64 N/A

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

   9. pow2 N/A

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

   10. lift-*.f64 N/A

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

   11. lower-*.f64 N/A

       \[\leadsto \frac{\frac{1}{2}}{\left(x \cdot x\right) \cdot \left(x \cdot \sqrt{\mathsf{PI}\left(\right)}\right)} \]

   12. lift-sqrt.f64 N/A

       \[\leadsto \frac{\frac{1}{2}}{\left(x \cdot x\right) \cdot \left(x \cdot \sqrt{\mathsf{PI}\left(\right)}\right)} \]

   13. lift-PI.f64 1.8

       \[\leadsto \frac{0.5}{\left(x \cdot x\right) \cdot \left(x \cdot \sqrt{\pi}\right)} \]

9. Applied rewrites 1.8%

   \[\leadsto \frac{0.5}{\left(x \cdot x\right) \cdot \left(x \cdot \color{blue}{\sqrt{\pi}}\right)} \]
10. Step-by-step derivation

    1. metadata-eval N/A

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

    2. lift-/.f64 N/A

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

    3. lift-*.f64 N/A

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

    4. lift-*.f64 N/A

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

    5. lift-*.f64 N/A

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

    6. lift-PI.f64 N/A

       \[\leadsto \frac{\frac{1}{2}}{\left(x \cdot x\right) \cdot \left(x \cdot \sqrt{\mathsf{PI}\left(\right)}\right)} \]

    7. lift-sqrt.f64 N/A

       \[\leadsto \frac{\frac{1}{2}}{\left(x \cdot x\right) \cdot \left(x \cdot \sqrt{\mathsf{PI}\left(\right)}\right)} \]

    8. pow2 N/A

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

    9. lift-sqrt.f64 N/A

       \[\leadsto \frac{\frac{1}{2}}{{x}^{2} \cdot \left(x \cdot \sqrt{\mathsf{PI}\left(\right)}\right)} \]

    10. lift-PI.f64 N/A

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

    11. lift-*.f64 N/A

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

    12. associate-/r* N/A

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

    13. metadata-eval N/A

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

    14. metadata-eval N/A

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

    15. metadata-eval N/A

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

    16. associate-*r/ N/A

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

    17. lower-/.f64 N/A

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

11. Applied rewrites 1.8%

    \[\leadsto \frac{\frac{0.5}{x \cdot x}}{\sqrt{\pi} \cdot \color{blue}{x}} \]
12. Add Preprocessing

Alternative 8: 1.8% accurate, 10.9× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{x \cdot \left(x \cdot \left(\sqrt{\pi} \cdot x\right)\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 0.5 (* x (* x (* (sqrt PI) x)))))

C

double code(double x) {
	return 0.5 / (x * (x * (sqrt(((double) M_PI)) * x)));
}

Java

public static double code(double x) {
	return 0.5 / (x * (x * (Math.sqrt(Math.PI) * x)));
}

Python

def code(x):
	return 0.5 / (x * (x * (math.sqrt(math.pi) * x)))

Julia

function code(x)
	return Float64(0.5 / Float64(x * Float64(x * Float64(sqrt(pi) * x))))
end

MATLAB

function tmp = code(x)
	tmp = 0.5 / (x * (x * (sqrt(pi) * x)));
end

Wolfram

code[x_] := N[(0.5 / N[(x * N[(x * N[(N[Sqrt[Pi], $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(\left(\left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{3}{4} \cdot \left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) + \frac{15}{8} \cdot \left(\left(\left(\left(\left(\left(\frac{1}{\left|x\right|} \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right) \cdot \frac{1}{\left|x\right|}\right)\right) \]

2. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\sqrt{\frac{1}{\mathsf{PI}\left(\right)}} \cdot \left(e^{{\left(\left|x\right|\right)}^{2}} \cdot \left(\frac{3}{4} \cdot \frac{1}{{\left(\left|x\right|\right)}^{5}} + \left(\frac{15}{8} \cdot \frac{1}{{\left(\left|x\right|\right)}^{7}} + \left(\frac{1}{\left|x\right|} + \frac{1}{2} \cdot \frac{1}{{\left(\left|x\right|\right)}^{3}}\right)\right)\right)\right)} \]

4. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{0.5}{x \cdot x} + 1, \frac{1}{\left|x\right|}, \mathsf{fma}\left({\left(\left|x\right|\right)}^{-7}, 1.875, {\left(\left|x\right|\right)}^{-5} \cdot 0.75\right)\right) \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}}} \]

5. Taylor expanded in x around 0

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

   1. metadata-eval N/A

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

   2. associate-*r* N/A

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

   3. associate-*r/ N/A

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

   4. metadata-eval N/A

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

   5. metadata-eval N/A

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

   6. metadata-eval N/A

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

   7. sqrt-div N/A

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

   8. metadata-eval N/A

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

   9. frac-times N/A

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

   10. metadata-eval N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. lower-/.f64 N/A

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

   14. metadata-eval N/A

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

7. Applied rewrites 1.8%

   \[\leadsto \frac{0.5}{\color{blue}{\left(\left(x \cdot x\right) \cdot x\right) \cdot \sqrt{\pi}}} \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. pow2 N/A

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

   5. lift-PI.f64 N/A

      \[\leadsto \frac{\frac{1}{2}}{\left({x}^{2} \cdot x\right) \cdot \sqrt{\mathsf{PI}\left(\right)}} \]

   6. lift-sqrt.f64 N/A

      \[\leadsto \frac{\frac{1}{2}}{\left({x}^{2} \cdot x\right) \cdot \sqrt{\mathsf{PI}\left(\right)}} \]

   7. associate-*l* N/A

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

   8. lower-*.f64 N/A

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

   9. pow2 N/A

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

   10. lift-*.f64 N/A

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

   11. lower-*.f64 N/A

       \[\leadsto \frac{\frac{1}{2}}{\left(x \cdot x\right) \cdot \left(x \cdot \sqrt{\mathsf{PI}\left(\right)}\right)} \]

   12. lift-sqrt.f64 N/A

       \[\leadsto \frac{\frac{1}{2}}{\left(x \cdot x\right) \cdot \left(x \cdot \sqrt{\mathsf{PI}\left(\right)}\right)} \]

   13. lift-PI.f64 1.8

       \[\leadsto \frac{0.5}{\left(x \cdot x\right) \cdot \left(x \cdot \sqrt{\pi}\right)} \]

9. Applied rewrites 1.8%

   \[\leadsto \frac{0.5}{\left(x \cdot x\right) \cdot \left(x \cdot \color{blue}{\sqrt{\pi}}\right)} \]
10. Step-by-step derivation

    1. lift-*.f64 N/A

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

    2. lift-*.f64 N/A

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

    3. associate-*l* N/A

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

    4. lower-*.f64 N/A

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

    5. lower-*.f64 1.8

       \[\leadsto \frac{0.5}{x \cdot \left(x \cdot \left(x \cdot \color{blue}{\sqrt{\pi}}\right)\right)} \]

    6. lift-*.f64 N/A

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

    7. *-commutative N/A

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

    8. lower-*.f64 1.8

       \[\leadsto \frac{0.5}{x \cdot \left(x \cdot \left(\sqrt{\pi} \cdot x\right)\right)} \]

11. Applied rewrites 1.8%

    \[\leadsto \frac{0.5}{x \cdot \left(x \cdot \color{blue}{\left(\sqrt{\pi} \cdot x\right)}\right)} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2025112 
(FPCore (x)
  :name "Jmat.Real.erfi, branch x greater than or equal to 5"
  :precision binary64
  :pre (>= x 0.5)
  (* (* (/ 1.0 (sqrt PI)) (exp (* (fabs x) (fabs x)))) (+ (+ (+ (/ 1.0 (fabs x)) (* (/ 1.0 2.0) (* (* (/ 1.0 (fabs x)) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))))) (* (/ 3.0 4.0) (* (* (* (* (/ 1.0 (fabs x)) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))))) (* (/ 15.0 8.0) (* (* (* (* (* (* (/ 1.0 (fabs x)) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))) (/ 1.0 (fabs x))) (/ 1.0 (fabs x)))))))