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

Percentage Accurate: 100.0% → 100.0%

Time: 4.5s

Alternatives: 8

Speedup: 2.4×

• 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, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x \cdot x\right) \cdot x\\ \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \left(-\left(\frac{\frac{\frac{1.875}{x \cdot x} + 0.75}{-t\_0 \cdot x} - 1}{x} - \frac{0.5}{t\_0}\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (* x x) x)))
   (*
    (* (/ 1.0 (sqrt PI)) (exp (* (fabs x) (fabs x))))
    (-
     (-
      (/ (- (/ (+ (/ 1.875 (* x x)) 0.75) (- (* t_0 x))) 1.0) x)
      (/ 0.5 t_0))))))

C

double code(double x) {
	double t_0 = (x * x) * x;
	return ((1.0 / sqrt(((double) M_PI))) * exp((fabs(x) * fabs(x)))) * -((((((1.875 / (x * x)) + 0.75) / -(t_0 * x)) - 1.0) / x) - (0.5 / t_0));
}

Java

public static double code(double x) {
	double t_0 = (x * x) * x;
	return ((1.0 / Math.sqrt(Math.PI)) * Math.exp((Math.abs(x) * Math.abs(x)))) * -((((((1.875 / (x * x)) + 0.75) / -(t_0 * x)) - 1.0) / x) - (0.5 / t_0));
}

Python

def code(x):
	t_0 = (x * x) * x
	return ((1.0 / math.sqrt(math.pi)) * math.exp((math.fabs(x) * math.fabs(x)))) * -((((((1.875 / (x * x)) + 0.75) / -(t_0 * x)) - 1.0) / x) - (0.5 / t_0))

Julia

function code(x)
	t_0 = Float64(Float64(x * x) * x)
	return Float64(Float64(Float64(1.0 / sqrt(pi)) * exp(Float64(abs(x) * abs(x)))) * Float64(-Float64(Float64(Float64(Float64(Float64(Float64(1.875 / Float64(x * x)) + 0.75) / Float64(-Float64(t_0 * x))) - 1.0) / x) - Float64(0.5 / t_0))))
end

MATLAB

function tmp = code(x)
	t_0 = (x * x) * x;
	tmp = ((1.0 / sqrt(pi)) * exp((abs(x) * abs(x)))) * -((((((1.875 / (x * x)) + 0.75) / -(t_0 * x)) - 1.0) / x) - (0.5 / t_0));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * x), $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[(N[(N[(N[(1.875 / N[(x * x), $MachinePrecision]), $MachinePrecision] + 0.75), $MachinePrecision] / (-N[(t$95$0 * x), $MachinePrecision])), $MachinePrecision] - 1.0), $MachinePrecision] / x), $MachinePrecision] - N[(0.5 / t$95$0), $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

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 100.0%

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

8. Taylor expanded in x around -inf

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

10. Applied rewrites 100.0%

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

12. Applied rewrites 100.0%

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

Alternative 2: 100.0% accurate, 2.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 100.0%

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

8. Taylor expanded in x around -inf

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

10. Applied rewrites 100.0%

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

12. Applied rewrites 100.0%

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

Alternative 3: 100.0% accurate, 2.7× speedup

Math

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

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

Wolfram

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

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

7. Applied rewrites 100.0%

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

8. Taylor expanded in x around -inf

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

10. Applied rewrites 100.0%

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

Alternative 4: 99.7% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\frac{0.75}{x \cdot x} + 0.5}{x \cdot x} + 1}{x} \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}} \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(N[(N[(N[(N[(0.75 / N[(x * x), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision] / 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%

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

5. Taylor expanded in x around inf

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

7. Applied rewrites 100.0%

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

8. Taylor expanded in x around -inf

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

   1. metadata-eval N/A

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

   2. metadata-eval N/A

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

   3. mul-1-neg N/A

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

   4. lower-neg.f64 N/A

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

   5. lower-/.f64 N/A

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

10. Applied rewrites 99.7%

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

11. Taylor expanded in x around inf

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

13. Applied rewrites 99.7%

    \[\leadsto \frac{\frac{\frac{0.75}{x \cdot x} + 0.5}{x \cdot x} + 1}{x} \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}} \]
14. Add Preprocessing

Alternative 5: 99.7% accurate, 4.7× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(N[(1.0 - N[(-0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $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

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

7. Applied rewrites 100.0%

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

8. Taylor expanded in x around -inf

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

   1. metadata-eval N/A

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

   2. metadata-eval N/A

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

   3. mul-1-neg N/A

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

   4. lower-neg.f64 N/A

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

   5. lower-/.f64 N/A

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

10. Applied rewrites 99.7%

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

11. Taylor expanded in x around inf

    \[\leadsto \frac{1 + \frac{1}{2} \cdot \frac{1}{{x}^{2}}}{x} \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}} \]
12. Step-by-step derivation

    1. metadata-eval N/A

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

    2. lower-/.f64 N/A

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

    3. fp-cancel-sign-sub-inv N/A

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

    4. metadata-eval N/A

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

    5. metadata-eval N/A

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

    6. metadata-eval N/A

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

    7. metadata-eval N/A

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

    8. times-frac N/A

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

    9. metadata-eval N/A

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

    10. metadata-eval N/A

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

    11. metadata-eval N/A

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

    12. mul-1-neg N/A

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

    13. frac-2neg N/A

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

    14. lower--.f64 N/A

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

    15. lower-/.f64 N/A

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

    16. pow2 N/A

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

    17. lift-*.f64 99.7

        \[\leadsto \frac{1 - \frac{-0.5}{x \cdot x}}{x} \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}} \]

13. Applied rewrites 99.7%

    \[\leadsto \frac{1 - \frac{-0.5}{x \cdot x}}{x} \cdot \frac{e^{x \cdot x}}{\sqrt{\pi}} \]
14. Add Preprocessing

Alternative 6: 99.7% accurate, 6.4× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

7. Applied rewrites 100.0%

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

8. Taylor expanded in x around inf

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

   1. lift-/.f64 99.7

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

10. Applied rewrites 99.7%

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

12. Applied rewrites 99.7%

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

Alternative 7: 52.6% accurate, 9.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

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

7. Applied rewrites 100.0%

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

8. Taylor expanded in x around inf

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

   1. lift-/.f64 99.7

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

10. Applied rewrites 99.7%

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

11. Taylor expanded in x around 0

    \[\leadsto \frac{1}{x} \cdot \frac{1 + {x}^{2}}{\sqrt{\color{blue}{\pi}}} \]
12. Step-by-step derivation

    1. +-commutative N/A

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

    2. pow2 N/A

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

    3. lower-fma.f64 52.6

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

13. Applied rewrites 52.6%

    \[\leadsto \frac{1}{x} \cdot \frac{\mathsf{fma}\left(x, x, 1\right)}{\sqrt{\color{blue}{\pi}}} \]
14. Add Preprocessing

Alternative 8: 2.3% accurate, 13.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

7. Applied rewrites 100.0%

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

8. Taylor expanded in x around inf

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

   1. lift-/.f64 99.7

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

10. Applied rewrites 99.7%

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

11. Taylor expanded in x around 0

    \[\leadsto \frac{1}{x} \cdot \frac{1}{\sqrt{\color{blue}{\pi}}} \]
12. Step-by-step derivation

13. Applied rewrites 2.3%

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

Reproduce

herbie shell --seed 2025117 
(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)))))))