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

Percentage Accurate: 100.0% → 100.0%

Time: 4.5s

Alternatives: 15

Speedup: 2.4×

• 99.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

\[x \geq 0.5\]

Math

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

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

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

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, 1.6× speedup

Math

\[\begin{array}{l} t_0 := \frac{1}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}\\ \left(\frac{1}{\sqrt{\pi}} \cdot {\left(e^{x + x}\right)}^{\left(\frac{x}{2}\right)}\right) \cdot \left(\frac{\mathsf{fma}\left(\frac{t\_0}{x \cdot x}, 1.875, 1\right)}{\left|x\right|} + \frac{\mathsf{fma}\left(t\_0, 0.75, \frac{0.5}{x \cdot x}\right)}{\left|x\right|}\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (* (* x x) (* x x)))))
   (*
    (* (/ 1.0 (sqrt PI)) (pow (exp (+ x x)) (/ x 2.0)))
    (+
     (/ (fma (/ t_0 (* x x)) 1.875 1.0) (fabs x))
     (/ (fma t_0 0.75 (/ 0.5 (* x x))) (fabs x))))))

C

double code(double x) {
	double t_0 = 1.0 / ((x * x) * (x * x));
	return ((1.0 / sqrt(((double) M_PI))) * pow(exp((x + x)), (x / 2.0))) * ((fma((t_0 / (x * x)), 1.875, 1.0) / fabs(x)) + (fma(t_0, 0.75, (0.5 / (x * x))) / fabs(x)));
}

Julia

function code(x)
	t_0 = Float64(1.0 / Float64(Float64(x * x) * Float64(x * x)))
	return Float64(Float64(Float64(1.0 / sqrt(pi)) * (exp(Float64(x + x)) ^ Float64(x / 2.0))) * Float64(Float64(fma(Float64(t_0 / Float64(x * x)), 1.875, 1.0) / abs(x)) + Float64(fma(t_0, 0.75, Float64(0.5 / Float64(x * x))) / abs(x))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 / N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[Power[N[Exp[N[(x + x), $MachinePrecision]], $MachinePrecision], N[(x / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(t$95$0 / N[(x * x), $MachinePrecision]), $MachinePrecision] * 1.875 + 1.0), $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$0 * 0.75 + N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 100.0%

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

3. Applied rewrites 100.0%

   \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \color{blue}{\left(\frac{\mathsf{fma}\left(\frac{\frac{1}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}}{x \cdot x}, 1.875, 1\right)}{\left|x\right|} + \frac{\mathsf{fma}\left(\frac{1}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}, 0.75, \frac{0.5}{x \cdot x}\right)}{\left|x\right|}\right)} \]
4. Step-by-step derivation

   1. lift-exp.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-fabs.f64 N/A

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

   4. lift-fabs.f64 N/A

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

   5. sqr-abs-rev N/A

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

   6. pow-exp N/A

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

   7. exp-fabs N/A

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

   8. lift-exp.f64 N/A

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

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

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

   10. sqrt-pow2 N/A

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

   11. lower-/.f32 N/A

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

   12. lower-unsound-/.f32 N/A

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

   13. lower-pow.f64 N/A

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

   14. lift-exp.f64 N/A

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

   15. lift-exp.f64 N/A

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

   16. prod-exp N/A

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

   17. lower-exp.f64 N/A

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

   18. lower-+.f64 N/A

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

   19. lower-unsound-/.f64 100.0

       \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot {\left(e^{x + x}\right)}^{\color{blue}{\left(\frac{x}{2}\right)}}\right) \cdot \left(\frac{\mathsf{fma}\left(\frac{\frac{1}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}}{x \cdot x}, 1.875, 1\right)}{\left|x\right|} + \frac{\mathsf{fma}\left(\frac{1}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}, 0.75, \frac{0.5}{x \cdot x}\right)}{\left|x\right|}\right) \]

5. Applied rewrites 100.0%

   \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot \color{blue}{{\left(e^{x + x}\right)}^{\left(\frac{x}{2}\right)}}\right) \cdot \left(\frac{\mathsf{fma}\left(\frac{\frac{1}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}}{x \cdot x}, 1.875, 1\right)}{\left|x\right|} + \frac{\mathsf{fma}\left(\frac{1}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}, 0.75, \frac{0.5}{x \cdot x}\right)}{\left|x\right|}\right) \]
6. Add Preprocessing

Alternative 2: 100.0% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 / N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[(N[(1.0 / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision] * N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[(t$95$0 / N[(x * x), $MachinePrecision]), $MachinePrecision] * 1.875 + 1.0), $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision] + N[(N[(t$95$0 * 0.75 + N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 100.0%

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

3. Applied rewrites 100.0%

   \[\leadsto \left(\frac{1}{\sqrt{\pi}} \cdot e^{\left|x\right| \cdot \left|x\right|}\right) \cdot \color{blue}{\left(\frac{\mathsf{fma}\left(\frac{\frac{1}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}}{x \cdot x}, 1.875, 1\right)}{\left|x\right|} + \frac{\mathsf{fma}\left(\frac{1}{\left(x \cdot x\right) \cdot \left(x \cdot x\right)}, 0.75, \frac{0.5}{x \cdot x}\right)}{\left|x\right|}\right)} \]
4. Step-by-step derivation

   1. lift-exp.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-fabs.f64 N/A

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

   4. lift-fabs.f64 N/A

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

   5. sqr-abs-rev N/A

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

   6. pow-exp N/A

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

   7. lift-exp.f64 N/A

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

   8. lift-pow.f64 100.0

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

5. Applied rewrites 100.0%

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

Alternative 3: 100.0% accurate, 2.1× speedup

Math

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

FPCore

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

C

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

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(fma(0.75, Float64(1.0 / Float64(x * x)), 0.5) / Float64(Float64(x * x) * abs(x))) + Float64(Float64(Float64(1.875 / Float64(t_0 * t_0)) - -1.0) / abs(x))))
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[(0.75 * N[(1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision] + 0.5), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(N[(1.875 / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 100.0%

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

3. Applied rewrites 100.0%

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

5. Applied rewrites 100.0%

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

Alternative 4: 100.0% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Applied rewrites 100.0%

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

5. Applied rewrites 100.0%

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

Alternative 5: 100.0% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Applied rewrites 100.0%

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

5. Applied rewrites 100.0%

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

Alternative 6: 100.0% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Applied rewrites 100.0%

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

5. Applied rewrites 100.0%

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

Alternative 7: 99.9% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Applied rewrites 100.0%

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

5. Applied rewrites 99.9%

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

Alternative 8: 99.7% accurate, 2.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision]}, N[(N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(N[(1.875 / N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] - N[(-1.0 - N[(0.5 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Abs[x], $MachinePrecision]), $MachinePrecision] / N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 100.0%

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

3. Applied rewrites 100.0%

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

5. Applied rewrites 100.0%

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

6. Taylor expanded in x around inf

   \[\leadsto e^{x \cdot x} \cdot \frac{\frac{\frac{\frac{15}{8}}{\left(\left(x \cdot x\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)} - \left(-1 - \frac{\color{blue}{\frac{1}{2}}}{x \cdot x}\right)}{\left|x\right|}}{\sqrt{\pi}} \]
7. Step-by-step derivation

8. Applied rewrites 99.7%

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

Alternative 9: 99.7% accurate, 3.8× speedup

Math

\[\frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \frac{\mathsf{fma}\left(\frac{0.5}{\left(\left|x\right| \cdot x\right) \cdot x}, \left|x\right|, 1\right)}{\left|x\right|} \]

FPCore

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

C

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

Julia

function code(x)
	return Float64(Float64(exp(Float64(x * x)) / sqrt(pi)) * Float64(fma(Float64(0.5 / Float64(Float64(abs(x) * x) * x)), abs(x), 1.0) / abs(x)))
end

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Applied rewrites 100.0%

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

4. Taylor expanded in x around inf

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

   1. lower-+.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-fabs.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-pow.f64 N/A

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

   8. lower-fabs.f64 99.7

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(\frac{1}{\left|x\right|} + 0.5 \cdot \frac{1}{{x}^{2} \cdot \left|x\right|}\right) \]

6. Applied rewrites 99.7%

   \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \color{blue}{\left(\frac{1}{\left|x\right|} + 0.5 \cdot \frac{1}{{x}^{2} \cdot \left|x\right|}\right)} \]
7. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. add-to-fraction N/A

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

   5. lower-/.f64 N/A

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

8. Applied rewrites 99.7%

   \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \frac{\mathsf{fma}\left(\frac{0.5}{\left(\left|x\right| \cdot x\right) \cdot x}, \left|x\right|, 1\right)}{\color{blue}{\left|x\right|}} \]
9. Add Preprocessing

Alternative 10: 99.7% accurate, 4.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Applied rewrites 100.0%

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

4. Taylor expanded in x around inf

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

   1. lower-+.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-fabs.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-pow.f64 N/A

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

   8. lower-fabs.f64 99.7

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(\frac{1}{\left|x\right|} + 0.5 \cdot \frac{1}{{x}^{2} \cdot \left|x\right|}\right) \]

6. Applied rewrites 99.7%

   \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \color{blue}{\left(\frac{1}{\left|x\right|} + 0.5 \cdot \frac{1}{{x}^{2} \cdot \left|x\right|}\right)} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. lower-/.f64 N/A

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

8. Applied rewrites 99.7%

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

Alternative 11: 99.7% accurate, 4.3× speedup

Math

\[\frac{\left(\frac{0.5}{\left(\left|x\right| \cdot x\right) \cdot x} - \frac{-1}{\left|x\right|}\right) \cdot e^{x \cdot x}}{1.772453850905516} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  (* (- (/ 0.5 (* (* (fabs x) x) x)) (/ -1.0 (fabs x))) (exp (* x x)))
  1.772453850905516))

C

double code(double x) {
	return (((0.5 / ((fabs(x) * x) * x)) - (-1.0 / fabs(x))) * exp((x * x))) / 1.772453850905516;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    code = (((0.5d0 / ((abs(x) * x) * x)) - ((-1.0d0) / abs(x))) * exp((x * x))) / 1.772453850905516d0
end function

Java

public static double code(double x) {
	return (((0.5 / ((Math.abs(x) * x) * x)) - (-1.0 / Math.abs(x))) * Math.exp((x * x))) / 1.772453850905516;
}

Python

def code(x):
	return (((0.5 / ((math.fabs(x) * x) * x)) - (-1.0 / math.fabs(x))) * math.exp((x * x))) / 1.772453850905516

Julia

function code(x)
	return Float64(Float64(Float64(Float64(0.5 / Float64(Float64(abs(x) * x) * x)) - Float64(-1.0 / abs(x))) * exp(Float64(x * x))) / 1.772453850905516)
end

MATLAB

function tmp = code(x)
	tmp = (((0.5 / ((abs(x) * x) * x)) - (-1.0 / abs(x))) * exp((x * x))) / 1.772453850905516;
end

Wolfram

code[x_] := N[(N[(N[(N[(0.5 / N[(N[(N[Abs[x], $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] - N[(-1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / 1.772453850905516), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Applied rewrites 100.0%

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

4. Taylor expanded in x around inf

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

   1. lower-+.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-fabs.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-pow.f64 N/A

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

   8. lower-fabs.f64 99.7

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(\frac{1}{\left|x\right|} + 0.5 \cdot \frac{1}{{x}^{2} \cdot \left|x\right|}\right) \]

6. Applied rewrites 99.7%

   \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \color{blue}{\left(\frac{1}{\left|x\right|} + 0.5 \cdot \frac{1}{{x}^{2} \cdot \left|x\right|}\right)} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. lower-/.f64 N/A

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

8. Applied rewrites 99.7%

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

9. Evaluated real constant 99.7%

   \[\leadsto \frac{\left(\frac{\frac{1}{2}}{\left(\left|x\right| \cdot x\right) \cdot x} - \frac{-1}{\left|x\right|}\right) \cdot e^{x \cdot x}}{\color{blue}{\frac{7982422502469483}{4503599627370496}}} \]
10. Add Preprocessing

Alternative 12: 99.6% accurate, 4.4× speedup

Math

\[\frac{\frac{e^{{x}^{2}}}{\left|x\right|}}{\sqrt{\pi}} \]

FPCore

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

C

double code(double x) {
	return (exp(pow(x, 2.0)) / fabs(x)) / sqrt(((double) M_PI));
}

Java

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

Python

def code(x):
	return (math.exp(math.pow(x, 2.0)) / math.fabs(x)) / math.sqrt(math.pi)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Applied rewrites 100.0%

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

4. Taylor expanded in x around inf

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

   1. lower-+.f64 N/A

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

   2. lower-/.f64 N/A

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

   3. lower-fabs.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-*.f64 N/A

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

   7. lower-pow.f64 N/A

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

   8. lower-fabs.f64 99.7

      \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(\frac{1}{\left|x\right|} + 0.5 \cdot \frac{1}{{x}^{2} \cdot \left|x\right|}\right) \]

6. Applied rewrites 99.7%

   \[\leadsto \frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \color{blue}{\left(\frac{1}{\left|x\right|} + 0.5 \cdot \frac{1}{{x}^{2} \cdot \left|x\right|}\right)} \]
7. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*l/ N/A

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

   4. lower-/.f64 N/A

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

8. Applied rewrites 99.7%

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

9. Taylor expanded in x around inf

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

    1. lower-/.f64 N/A

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

    2. lower-exp.f64 N/A

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

    3. lower-pow.f64 N/A

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

    4. lower-fabs.f64 99.6

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

11. Applied rewrites 99.6%

    \[\leadsto \frac{\color{blue}{\frac{e^{{x}^{2}}}{\left|x\right|}}}{\sqrt{\pi}} \]
12. Add Preprocessing

Alternative 13: 99.6% accurate, 6.2× speedup

Math

\[e^{x \cdot x} \cdot \frac{1}{\left|x\right| \cdot \sqrt{\pi}} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Applied rewrites 100.0%

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

5. Applied rewrites 100.0%

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

6. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-fabs.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-PI.f64 99.6

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

8. Applied rewrites 99.6%

   \[\leadsto e^{x \cdot x} \cdot \color{blue}{\frac{1}{\left|x\right| \cdot \sqrt{\pi}}} \]
9. Add Preprocessing

Alternative 14: 99.6% accurate, 6.6× speedup

Math

\[e^{x \cdot x} \cdot \frac{1}{\left|x\right| \cdot 1.772453850905516} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (exp (* x x)) (/ 1.0 (* (fabs x) 1.772453850905516))))

C

double code(double x) {
	return exp((x * x)) * (1.0 / (fabs(x) * 1.772453850905516));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    code = exp((x * x)) * (1.0d0 / (abs(x) * 1.772453850905516d0))
end function

Java

public static double code(double x) {
	return Math.exp((x * x)) * (1.0 / (Math.abs(x) * 1.772453850905516));
}

Python

def code(x):
	return math.exp((x * x)) * (1.0 / (math.fabs(x) * 1.772453850905516))

Julia

function code(x)
	return Float64(exp(Float64(x * x)) * Float64(1.0 / Float64(abs(x) * 1.772453850905516)))
end

MATLAB

function tmp = code(x)
	tmp = exp((x * x)) * (1.0 / (abs(x) * 1.772453850905516));
end

Wolfram

code[x_] := N[(N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision] * N[(1.0 / N[(N[Abs[x], $MachinePrecision] * 1.772453850905516), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Applied rewrites 100.0%

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

5. Applied rewrites 100.0%

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

6. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-fabs.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-PI.f64 99.6

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

8. Applied rewrites 99.6%

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

9. Evaluated real constant 99.6%

   \[\leadsto e^{x \cdot x} \cdot \frac{1}{\left|x\right| \cdot \frac{7982422502469483}{4503599627370496}} \]
10. Add Preprocessing

Alternative 15: 2.3% accurate, 12.3× speedup

Math

\[1 \cdot \frac{1}{\left|x\right| \cdot \sqrt{\pi}} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Applied rewrites 100.0%

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

5. Applied rewrites 100.0%

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

6. Taylor expanded in x around inf

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower-fabs.f64 N/A

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

   4. lower-sqrt.f64 N/A

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

   5. lower-PI.f64 99.6

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

8. Applied rewrites 99.6%

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

9. Taylor expanded in x around 0

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

11. Applied rewrites 2.3%

    \[\leadsto \color{blue}{1} \cdot \frac{1}{\left|x\right| \cdot \sqrt{\pi}} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2025170 
(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)))))))