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

Average Error: 2.8 → 1.3

Time: 15.8s

Precision: binary64

Cost: 39872

\[x \geq 0.5\]

Math

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

↓

\[\frac{{\left(e^{x}\right)}^{x} \cdot \left(1 + \left(1.875 \cdot {x}^{-6} + \left(\frac{0.75}{{x}^{4}} + \frac{\frac{0.5}{x}}{x}\right)\right)\right)}{x \cdot \sqrt{\pi}} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (* (/ 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)))))))

↓

(FPCore (x)
 :precision binary64
 (/
  (*
   (pow (exp x) x)
   (+ 1.0 (+ (* 1.875 (pow x -6.0)) (+ (/ 0.75 (pow x 4.0)) (/ (/ 0.5 x) x)))))
  (* x (sqrt PI))))

C

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

↓

double code(double x) {
	return (pow(exp(x), x) * (1.0 + ((1.875 * pow(x, -6.0)) + ((0.75 / pow(x, 4.0)) + ((0.5 / x) / x))))) / (x * sqrt(((double) M_PI)));
}

Java

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

↓

public static double code(double x) {
	return (Math.pow(Math.exp(x), x) * (1.0 + ((1.875 * Math.pow(x, -6.0)) + ((0.75 / Math.pow(x, 4.0)) + ((0.5 / x) / x))))) / (x * Math.sqrt(Math.PI));
}

Python

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

↓

def code(x):
	return (math.pow(math.exp(x), x) * (1.0 + ((1.875 * math.pow(x, -6.0)) + ((0.75 / math.pow(x, 4.0)) + ((0.5 / x) / x))))) / (x * math.sqrt(math.pi))

Julia

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

↓

function code(x)
	return Float64(Float64((exp(x) ^ x) * Float64(1.0 + Float64(Float64(1.875 * (x ^ -6.0)) + Float64(Float64(0.75 / (x ^ 4.0)) + Float64(Float64(0.5 / x) / x))))) / Float64(x * sqrt(pi)))
end

MATLAB

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

↓

function tmp = code(x)
	tmp = ((exp(x) ^ x) * (1.0 + ((1.875 * (x ^ -6.0)) + ((0.75 / (x ^ 4.0)) + ((0.5 / x) / x))))) / (x * sqrt(pi));
end

Wolfram

code[x_] := 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[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 / 2.0), $MachinePrecision] * N[(N[(N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(3.0 / 4.0), $MachinePrecision] * N[(N[(N[(N[(N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(15.0 / 8.0), $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_] := N[(N[(N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision] * N[(1.0 + N[(N[(1.875 * N[Power[x, -6.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.75 / N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.5 / x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * N[Sqrt[Pi], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 2.8

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

2. Simplified 2.7

   \[\leadsto \color{blue}{\frac{\frac{e^{x \cdot x}}{\left|x\right|}}{\sqrt{\pi}} \cdot \left(1 + \left(\frac{1.875}{{x}^{6}} + \frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x}\right)\right)} \]
   Proof

   [Start] 2.8

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

   associate-+l+ [=>] 2.8

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

3. Applied egg-rr 1.3

   \[\leadsto \color{blue}{\frac{{\left(e^{x}\right)}^{x} \cdot \left(1 + \left(1.875 \cdot {x}^{-6} + \left(0.5 + 0.75 \cdot {x}^{-2}\right) \cdot {x}^{-2}\right)\right)}{x \cdot \sqrt{\pi}}} \]

4. Taylor expanded in x around 0 1.3

   \[\leadsto \frac{{\left(e^{x}\right)}^{x} \cdot \left(1 + \left(1.875 \cdot {x}^{-6} + \color{blue}{\left(0.75 \cdot \frac{1}{{x}^{4}} + 0.5 \cdot \frac{1}{{x}^{2}}\right)}\right)\right)}{x \cdot \sqrt{\pi}} \]

5. Simplified 1.3

   \[\leadsto \frac{{\left(e^{x}\right)}^{x} \cdot \left(1 + \left(1.875 \cdot {x}^{-6} + \color{blue}{\left(\frac{0.75}{{x}^{4}} + \frac{\frac{0.5}{x}}{x}\right)}\right)\right)}{x \cdot \sqrt{\pi}} \]
   Proof

   [Start] 1.3	\[ \frac{{\left(e^{x}\right)}^{x} \cdot \left(1 + \left(1.875 \cdot {x}^{-6} + \left(0.75 \cdot \frac{1}{{x}^{4}} + 0.5 \cdot \frac{1}{{x}^{2}}\right)\right)\right)}{x \cdot \sqrt{\pi}} \]
   associate-*r/ [=>] 1.3	\[ \frac{{\left(e^{x}\right)}^{x} \cdot \left(1 + \left(1.875 \cdot {x}^{-6} + \left(\color{blue}{\frac{0.75 \cdot 1}{{x}^{4}}} + 0.5 \cdot \frac{1}{{x}^{2}}\right)\right)\right)}{x \cdot \sqrt{\pi}} \]
   metadata-eval [=>] 1.3	\[ \frac{{\left(e^{x}\right)}^{x} \cdot \left(1 + \left(1.875 \cdot {x}^{-6} + \left(\frac{\color{blue}{0.75}}{{x}^{4}} + 0.5 \cdot \frac{1}{{x}^{2}}\right)\right)\right)}{x \cdot \sqrt{\pi}} \]
   associate-*r/ [=>] 1.3	\[ \frac{{\left(e^{x}\right)}^{x} \cdot \left(1 + \left(1.875 \cdot {x}^{-6} + \left(\frac{0.75}{{x}^{4}} + \color{blue}{\frac{0.5 \cdot 1}{{x}^{2}}}\right)\right)\right)}{x \cdot \sqrt{\pi}} \]
   metadata-eval [=>] 1.3	\[ \frac{{\left(e^{x}\right)}^{x} \cdot \left(1 + \left(1.875 \cdot {x}^{-6} + \left(\frac{0.75}{{x}^{4}} + \frac{\color{blue}{0.5}}{{x}^{2}}\right)\right)\right)}{x \cdot \sqrt{\pi}} \]
   unpow2 [=>] 1.3	\[ \frac{{\left(e^{x}\right)}^{x} \cdot \left(1 + \left(1.875 \cdot {x}^{-6} + \left(\frac{0.75}{{x}^{4}} + \frac{0.5}{\color{blue}{x \cdot x}}\right)\right)\right)}{x \cdot \sqrt{\pi}} \]
   associate-/r* [=>] 1.3	\[ \frac{{\left(e^{x}\right)}^{x} \cdot \left(1 + \left(1.875 \cdot {x}^{-6} + \left(\frac{0.75}{{x}^{4}} + \color{blue}{\frac{\frac{0.5}{x}}{x}}\right)\right)\right)}{x \cdot \sqrt{\pi}} \]

6. Final simplification 1.3

   \[\leadsto \frac{{\left(e^{x}\right)}^{x} \cdot \left(1 + \left(1.875 \cdot {x}^{-6} + \left(\frac{0.75}{{x}^{4}} + \frac{\frac{0.5}{x}}{x}\right)\right)\right)}{x \cdot \sqrt{\pi}} \]

Alternatives

Alternative 1
Error	1.3
Cost	33536

\[\left(\frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x} + \left(1 + \frac{1.875}{{x}^{6}}\right)\right) \cdot \frac{\frac{{\left(e^{x}\right)}^{x}}{x}}{\sqrt{\pi}} \]

Alternative 2
Error	1.3
Cost	33536

\[\frac{{\left(e^{x}\right)}^{x} \cdot \left(\frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x} + \left(1 + \frac{1.875}{{x}^{6}}\right)\right)}{x \cdot \sqrt{\pi}} \]

Alternative 3
Error	2.7
Cost	27200

\[\frac{\left(\frac{0.5 + \frac{0.75}{x \cdot x}}{x \cdot x} + \left(1 + \frac{1.875}{{x}^{6}}\right)\right) \cdot e^{x \cdot x}}{x \cdot \sqrt{\pi}} \]

Alternative 4
Error	44.7
Cost	26560

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

Alternative 5
Error	44.7
Cost	26560

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

Alternative 6
Error	48.0
Cost	26432

\[\frac{e^{x \cdot x}}{\sqrt{\pi}} \cdot \left(\frac{0.75}{{x}^{5}} + \frac{1}{x}\right) \]

Alternative 7
Error	48.3
Cost	19584

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

Alternative 8
Error	56.6
Cost	13184

\[\sqrt{\frac{1}{\pi}} \cdot \frac{2.1875}{x} \]

Reproduce

herbie shell --seed 2023237 
(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)))))))