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

Average Error: 2.8 → 1.3

Time: 1.9min

Precision: binary64

Cost: 46336

\[x \geq 0.5\]

Math

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

↓

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

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
 (/
  (/
   (+ 1.0 (/ (/ (+ 0.5 (/ (+ 0.75 (/ 1.875 (pow x 2.0))) (pow x 2.0))) x) x))
   (* (sqrt PI) (fabs x)))
  (pow (exp x) (- x))))

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 ((1.0 + (((0.5 + ((0.75 + (1.875 / pow(x, 2.0))) / pow(x, 2.0))) / x) / x)) / (sqrt(((double) M_PI)) * fabs(x))) / pow(exp(x), -x);
}

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 ((1.0 + (((0.5 + ((0.75 + (1.875 / Math.pow(x, 2.0))) / Math.pow(x, 2.0))) / x) / x)) / (Math.sqrt(Math.PI) * Math.abs(x))) / Math.pow(Math.exp(x), -x);
}

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 ((1.0 + (((0.5 + ((0.75 + (1.875 / math.pow(x, 2.0))) / math.pow(x, 2.0))) / x) / x)) / (math.sqrt(math.pi) * math.fabs(x))) / math.pow(math.exp(x), -x)

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(Float64(1.0 + Float64(Float64(Float64(0.5 + Float64(Float64(0.75 + Float64(1.875 / (x ^ 2.0))) / (x ^ 2.0))) / x) / x)) / Float64(sqrt(pi) * abs(x))) / (exp(x) ^ Float64(-x)))
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 = ((1.0 + (((0.5 + ((0.75 + (1.875 / (x ^ 2.0))) / (x ^ 2.0))) / x) / x)) / (sqrt(pi) * abs(x))) / (exp(x) ^ -x);
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[(1.0 + N[(N[(N[(0.5 + N[(N[(0.75 + N[(1.875 / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision] / N[(N[Sqrt[Pi], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[Power[N[Exp[x], $MachinePrecision], (-x)], $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. Applied egg-rr 3.0

   \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{e^{{x}^{2}}}{\sqrt{\pi}} \cdot \left(\frac{1}{\left|x\right|} \cdot \left(\frac{1}{{\left({x}^{2}\right)}^{2}} \cdot \left(0.75 + 1.875 \cdot {\left({x}^{-1}\right)}^{2}\right) + \left(1 + \frac{0.5}{{x}^{2}}\right)\right)\right)}\right)}^{3}} \]

3. Applied egg-rr 2.7

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

4. Simplified 1.3

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

5. Applied egg-rr 2.7

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

6. Applied egg-rr 1.3

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

Alternatives

Alternative 1
Error	43.3
Cost	39556

\[\begin{array}{l} \mathbf{if}\;\left|x\right| \leq 1.3:\\ \;\;\;\;\frac{\frac{0.5 + \frac{0.75 + \frac{1.875}{x \cdot x}}{x \cdot x}}{x \cdot x} - -1}{\frac{\sqrt{\pi}}{\left(x \cdot x - -1\right) + {x}^{4} \cdot 0.5} \cdot \left|x\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{0.5 \cdot \frac{\frac{1}{x}}{x} + 1}{\sqrt{\pi} \cdot \left|x\right|}}{e^{-{x}^{2}}}\\ \end{array} \]

Alternative 2
Error	43.3
Cost	39492

\[\begin{array}{l} \mathbf{if}\;\left|x\right| \leq 1.3:\\ \;\;\;\;\frac{\frac{0.5 + \frac{0.75 + \frac{1.875}{x \cdot x}}{x \cdot x}}{x \cdot x} - -1}{\frac{\sqrt{\pi}}{\left(x \cdot x - -1\right) + {x}^{4} \cdot 0.5} \cdot \left|x\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}}{\frac{\sqrt{\pi}}{{\left(e^{-x}\right)}^{\left(-x\right)}}}\\ \end{array} \]

Alternative 3
Error	43.3
Cost	39364

\[\begin{array}{l} \mathbf{if}\;\left|x\right| \leq 1.3:\\ \;\;\;\;\frac{\frac{0.5 + \frac{0.75 + \frac{1.875}{x \cdot x}}{x \cdot x}}{x \cdot x} - -1}{\frac{\sqrt{\pi}}{\left(x \cdot x - -1\right) + {x}^{4} \cdot 0.5} \cdot \left|x\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 + \frac{0.5}{x \cdot x}}{\left|x\right|}}{\frac{\sqrt{\pi}}{e^{{x}^{2}}}}\\ \end{array} \]

Alternative 4
Error	46.8
Cost	39044

\[\begin{array}{l} \mathbf{if}\;\left|x\right| \leq 1.46:\\ \;\;\;\;\frac{\frac{0.5 + \frac{0.75 + \frac{1.875}{x \cdot x}}{x \cdot x}}{x \cdot x} - -1}{\frac{\sqrt{\pi}}{\left(x \cdot x - -1\right) + {x}^{4} \cdot 0.5} \cdot \left|x\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{\sqrt{\pi} \cdot \left|x\right|}}{e^{-{x}^{2}}}\\ \end{array} \]

Alternative 5
Error	46.8
Cost	38852

\[\begin{array}{l} \mathbf{if}\;\left|x\right| \leq 1.46:\\ \;\;\;\;\frac{\frac{0.5 + \frac{0.75 + \frac{1.875}{x \cdot x}}{x \cdot x}}{x \cdot x} - -1}{\frac{\sqrt{\pi}}{\left(x \cdot x - -1\right) + {x}^{4} \cdot 0.5} \cdot \left|x\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{{x}^{2}}}{\sqrt{\pi} \cdot \left|x\right|}\\ \end{array} \]

Alternative 6
Error	46.8
Cost	38852

\[\begin{array}{l} \mathbf{if}\;\left|x\right| \leq 1.46:\\ \;\;\;\;\frac{\frac{0.5 + \frac{0.75 + \frac{1.875}{x \cdot x}}{x \cdot x}}{x \cdot x} - -1}{\frac{\sqrt{\pi}}{\left(x \cdot x - -1\right) + {x}^{4} \cdot 0.5} \cdot \left|x\right|}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{\left(e^{x}\right)}^{x}}{\sqrt{\pi}}}{\left|x\right|}\\ \end{array} \]

Alternative 7
Error	1.3
Cost	33664

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

Alternative 8
Error	1.3
Cost	33664

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

Alternative 9
Error	2.7
Cost	33600

\[\frac{\frac{0.5 + \frac{0.75 + \frac{1.875}{x \cdot x}}{x \cdot x}}{x \cdot x} - -1}{\frac{\sqrt{\pi}}{e^{{x}^{2}}} \cdot \left|x\right|} \]

Alternative 10
Error	1.3
Cost	33600

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

Alternative 11
Error	1.3
Cost	33600

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

Alternative 12
Error	1.3
Cost	33600

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

Alternative 13
Error	54.8
Cost	27712

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

Alternative 14
Error	55.6
Cost	20992

\[\frac{\frac{0.5 + \frac{0.75 + \frac{1.875}{x \cdot x}}{x \cdot x}}{x \cdot x} - -1}{\frac{\sqrt{\pi}}{x \cdot x - -1} \cdot \left|x\right|} \]

Alternative 15
Error	56.3
Cost	20608

\[\frac{\frac{\frac{0.5 + \frac{\frac{0.75 + \frac{\frac{1.875}{x}}{x}}{x}}{x}}{x}}{x} + 1}{\sqrt{\pi} \cdot \left|x\right|} \]

Alternative 16
Error	56.9
Cost	19456

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

Reproduce

herbie shell --seed 2023033 
(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)))))))