Jmat.Real.erf

Average Error: 13.5 → 13.5

Time: 17.0s

Precision: binary64

Cost: 48064

Math

\[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

↓

\[\begin{array}{l} t_0 := 0.3275911 \cdot \left|x\right|\\ t_1 := t_0 + 1\\ t_2 := 1 + t_0\\ 1 - \left(0.254829592 + \frac{-0.284496736 + \frac{\left(1.421413741 + 1.061405429 \cdot \frac{1}{{t_1}^{2}}\right) - 1.453152027 \cdot \frac{1}{t_1}}{t_2}}{t_2}\right) \cdot \frac{e^{x \cdot \left(-x\right)}}{t_2} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (-
  1.0
  (*
   (*
    (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x))))
    (+
     0.254829592
     (*
      (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x))))
      (+
       -0.284496736
       (*
        (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x))))
        (+
         1.421413741
         (*
          (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x))))
          (+
           -1.453152027
           (* (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x)))) 1.061405429)))))))))
   (exp (- (* (fabs x) (fabs x)))))))

↓

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* 0.3275911 (fabs x))) (t_1 (+ t_0 1.0)) (t_2 (+ 1.0 t_0)))
   (-
    1.0
    (*
     (+
      0.254829592
      (/
       (+
        -0.284496736
        (/
         (-
          (+ 1.421413741 (* 1.061405429 (/ 1.0 (pow t_1 2.0))))
          (* 1.453152027 (/ 1.0 t_1)))
         t_2))
       t_2))
     (/ (exp (* x (- x))) t_2)))))

C

double code(double x) {
	return 1.0 - (((1.0 / (1.0 + (0.3275911 * fabs(x)))) * (0.254829592 + ((1.0 / (1.0 + (0.3275911 * fabs(x)))) * (-0.284496736 + ((1.0 / (1.0 + (0.3275911 * fabs(x)))) * (1.421413741 + ((1.0 / (1.0 + (0.3275911 * fabs(x)))) * (-1.453152027 + ((1.0 / (1.0 + (0.3275911 * fabs(x)))) * 1.061405429))))))))) * exp(-(fabs(x) * fabs(x))));
}

↓

double code(double x) {
	double t_0 = 0.3275911 * fabs(x);
	double t_1 = t_0 + 1.0;
	double t_2 = 1.0 + t_0;
	return 1.0 - ((0.254829592 + ((-0.284496736 + (((1.421413741 + (1.061405429 * (1.0 / pow(t_1, 2.0)))) - (1.453152027 * (1.0 / t_1))) / t_2)) / t_2)) * (exp((x * -x)) / t_2));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 - (((1.0d0 / (1.0d0 + (0.3275911d0 * abs(x)))) * (0.254829592d0 + ((1.0d0 / (1.0d0 + (0.3275911d0 * abs(x)))) * ((-0.284496736d0) + ((1.0d0 / (1.0d0 + (0.3275911d0 * abs(x)))) * (1.421413741d0 + ((1.0d0 / (1.0d0 + (0.3275911d0 * abs(x)))) * ((-1.453152027d0) + ((1.0d0 / (1.0d0 + (0.3275911d0 * abs(x)))) * 1.061405429d0))))))))) * exp(-(abs(x) * abs(x))))
end function

↓

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    t_0 = 0.3275911d0 * abs(x)
    t_1 = t_0 + 1.0d0
    t_2 = 1.0d0 + t_0
    code = 1.0d0 - ((0.254829592d0 + (((-0.284496736d0) + (((1.421413741d0 + (1.061405429d0 * (1.0d0 / (t_1 ** 2.0d0)))) - (1.453152027d0 * (1.0d0 / t_1))) / t_2)) / t_2)) * (exp((x * -x)) / t_2))
end function

Java

public static double code(double x) {
	return 1.0 - (((1.0 / (1.0 + (0.3275911 * Math.abs(x)))) * (0.254829592 + ((1.0 / (1.0 + (0.3275911 * Math.abs(x)))) * (-0.284496736 + ((1.0 / (1.0 + (0.3275911 * Math.abs(x)))) * (1.421413741 + ((1.0 / (1.0 + (0.3275911 * Math.abs(x)))) * (-1.453152027 + ((1.0 / (1.0 + (0.3275911 * Math.abs(x)))) * 1.061405429))))))))) * Math.exp(-(Math.abs(x) * Math.abs(x))));
}

↓

public static double code(double x) {
	double t_0 = 0.3275911 * Math.abs(x);
	double t_1 = t_0 + 1.0;
	double t_2 = 1.0 + t_0;
	return 1.0 - ((0.254829592 + ((-0.284496736 + (((1.421413741 + (1.061405429 * (1.0 / Math.pow(t_1, 2.0)))) - (1.453152027 * (1.0 / t_1))) / t_2)) / t_2)) * (Math.exp((x * -x)) / t_2));
}

Python

def code(x):
	return 1.0 - (((1.0 / (1.0 + (0.3275911 * math.fabs(x)))) * (0.254829592 + ((1.0 / (1.0 + (0.3275911 * math.fabs(x)))) * (-0.284496736 + ((1.0 / (1.0 + (0.3275911 * math.fabs(x)))) * (1.421413741 + ((1.0 / (1.0 + (0.3275911 * math.fabs(x)))) * (-1.453152027 + ((1.0 / (1.0 + (0.3275911 * math.fabs(x)))) * 1.061405429))))))))) * math.exp(-(math.fabs(x) * math.fabs(x))))

↓

def code(x):
	t_0 = 0.3275911 * math.fabs(x)
	t_1 = t_0 + 1.0
	t_2 = 1.0 + t_0
	return 1.0 - ((0.254829592 + ((-0.284496736 + (((1.421413741 + (1.061405429 * (1.0 / math.pow(t_1, 2.0)))) - (1.453152027 * (1.0 / t_1))) / t_2)) / t_2)) * (math.exp((x * -x)) / t_2))

Julia

function code(x)
	return Float64(1.0 - Float64(Float64(Float64(1.0 / Float64(1.0 + Float64(0.3275911 * abs(x)))) * Float64(0.254829592 + Float64(Float64(1.0 / Float64(1.0 + Float64(0.3275911 * abs(x)))) * Float64(-0.284496736 + Float64(Float64(1.0 / Float64(1.0 + Float64(0.3275911 * abs(x)))) * Float64(1.421413741 + Float64(Float64(1.0 / Float64(1.0 + Float64(0.3275911 * abs(x)))) * Float64(-1.453152027 + Float64(Float64(1.0 / Float64(1.0 + Float64(0.3275911 * abs(x)))) * 1.061405429))))))))) * exp(Float64(-Float64(abs(x) * abs(x))))))
end

↓

function code(x)
	t_0 = Float64(0.3275911 * abs(x))
	t_1 = Float64(t_0 + 1.0)
	t_2 = Float64(1.0 + t_0)
	return Float64(1.0 - Float64(Float64(0.254829592 + Float64(Float64(-0.284496736 + Float64(Float64(Float64(1.421413741 + Float64(1.061405429 * Float64(1.0 / (t_1 ^ 2.0)))) - Float64(1.453152027 * Float64(1.0 / t_1))) / t_2)) / t_2)) * Float64(exp(Float64(x * Float64(-x))) / t_2)))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 - (((1.0 / (1.0 + (0.3275911 * abs(x)))) * (0.254829592 + ((1.0 / (1.0 + (0.3275911 * abs(x)))) * (-0.284496736 + ((1.0 / (1.0 + (0.3275911 * abs(x)))) * (1.421413741 + ((1.0 / (1.0 + (0.3275911 * abs(x)))) * (-1.453152027 + ((1.0 / (1.0 + (0.3275911 * abs(x)))) * 1.061405429))))))))) * exp(-(abs(x) * abs(x))));
end

↓

function tmp = code(x)
	t_0 = 0.3275911 * abs(x);
	t_1 = t_0 + 1.0;
	t_2 = 1.0 + t_0;
	tmp = 1.0 - ((0.254829592 + ((-0.284496736 + (((1.421413741 + (1.061405429 * (1.0 / (t_1 ^ 2.0)))) - (1.453152027 * (1.0 / t_1))) / t_2)) / t_2)) * (exp((x * -x)) / t_2));
end

Wolfram

code[x_] := N[(1.0 - N[(N[(N[(1.0 / N[(1.0 + N[(0.3275911 * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(0.254829592 + N[(N[(1.0 / N[(1.0 + N[(0.3275911 * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-0.284496736 + N[(N[(1.0 / N[(1.0 + N[(0.3275911 * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.421413741 + N[(N[(1.0 / N[(1.0 + N[(0.3275911 * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.453152027 + N[(N[(1.0 / N[(1.0 + N[(0.3275911 * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * 1.061405429), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Exp[(-N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

↓

code[x_] := Block[{t$95$0 = N[(0.3275911 * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 + t$95$0), $MachinePrecision]}, N[(1.0 - N[(N[(0.254829592 + N[(N[(-0.284496736 + N[(N[(N[(1.421413741 + N[(1.061405429 * N[(1.0 / N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(1.453152027 * N[(1.0 / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision] * N[(N[Exp[N[(x * (-x)), $MachinePrecision]], $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Initial program 13.5

   \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

2. Simplified 13.5

   \[\leadsto \color{blue}{1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right) \cdot e^{x \cdot \left(-x\right)}\right)} \]
   Proof

   [Start] 13.5	\[ 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]
   rational.json-simplify-2 [=>] 13.5	\[ 1 - \color{blue}{e^{-\left|x\right| \cdot \left|x\right|} \cdot \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right)} \]
   rational.json-simplify-43 [=>] 13.5	\[ 1 - \color{blue}{\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\right)} \]

3. Applied egg-rr 13.5

   \[\leadsto \color{blue}{\left(1 - \left(0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}\right) \cdot \frac{e^{x \cdot \left(-x\right)}}{1 + 0.3275911 \cdot \left|x\right|}\right) + 0} \]

4. Simplified 13.5

   \[\leadsto \color{blue}{1 - \left(0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}\right) \cdot \frac{e^{x \cdot \left(-x\right)}}{1 + 0.3275911 \cdot \left|x\right|}} \]
   Proof

   [Start] 13.5	\[ \left(1 - \left(0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}\right) \cdot \frac{e^{x \cdot \left(-x\right)}}{1 + 0.3275911 \cdot \left|x\right|}\right) + 0 \]
   rational.json-simplify-4 [=>] 13.5	\[ \color{blue}{1 - \left(0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}\right) \cdot \frac{e^{x \cdot \left(-x\right)}}{1 + 0.3275911 \cdot \left|x\right|}} \]

5. Taylor expanded in x around 0 13.5

   \[\leadsto 1 - \left(0.254829592 + \frac{-0.284496736 + \frac{\color{blue}{\left(1.421413741 + 1.061405429 \cdot \frac{1}{{\left(0.3275911 \cdot \left|x\right| + 1\right)}^{2}}\right) - 1.453152027 \cdot \frac{1}{0.3275911 \cdot \left|x\right| + 1}}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}\right) \cdot \frac{e^{x \cdot \left(-x\right)}}{1 + 0.3275911 \cdot \left|x\right|} \]

6. Final simplification 13.5

   \[\leadsto 1 - \left(0.254829592 + \frac{-0.284496736 + \frac{\left(1.421413741 + 1.061405429 \cdot \frac{1}{{\left(0.3275911 \cdot \left|x\right| + 1\right)}^{2}}\right) - 1.453152027 \cdot \frac{1}{0.3275911 \cdot \left|x\right| + 1}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}\right) \cdot \frac{e^{x \cdot \left(-x\right)}}{1 + 0.3275911 \cdot \left|x\right|} \]

Alternatives

Alternative 1
Error	13.5
Cost	41472

\[\begin{array}{l} t_0 := 1 + 0.3275911 \cdot \left|x\right|\\ 1 - \left(0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1}{t_0} \cdot 1.061405429}{t_0}}{t_0}}{t_0}\right) \cdot \frac{e^{x \cdot \left(-x\right)}}{t_0} \end{array} \]

Alternative 2
Error	13.5
Cost	41408

\[\begin{array}{l} t_0 := 1 + 0.3275911 \cdot \left|x\right|\\ 1 - \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{2}{\left|x\right| \cdot 0.6551822 + 2} \cdot 1.061405429}{t_0}}{t_0}}{t_0}}{t_0 \cdot e^{x \cdot x}} \end{array} \]

Reproduce

herbie shell --seed 2023067 
(FPCore (x)
  :name "Jmat.Real.erf"
  :precision binary64
  (- 1.0 (* (* (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x)))) (+ 0.254829592 (* (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x)))) (+ -0.284496736 (* (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x)))) (+ 1.421413741 (* (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x)))) (+ -1.453152027 (* (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x)))) 1.061405429))))))))) (exp (- (* (fabs x) (fabs x)))))))