Jmat.Real.erf

Average Error: 13.1 → 13.2

Time: 18.3s

Precision: binary64

Cost: 60608

Math

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

↓

\[\begin{array}{l} t_0 := 0.3275911 \cdot \left|x\right|\\ t_1 := 1 + t_0\\ 1 - e^{-{x}^{2}} \cdot \frac{0.254829592 + \left(\frac{\frac{-1.061405429}{-1 - t_0} - 1.453152027}{{t_1}^{3}} + \left(\frac{1.421413741}{{t_1}^{2}} - \frac{0.284496736}{t_1}\right)\right)}{t_1} \end{array} \]

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 (+ 1.0 t_0)))
   (-
    1.0
    (*
     (exp (- (pow x 2.0)))
     (/
      (+
       0.254829592
       (+
        (/ (- (/ -1.061405429 (- -1.0 t_0)) 1.453152027) (pow t_1 3.0))
        (- (/ 1.421413741 (pow t_1 2.0)) (/ 0.284496736 t_1))))
      t_1)))))

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 = 1.0 + t_0;
	return 1.0 - (exp(-pow(x, 2.0)) * ((0.254829592 + ((((-1.061405429 / (-1.0 - t_0)) - 1.453152027) / pow(t_1, 3.0)) + ((1.421413741 / pow(t_1, 2.0)) - (0.284496736 / t_1)))) / t_1));
}

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
    t_0 = 0.3275911d0 * abs(x)
    t_1 = 1.0d0 + t_0
    code = 1.0d0 - (exp(-(x ** 2.0d0)) * ((0.254829592d0 + (((((-1.061405429d0) / ((-1.0d0) - t_0)) - 1.453152027d0) / (t_1 ** 3.0d0)) + ((1.421413741d0 / (t_1 ** 2.0d0)) - (0.284496736d0 / t_1)))) / t_1))
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 = 1.0 + t_0;
	return 1.0 - (Math.exp(-Math.pow(x, 2.0)) * ((0.254829592 + ((((-1.061405429 / (-1.0 - t_0)) - 1.453152027) / Math.pow(t_1, 3.0)) + ((1.421413741 / Math.pow(t_1, 2.0)) - (0.284496736 / t_1)))) / t_1));
}

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 = 1.0 + t_0
	return 1.0 - (math.exp(-math.pow(x, 2.0)) * ((0.254829592 + ((((-1.061405429 / (-1.0 - t_0)) - 1.453152027) / math.pow(t_1, 3.0)) + ((1.421413741 / math.pow(t_1, 2.0)) - (0.284496736 / t_1)))) / t_1))

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(1.0 + t_0)
	return Float64(1.0 - Float64(exp(Float64(-(x ^ 2.0))) * Float64(Float64(0.254829592 + Float64(Float64(Float64(Float64(-1.061405429 / Float64(-1.0 - t_0)) - 1.453152027) / (t_1 ^ 3.0)) + Float64(Float64(1.421413741 / (t_1 ^ 2.0)) - Float64(0.284496736 / t_1)))) / t_1)))
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 = 1.0 + t_0;
	tmp = 1.0 - (exp(-(x ^ 2.0)) * ((0.254829592 + ((((-1.061405429 / (-1.0 - t_0)) - 1.453152027) / (t_1 ^ 3.0)) + ((1.421413741 / (t_1 ^ 2.0)) - (0.284496736 / t_1)))) / t_1));
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[(1.0 + t$95$0), $MachinePrecision]}, N[(1.0 - N[(N[Exp[(-N[Power[x, 2.0], $MachinePrecision])], $MachinePrecision] * N[(N[(0.254829592 + N[(N[(N[(N[(-1.061405429 / N[(-1.0 - t$95$0), $MachinePrecision]), $MachinePrecision] - 1.453152027), $MachinePrecision] / N[Power[t$95$1, 3.0], $MachinePrecision]), $MachinePrecision] + N[(N[(1.421413741 / N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision] - N[(0.284496736 / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 13.1

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

2. Simplified 13.1

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

   [Start] 13.1	\[ 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]
   rational.json-simplify-39 [=>] 13.1	\[ 1 - \color{blue}{e^{-\left|x\right| \cdot \left|x\right|} \cdot \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right)} \]
   rational.json-simplify-3 [=>] 13.1	\[ 1 - \color{blue}{\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(e^{-\left|x\right| \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right)} \]

3. Applied egg-rr 13.1

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

4. Simplified 13.9

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

   [Start] 13.1	\[ 1 + \left(-e^{x \cdot \left(-x\right)} \cdot \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}\right) \]
   rational.json-simplify-67 [=>] 13.1	\[ 1 + \color{blue}{\left(0 - e^{x \cdot \left(-x\right)} \cdot \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}\right)} \]
   rational.json-simplify-14 [=>] 13.1	\[ \color{blue}{\left(1 + 0\right) - e^{x \cdot \left(-x\right)} \cdot \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}} \]
   metadata-eval [=>] 13.1	\[ \color{blue}{1} - e^{x \cdot \left(-x\right)} \cdot \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|} \]
   rational.json-simplify-63 [<=] 13.1	\[ \color{blue}{\left(1 - e^{x \cdot \left(-x\right)} \cdot \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}}{1 + 0.3275911 \cdot \left|x\right|}\right) \cdot 1} \]

5. Taylor expanded in x around inf 13.9

   \[\leadsto \color{blue}{1 - \frac{\left(\left(0.254829592 + \left(-1 \cdot \frac{1.453152027 - 1.061405429 \cdot \frac{1}{0.3275911 \cdot \left|x\right| + 1}}{{\left(0.3275911 \cdot \left|x\right| + 1\right)}^{3}} + 1.421413741 \cdot \frac{1}{{\left(0.3275911 \cdot \left|x\right| + 1\right)}^{2}}\right)\right) - 0.284496736 \cdot \frac{1}{0.3275911 \cdot \left|x\right| + 1}\right) \cdot e^{-1 \cdot {x}^{2}}}{0.3275911 \cdot \left|x\right| + 1}} \]

6. Simplified 13.2

   \[\leadsto \color{blue}{1 - e^{-{x}^{2}} \cdot \frac{0.254829592 + \left(\frac{\frac{-1.061405429}{-1 - 0.3275911 \cdot \left|x\right|} - 1.453152027}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{3}} + \left(\frac{1.421413741}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{2}} - \frac{0.284496736}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)}{1 + 0.3275911 \cdot \left|x\right|}} \]
   Proof

   [Start] 13.9

   \[ 1 - \frac{\left(\left(0.254829592 + \left(-1 \cdot \frac{1.453152027 - 1.061405429 \cdot \frac{1}{0.3275911 \cdot \left|x\right| + 1}}{{\left(0.3275911 \cdot \left|x\right| + 1\right)}^{3}} + 1.421413741 \cdot \frac{1}{{\left(0.3275911 \cdot \left|x\right| + 1\right)}^{2}}\right)\right) - 0.284496736 \cdot \frac{1}{0.3275911 \cdot \left|x\right| + 1}\right) \cdot e^{-1 \cdot {x}^{2}}}{0.3275911 \cdot \left|x\right| + 1} \]

7. Final simplification 13.2

   \[\leadsto 1 - e^{-{x}^{2}} \cdot \frac{0.254829592 + \left(\frac{\frac{-1.061405429}{-1 - 0.3275911 \cdot \left|x\right|} - 1.453152027}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{3}} + \left(\frac{1.421413741}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{2}} - \frac{0.284496736}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)}{1 + 0.3275911 \cdot \left|x\right|} \]

Alternatives

Alternative 1
Error	13.1
Cost	54848

\[\begin{array}{l} t_0 := 0.3275911 \cdot \left|x\right|\\ t_1 := 1 + t_0\\ t_2 := e^{x \cdot \left(-x\right)}\\ t_3 := -1 - t_0\\ 1 - \left(\frac{-0.284496736 + \frac{-1.421413741 + \frac{1.453152027 + \frac{1.061405429}{t_3}}{t_1}}{t_3}}{t_1 \cdot \frac{t_1}{t_2}} + 0.254829592 \cdot \frac{t_2}{t_1}\right) \end{array} \]

Alternative 2
Error	13.1
Cost	41344

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

Alternative 3
Error	14.8
Cost	41088

\[\begin{array}{l} t_0 := 1 + 0.3275911 \cdot \left|x\right|\\ t_1 := -1 + \left|x\right| \cdot -0.3275911\\ 1 - \left(\frac{1.421413741 + \frac{\frac{-1.061405429}{t_1} + -1.453152027}{t_0}}{{t_0}^{3}} - \frac{\frac{0.284496736}{t_1} - -0.254829592}{t_1}\right) \end{array} \]

Reproduce

herbie shell --seed 2023066 
(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)))))))