Jmat.Real.erf

Average Error: 13.2 → 13.2

Time: 16.9s

Precision: binary64

Cost: 54848

Math

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

↓

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

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 (+ 1.0 (* 0.3275911 (fabs x)))) (t_1 (/ 1.0 t_0)))
   (-
    1.0
    (*
     (*
      t_1
      (+
       0.254829592
       (*
        t_1
        (+
         -0.284496736
         (*
          t_1
          (+
           (* 1.061405429 (/ 1.0 (pow t_0 2.0)))
           (- 1.421413741 (* 1.453152027 t_1))))))))
     (exp (- (fabs (* x x))))))))

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

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

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

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(1.0 + Float64(0.3275911 * abs(x)))
	t_1 = Float64(1.0 / t_0)
	return Float64(1.0 - Float64(Float64(t_1 * Float64(0.254829592 + Float64(t_1 * Float64(-0.284496736 + Float64(t_1 * Float64(Float64(1.061405429 * Float64(1.0 / (t_0 ^ 2.0))) + Float64(1.421413741 - Float64(1.453152027 * t_1)))))))) * exp(Float64(-abs(Float64(x * x))))))
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 = 1.0 + (0.3275911 * abs(x));
	t_1 = 1.0 / t_0;
	tmp = 1.0 - ((t_1 * (0.254829592 + (t_1 * (-0.284496736 + (t_1 * ((1.061405429 * (1.0 / (t_0 ^ 2.0))) + (1.421413741 - (1.453152027 * t_1)))))))) * exp(-abs((x * x))));
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[(1.0 + N[(0.3275911 * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / t$95$0), $MachinePrecision]}, N[(1.0 - N[(N[(t$95$1 * N[(0.254829592 + N[(t$95$1 * N[(-0.284496736 + N[(t$95$1 * N[(N[(1.061405429 * N[(1.0 / N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.421413741 - N[(1.453152027 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Exp[(-N[Abs[N[(x * x), $MachinePrecision]], $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 13.2

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

2. Simplified 13.2

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

   [Start] 13.2

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

   rational.json-simplify-38 [=>] 13.2

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

3. Taylor expanded in x around 0 13.2

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

4. Simplified 13.2

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

   [Start] 13.2	\[ 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(1.421413741 + 1.061405429 \cdot \frac{1}{{\left(0.3275911 \cdot \left|x\right| + 1\right)}^{2}}\right) - 1.453152027 \cdot \frac{1}{0.3275911 \cdot \left|x\right| + 1}\right)\right)\right)\right) \cdot e^{-\left|x \cdot x\right|} \]
   rational.json-simplify-48 [=>] 13.2	\[ 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \color{blue}{\left(1.061405429 \cdot \frac{1}{{\left(0.3275911 \cdot \left|x\right| + 1\right)}^{2}} + \left(1.421413741 - 1.453152027 \cdot \frac{1}{0.3275911 \cdot \left|x\right| + 1}\right)\right)}\right)\right)\right) \cdot e^{-\left|x \cdot x\right|} \]
   rational.json-simplify-1 [<=] 13.2	\[ 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.061405429 \cdot \frac{1}{{\color{blue}{\left(1 + 0.3275911 \cdot \left|x\right|\right)}}^{2}} + \left(1.421413741 - 1.453152027 \cdot \frac{1}{0.3275911 \cdot \left|x\right| + 1}\right)\right)\right)\right)\right) \cdot e^{-\left|x \cdot x\right|} \]
   rational.json-simplify-1 [<=] 13.2	\[ 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.061405429 \cdot \frac{1}{{\left(1 + 0.3275911 \cdot \left|x\right|\right)}^{2}} + \left(1.421413741 - 1.453152027 \cdot \frac{1}{\color{blue}{1 + 0.3275911 \cdot \left|x\right|}}\right)\right)\right)\right)\right) \cdot e^{-\left|x \cdot x\right|} \]

5. Final simplification 13.2

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

Alternatives

Alternative 1
Error	13.2
Cost	48000

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

Alternative 2
Error	13.2
Cost	41408

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

Reproduce

herbie shell --seed 2023068 
(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)))))))