Jmat.Real.erf

Average Error: 13.5 → 0.2

Time: 15.1s

Precision: binary64

Cost: 54980

Math

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

↓

\[\begin{array}{l} t_0 := 1 + \left|x\right| \cdot 0.3275911\\ t_1 := \frac{1}{t_0}\\ \mathbf{if}\;\left|x\right| \leq 10^{-8}:\\ \;\;\;\;10^{-9} + \sqrt{x \cdot \left(x \cdot 1.2732557730789702\right)}\\ \mathbf{else}:\\ \;\;\;\;1 - \left(\frac{-1}{t_0} \cdot \left(-0.254829592 - t_1 \cdot \left(-0.284496736 - t_1 \cdot \left(t_1 \cdot 1.453152027 - \left(1.421413741 + 1.061405429 \cdot \frac{1}{{t_0}^{2}}\right)\right)\right)\right)\right) \cdot e^{x \cdot \left(-x\right)}\\ \end{array} \]

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 (* (fabs x) 0.3275911))) (t_1 (/ 1.0 t_0)))
   (if (<= (fabs x) 1e-8)
     (+ 1e-9 (sqrt (* x (* x 1.2732557730789702))))
     (-
      1.0
      (*
       (*
        (/ -1.0 t_0)
        (-
         -0.254829592
         (*
          t_1
          (-
           -0.284496736
           (*
            t_1
            (-
             (* t_1 1.453152027)
             (+ 1.421413741 (* 1.061405429 (/ 1.0 (pow t_0 2.0))))))))))
       (exp (* 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 + (fabs(x) * 0.3275911);
	double t_1 = 1.0 / t_0;
	double tmp;
	if (fabs(x) <= 1e-8) {
		tmp = 1e-9 + sqrt((x * (x * 1.2732557730789702)));
	} else {
		tmp = 1.0 - (((-1.0 / t_0) * (-0.254829592 - (t_1 * (-0.284496736 - (t_1 * ((t_1 * 1.453152027) - (1.421413741 + (1.061405429 * (1.0 / pow(t_0, 2.0)))))))))) * exp((x * -x)));
	}
	return tmp;
}

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) :: tmp
    t_0 = 1.0d0 + (abs(x) * 0.3275911d0)
    t_1 = 1.0d0 / t_0
    if (abs(x) <= 1d-8) then
        tmp = 1d-9 + sqrt((x * (x * 1.2732557730789702d0)))
    else
        tmp = 1.0d0 - ((((-1.0d0) / t_0) * ((-0.254829592d0) - (t_1 * ((-0.284496736d0) - (t_1 * ((t_1 * 1.453152027d0) - (1.421413741d0 + (1.061405429d0 * (1.0d0 / (t_0 ** 2.0d0)))))))))) * exp((x * -x)))
    end if
    code = tmp
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 + (Math.abs(x) * 0.3275911);
	double t_1 = 1.0 / t_0;
	double tmp;
	if (Math.abs(x) <= 1e-8) {
		tmp = 1e-9 + Math.sqrt((x * (x * 1.2732557730789702)));
	} else {
		tmp = 1.0 - (((-1.0 / t_0) * (-0.254829592 - (t_1 * (-0.284496736 - (t_1 * ((t_1 * 1.453152027) - (1.421413741 + (1.061405429 * (1.0 / Math.pow(t_0, 2.0)))))))))) * Math.exp((x * -x)));
	}
	return tmp;
}

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 + (math.fabs(x) * 0.3275911)
	t_1 = 1.0 / t_0
	tmp = 0
	if math.fabs(x) <= 1e-8:
		tmp = 1e-9 + math.sqrt((x * (x * 1.2732557730789702)))
	else:
		tmp = 1.0 - (((-1.0 / t_0) * (-0.254829592 - (t_1 * (-0.284496736 - (t_1 * ((t_1 * 1.453152027) - (1.421413741 + (1.061405429 * (1.0 / math.pow(t_0, 2.0)))))))))) * math.exp((x * -x)))
	return tmp

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(abs(x) * 0.3275911))
	t_1 = Float64(1.0 / t_0)
	tmp = 0.0
	if (abs(x) <= 1e-8)
		tmp = Float64(1e-9 + sqrt(Float64(x * Float64(x * 1.2732557730789702))));
	else
		tmp = Float64(1.0 - Float64(Float64(Float64(-1.0 / t_0) * Float64(-0.254829592 - Float64(t_1 * Float64(-0.284496736 - Float64(t_1 * Float64(Float64(t_1 * 1.453152027) - Float64(1.421413741 + Float64(1.061405429 * Float64(1.0 / (t_0 ^ 2.0)))))))))) * exp(Float64(x * Float64(-x)))));
	end
	return tmp
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_2 = code(x)
	t_0 = 1.0 + (abs(x) * 0.3275911);
	t_1 = 1.0 / t_0;
	tmp = 0.0;
	if (abs(x) <= 1e-8)
		tmp = 1e-9 + sqrt((x * (x * 1.2732557730789702)));
	else
		tmp = 1.0 - (((-1.0 / t_0) * (-0.254829592 - (t_1 * (-0.284496736 - (t_1 * ((t_1 * 1.453152027) - (1.421413741 + (1.061405429 * (1.0 / (t_0 ^ 2.0)))))))))) * exp((x * -x)));
	end
	tmp_2 = tmp;
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[(N[Abs[x], $MachinePrecision] * 0.3275911), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / t$95$0), $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 1e-8], N[(1e-9 + N[Sqrt[N[(x * N[(x * 1.2732557730789702), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(N[(N[(-1.0 / t$95$0), $MachinePrecision] * N[(-0.254829592 - N[(t$95$1 * N[(-0.284496736 - N[(t$95$1 * N[(N[(t$95$1 * 1.453152027), $MachinePrecision] - N[(1.421413741 + N[(1.061405429 * N[(1.0 / N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(x * (-x)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 1e-8

   1. Initial program 27.1

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

   2. Applied egg-rr 27.3

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

   3. Taylor expanded in x around 0 0.9

      \[\leadsto \color{blue}{10^{-9} + 1.128386358070218 \cdot x} \]

   4. Applied egg-rr 0.1

      \[\leadsto 10^{-9} + \color{blue}{\sqrt{{\left(1.128386358070218 \cdot x\right)}^{2}}} \]

   5. Taylor expanded in x around 0 0.1

      \[\leadsto 10^{-9} + \sqrt{\color{blue}{1.2732557730789702 \cdot {x}^{2}}} \]

   6. Simplified 0.1

      \[\leadsto 10^{-9} + \sqrt{\color{blue}{x \cdot \left(x \cdot 1.2732557730789702\right)}} \]
      Proof
      (*.f64 x (*.f64 x 318313943269742557644254641881/250000000000000000000000000000)): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 x x) 318313943269742557644254641881/250000000000000000000000000000)): 21 points increase in error, 34 points decrease in error
      (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 x 2)) 318313943269742557644254641881/250000000000000000000000000000): 0 points increase in error, 0 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 318313943269742557644254641881/250000000000000000000000000000 (pow.f64 x 2))): 0 points increase in error, 0 points decrease in error

   if 1e-8 < (fabs.f64 x)

   1. Initial program 0.4

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

   2. Taylor expanded in x around 0 0.4

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

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 10^{-8}:\\ \;\;\;\;10^{-9} + \sqrt{x \cdot \left(x \cdot 1.2732557730789702\right)}\\ \mathbf{else}:\\ \;\;\;\;1 - \left(\frac{-1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(-0.254829592 - \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(-0.284496736 - \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(\frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot 1.453152027 - \left(1.421413741 + 1.061405429 \cdot \frac{1}{{\left(1 + \left|x\right| \cdot 0.3275911\right)}^{2}}\right)\right)\right)\right)\right) \cdot e^{x \cdot \left(-x\right)}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.2
Cost	48516

\[\begin{array}{l} t_0 := 1 + \left|x\right| \cdot 0.3275911\\ t_1 := \frac{1}{t_0}\\ t_2 := \frac{-1}{t_0}\\ \mathbf{if}\;\left|x\right| \leq 10^{-8}:\\ \;\;\;\;10^{-9} + \sqrt{x \cdot \left(x \cdot 1.2732557730789702\right)}\\ \mathbf{else}:\\ \;\;\;\;1 + e^{x \cdot \left(-x\right)} \cdot \left(t_1 \cdot \left(-0.254829592 + \left(-0.284496736 - \left(1.421413741 + t_1 \cdot \left(-1.453152027 - t_1 \cdot -1.061405429\right)\right) \cdot t_2\right) \cdot t_2\right)\right)\\ \end{array} \]

Alternative 2
Error	0.5
Cost	13380

\[\begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.002:\\ \;\;\;\;10^{-9} + \sqrt{1.2732557730789702 \cdot \left(x \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 3
Error	1.1
Cost	7112

\[\begin{array}{l} \mathbf{if}\;x \leq -18.36429154204393:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2.4523536641839257 \cdot 10^{-17}:\\ \;\;\;\;10^{-9} + \sqrt{x \cdot \left(x \cdot 1.2732557730789702\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 4
Error	1.2
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -18.36429154204393:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2.4523536641839257 \cdot 10^{-17}:\\ \;\;\;\;10^{-9} + x \cdot -1.128386358070218\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 5
Error	1.7
Cost	328

\[\begin{array}{l} \mathbf{if}\;x \leq -0.0002572997791114476:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 2.4523536641839257 \cdot 10^{-17}:\\ \;\;\;\;10^{-9}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 6
Error	29.7
Cost	64

\[10^{-9} \]

Reproduce

herbie shell --seed 2022290 
(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)))))))