Jmat.Real.erf

Percentage Accurate: 79.4% → 79.4%

Time: 13.1s

Alternatives: 3

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{1 + 0.3275911 \cdot \left|x\right|}\\ 1 - \left(t\_0 \cdot \left(0.254829592 + t\_0 \cdot \left(-0.284496736 + t\_0 \cdot \left(1.421413741 + t\_0 \cdot \left(-1.453152027 + t\_0 \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 79.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{1 + 0.3275911 \cdot \left|x\right|}\\ 1 - \left(t\_0 \cdot \left(0.254829592 + t\_0 \cdot \left(-0.284496736 + t\_0 \cdot \left(1.421413741 + t\_0 \cdot \left(-1.453152027 + t\_0 \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \end{array} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 79.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.3275911 \cdot \left|x\right|\\ t_1 := \frac{1}{1 + t\_0}\\ 1 + e^{\left|x\right| \cdot \left(-\left|x\right|\right)} \cdot \left(t\_1 \cdot \left(t\_1 \cdot \left(t\_1 \cdot \left(\left(\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -1.453152027\right) \cdot \frac{1}{-1 - t\_0} - 1.421413741\right) - -0.284496736\right) - 0.254829592\right)\right) \end{array} \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.1%

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

3. Taylor expanded in x around 0

   \[\leadsto 1 - \left(\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{31853699}{125000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{-8890523}{31250000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{1421413741}{1000000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \color{blue}{\left(\frac{1061405429}{1000000000} \cdot \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} - \frac{1453152027}{1000000000}\right)}\right)\right)\right)\right) \cdot e^{\mathsf{neg}\left(\left|x\right| \cdot \left|x\right|\right)} \]
4. Step-by-step derivation

   1. sub-neg N/A

      \[\leadsto 1 - \left(\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{31853699}{125000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{-8890523}{31250000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{1421413741}{1000000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \color{blue}{\left(\frac{1061405429}{1000000000} \cdot \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} + \left(\mathsf{neg}\left(\frac{1453152027}{1000000000}\right)\right)\right)}\right)\right)\right)\right) \cdot e^{\mathsf{neg}\left(\left|x\right| \cdot \left|x\right|\right)} \]

   2. lower-+.f64 N/A

      \[\leadsto 1 - \left(\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{31853699}{125000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{-8890523}{31250000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{1421413741}{1000000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \color{blue}{\left(\frac{1061405429}{1000000000} \cdot \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} + \left(\mathsf{neg}\left(\frac{1453152027}{1000000000}\right)\right)\right)}\right)\right)\right)\right) \cdot e^{\mathsf{neg}\left(\left|x\right| \cdot \left|x\right|\right)} \]

   3. associate-*r/ N/A

      \[\leadsto 1 - \left(\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{31853699}{125000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{-8890523}{31250000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{1421413741}{1000000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\color{blue}{\frac{\frac{1061405429}{1000000000} \cdot 1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}} + \left(\mathsf{neg}\left(\frac{1453152027}{1000000000}\right)\right)\right)\right)\right)\right)\right) \cdot e^{\mathsf{neg}\left(\left|x\right| \cdot \left|x\right|\right)} \]

   4. metadata-eval N/A

      \[\leadsto 1 - \left(\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{31853699}{125000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{-8890523}{31250000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{1421413741}{1000000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{\color{blue}{\frac{1061405429}{1000000000}}}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} + \left(\mathsf{neg}\left(\frac{1453152027}{1000000000}\right)\right)\right)\right)\right)\right)\right) \cdot e^{\mathsf{neg}\left(\left|x\right| \cdot \left|x\right|\right)} \]

   5. lower-/.f64 N/A

      \[\leadsto 1 - \left(\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{31853699}{125000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{-8890523}{31250000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{1421413741}{1000000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\color{blue}{\frac{\frac{1061405429}{1000000000}}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}} + \left(\mathsf{neg}\left(\frac{1453152027}{1000000000}\right)\right)\right)\right)\right)\right)\right) \cdot e^{\mathsf{neg}\left(\left|x\right| \cdot \left|x\right|\right)} \]

   6. +-commutative N/A

      \[\leadsto 1 - \left(\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{31853699}{125000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{-8890523}{31250000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{1421413741}{1000000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{\frac{1061405429}{1000000000}}{\color{blue}{\frac{3275911}{10000000} \cdot \left|x\right| + 1}} + \left(\mathsf{neg}\left(\frac{1453152027}{1000000000}\right)\right)\right)\right)\right)\right)\right) \cdot e^{\mathsf{neg}\left(\left|x\right| \cdot \left|x\right|\right)} \]

   7. *-commutative N/A

      \[\leadsto 1 - \left(\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{31853699}{125000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{-8890523}{31250000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{1421413741}{1000000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{\frac{1061405429}{1000000000}}{\color{blue}{\left|x\right| \cdot \frac{3275911}{10000000}} + 1} + \left(\mathsf{neg}\left(\frac{1453152027}{1000000000}\right)\right)\right)\right)\right)\right)\right) \cdot e^{\mathsf{neg}\left(\left|x\right| \cdot \left|x\right|\right)} \]

   8. lower-fma.f64 N/A

      \[\leadsto 1 - \left(\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{31853699}{125000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{-8890523}{31250000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{1421413741}{1000000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{\frac{1061405429}{1000000000}}{\color{blue}{\mathsf{fma}\left(\left|x\right|, \frac{3275911}{10000000}, 1\right)}} + \left(\mathsf{neg}\left(\frac{1453152027}{1000000000}\right)\right)\right)\right)\right)\right)\right) \cdot e^{\mathsf{neg}\left(\left|x\right| \cdot \left|x\right|\right)} \]

   9. lower-fabs.f64 N/A

      \[\leadsto 1 - \left(\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{31853699}{125000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{-8890523}{31250000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{1421413741}{1000000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\color{blue}{\left|x\right|}, \frac{3275911}{10000000}, 1\right)} + \left(\mathsf{neg}\left(\frac{1453152027}{1000000000}\right)\right)\right)\right)\right)\right)\right) \cdot e^{\mathsf{neg}\left(\left|x\right| \cdot \left|x\right|\right)} \]

   10. metadata-eval 79.1

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

5. Simplified 79.1%

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

6. Final simplification 79.1%

   \[\leadsto 1 + e^{\left|x\right| \cdot \left(-\left|x\right|\right)} \cdot \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(\frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -1.453152027\right) \cdot \frac{1}{-1 - 0.3275911 \cdot \left|x\right|} - 1.421413741\right) - -0.284496736\right) - 0.254829592\right)\right) \]
7. Add Preprocessing

Alternative 2: 79.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\\ t_1 := 0.3275911 \cdot \left|x\right|\\ t_2 := \frac{1}{1 + t\_1}\\ 1 + e^{\left|x\right| \cdot \left(-\left|x\right|\right)} \cdot \left(t\_2 \cdot \left(t\_2 \cdot \left(\mathsf{fma}\left(\frac{1}{t\_0}, \frac{1.061405429}{t\_0} + -1.453152027, 1.421413741\right) \cdot \frac{1}{-1 - t\_1} - -0.284496736\right) - 0.254829592\right)\right) \end{array} \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 79.1%

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

3. Taylor expanded in x around 0

   \[\leadsto 1 - \left(\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{31853699}{125000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{-8890523}{31250000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \color{blue}{\left(\left(\frac{1421413741}{1000000000} + \frac{1061405429}{1000000000} \cdot \frac{1}{{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right)}^{2}}\right) - \frac{1453152027}{1000000000} \cdot \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}\right)}\right)\right)\right) \cdot e^{\mathsf{neg}\left(\left|x\right| \cdot \left|x\right|\right)} \]
4. Step-by-step derivation

   1. associate--l+ N/A

      \[\leadsto 1 - \left(\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{31853699}{125000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{-8890523}{31250000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \color{blue}{\left(\frac{1421413741}{1000000000} + \left(\frac{1061405429}{1000000000} \cdot \frac{1}{{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right)}^{2}} - \frac{1453152027}{1000000000} \cdot \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}\right)\right)}\right)\right)\right) \cdot e^{\mathsf{neg}\left(\left|x\right| \cdot \left|x\right|\right)} \]

   2. +-commutative N/A

      \[\leadsto 1 - \left(\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{31853699}{125000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{-8890523}{31250000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \color{blue}{\left(\left(\frac{1061405429}{1000000000} \cdot \frac{1}{{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right)}^{2}} - \frac{1453152027}{1000000000} \cdot \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}\right) + \frac{1421413741}{1000000000}\right)}\right)\right)\right) \cdot e^{\mathsf{neg}\left(\left|x\right| \cdot \left|x\right|\right)} \]

   3. sub-neg N/A

      \[\leadsto 1 - \left(\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{31853699}{125000000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{-8890523}{31250000} + \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\color{blue}{\left(\frac{1061405429}{1000000000} \cdot \frac{1}{{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right)}^{2}} + \left(\mathsf{neg}\left(\frac{1453152027}{1000000000} \cdot \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}\right)\right)\right)} + \frac{1421413741}{1000000000}\right)\right)\right)\right) \cdot e^{\mathsf{neg}\left(\left|x\right| \cdot \left|x\right|\right)} \]

5. Simplified 79.0%

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

6. Final simplification 79.0%

   \[\leadsto 1 + e^{\left|x\right| \cdot \left(-\left|x\right|\right)} \cdot \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\mathsf{fma}\left(\frac{1}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)}, \frac{1.061405429}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)} + -1.453152027, 1.421413741\right) \cdot \frac{1}{-1 - 0.3275911 \cdot \left|x\right|} - -0.284496736\right) - 0.254829592\right)\right) \]
7. Add Preprocessing

Alternative 3: 77.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\\ t_1 := \mathsf{fma}\left(\left|x\right|, -0.3275911, -1\right)\\ t_2 := t\_0 \cdot t\_1\\ t_3 := t\_0 \cdot t\_2\\ \left(\frac{1.421413741}{t\_3} + \left(1 + \frac{0.254829592}{t\_1}\right)\right) + \left(\frac{-0.284496736}{t\_2} - \left(\frac{1.061405429}{\left(t\_0 \cdot \left(t\_0 \cdot t\_0\right)\right) \cdot \left(t\_1 \cdot t\_1\right)} + \frac{1.453152027}{t\_0 \cdot t\_3}\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma (fabs x) 0.3275911 1.0))
        (t_1 (fma (fabs x) -0.3275911 -1.0))
        (t_2 (* t_0 t_1))
        (t_3 (* t_0 t_2)))
   (+
    (+ (/ 1.421413741 t_3) (+ 1.0 (/ 0.254829592 t_1)))
    (-
     (/ -0.284496736 t_2)
     (+
      (/ 1.061405429 (* (* t_0 (* t_0 t_0)) (* t_1 t_1)))
      (/ 1.453152027 (* t_0 t_3)))))))

C

double code(double x) {
	double t_0 = fma(fabs(x), 0.3275911, 1.0);
	double t_1 = fma(fabs(x), -0.3275911, -1.0);
	double t_2 = t_0 * t_1;
	double t_3 = t_0 * t_2;
	return ((1.421413741 / t_3) + (1.0 + (0.254829592 / t_1))) + ((-0.284496736 / t_2) - ((1.061405429 / ((t_0 * (t_0 * t_0)) * (t_1 * t_1))) + (1.453152027 / (t_0 * t_3))));
}

Julia

function code(x)
	t_0 = fma(abs(x), 0.3275911, 1.0)
	t_1 = fma(abs(x), -0.3275911, -1.0)
	t_2 = Float64(t_0 * t_1)
	t_3 = Float64(t_0 * t_2)
	return Float64(Float64(Float64(1.421413741 / t_3) + Float64(1.0 + Float64(0.254829592 / t_1))) + Float64(Float64(-0.284496736 / t_2) - Float64(Float64(1.061405429 / Float64(Float64(t_0 * Float64(t_0 * t_0)) * Float64(t_1 * t_1))) + Float64(1.453152027 / Float64(t_0 * t_3)))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[Abs[x], $MachinePrecision] * 0.3275911 + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(N[Abs[x], $MachinePrecision] * -0.3275911 + -1.0), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 * t$95$1), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$0 * t$95$2), $MachinePrecision]}, N[(N[(N[(1.421413741 / t$95$3), $MachinePrecision] + N[(1.0 + N[(0.254829592 / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.284496736 / t$95$2), $MachinePrecision] - N[(N[(1.061405429 / N[(N[(t$95$0 * N[(t$95$0 * t$95$0), $MachinePrecision]), $MachinePrecision] * N[(t$95$1 * t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.453152027 / N[(t$95$0 * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Initial program 79.1%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1 - \frac{e^{\mathsf{neg}\left({\left(\left|x\right|\right)}^{2}\right)} \cdot \left(\left(\frac{31853699}{125000000} + \left(\frac{1061405429}{1000000000} \cdot \frac{1}{{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right)}^{4}} + \frac{1421413741}{1000000000} \cdot \frac{1}{{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right)}^{2}}\right)\right) - \left(\frac{8890523}{31250000} \cdot \frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} + \frac{1453152027}{1000000000} \cdot \frac{1}{{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right)}^{3}}\right)\right)}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}} \]

5. Simplified 79.0%

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

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\left(1 + \left(\frac{31853699}{125000000} \cdot \frac{1}{\frac{-3275911}{10000000} \cdot \left|x\right| - 1} + \frac{1421413741}{1000000000} \cdot \frac{1}{{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right)}^{2} \cdot \left(\frac{-3275911}{10000000} \cdot \left|x\right| - 1\right)}\right)\right) - \left(\frac{8890523}{31250000} \cdot \frac{1}{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right) \cdot \left(\frac{-3275911}{10000000} \cdot \left|x\right| - 1\right)} + \left(\frac{1061405429}{1000000000} \cdot \frac{1}{{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right)}^{3} \cdot {\left(\frac{-3275911}{10000000} \cdot \left|x\right| - 1\right)}^{2}} + \frac{1453152027}{1000000000} \cdot \frac{1}{{\left(1 + \frac{3275911}{10000000} \cdot \left|x\right|\right)}^{3} \cdot \left(\frac{-3275911}{10000000} \cdot \left|x\right| - 1\right)}\right)\right)} \]

8. Simplified 78.0%

   \[\leadsto \color{blue}{\left(\frac{1.421413741}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right) \cdot \left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right) \cdot \mathsf{fma}\left(\left|x\right|, -0.3275911, -1\right)\right)} + \left(1 + \frac{0.254829592}{\mathsf{fma}\left(\left|x\right|, -0.3275911, -1\right)}\right)\right) + \left(\frac{-0.284496736}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right) \cdot \mathsf{fma}\left(\left|x\right|, -0.3275911, -1\right)} - \left(\frac{1.061405429}{\left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right) \cdot \left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right) \cdot \mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right)\right)\right) \cdot \left(\mathsf{fma}\left(\left|x\right|, -0.3275911, -1\right) \cdot \mathsf{fma}\left(\left|x\right|, -0.3275911, -1\right)\right)} + \frac{1.453152027}{\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right) \cdot \left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right) \cdot \left(\mathsf{fma}\left(\left|x\right|, 0.3275911, 1\right) \cdot \mathsf{fma}\left(\left|x\right|, -0.3275911, -1\right)\right)\right)}\right)\right)} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024215 
(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)))))))