Jmat.Real.erf

Percentage Accurate: 79.3% → 79.3%

Time: 16.4s

Alternatives: 10

Speedup: 1.2×

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: 79.3% 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: 79.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 0.3275911 \cdot \left|x\right|\\ t_1 := \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\\ 1 + e^{-\left|x\right| \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \mathsf{fma}\left(\frac{\frac{1.421413741 + \frac{\mathsf{fma}\left(\frac{1.061405429}{\mathsf{fma}\left(x \cdot x, -0.10731592879921, 1\right)}, \mathsf{fma}\left(\left|x\right|, -0.3275911, 1\right), -1.453152027\right)}{t\_1}}{t\_1}}{1 - \left(x \cdot x\right) \cdot 0.10731592879921}, \frac{1}{\frac{1}{1 - t\_0}}, \frac{-0.284496736}{t\_1}\right)\right) \cdot \frac{1}{-1 - t\_0}\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* 0.3275911 (fabs x))) (t_1 (fma 0.3275911 (fabs x) 1.0)))
   (+
    1.0
    (*
     (exp (- (* (fabs x) (fabs x))))
     (*
      (+
       0.254829592
       (fma
        (/
         (/
          (+
           1.421413741
           (/
            (fma
             (/ 1.061405429 (fma (* x x) -0.10731592879921 1.0))
             (fma (fabs x) -0.3275911 1.0)
             -1.453152027)
            t_1))
          t_1)
         (- 1.0 (* (* x x) 0.10731592879921)))
        (/ 1.0 (/ 1.0 (- 1.0 t_0)))
        (/ -0.284496736 t_1)))
      (/ 1.0 (- -1.0 t_0)))))))

C

double code(double x) {
	double t_0 = 0.3275911 * fabs(x);
	double t_1 = fma(0.3275911, fabs(x), 1.0);
	return 1.0 + (exp(-(fabs(x) * fabs(x))) * ((0.254829592 + fma((((1.421413741 + (fma((1.061405429 / fma((x * x), -0.10731592879921, 1.0)), fma(fabs(x), -0.3275911, 1.0), -1.453152027) / t_1)) / t_1) / (1.0 - ((x * x) * 0.10731592879921))), (1.0 / (1.0 / (1.0 - t_0))), (-0.284496736 / t_1))) * (1.0 / (-1.0 - t_0))));
}

Julia

function code(x)
	t_0 = Float64(0.3275911 * abs(x))
	t_1 = fma(0.3275911, abs(x), 1.0)
	return Float64(1.0 + Float64(exp(Float64(-Float64(abs(x) * abs(x)))) * Float64(Float64(0.254829592 + fma(Float64(Float64(Float64(1.421413741 + Float64(fma(Float64(1.061405429 / fma(Float64(x * x), -0.10731592879921, 1.0)), fma(abs(x), -0.3275911, 1.0), -1.453152027) / t_1)) / t_1) / Float64(1.0 - Float64(Float64(x * x) * 0.10731592879921))), Float64(1.0 / Float64(1.0 / Float64(1.0 - t_0))), Float64(-0.284496736 / t_1))) * Float64(1.0 / Float64(-1.0 - t_0)))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.3275911 * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(0.3275911 * N[Abs[x], $MachinePrecision] + 1.0), $MachinePrecision]}, N[(1.0 + N[(N[Exp[(-N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision])], $MachinePrecision] * N[(N[(0.254829592 + N[(N[(N[(N[(1.421413741 + N[(N[(N[(1.061405429 / N[(N[(x * x), $MachinePrecision] * -0.10731592879921 + 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[Abs[x], $MachinePrecision] * -0.3275911 + 1.0), $MachinePrecision] + -1.453152027), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision] / N[(1.0 - N[(N[(x * x), $MachinePrecision] * 0.10731592879921), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(1.0 / N[(1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.284496736 / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(-1.0 - t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 78.9%

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

4. Applied egg-rr 78.9%

   \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \color{blue}{\mathsf{fma}\left(\frac{\frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{1 - \left(x \cdot x\right) \cdot 0.10731592879921}, \frac{1}{\frac{1}{1 - 0.3275911 \cdot \left|x\right|}}, \frac{-0.284496736}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]
5. Step-by-step derivation

   1. lift-fabs.f64 N/A

      \[\leadsto 1 - \left(\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{31853699}{125000000} + \mathsf{fma}\left(\frac{\frac{\frac{1421413741}{1000000000} + \frac{\frac{-1453152027}{1000000000} + \frac{\frac{1061405429}{1000000000}}{\frac{3275911}{10000000} \cdot \color{blue}{\left|x\right|} + 1}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{1 - \left(x \cdot x\right) \cdot \frac{10731592879921}{100000000000000}}, \frac{1}{\frac{1}{1 - \frac{3275911}{10000000} \cdot \left|x\right|}}, \frac{\frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}\right)\right)\right) \cdot e^{\mathsf{neg}\left(\left|x\right| \cdot \left|x\right|\right)} \]

   2. lift-fma.f64 N/A

      \[\leadsto 1 - \left(\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{31853699}{125000000} + \mathsf{fma}\left(\frac{\frac{\frac{1421413741}{1000000000} + \frac{\frac{-1453152027}{1000000000} + \frac{\frac{1061405429}{1000000000}}{\color{blue}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{1 - \left(x \cdot x\right) \cdot \frac{10731592879921}{100000000000000}}, \frac{1}{\frac{1}{1 - \frac{3275911}{10000000} \cdot \left|x\right|}}, \frac{\frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}\right)\right)\right) \cdot e^{\mathsf{neg}\left(\left|x\right| \cdot \left|x\right|\right)} \]

   3. lift-/.f64 N/A

      \[\leadsto 1 - \left(\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{31853699}{125000000} + \mathsf{fma}\left(\frac{\frac{\frac{1421413741}{1000000000} + \frac{\frac{-1453152027}{1000000000} + \color{blue}{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{1 - \left(x \cdot x\right) \cdot \frac{10731592879921}{100000000000000}}, \frac{1}{\frac{1}{1 - \frac{3275911}{10000000} \cdot \left|x\right|}}, \frac{\frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}\right)\right)\right) \cdot e^{\mathsf{neg}\left(\left|x\right| \cdot \left|x\right|\right)} \]

   4. +-commutative N/A

      \[\leadsto 1 - \left(\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{31853699}{125000000} + \mathsf{fma}\left(\frac{\frac{\frac{1421413741}{1000000000} + \frac{\color{blue}{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)} + \frac{-1453152027}{1000000000}}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{1 - \left(x \cdot x\right) \cdot \frac{10731592879921}{100000000000000}}, \frac{1}{\frac{1}{1 - \frac{3275911}{10000000} \cdot \left|x\right|}}, \frac{\frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}\right)\right)\right) \cdot e^{\mathsf{neg}\left(\left|x\right| \cdot \left|x\right|\right)} \]

   5. lift-/.f64 N/A

      \[\leadsto 1 - \left(\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{31853699}{125000000} + \mathsf{fma}\left(\frac{\frac{\frac{1421413741}{1000000000} + \frac{\color{blue}{\frac{\frac{1061405429}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{1 - \left(x \cdot x\right) \cdot \frac{10731592879921}{100000000000000}}, \frac{1}{\frac{1}{1 - \frac{3275911}{10000000} \cdot \left|x\right|}}, \frac{\frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}\right)\right)\right) \cdot e^{\mathsf{neg}\left(\left|x\right| \cdot \left|x\right|\right)} \]

   6. lift-fma.f64 N/A

      \[\leadsto 1 - \left(\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{31853699}{125000000} + \mathsf{fma}\left(\frac{\frac{\frac{1421413741}{1000000000} + \frac{\frac{\frac{1061405429}{1000000000}}{\color{blue}{\frac{3275911}{10000000} \cdot \left|x\right| + 1}} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{1 - \left(x \cdot x\right) \cdot \frac{10731592879921}{100000000000000}}, \frac{1}{\frac{1}{1 - \frac{3275911}{10000000} \cdot \left|x\right|}}, \frac{\frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}\right)\right)\right) \cdot e^{\mathsf{neg}\left(\left|x\right| \cdot \left|x\right|\right)} \]

   7. lift-*.f64 N/A

      \[\leadsto 1 - \left(\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{31853699}{125000000} + \mathsf{fma}\left(\frac{\frac{\frac{1421413741}{1000000000} + \frac{\frac{\frac{1061405429}{1000000000}}{\color{blue}{\frac{3275911}{10000000} \cdot \left|x\right|} + 1} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{1 - \left(x \cdot x\right) \cdot \frac{10731592879921}{100000000000000}}, \frac{1}{\frac{1}{1 - \frac{3275911}{10000000} \cdot \left|x\right|}}, \frac{\frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}\right)\right)\right) \cdot e^{\mathsf{neg}\left(\left|x\right| \cdot \left|x\right|\right)} \]

   8. +-commutative N/A

      \[\leadsto 1 - \left(\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{31853699}{125000000} + \mathsf{fma}\left(\frac{\frac{\frac{1421413741}{1000000000} + \frac{\frac{\frac{1061405429}{1000000000}}{\color{blue}{1 + \frac{3275911}{10000000} \cdot \left|x\right|}} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{1 - \left(x \cdot x\right) \cdot \frac{10731592879921}{100000000000000}}, \frac{1}{\frac{1}{1 - \frac{3275911}{10000000} \cdot \left|x\right|}}, \frac{\frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}\right)\right)\right) \cdot e^{\mathsf{neg}\left(\left|x\right| \cdot \left|x\right|\right)} \]

   9. flip-+ N/A

      \[\leadsto 1 - \left(\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{31853699}{125000000} + \mathsf{fma}\left(\frac{\frac{\frac{1421413741}{1000000000} + \frac{\frac{\frac{1061405429}{1000000000}}{\color{blue}{\frac{1 \cdot 1 - \left(\frac{3275911}{10000000} \cdot \left|x\right|\right) \cdot \left(\frac{3275911}{10000000} \cdot \left|x\right|\right)}{1 - \frac{3275911}{10000000} \cdot \left|x\right|}}} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{1 - \left(x \cdot x\right) \cdot \frac{10731592879921}{100000000000000}}, \frac{1}{\frac{1}{1 - \frac{3275911}{10000000} \cdot \left|x\right|}}, \frac{\frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}\right)\right)\right) \cdot e^{\mathsf{neg}\left(\left|x\right| \cdot \left|x\right|\right)} \]

   10. lift--.f64 N/A

       \[\leadsto 1 - \left(\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{31853699}{125000000} + \mathsf{fma}\left(\frac{\frac{\frac{1421413741}{1000000000} + \frac{\frac{\frac{1061405429}{1000000000}}{\frac{1 \cdot 1 - \left(\frac{3275911}{10000000} \cdot \left|x\right|\right) \cdot \left(\frac{3275911}{10000000} \cdot \left|x\right|\right)}{\color{blue}{1 - \frac{3275911}{10000000} \cdot \left|x\right|}}} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{1 - \left(x \cdot x\right) \cdot \frac{10731592879921}{100000000000000}}, \frac{1}{\frac{1}{1 - \frac{3275911}{10000000} \cdot \left|x\right|}}, \frac{\frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}\right)\right)\right) \cdot e^{\mathsf{neg}\left(\left|x\right| \cdot \left|x\right|\right)} \]

   11. associate-/r/ N/A

       \[\leadsto 1 - \left(\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{31853699}{125000000} + \mathsf{fma}\left(\frac{\frac{\frac{1421413741}{1000000000} + \frac{\color{blue}{\frac{\frac{1061405429}{1000000000}}{1 \cdot 1 - \left(\frac{3275911}{10000000} \cdot \left|x\right|\right) \cdot \left(\frac{3275911}{10000000} \cdot \left|x\right|\right)} \cdot \left(1 - \frac{3275911}{10000000} \cdot \left|x\right|\right)} + \frac{-1453152027}{1000000000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{1 - \left(x \cdot x\right) \cdot \frac{10731592879921}{100000000000000}}, \frac{1}{\frac{1}{1 - \frac{3275911}{10000000} \cdot \left|x\right|}}, \frac{\frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}\right)\right)\right) \cdot e^{\mathsf{neg}\left(\left|x\right| \cdot \left|x\right|\right)} \]

   12. lower-fma.f64 N/A

       \[\leadsto 1 - \left(\frac{1}{1 + \frac{3275911}{10000000} \cdot \left|x\right|} \cdot \left(\frac{31853699}{125000000} + \mathsf{fma}\left(\frac{\frac{\frac{1421413741}{1000000000} + \frac{\color{blue}{\mathsf{fma}\left(\frac{\frac{1061405429}{1000000000}}{1 \cdot 1 - \left(\frac{3275911}{10000000} \cdot \left|x\right|\right) \cdot \left(\frac{3275911}{10000000} \cdot \left|x\right|\right)}, 1 - \frac{3275911}{10000000} \cdot \left|x\right|, \frac{-1453152027}{1000000000}\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}}{1 - \left(x \cdot x\right) \cdot \frac{10731592879921}{100000000000000}}, \frac{1}{\frac{1}{1 - \frac{3275911}{10000000} \cdot \left|x\right|}}, \frac{\frac{-8890523}{31250000}}{\mathsf{fma}\left(\frac{3275911}{10000000}, \left|x\right|, 1\right)}\right)\right)\right) \cdot e^{\mathsf{neg}\left(\left|x\right| \cdot \left|x\right|\right)} \]

6. Applied egg-rr 78.9%

   \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \mathsf{fma}\left(\frac{\frac{1.421413741 + \frac{\color{blue}{\mathsf{fma}\left(\frac{1.061405429}{\mathsf{fma}\left(x \cdot x, -0.10731592879921, 1\right)}, \mathsf{fma}\left(\left|x\right|, -0.3275911, 1\right), -1.453152027\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{1 - \left(x \cdot x\right) \cdot 0.10731592879921}, \frac{1}{\frac{1}{1 - 0.3275911 \cdot \left|x\right|}}, \frac{-0.284496736}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

7. Final simplification 78.9%

   \[\leadsto 1 + e^{-\left|x\right| \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \mathsf{fma}\left(\frac{\frac{1.421413741 + \frac{\mathsf{fma}\left(\frac{1.061405429}{\mathsf{fma}\left(x \cdot x, -0.10731592879921, 1\right)}, \mathsf{fma}\left(\left|x\right|, -0.3275911, 1\right), -1.453152027\right)}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{1 - \left(x \cdot x\right) \cdot 0.10731592879921}, \frac{1}{\frac{1}{1 - 0.3275911 \cdot \left|x\right|}}, \frac{-0.284496736}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)\right) \cdot \frac{1}{-1 - 0.3275911 \cdot \left|x\right|}\right) \]
8. Add Preprocessing

Alternative 2: 79.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\\ t_1 := 0.3275911 \cdot \left|x\right|\\ 1 + e^{-\left|x\right| \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \mathsf{fma}\left(\frac{\frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{t\_0}}{t\_0}}{t\_0}}{1 - \left(x \cdot x\right) \cdot 0.10731592879921}, \frac{1}{\frac{1}{1 - t\_1}}, \frac{-0.284496736}{t\_0}\right)\right) \cdot \frac{1}{-1 - t\_1}\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma 0.3275911 (fabs x) 1.0)) (t_1 (* 0.3275911 (fabs x))))
   (+
    1.0
    (*
     (exp (- (* (fabs x) (fabs x))))
     (*
      (+
       0.254829592
       (fma
        (/
         (/ (+ 1.421413741 (/ (+ -1.453152027 (/ 1.061405429 t_0)) t_0)) t_0)
         (- 1.0 (* (* x x) 0.10731592879921)))
        (/ 1.0 (/ 1.0 (- 1.0 t_1)))
        (/ -0.284496736 t_0)))
      (/ 1.0 (- -1.0 t_1)))))))

C

double code(double x) {
	double t_0 = fma(0.3275911, fabs(x), 1.0);
	double t_1 = 0.3275911 * fabs(x);
	return 1.0 + (exp(-(fabs(x) * fabs(x))) * ((0.254829592 + fma((((1.421413741 + ((-1.453152027 + (1.061405429 / t_0)) / t_0)) / t_0) / (1.0 - ((x * x) * 0.10731592879921))), (1.0 / (1.0 / (1.0 - t_1))), (-0.284496736 / t_0))) * (1.0 / (-1.0 - t_1))));
}

Julia

function code(x)
	t_0 = fma(0.3275911, abs(x), 1.0)
	t_1 = Float64(0.3275911 * abs(x))
	return Float64(1.0 + Float64(exp(Float64(-Float64(abs(x) * abs(x)))) * Float64(Float64(0.254829592 + fma(Float64(Float64(Float64(1.421413741 + Float64(Float64(-1.453152027 + Float64(1.061405429 / t_0)) / t_0)) / t_0) / Float64(1.0 - Float64(Float64(x * x) * 0.10731592879921))), Float64(1.0 / Float64(1.0 / Float64(1.0 - t_1))), Float64(-0.284496736 / t_0))) * Float64(1.0 / Float64(-1.0 - t_1)))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.3275911 * N[Abs[x], $MachinePrecision] + 1.0), $MachinePrecision]}, Block[{t$95$1 = N[(0.3275911 * N[Abs[x], $MachinePrecision]), $MachinePrecision]}, N[(1.0 + N[(N[Exp[(-N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision])], $MachinePrecision] * N[(N[(0.254829592 + N[(N[(N[(N[(1.421413741 + N[(N[(-1.453152027 + N[(1.061405429 / t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] / N[(1.0 - N[(N[(x * x), $MachinePrecision] * 0.10731592879921), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(1.0 / N[(1.0 - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(-0.284496736 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(-1.0 - t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 78.9%

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

4. Applied egg-rr 78.9%

   \[\leadsto 1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \color{blue}{\mathsf{fma}\left(\frac{\frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{1 - \left(x \cdot x\right) \cdot 0.10731592879921}, \frac{1}{\frac{1}{1 - 0.3275911 \cdot \left|x\right|}}, \frac{-0.284496736}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

5. Final simplification 78.9%

   \[\leadsto 1 + e^{-\left|x\right| \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \mathsf{fma}\left(\frac{\frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{1 - \left(x \cdot x\right) \cdot 0.10731592879921}, \frac{1}{\frac{1}{1 - 0.3275911 \cdot \left|x\right|}}, \frac{-0.284496736}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)\right) \cdot \frac{1}{-1 - 0.3275911 \cdot \left|x\right|}\right) \]
6. Add Preprocessing

Alternative 3: 79.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\\ 1 - \frac{\mathsf{fma}\left(\frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{t\_0}}{t\_0}}{t\_0}}{\mathsf{fma}\left(x \cdot x, -0.10731592879921, 1\right)}, \mathsf{fma}\left(\left|x\right|, -0.3275911, 1\right), 0.254829592\right)}{t\_0 \cdot e^{x \cdot x}} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma 0.3275911 (fabs x) 1.0)))
   (-
    1.0
    (/
     (fma
      (/
       (+
        -0.284496736
        (/ (+ 1.421413741 (/ (+ -1.453152027 (/ 1.061405429 t_0)) t_0)) t_0))
       (fma (* x x) -0.10731592879921 1.0))
      (fma (fabs x) -0.3275911 1.0)
      0.254829592)
     (* t_0 (exp (* x x)))))))

C

double code(double x) {
	double t_0 = fma(0.3275911, fabs(x), 1.0);
	return 1.0 - (fma(((-0.284496736 + ((1.421413741 + ((-1.453152027 + (1.061405429 / t_0)) / t_0)) / t_0)) / fma((x * x), -0.10731592879921, 1.0)), fma(fabs(x), -0.3275911, 1.0), 0.254829592) / (t_0 * exp((x * x))));
}

Julia

function code(x)
	t_0 = fma(0.3275911, abs(x), 1.0)
	return Float64(1.0 - Float64(fma(Float64(Float64(-0.284496736 + Float64(Float64(1.421413741 + Float64(Float64(-1.453152027 + Float64(1.061405429 / t_0)) / t_0)) / t_0)) / fma(Float64(x * x), -0.10731592879921, 1.0)), fma(abs(x), -0.3275911, 1.0), 0.254829592) / Float64(t_0 * exp(Float64(x * x)))))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.3275911 * N[Abs[x], $MachinePrecision] + 1.0), $MachinePrecision]}, N[(1.0 - N[(N[(N[(N[(-0.284496736 + N[(N[(1.421413741 + N[(N[(-1.453152027 + N[(1.061405429 / t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] * -0.10731592879921 + 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[Abs[x], $MachinePrecision] * -0.3275911 + 1.0), $MachinePrecision] + 0.254829592), $MachinePrecision] / N[(t$95$0 * N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 78.9%

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

4. Applied egg-rr 78.9%

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

6. Applied egg-rr 78.9%

   \[\leadsto 1 - \frac{\color{blue}{\mathsf{fma}\left(\frac{\frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + -0.284496736}{\mathsf{fma}\left(x \cdot x, -0.10731592879921, 1\right)}, \mathsf{fma}\left(\left|x\right|, -0.3275911, 1\right), 0.254829592\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) \cdot e^{x \cdot x}} \]

7. Final simplification 78.9%

   \[\leadsto 1 - \frac{\mathsf{fma}\left(\frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(x \cdot x, -0.10731592879921, 1\right)}, \mathsf{fma}\left(\left|x\right|, -0.3275911, 1\right), 0.254829592\right)}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) \cdot e^{x \cdot x}} \]
8. Add Preprocessing

Alternative 4: 79.3% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 78.9%

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

4. Applied egg-rr 78.9%

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

5. Final simplification 78.9%

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

Alternative 5: 79.3% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 78.9%

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

4. Applied egg-rr 78.9%

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

Alternative 6: 78.9% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 78.9%

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

4. Applied egg-rr 78.9%

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

   1. lift-fabs.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. +-commutative N/A

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

   5. lift-/.f64 N/A

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

   6. lift-fma.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. +-commutative N/A

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

   9. flip-+ N/A

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

   10. lift--.f64 N/A

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

   11. associate-/r/ N/A

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

   12. lower-fma.f64 N/A

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

6. Applied egg-rr 78.9%

   \[\leadsto 1 - \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{\color{blue}{\mathsf{fma}\left(\frac{1.061405429}{\mathsf{fma}\left(x \cdot x, -0.10731592879921, 1\right)}, \mathsf{fma}\left(\left|x\right|, -0.3275911, 1\right), -1.453152027\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) \cdot e^{x \cdot x}} \]

7. Taylor expanded in x around 0

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

   1. sub-neg N/A

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

   2. lower-fma.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-fabs.f64 N/A

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

   7. metadata-eval 78.3

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

9. Simplified 78.3%

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

Alternative 7: 77.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 78.9%

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

4. Applied egg-rr 78.9%

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

   1. sqr-abs N/A

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

   2. lift-*.f64 78.9

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

6. Applied egg-rr 78.9%

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

7. Taylor expanded in x around 0

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

9. Simplified 76.8%

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

10. Final simplification 76.8%

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

Alternative 8: 77.7% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 78.9%

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

4. Applied egg-rr 78.9%

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

6. Applied egg-rr 77.7%

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

7. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. lower-fabs.f64 75.5

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

9. Simplified 75.5%

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

11. Applied egg-rr 76.7%

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

12. Final simplification 76.7%

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

Alternative 9: 77.7% accurate, 2.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 78.9%

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

4. Applied egg-rr 78.9%

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

6. Applied egg-rr 77.7%

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

7. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. lower-fabs.f64 75.5

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

9. Simplified 75.5%

   \[\leadsto 1 - \frac{1}{\frac{\color{blue}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{0.254829592 + \frac{\frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} + -0.284496736}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}} \]
10. Step-by-step derivation

    1. lift-fabs.f64 N/A

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

    2. lift-fma.f64 N/A

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

11. Applied egg-rr 76.7%

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

12. Final simplification 76.7%

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

Alternative 10: 56.0% accurate, 262.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x) :precision binary64 1.0)

C

double code(double x) {
	return 1.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0
end function

Java

public static double code(double x) {
	return 1.0;
}

Python

def code(x):
	return 1.0

Julia

function code(x)
	return 1.0
end

MATLAB

function tmp = code(x)
	tmp = 1.0;
end

Wolfram

code[x_] := 1.0

Derivation

1. Initial program 78.9%

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

4. Applied egg-rr 51.6%

   \[\leadsto 1 - \color{blue}{\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.035155743162854886, \left|x\right| \cdot \left(x \cdot x\right), 1\right)} \cdot \frac{e^{-x \cdot x}}{\frac{1}{\mathsf{fma}\left(0.3275911 \cdot \left|x\right|, \mathsf{fma}\left(0.3275911, \left|x\right|, -1\right), 1\right)}}} \]

5. Taylor expanded in x around inf

   \[\leadsto \color{blue}{1} \]
6. Step-by-step derivation

7. Simplified 54.9%

   \[\leadsto \color{blue}{1} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024214 
(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)))))))