Logistic function from Lakshay Garg

Percentage Accurate: 54.2% → 100.0%

Time: 9.0s

Alternatives: 15

Speedup: 5.1×

Specification

Math

\[\begin{array}{l} \\ \frac{2}{1 + e^{-2 \cdot x}} - 1 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))

C

double code(double x) {
	return (2.0 / (1.0 + exp((-2.0 * x)))) - 1.0;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    code = (2.0d0 / (1.0d0 + exp(((-2.0d0) * x)))) - 1.0d0
end function

Java

public static double code(double x) {
	return (2.0 / (1.0 + Math.exp((-2.0 * x)))) - 1.0;
}

Python

def code(x):
	return (2.0 / (1.0 + math.exp((-2.0 * x)))) - 1.0

Julia

function code(x)
	return Float64(Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x)))) - 1.0)
end

MATLAB

function tmp = code(x)
	tmp = (2.0 / (1.0 + exp((-2.0 * x)))) - 1.0;
end

Wolfram

code[x_] := N[(N[(2.0 / N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

Initial Program: 54.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{2}{1 + e^{-2 \cdot x}} - 1 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))

C

double code(double x) {
	return (2.0 / (1.0 + exp((-2.0 * x)))) - 1.0;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    code = (2.0d0 / (1.0d0 + exp(((-2.0d0) * x)))) - 1.0d0
end function

Java

public static double code(double x) {
	return (2.0 / (1.0 + Math.exp((-2.0 * x)))) - 1.0;
}

Python

def code(x):
	return (2.0 / (1.0 + math.exp((-2.0 * x)))) - 1.0

Julia

function code(x)
	return Float64(Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x)))) - 1.0)
end

MATLAB

function tmp = code(x)
	tmp = (2.0 / (1.0 + exp((-2.0 * x)))) - 1.0;
end

Wolfram

code[x_] := N[(N[(2.0 / N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]

Alternative 1: 100.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.0255:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} - 1\\ \mathbf{elif}\;x \leq 0.01:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.05396825396825397, x \cdot x, 0.13333333333333333\right), x \cdot x, -0.3333333333333333\right) \cdot \left(x \cdot x\right), x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{2}{{\left(e^{-2}\right)}^{\left(x \cdot 3\right)} - -1}, \mathsf{fma}\left({\left(e^{x}\right)}^{-2}, \mathsf{expm1}\left(x \cdot -2\right), 1\right), -1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -0.0255)
   (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0)
   (if (<= x 0.01)
     (fma
      (*
       (fma
        (fma -0.05396825396825397 (* x x) 0.13333333333333333)
        (* x x)
        -0.3333333333333333)
       (* x x))
      x
      x)
     (fma
      (/ 2.0 (- (pow (exp -2.0) (* x 3.0)) -1.0))
      (fma (pow (exp x) -2.0) (expm1 (* x -2.0)) 1.0)
      -1.0))))

C

double code(double x) {
	double tmp;
	if (x <= -0.0255) {
		tmp = (2.0 / (1.0 + exp((-2.0 * x)))) - 1.0;
	} else if (x <= 0.01) {
		tmp = fma((fma(fma(-0.05396825396825397, (x * x), 0.13333333333333333), (x * x), -0.3333333333333333) * (x * x)), x, x);
	} else {
		tmp = fma((2.0 / (pow(exp(-2.0), (x * 3.0)) - -1.0)), fma(pow(exp(x), -2.0), expm1((x * -2.0)), 1.0), -1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -0.0255)
		tmp = Float64(Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x)))) - 1.0);
	elseif (x <= 0.01)
		tmp = fma(Float64(fma(fma(-0.05396825396825397, Float64(x * x), 0.13333333333333333), Float64(x * x), -0.3333333333333333) * Float64(x * x)), x, x);
	else
		tmp = fma(Float64(2.0 / Float64((exp(-2.0) ^ Float64(x * 3.0)) - -1.0)), fma((exp(x) ^ -2.0), expm1(Float64(x * -2.0)), 1.0), -1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -0.0255], N[(N[(2.0 / N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], If[LessEqual[x, 0.01], N[(N[(N[(N[(-0.05396825396825397 * N[(x * x), $MachinePrecision] + 0.13333333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * x + x), $MachinePrecision], N[(N[(2.0 / N[(N[Power[N[Exp[-2.0], $MachinePrecision], N[(x * 3.0), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Exp[x], $MachinePrecision], -2.0], $MachinePrecision] * N[(Exp[N[(x * -2.0), $MachinePrecision]] - 1), $MachinePrecision] + 1.0), $MachinePrecision] + -1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.0254999999999999984

   1. Initial program 100.0%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   2. Add Preprocessing

   if -0.0254999999999999984 < x < 0.0100000000000000002

   1. Initial program 8.2%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) - \frac{1}{3}\right)\right)} \]
   4. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto \color{blue}{x \cdot 1 + x \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) - \frac{1}{3}\right)\right)} \]

      2. *-rgt-identity N/A

         \[\leadsto \color{blue}{x} + x \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) - \frac{1}{3}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{x \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) - \frac{1}{3}\right)\right) + x} \]

      4. associate-*r* N/A

         \[\leadsto \color{blue}{\left(x \cdot {x}^{2}\right) \cdot \left({x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) - \frac{1}{3}\right)} + x \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot {x}^{2}, {x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) - \frac{1}{3}, x\right)} \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\mathsf{fma}\left(-0.05396825396825397, x \cdot x, 0.13333333333333333\right), x \cdot x, -0.3333333333333333\right), x\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 100.0%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.05396825396825397, x \cdot x, 0.13333333333333333\right), x \cdot x, -0.3333333333333333\right) \cdot \left(x \cdot x\right), \color{blue}{x}, x\right) \]

   if 0.0100000000000000002 < x

   1. Initial program 99.9%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{1 + e^{-2 \cdot x}} - 1} \]

      2. metadata-eval N/A

         \[\leadsto \frac{2}{1 + e^{-2 \cdot x}} - \color{blue}{1 \cdot 1} \]

      3. fp-cancel-sub-sign-inv N/A

         \[\leadsto \color{blue}{\frac{2}{1 + e^{-2 \cdot x}} + \left(\mathsf{neg}\left(1\right)\right) \cdot 1} \]

      4. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2}{1 + e^{-2 \cdot x}}} + \left(\mathsf{neg}\left(1\right)\right) \cdot 1 \]

      5. lift-+.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{1 + e^{-2 \cdot x}}} + \left(\mathsf{neg}\left(1\right)\right) \cdot 1 \]

      6. flip3-+ N/A

         \[\leadsto \frac{2}{\color{blue}{\frac{{1}^{3} + {\left(e^{-2 \cdot x}\right)}^{3}}{1 \cdot 1 + \left(e^{-2 \cdot x} \cdot e^{-2 \cdot x} - 1 \cdot e^{-2 \cdot x}\right)}}} + \left(\mathsf{neg}\left(1\right)\right) \cdot 1 \]

      7. associate-/r/ N/A

         \[\leadsto \color{blue}{\frac{2}{{1}^{3} + {\left(e^{-2 \cdot x}\right)}^{3}} \cdot \left(1 \cdot 1 + \left(e^{-2 \cdot x} \cdot e^{-2 \cdot x} - 1 \cdot e^{-2 \cdot x}\right)\right)} + \left(\mathsf{neg}\left(1\right)\right) \cdot 1 \]

      8. metadata-eval N/A

         \[\leadsto \frac{2}{{1}^{3} + {\left(e^{-2 \cdot x}\right)}^{3}} \cdot \left(1 \cdot 1 + \left(e^{-2 \cdot x} \cdot e^{-2 \cdot x} - 1 \cdot e^{-2 \cdot x}\right)\right) + \color{blue}{-1} \cdot 1 \]

      9. metadata-eval N/A

         \[\leadsto \frac{2}{{1}^{3} + {\left(e^{-2 \cdot x}\right)}^{3}} \cdot \left(1 \cdot 1 + \left(e^{-2 \cdot x} \cdot e^{-2 \cdot x} - 1 \cdot e^{-2 \cdot x}\right)\right) + \color{blue}{-1} \]

      10. metadata-eval N/A

          \[\leadsto \frac{2}{{1}^{3} + {\left(e^{-2 \cdot x}\right)}^{3}} \cdot \left(1 \cdot 1 + \left(e^{-2 \cdot x} \cdot e^{-2 \cdot x} - 1 \cdot e^{-2 \cdot x}\right)\right) + \color{blue}{\left(\mathsf{neg}\left(1\right)\right)} \]

      11. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{2}{{1}^{3} + {\left(e^{-2 \cdot x}\right)}^{3}}, 1 \cdot 1 + \left(e^{-2 \cdot x} \cdot e^{-2 \cdot x} - 1 \cdot e^{-2 \cdot x}\right), \mathsf{neg}\left(1\right)\right)} \]

   4. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{2}{{\left(e^{-2}\right)}^{\left(x \cdot 3\right)} + 1}, \mathsf{fma}\left({\left(e^{x}\right)}^{-2}, \mathsf{expm1}\left(x \cdot -2\right), 1\right), -1\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0255:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} - 1\\ \mathbf{elif}\;x \leq 0.01:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.05396825396825397, x \cdot x, 0.13333333333333333\right), x \cdot x, -0.3333333333333333\right) \cdot \left(x \cdot x\right), x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{2}{{\left(e^{-2}\right)}^{\left(x \cdot 3\right)} - -1}, \mathsf{fma}\left({\left(e^{x}\right)}^{-2}, \mathsf{expm1}\left(x \cdot -2\right), 1\right), -1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 100.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.0255 \lor \neg \left(x \leq 0.023\right):\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.05396825396825397, x \cdot x, 0.13333333333333333\right), x \cdot x, -0.3333333333333333\right) \cdot \left(x \cdot x\right), x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -0.0255) (not (<= x 0.023)))
   (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0)
   (fma
    (*
     (fma
      (fma -0.05396825396825397 (* x x) 0.13333333333333333)
      (* x x)
      -0.3333333333333333)
     (* x x))
    x
    x)))

C

double code(double x) {
	double tmp;
	if ((x <= -0.0255) || !(x <= 0.023)) {
		tmp = (2.0 / (1.0 + exp((-2.0 * x)))) - 1.0;
	} else {
		tmp = fma((fma(fma(-0.05396825396825397, (x * x), 0.13333333333333333), (x * x), -0.3333333333333333) * (x * x)), x, x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if ((x <= -0.0255) || !(x <= 0.023))
		tmp = Float64(Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x)))) - 1.0);
	else
		tmp = fma(Float64(fma(fma(-0.05396825396825397, Float64(x * x), 0.13333333333333333), Float64(x * x), -0.3333333333333333) * Float64(x * x)), x, x);
	end
	return tmp
end

Wolfram

code[x_] := If[Or[LessEqual[x, -0.0255], N[Not[LessEqual[x, 0.023]], $MachinePrecision]], N[(N[(2.0 / N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(N[(-0.05396825396825397 * N[(x * x), $MachinePrecision] + 0.13333333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * x + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.0254999999999999984 or 0.023 < x

   1. Initial program 99.9%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   2. Add Preprocessing

   if -0.0254999999999999984 < x < 0.023

   1. Initial program 8.2%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) - \frac{1}{3}\right)\right)} \]
   4. Step-by-step derivation

      1. distribute-lft-in N/A

         \[\leadsto \color{blue}{x \cdot 1 + x \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) - \frac{1}{3}\right)\right)} \]

      2. *-rgt-identity N/A

         \[\leadsto \color{blue}{x} + x \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) - \frac{1}{3}\right)\right) \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{x \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) - \frac{1}{3}\right)\right) + x} \]

      4. associate-*r* N/A

         \[\leadsto \color{blue}{\left(x \cdot {x}^{2}\right) \cdot \left({x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) - \frac{1}{3}\right)} + x \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot {x}^{2}, {x}^{2} \cdot \left(\frac{2}{15} + \frac{-17}{315} \cdot {x}^{2}\right) - \frac{1}{3}, x\right)} \]

   5. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\mathsf{fma}\left(-0.05396825396825397, x \cdot x, 0.13333333333333333\right), x \cdot x, -0.3333333333333333\right), x\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 100.0%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.05396825396825397, x \cdot x, 0.13333333333333333\right), x \cdot x, -0.3333333333333333\right) \cdot \left(x \cdot x\right), \color{blue}{x}, x\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0255 \lor \neg \left(x \leq 0.023\right):\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.05396825396825397, x \cdot x, 0.13333333333333333\right), x \cdot x, -0.3333333333333333\right) \cdot \left(x \cdot x\right), x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.0071:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.08888888888888889, x \cdot x, 0.6666666666666666\right), x \cdot x, 2\right), x \cdot x, 2\right) + \mathsf{fma}\left(\left(\left(-0.025396825396825397 \cdot \left(x \cdot x\right) - 0.26666666666666666\right) \cdot x\right) \cdot x - 1.3333333333333333, x \cdot x, -2\right) \cdot x} - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.13333333333333333, -0.3333333333333333\right) \cdot \left(x \cdot x\right), x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -0.0071)
   (-
    (/
     2.0
     (+
      (fma
       (fma (fma 0.08888888888888889 (* x x) 0.6666666666666666) (* x x) 2.0)
       (* x x)
       2.0)
      (*
       (fma
        (-
         (* (* (- (* -0.025396825396825397 (* x x)) 0.26666666666666666) x) x)
         1.3333333333333333)
        (* x x)
        -2.0)
       x)))
    1.0)
   (fma
    (* (fma (* x x) 0.13333333333333333 -0.3333333333333333) (* x x))
    x
    x)))

C

double code(double x) {
	double tmp;
	if (x <= -0.0071) {
		tmp = (2.0 / (fma(fma(fma(0.08888888888888889, (x * x), 0.6666666666666666), (x * x), 2.0), (x * x), 2.0) + (fma((((((-0.025396825396825397 * (x * x)) - 0.26666666666666666) * x) * x) - 1.3333333333333333), (x * x), -2.0) * x))) - 1.0;
	} else {
		tmp = fma((fma((x * x), 0.13333333333333333, -0.3333333333333333) * (x * x)), x, x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -0.0071)
		tmp = Float64(Float64(2.0 / Float64(fma(fma(fma(0.08888888888888889, Float64(x * x), 0.6666666666666666), Float64(x * x), 2.0), Float64(x * x), 2.0) + Float64(fma(Float64(Float64(Float64(Float64(Float64(-0.025396825396825397 * Float64(x * x)) - 0.26666666666666666) * x) * x) - 1.3333333333333333), Float64(x * x), -2.0) * x))) - 1.0);
	else
		tmp = fma(Float64(fma(Float64(x * x), 0.13333333333333333, -0.3333333333333333) * Float64(x * x)), x, x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -0.0071], N[(N[(2.0 / N[(N[(N[(N[(0.08888888888888889 * N[(x * x), $MachinePrecision] + 0.6666666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision] * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision] + N[(N[(N[(N[(N[(N[(N[(-0.025396825396825397 * N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.26666666666666666), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] - 1.3333333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision] + -2.0), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(N[(x * x), $MachinePrecision] * 0.13333333333333333 + -0.3333333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * x + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.0071000000000000004

   1. Initial program 99.9%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{1 + e^{-2 \cdot x}}} - 1 \]

      2. lift-exp.f64 N/A

         \[\leadsto \frac{2}{1 + \color{blue}{e^{-2 \cdot x}}} - 1 \]

      3. sinh-+-cosh-rev N/A

         \[\leadsto \frac{2}{1 + \color{blue}{\left(\cosh \left(-2 \cdot x\right) + \sinh \left(-2 \cdot x\right)\right)}} - 1 \]

      4. associate-+r+ N/A

         \[\leadsto \frac{2}{\color{blue}{\left(1 + \cosh \left(-2 \cdot x\right)\right) + \sinh \left(-2 \cdot x\right)}} - 1 \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(1 + \cosh \left(-2 \cdot x\right)\right) + \sinh \left(-2 \cdot x\right)}} - 1 \]

      6. lower-+.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(1 + \cosh \left(-2 \cdot x\right)\right)} + \sinh \left(-2 \cdot x\right)} - 1 \]

      7. lower-cosh.f64 N/A

         \[\leadsto \frac{2}{\left(1 + \color{blue}{\cosh \left(-2 \cdot x\right)}\right) + \sinh \left(-2 \cdot x\right)} - 1 \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(1 + \cosh \color{blue}{\left(-2 \cdot x\right)}\right) + \sinh \left(-2 \cdot x\right)} - 1 \]

      9. *-commutative N/A

         \[\leadsto \frac{2}{\left(1 + \cosh \color{blue}{\left(x \cdot -2\right)}\right) + \sinh \left(-2 \cdot x\right)} - 1 \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(1 + \cosh \color{blue}{\left(x \cdot -2\right)}\right) + \sinh \left(-2 \cdot x\right)} - 1 \]

      11. lower-sinh.f64 99.9

          \[\leadsto \frac{2}{\left(1 + \cosh \left(x \cdot -2\right)\right) + \color{blue}{\sinh \left(-2 \cdot x\right)}} - 1 \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(1 + \cosh \left(x \cdot -2\right)\right) + \sinh \color{blue}{\left(-2 \cdot x\right)}} - 1 \]

      13. *-commutative N/A

          \[\leadsto \frac{2}{\left(1 + \cosh \left(x \cdot -2\right)\right) + \sinh \color{blue}{\left(x \cdot -2\right)}} - 1 \]

      14. lower-*.f64 99.9

          \[\leadsto \frac{2}{\left(1 + \cosh \left(x \cdot -2\right)\right) + \sinh \color{blue}{\left(x \cdot -2\right)}} - 1 \]

   4. Applied rewrites 99.9%

      \[\leadsto \frac{2}{\color{blue}{\left(1 + \cosh \left(x \cdot -2\right)\right) + \sinh \left(x \cdot -2\right)}} - 1 \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{2}{\left(1 + \cosh \left(x \cdot -2\right)\right) + \color{blue}{x \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-8}{315} \cdot {x}^{2} - \frac{4}{15}\right) - \frac{4}{3}\right) - 2\right)}} - 1 \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\left(1 + \cosh \left(x \cdot -2\right)\right) + \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-8}{315} \cdot {x}^{2} - \frac{4}{15}\right) - \frac{4}{3}\right) - 2\right) \cdot x}} - 1 \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(1 + \cosh \left(x \cdot -2\right)\right) + \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-8}{315} \cdot {x}^{2} - \frac{4}{15}\right) - \frac{4}{3}\right) - 2\right) \cdot x}} - 1 \]

   7. Applied rewrites 99.0%

      \[\leadsto \frac{2}{\left(1 + \cosh \left(x \cdot -2\right)\right) + \color{blue}{\mathsf{fma}\left(\left(\left(-0.025396825396825397 \cdot \left(x \cdot x\right) - 0.26666666666666666\right) \cdot x\right) \cdot x - 1.3333333333333333, x \cdot x, -2\right) \cdot x}} - 1 \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{2}{\color{blue}{\left(2 + {x}^{2} \cdot \left(2 + {x}^{2} \cdot \left(\frac{2}{3} + \frac{4}{45} \cdot {x}^{2}\right)\right)\right)} + \mathsf{fma}\left(\left(\left(\frac{-8}{315} \cdot \left(x \cdot x\right) - \frac{4}{15}\right) \cdot x\right) \cdot x - \frac{4}{3}, x \cdot x, -2\right) \cdot x} - 1 \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2}{\color{blue}{\left({x}^{2} \cdot \left(2 + {x}^{2} \cdot \left(\frac{2}{3} + \frac{4}{45} \cdot {x}^{2}\right)\right) + 2\right)} + \mathsf{fma}\left(\left(\left(\frac{-8}{315} \cdot \left(x \cdot x\right) - \frac{4}{15}\right) \cdot x\right) \cdot x - \frac{4}{3}, x \cdot x, -2\right) \cdot x} - 1 \]

      2. *-commutative N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(2 + {x}^{2} \cdot \left(\frac{2}{3} + \frac{4}{45} \cdot {x}^{2}\right)\right) \cdot {x}^{2}} + 2\right) + \mathsf{fma}\left(\left(\left(\frac{-8}{315} \cdot \left(x \cdot x\right) - \frac{4}{15}\right) \cdot x\right) \cdot x - \frac{4}{3}, x \cdot x, -2\right) \cdot x} - 1 \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(2 + {x}^{2} \cdot \left(\frac{2}{3} + \frac{4}{45} \cdot {x}^{2}\right), {x}^{2}, 2\right)} + \mathsf{fma}\left(\left(\left(\frac{-8}{315} \cdot \left(x \cdot x\right) - \frac{4}{15}\right) \cdot x\right) \cdot x - \frac{4}{3}, x \cdot x, -2\right) \cdot x} - 1 \]

      4. +-commutative N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(\color{blue}{{x}^{2} \cdot \left(\frac{2}{3} + \frac{4}{45} \cdot {x}^{2}\right) + 2}, {x}^{2}, 2\right) + \mathsf{fma}\left(\left(\left(\frac{-8}{315} \cdot \left(x \cdot x\right) - \frac{4}{15}\right) \cdot x\right) \cdot x - \frac{4}{3}, x \cdot x, -2\right) \cdot x} - 1 \]

      5. *-commutative N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(\color{blue}{\left(\frac{2}{3} + \frac{4}{45} \cdot {x}^{2}\right) \cdot {x}^{2}} + 2, {x}^{2}, 2\right) + \mathsf{fma}\left(\left(\left(\frac{-8}{315} \cdot \left(x \cdot x\right) - \frac{4}{15}\right) \cdot x\right) \cdot x - \frac{4}{3}, x \cdot x, -2\right) \cdot x} - 1 \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{2}{3} + \frac{4}{45} \cdot {x}^{2}, {x}^{2}, 2\right)}, {x}^{2}, 2\right) + \mathsf{fma}\left(\left(\left(\frac{-8}{315} \cdot \left(x \cdot x\right) - \frac{4}{15}\right) \cdot x\right) \cdot x - \frac{4}{3}, x \cdot x, -2\right) \cdot x} - 1 \]

      7. +-commutative N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{4}{45} \cdot {x}^{2} + \frac{2}{3}}, {x}^{2}, 2\right), {x}^{2}, 2\right) + \mathsf{fma}\left(\left(\left(\frac{-8}{315} \cdot \left(x \cdot x\right) - \frac{4}{15}\right) \cdot x\right) \cdot x - \frac{4}{3}, x \cdot x, -2\right) \cdot x} - 1 \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{4}{45}, {x}^{2}, \frac{2}{3}\right)}, {x}^{2}, 2\right), {x}^{2}, 2\right) + \mathsf{fma}\left(\left(\left(\frac{-8}{315} \cdot \left(x \cdot x\right) - \frac{4}{15}\right) \cdot x\right) \cdot x - \frac{4}{3}, x \cdot x, -2\right) \cdot x} - 1 \]

      9. unpow2 N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{4}{45}, \color{blue}{x \cdot x}, \frac{2}{3}\right), {x}^{2}, 2\right), {x}^{2}, 2\right) + \mathsf{fma}\left(\left(\left(\frac{-8}{315} \cdot \left(x \cdot x\right) - \frac{4}{15}\right) \cdot x\right) \cdot x - \frac{4}{3}, x \cdot x, -2\right) \cdot x} - 1 \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{4}{45}, \color{blue}{x \cdot x}, \frac{2}{3}\right), {x}^{2}, 2\right), {x}^{2}, 2\right) + \mathsf{fma}\left(\left(\left(\frac{-8}{315} \cdot \left(x \cdot x\right) - \frac{4}{15}\right) \cdot x\right) \cdot x - \frac{4}{3}, x \cdot x, -2\right) \cdot x} - 1 \]

      11. unpow2 N/A

          \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{4}{45}, x \cdot x, \frac{2}{3}\right), \color{blue}{x \cdot x}, 2\right), {x}^{2}, 2\right) + \mathsf{fma}\left(\left(\left(\frac{-8}{315} \cdot \left(x \cdot x\right) - \frac{4}{15}\right) \cdot x\right) \cdot x - \frac{4}{3}, x \cdot x, -2\right) \cdot x} - 1 \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{4}{45}, x \cdot x, \frac{2}{3}\right), \color{blue}{x \cdot x}, 2\right), {x}^{2}, 2\right) + \mathsf{fma}\left(\left(\left(\frac{-8}{315} \cdot \left(x \cdot x\right) - \frac{4}{15}\right) \cdot x\right) \cdot x - \frac{4}{3}, x \cdot x, -2\right) \cdot x} - 1 \]

      13. unpow2 N/A

          \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{4}{45}, x \cdot x, \frac{2}{3}\right), x \cdot x, 2\right), \color{blue}{x \cdot x}, 2\right) + \mathsf{fma}\left(\left(\left(\frac{-8}{315} \cdot \left(x \cdot x\right) - \frac{4}{15}\right) \cdot x\right) \cdot x - \frac{4}{3}, x \cdot x, -2\right) \cdot x} - 1 \]

      14. lower-*.f64 98.9

          \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.08888888888888889, x \cdot x, 0.6666666666666666\right), x \cdot x, 2\right), \color{blue}{x \cdot x}, 2\right) + \mathsf{fma}\left(\left(\left(-0.025396825396825397 \cdot \left(x \cdot x\right) - 0.26666666666666666\right) \cdot x\right) \cdot x - 1.3333333333333333, x \cdot x, -2\right) \cdot x} - 1 \]

   10. Applied rewrites 98.9%

       \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.08888888888888889, x \cdot x, 0.6666666666666666\right), x \cdot x, 2\right), x \cdot x, 2\right)} + \mathsf{fma}\left(\left(\left(-0.025396825396825397 \cdot \left(x \cdot x\right) - 0.26666666666666666\right) \cdot x\right) \cdot x - 1.3333333333333333, x \cdot x, -2\right) \cdot x} - 1 \]

   if -0.0071000000000000004 < x

   1. Initial program 37.1%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right) + 1\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{x \cdot \left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)\right) + x \cdot 1} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(x \cdot {x}^{2}\right) \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)} + x \cdot 1 \]

      4. *-rgt-identity N/A

         \[\leadsto \left(x \cdot {x}^{2}\right) \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right) + \color{blue}{x} \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot {x}^{2}, \frac{2}{15} \cdot {x}^{2} - \frac{1}{3}, x\right)} \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{2} \cdot x}, \frac{2}{15} \cdot {x}^{2} - \frac{1}{3}, x\right) \]

      7. pow-plus N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{\left(2 + 1\right)}}, \frac{2}{15} \cdot {x}^{2} - \frac{1}{3}, x\right) \]

      8. lower-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{\left(2 + 1\right)}}, \frac{2}{15} \cdot {x}^{2} - \frac{1}{3}, x\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left({x}^{\color{blue}{3}}, \frac{2}{15} \cdot {x}^{2} - \frac{1}{3}, x\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, \frac{2}{15} \cdot {x}^{2} - \color{blue}{\frac{1}{3} \cdot 1}, x\right) \]

      11. fp-cancel-sub-sign-inv N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{\frac{2}{15} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot 1}, x\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, \frac{2}{15} \cdot {x}^{2} + \color{blue}{\frac{-1}{3}} \cdot 1, x\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, \frac{2}{15} \cdot {x}^{2} + \color{blue}{\frac{-1}{3}}, x\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{\mathsf{fma}\left(\frac{2}{15}, {x}^{2}, \frac{-1}{3}\right)}, x\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\frac{2}{15}, \color{blue}{x \cdot x}, \frac{-1}{3}\right), x\right) \]

      16. lower-*.f64 69.4

          \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(0.13333333333333333, \color{blue}{x \cdot x}, -0.3333333333333333\right), x\right) \]

   5. Applied rewrites 69.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(0.13333333333333333, x \cdot x, -0.3333333333333333\right), x\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 69.4%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.13333333333333333, -0.3333333333333333\right) \cdot \left(x \cdot x\right), \color{blue}{x}, x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 75.4% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.365:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right), x \cdot x, 2\right) + \mathsf{fma}\left(\left(\left(-0.025396825396825397 \cdot \left(x \cdot x\right) - 0.26666666666666666\right) \cdot x\right) \cdot x - 1.3333333333333333, x \cdot x, -2\right) \cdot x} - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.13333333333333333, -0.3333333333333333\right) \cdot \left(x \cdot x\right), x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -0.365)
   (-
    (/
     2.0
     (+
      (fma (fma 0.6666666666666666 (* x x) 2.0) (* x x) 2.0)
      (*
       (fma
        (-
         (* (* (- (* -0.025396825396825397 (* x x)) 0.26666666666666666) x) x)
         1.3333333333333333)
        (* x x)
        -2.0)
       x)))
    1.0)
   (fma
    (* (fma (* x x) 0.13333333333333333 -0.3333333333333333) (* x x))
    x
    x)))

C

double code(double x) {
	double tmp;
	if (x <= -0.365) {
		tmp = (2.0 / (fma(fma(0.6666666666666666, (x * x), 2.0), (x * x), 2.0) + (fma((((((-0.025396825396825397 * (x * x)) - 0.26666666666666666) * x) * x) - 1.3333333333333333), (x * x), -2.0) * x))) - 1.0;
	} else {
		tmp = fma((fma((x * x), 0.13333333333333333, -0.3333333333333333) * (x * x)), x, x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -0.365)
		tmp = Float64(Float64(2.0 / Float64(fma(fma(0.6666666666666666, Float64(x * x), 2.0), Float64(x * x), 2.0) + Float64(fma(Float64(Float64(Float64(Float64(Float64(-0.025396825396825397 * Float64(x * x)) - 0.26666666666666666) * x) * x) - 1.3333333333333333), Float64(x * x), -2.0) * x))) - 1.0);
	else
		tmp = fma(Float64(fma(Float64(x * x), 0.13333333333333333, -0.3333333333333333) * Float64(x * x)), x, x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -0.365], N[(N[(2.0 / N[(N[(N[(0.6666666666666666 * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision] * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision] + N[(N[(N[(N[(N[(N[(N[(-0.025396825396825397 * N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.26666666666666666), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] - 1.3333333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision] + -2.0), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(N[(x * x), $MachinePrecision] * 0.13333333333333333 + -0.3333333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * x + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.36499999999999999

   1. Initial program 100.0%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{1 + e^{-2 \cdot x}}} - 1 \]

      2. lift-exp.f64 N/A

         \[\leadsto \frac{2}{1 + \color{blue}{e^{-2 \cdot x}}} - 1 \]

      3. sinh-+-cosh-rev N/A

         \[\leadsto \frac{2}{1 + \color{blue}{\left(\cosh \left(-2 \cdot x\right) + \sinh \left(-2 \cdot x\right)\right)}} - 1 \]

      4. associate-+r+ N/A

         \[\leadsto \frac{2}{\color{blue}{\left(1 + \cosh \left(-2 \cdot x\right)\right) + \sinh \left(-2 \cdot x\right)}} - 1 \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(1 + \cosh \left(-2 \cdot x\right)\right) + \sinh \left(-2 \cdot x\right)}} - 1 \]

      6. lower-+.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(1 + \cosh \left(-2 \cdot x\right)\right)} + \sinh \left(-2 \cdot x\right)} - 1 \]

      7. lower-cosh.f64 N/A

         \[\leadsto \frac{2}{\left(1 + \color{blue}{\cosh \left(-2 \cdot x\right)}\right) + \sinh \left(-2 \cdot x\right)} - 1 \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(1 + \cosh \color{blue}{\left(-2 \cdot x\right)}\right) + \sinh \left(-2 \cdot x\right)} - 1 \]

      9. *-commutative N/A

         \[\leadsto \frac{2}{\left(1 + \cosh \color{blue}{\left(x \cdot -2\right)}\right) + \sinh \left(-2 \cdot x\right)} - 1 \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(1 + \cosh \color{blue}{\left(x \cdot -2\right)}\right) + \sinh \left(-2 \cdot x\right)} - 1 \]

      11. lower-sinh.f64 100.0

          \[\leadsto \frac{2}{\left(1 + \cosh \left(x \cdot -2\right)\right) + \color{blue}{\sinh \left(-2 \cdot x\right)}} - 1 \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(1 + \cosh \left(x \cdot -2\right)\right) + \sinh \color{blue}{\left(-2 \cdot x\right)}} - 1 \]

      13. *-commutative N/A

          \[\leadsto \frac{2}{\left(1 + \cosh \left(x \cdot -2\right)\right) + \sinh \color{blue}{\left(x \cdot -2\right)}} - 1 \]

      14. lower-*.f64 100.0

          \[\leadsto \frac{2}{\left(1 + \cosh \left(x \cdot -2\right)\right) + \sinh \color{blue}{\left(x \cdot -2\right)}} - 1 \]

   4. Applied rewrites 100.0%

      \[\leadsto \frac{2}{\color{blue}{\left(1 + \cosh \left(x \cdot -2\right)\right) + \sinh \left(x \cdot -2\right)}} - 1 \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{2}{\left(1 + \cosh \left(x \cdot -2\right)\right) + \color{blue}{x \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-8}{315} \cdot {x}^{2} - \frac{4}{15}\right) - \frac{4}{3}\right) - 2\right)}} - 1 \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\left(1 + \cosh \left(x \cdot -2\right)\right) + \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-8}{315} \cdot {x}^{2} - \frac{4}{15}\right) - \frac{4}{3}\right) - 2\right) \cdot x}} - 1 \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(1 + \cosh \left(x \cdot -2\right)\right) + \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-8}{315} \cdot {x}^{2} - \frac{4}{15}\right) - \frac{4}{3}\right) - 2\right) \cdot x}} - 1 \]

   7. Applied rewrites 99.2%

      \[\leadsto \frac{2}{\left(1 + \cosh \left(x \cdot -2\right)\right) + \color{blue}{\mathsf{fma}\left(\left(\left(-0.025396825396825397 \cdot \left(x \cdot x\right) - 0.26666666666666666\right) \cdot x\right) \cdot x - 1.3333333333333333, x \cdot x, -2\right) \cdot x}} - 1 \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{2}{\color{blue}{\left(2 + {x}^{2} \cdot \left(2 + \frac{2}{3} \cdot {x}^{2}\right)\right)} + \mathsf{fma}\left(\left(\left(\frac{-8}{315} \cdot \left(x \cdot x\right) - \frac{4}{15}\right) \cdot x\right) \cdot x - \frac{4}{3}, x \cdot x, -2\right) \cdot x} - 1 \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2}{\color{blue}{\left({x}^{2} \cdot \left(2 + \frac{2}{3} \cdot {x}^{2}\right) + 2\right)} + \mathsf{fma}\left(\left(\left(\frac{-8}{315} \cdot \left(x \cdot x\right) - \frac{4}{15}\right) \cdot x\right) \cdot x - \frac{4}{3}, x \cdot x, -2\right) \cdot x} - 1 \]

      2. *-commutative N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(2 + \frac{2}{3} \cdot {x}^{2}\right) \cdot {x}^{2}} + 2\right) + \mathsf{fma}\left(\left(\left(\frac{-8}{315} \cdot \left(x \cdot x\right) - \frac{4}{15}\right) \cdot x\right) \cdot x - \frac{4}{3}, x \cdot x, -2\right) \cdot x} - 1 \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(2 + \frac{2}{3} \cdot {x}^{2}, {x}^{2}, 2\right)} + \mathsf{fma}\left(\left(\left(\frac{-8}{315} \cdot \left(x \cdot x\right) - \frac{4}{15}\right) \cdot x\right) \cdot x - \frac{4}{3}, x \cdot x, -2\right) \cdot x} - 1 \]

      4. +-commutative N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(\color{blue}{\frac{2}{3} \cdot {x}^{2} + 2}, {x}^{2}, 2\right) + \mathsf{fma}\left(\left(\left(\frac{-8}{315} \cdot \left(x \cdot x\right) - \frac{4}{15}\right) \cdot x\right) \cdot x - \frac{4}{3}, x \cdot x, -2\right) \cdot x} - 1 \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{2}{3}, {x}^{2}, 2\right)}, {x}^{2}, 2\right) + \mathsf{fma}\left(\left(\left(\frac{-8}{315} \cdot \left(x \cdot x\right) - \frac{4}{15}\right) \cdot x\right) \cdot x - \frac{4}{3}, x \cdot x, -2\right) \cdot x} - 1 \]

      6. unpow2 N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{3}, \color{blue}{x \cdot x}, 2\right), {x}^{2}, 2\right) + \mathsf{fma}\left(\left(\left(\frac{-8}{315} \cdot \left(x \cdot x\right) - \frac{4}{15}\right) \cdot x\right) \cdot x - \frac{4}{3}, x \cdot x, -2\right) \cdot x} - 1 \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{3}, \color{blue}{x \cdot x}, 2\right), {x}^{2}, 2\right) + \mathsf{fma}\left(\left(\left(\frac{-8}{315} \cdot \left(x \cdot x\right) - \frac{4}{15}\right) \cdot x\right) \cdot x - \frac{4}{3}, x \cdot x, -2\right) \cdot x} - 1 \]

      8. unpow2 N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{2}{3}, x \cdot x, 2\right), \color{blue}{x \cdot x}, 2\right) + \mathsf{fma}\left(\left(\left(\frac{-8}{315} \cdot \left(x \cdot x\right) - \frac{4}{15}\right) \cdot x\right) \cdot x - \frac{4}{3}, x \cdot x, -2\right) \cdot x} - 1 \]

      9. lower-*.f64 99.0

         \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right), \color{blue}{x \cdot x}, 2\right) + \mathsf{fma}\left(\left(\left(-0.025396825396825397 \cdot \left(x \cdot x\right) - 0.26666666666666666\right) \cdot x\right) \cdot x - 1.3333333333333333, x \cdot x, -2\right) \cdot x} - 1 \]

   10. Applied rewrites 99.0%

       \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.6666666666666666, x \cdot x, 2\right), x \cdot x, 2\right)} + \mathsf{fma}\left(\left(\left(-0.025396825396825397 \cdot \left(x \cdot x\right) - 0.26666666666666666\right) \cdot x\right) \cdot x - 1.3333333333333333, x \cdot x, -2\right) \cdot x} - 1 \]

   if -0.36499999999999999 < x

   1. Initial program 37.4%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right) + 1\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{x \cdot \left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)\right) + x \cdot 1} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(x \cdot {x}^{2}\right) \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)} + x \cdot 1 \]

      4. *-rgt-identity N/A

         \[\leadsto \left(x \cdot {x}^{2}\right) \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right) + \color{blue}{x} \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot {x}^{2}, \frac{2}{15} \cdot {x}^{2} - \frac{1}{3}, x\right)} \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{2} \cdot x}, \frac{2}{15} \cdot {x}^{2} - \frac{1}{3}, x\right) \]

      7. pow-plus N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{\left(2 + 1\right)}}, \frac{2}{15} \cdot {x}^{2} - \frac{1}{3}, x\right) \]

      8. lower-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{\left(2 + 1\right)}}, \frac{2}{15} \cdot {x}^{2} - \frac{1}{3}, x\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left({x}^{\color{blue}{3}}, \frac{2}{15} \cdot {x}^{2} - \frac{1}{3}, x\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, \frac{2}{15} \cdot {x}^{2} - \color{blue}{\frac{1}{3} \cdot 1}, x\right) \]

      11. fp-cancel-sub-sign-inv N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{\frac{2}{15} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot 1}, x\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, \frac{2}{15} \cdot {x}^{2} + \color{blue}{\frac{-1}{3}} \cdot 1, x\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, \frac{2}{15} \cdot {x}^{2} + \color{blue}{\frac{-1}{3}}, x\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{\mathsf{fma}\left(\frac{2}{15}, {x}^{2}, \frac{-1}{3}\right)}, x\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\frac{2}{15}, \color{blue}{x \cdot x}, \frac{-1}{3}\right), x\right) \]

      16. lower-*.f64 69.5

          \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(0.13333333333333333, \color{blue}{x \cdot x}, -0.3333333333333333\right), x\right) \]

   5. Applied rewrites 69.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(0.13333333333333333, x \cdot x, -0.3333333333333333\right), x\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 69.5%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.13333333333333333, -0.3333333333333333\right) \cdot \left(x \cdot x\right), \color{blue}{x}, x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 75.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.9:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(2 \cdot x, x, 2\right) + \mathsf{fma}\left(\left(\left(-0.025396825396825397 \cdot \left(x \cdot x\right) - 0.26666666666666666\right) \cdot x\right) \cdot x - 1.3333333333333333, x \cdot x, -2\right) \cdot x} - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.13333333333333333, -0.3333333333333333\right) \cdot \left(x \cdot x\right), x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -0.9)
   (-
    (/
     2.0
     (+
      (fma (* 2.0 x) x 2.0)
      (*
       (fma
        (-
         (* (* (- (* -0.025396825396825397 (* x x)) 0.26666666666666666) x) x)
         1.3333333333333333)
        (* x x)
        -2.0)
       x)))
    1.0)
   (fma
    (* (fma (* x x) 0.13333333333333333 -0.3333333333333333) (* x x))
    x
    x)))

C

double code(double x) {
	double tmp;
	if (x <= -0.9) {
		tmp = (2.0 / (fma((2.0 * x), x, 2.0) + (fma((((((-0.025396825396825397 * (x * x)) - 0.26666666666666666) * x) * x) - 1.3333333333333333), (x * x), -2.0) * x))) - 1.0;
	} else {
		tmp = fma((fma((x * x), 0.13333333333333333, -0.3333333333333333) * (x * x)), x, x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -0.9)
		tmp = Float64(Float64(2.0 / Float64(fma(Float64(2.0 * x), x, 2.0) + Float64(fma(Float64(Float64(Float64(Float64(Float64(-0.025396825396825397 * Float64(x * x)) - 0.26666666666666666) * x) * x) - 1.3333333333333333), Float64(x * x), -2.0) * x))) - 1.0);
	else
		tmp = fma(Float64(fma(Float64(x * x), 0.13333333333333333, -0.3333333333333333) * Float64(x * x)), x, x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -0.9], N[(N[(2.0 / N[(N[(N[(2.0 * x), $MachinePrecision] * x + 2.0), $MachinePrecision] + N[(N[(N[(N[(N[(N[(N[(-0.025396825396825397 * N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.26666666666666666), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] - 1.3333333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision] + -2.0), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(N[(x * x), $MachinePrecision] * 0.13333333333333333 + -0.3333333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * x + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.900000000000000022

   1. Initial program 100.0%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{1 + e^{-2 \cdot x}}} - 1 \]

      2. lift-exp.f64 N/A

         \[\leadsto \frac{2}{1 + \color{blue}{e^{-2 \cdot x}}} - 1 \]

      3. sinh-+-cosh-rev N/A

         \[\leadsto \frac{2}{1 + \color{blue}{\left(\cosh \left(-2 \cdot x\right) + \sinh \left(-2 \cdot x\right)\right)}} - 1 \]

      4. associate-+r+ N/A

         \[\leadsto \frac{2}{\color{blue}{\left(1 + \cosh \left(-2 \cdot x\right)\right) + \sinh \left(-2 \cdot x\right)}} - 1 \]

      5. lower-+.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(1 + \cosh \left(-2 \cdot x\right)\right) + \sinh \left(-2 \cdot x\right)}} - 1 \]

      6. lower-+.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\left(1 + \cosh \left(-2 \cdot x\right)\right)} + \sinh \left(-2 \cdot x\right)} - 1 \]

      7. lower-cosh.f64 N/A

         \[\leadsto \frac{2}{\left(1 + \color{blue}{\cosh \left(-2 \cdot x\right)}\right) + \sinh \left(-2 \cdot x\right)} - 1 \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(1 + \cosh \color{blue}{\left(-2 \cdot x\right)}\right) + \sinh \left(-2 \cdot x\right)} - 1 \]

      9. *-commutative N/A

         \[\leadsto \frac{2}{\left(1 + \cosh \color{blue}{\left(x \cdot -2\right)}\right) + \sinh \left(-2 \cdot x\right)} - 1 \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(1 + \cosh \color{blue}{\left(x \cdot -2\right)}\right) + \sinh \left(-2 \cdot x\right)} - 1 \]

      11. lower-sinh.f64 100.0

          \[\leadsto \frac{2}{\left(1 + \cosh \left(x \cdot -2\right)\right) + \color{blue}{\sinh \left(-2 \cdot x\right)}} - 1 \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(1 + \cosh \left(x \cdot -2\right)\right) + \sinh \color{blue}{\left(-2 \cdot x\right)}} - 1 \]

      13. *-commutative N/A

          \[\leadsto \frac{2}{\left(1 + \cosh \left(x \cdot -2\right)\right) + \sinh \color{blue}{\left(x \cdot -2\right)}} - 1 \]

      14. lower-*.f64 100.0

          \[\leadsto \frac{2}{\left(1 + \cosh \left(x \cdot -2\right)\right) + \sinh \color{blue}{\left(x \cdot -2\right)}} - 1 \]

   4. Applied rewrites 100.0%

      \[\leadsto \frac{2}{\color{blue}{\left(1 + \cosh \left(x \cdot -2\right)\right) + \sinh \left(x \cdot -2\right)}} - 1 \]

   5. Taylor expanded in x around 0

      \[\leadsto \frac{2}{\left(1 + \cosh \left(x \cdot -2\right)\right) + \color{blue}{x \cdot \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-8}{315} \cdot {x}^{2} - \frac{4}{15}\right) - \frac{4}{3}\right) - 2\right)}} - 1 \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\left(1 + \cosh \left(x \cdot -2\right)\right) + \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-8}{315} \cdot {x}^{2} - \frac{4}{15}\right) - \frac{4}{3}\right) - 2\right) \cdot x}} - 1 \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(1 + \cosh \left(x \cdot -2\right)\right) + \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-8}{315} \cdot {x}^{2} - \frac{4}{15}\right) - \frac{4}{3}\right) - 2\right) \cdot x}} - 1 \]

   7. Applied rewrites 99.2%

      \[\leadsto \frac{2}{\left(1 + \cosh \left(x \cdot -2\right)\right) + \color{blue}{\mathsf{fma}\left(\left(\left(-0.025396825396825397 \cdot \left(x \cdot x\right) - 0.26666666666666666\right) \cdot x\right) \cdot x - 1.3333333333333333, x \cdot x, -2\right) \cdot x}} - 1 \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{2}{\color{blue}{\left(2 + 2 \cdot {x}^{2}\right)} + \mathsf{fma}\left(\left(\left(\frac{-8}{315} \cdot \left(x \cdot x\right) - \frac{4}{15}\right) \cdot x\right) \cdot x - \frac{4}{3}, x \cdot x, -2\right) \cdot x} - 1 \]
   9. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot {x}^{2} + 2\right)} + \mathsf{fma}\left(\left(\left(\frac{-8}{315} \cdot \left(x \cdot x\right) - \frac{4}{15}\right) \cdot x\right) \cdot x - \frac{4}{3}, x \cdot x, -2\right) \cdot x} - 1 \]

      2. unpow2 N/A

         \[\leadsto \frac{2}{\left(2 \cdot \color{blue}{\left(x \cdot x\right)} + 2\right) + \mathsf{fma}\left(\left(\left(\frac{-8}{315} \cdot \left(x \cdot x\right) - \frac{4}{15}\right) \cdot x\right) \cdot x - \frac{4}{3}, x \cdot x, -2\right) \cdot x} - 1 \]

      3. associate-*r* N/A

         \[\leadsto \frac{2}{\left(\color{blue}{\left(2 \cdot x\right) \cdot x} + 2\right) + \mathsf{fma}\left(\left(\left(\frac{-8}{315} \cdot \left(x \cdot x\right) - \frac{4}{15}\right) \cdot x\right) \cdot x - \frac{4}{3}, x \cdot x, -2\right) \cdot x} - 1 \]

      4. lower-fma.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(2 \cdot x, x, 2\right)} + \mathsf{fma}\left(\left(\left(\frac{-8}{315} \cdot \left(x \cdot x\right) - \frac{4}{15}\right) \cdot x\right) \cdot x - \frac{4}{3}, x \cdot x, -2\right) \cdot x} - 1 \]

      5. lower-*.f64 99.0

         \[\leadsto \frac{2}{\mathsf{fma}\left(\color{blue}{2 \cdot x}, x, 2\right) + \mathsf{fma}\left(\left(\left(-0.025396825396825397 \cdot \left(x \cdot x\right) - 0.26666666666666666\right) \cdot x\right) \cdot x - 1.3333333333333333, x \cdot x, -2\right) \cdot x} - 1 \]

   10. Applied rewrites 99.0%

       \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(2 \cdot x, x, 2\right)} + \mathsf{fma}\left(\left(\left(-0.025396825396825397 \cdot \left(x \cdot x\right) - 0.26666666666666666\right) \cdot x\right) \cdot x - 1.3333333333333333, x \cdot x, -2\right) \cdot x} - 1 \]

   if -0.900000000000000022 < x

   1. Initial program 37.4%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right) + 1\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{x \cdot \left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)\right) + x \cdot 1} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(x \cdot {x}^{2}\right) \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)} + x \cdot 1 \]

      4. *-rgt-identity N/A

         \[\leadsto \left(x \cdot {x}^{2}\right) \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right) + \color{blue}{x} \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot {x}^{2}, \frac{2}{15} \cdot {x}^{2} - \frac{1}{3}, x\right)} \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{2} \cdot x}, \frac{2}{15} \cdot {x}^{2} - \frac{1}{3}, x\right) \]

      7. pow-plus N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{\left(2 + 1\right)}}, \frac{2}{15} \cdot {x}^{2} - \frac{1}{3}, x\right) \]

      8. lower-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{\left(2 + 1\right)}}, \frac{2}{15} \cdot {x}^{2} - \frac{1}{3}, x\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left({x}^{\color{blue}{3}}, \frac{2}{15} \cdot {x}^{2} - \frac{1}{3}, x\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, \frac{2}{15} \cdot {x}^{2} - \color{blue}{\frac{1}{3} \cdot 1}, x\right) \]

      11. fp-cancel-sub-sign-inv N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{\frac{2}{15} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot 1}, x\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, \frac{2}{15} \cdot {x}^{2} + \color{blue}{\frac{-1}{3}} \cdot 1, x\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, \frac{2}{15} \cdot {x}^{2} + \color{blue}{\frac{-1}{3}}, x\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{\mathsf{fma}\left(\frac{2}{15}, {x}^{2}, \frac{-1}{3}\right)}, x\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\frac{2}{15}, \color{blue}{x \cdot x}, \frac{-1}{3}\right), x\right) \]

      16. lower-*.f64 69.5

          \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(0.13333333333333333, \color{blue}{x \cdot x}, -0.3333333333333333\right), x\right) \]

   5. Applied rewrites 69.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(0.13333333333333333, x \cdot x, -0.3333333333333333\right), x\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 69.5%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.13333333333333333, -0.3333333333333333\right) \cdot \left(x \cdot x\right), \color{blue}{x}, x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 75.3% accurate, 3.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.98:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-1.3333333333333333, x, 2\right), x, -2\right), x, 2\right)} - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.13333333333333333, -0.3333333333333333\right) \cdot \left(x \cdot x\right), x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -0.98)
   (- (/ 2.0 (fma (fma (fma -1.3333333333333333 x 2.0) x -2.0) x 2.0)) 1.0)
   (fma
    (* (fma (* x x) 0.13333333333333333 -0.3333333333333333) (* x x))
    x
    x)))

C

double code(double x) {
	double tmp;
	if (x <= -0.98) {
		tmp = (2.0 / fma(fma(fma(-1.3333333333333333, x, 2.0), x, -2.0), x, 2.0)) - 1.0;
	} else {
		tmp = fma((fma((x * x), 0.13333333333333333, -0.3333333333333333) * (x * x)), x, x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -0.98)
		tmp = Float64(Float64(2.0 / fma(fma(fma(-1.3333333333333333, x, 2.0), x, -2.0), x, 2.0)) - 1.0);
	else
		tmp = fma(Float64(fma(Float64(x * x), 0.13333333333333333, -0.3333333333333333) * Float64(x * x)), x, x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -0.98], N[(N[(2.0 / N[(N[(N[(-1.3333333333333333 * x + 2.0), $MachinePrecision] * x + -2.0), $MachinePrecision] * x + 2.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(N[(x * x), $MachinePrecision] * 0.13333333333333333 + -0.3333333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * x + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.97999999999999998

   1. Initial program 100.0%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{2}{\color{blue}{2 + x \cdot \left(x \cdot \left(2 + \frac{-4}{3} \cdot x\right) - 2\right)}} - 1 \]
   4. Step-by-step derivation

      1. fp-cancel-sign-sub-inv N/A

         \[\leadsto \frac{2}{\color{blue}{2 - \left(\mathsf{neg}\left(x\right)\right) \cdot \left(x \cdot \left(2 + \frac{-4}{3} \cdot x\right) - 2\right)}} - 1 \]

      2. fp-cancel-sub-sign-inv N/A

         \[\leadsto \frac{2}{\color{blue}{2 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \cdot \left(x \cdot \left(2 + \frac{-4}{3} \cdot x\right) - 2\right)}} - 1 \]

      3. +-commutative N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \cdot \left(x \cdot \left(2 + \frac{-4}{3} \cdot x\right) - 2\right) + 2}} - 1 \]

      4. remove-double-neg N/A

         \[\leadsto \frac{2}{\color{blue}{x} \cdot \left(x \cdot \left(2 + \frac{-4}{3} \cdot x\right) - 2\right) + 2} - 1 \]

      5. *-commutative N/A

         \[\leadsto \frac{2}{\color{blue}{\left(x \cdot \left(2 + \frac{-4}{3} \cdot x\right) - 2\right) \cdot x} + 2} - 1 \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(x \cdot \left(2 + \frac{-4}{3} \cdot x\right) - 2, x, 2\right)}} - 1 \]

      7. metadata-eval N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(x \cdot \left(2 + \frac{-4}{3} \cdot x\right) - \color{blue}{2 \cdot 1}, x, 2\right)} - 1 \]

      8. fp-cancel-sub-sign-inv N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(\color{blue}{x \cdot \left(2 + \frac{-4}{3} \cdot x\right) + \left(\mathsf{neg}\left(2\right)\right) \cdot 1}, x, 2\right)} - 1 \]

      9. *-commutative N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(\color{blue}{\left(2 + \frac{-4}{3} \cdot x\right) \cdot x} + \left(\mathsf{neg}\left(2\right)\right) \cdot 1, x, 2\right)} - 1 \]

      10. metadata-eval N/A

          \[\leadsto \frac{2}{\mathsf{fma}\left(\left(2 + \frac{-4}{3} \cdot x\right) \cdot x + \color{blue}{-2} \cdot 1, x, 2\right)} - 1 \]

      11. metadata-eval N/A

          \[\leadsto \frac{2}{\mathsf{fma}\left(\left(2 + \frac{-4}{3} \cdot x\right) \cdot x + \color{blue}{-2}, x, 2\right)} - 1 \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{2}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(2 + \frac{-4}{3} \cdot x, x, -2\right)}, x, 2\right)} - 1 \]

      13. +-commutative N/A

          \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{-4}{3} \cdot x + 2}, x, -2\right), x, 2\right)} - 1 \]

      14. lower-fma.f64 98.3

          \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(-1.3333333333333333, x, 2\right)}, x, -2\right), x, 2\right)} - 1 \]

   5. Applied rewrites 98.3%

      \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-1.3333333333333333, x, 2\right), x, -2\right), x, 2\right)}} - 1 \]

   if -0.97999999999999998 < x

   1. Initial program 37.4%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right) + 1\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{x \cdot \left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)\right) + x \cdot 1} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(x \cdot {x}^{2}\right) \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)} + x \cdot 1 \]

      4. *-rgt-identity N/A

         \[\leadsto \left(x \cdot {x}^{2}\right) \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right) + \color{blue}{x} \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot {x}^{2}, \frac{2}{15} \cdot {x}^{2} - \frac{1}{3}, x\right)} \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{2} \cdot x}, \frac{2}{15} \cdot {x}^{2} - \frac{1}{3}, x\right) \]

      7. pow-plus N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{\left(2 + 1\right)}}, \frac{2}{15} \cdot {x}^{2} - \frac{1}{3}, x\right) \]

      8. lower-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{\left(2 + 1\right)}}, \frac{2}{15} \cdot {x}^{2} - \frac{1}{3}, x\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left({x}^{\color{blue}{3}}, \frac{2}{15} \cdot {x}^{2} - \frac{1}{3}, x\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, \frac{2}{15} \cdot {x}^{2} - \color{blue}{\frac{1}{3} \cdot 1}, x\right) \]

      11. fp-cancel-sub-sign-inv N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{\frac{2}{15} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot 1}, x\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, \frac{2}{15} \cdot {x}^{2} + \color{blue}{\frac{-1}{3}} \cdot 1, x\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, \frac{2}{15} \cdot {x}^{2} + \color{blue}{\frac{-1}{3}}, x\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{\mathsf{fma}\left(\frac{2}{15}, {x}^{2}, \frac{-1}{3}\right)}, x\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\frac{2}{15}, \color{blue}{x \cdot x}, \frac{-1}{3}\right), x\right) \]

      16. lower-*.f64 69.5

          \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(0.13333333333333333, \color{blue}{x \cdot x}, -0.3333333333333333\right), x\right) \]

   5. Applied rewrites 69.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(0.13333333333333333, x \cdot x, -0.3333333333333333\right), x\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 69.5%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.13333333333333333, -0.3333333333333333\right) \cdot \left(x \cdot x\right), \color{blue}{x}, x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 75.3% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.32:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(-1.3333333333333333, x, 2\right) \cdot \left(x \cdot x\right)} - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.13333333333333333, -0.3333333333333333\right) \cdot \left(x \cdot x\right), x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.32)
   (- (/ 2.0 (* (fma -1.3333333333333333 x 2.0) (* x x))) 1.0)
   (fma
    (* (fma (* x x) 0.13333333333333333 -0.3333333333333333) (* x x))
    x
    x)))

C

double code(double x) {
	double tmp;
	if (x <= -1.32) {
		tmp = (2.0 / (fma(-1.3333333333333333, x, 2.0) * (x * x))) - 1.0;
	} else {
		tmp = fma((fma((x * x), 0.13333333333333333, -0.3333333333333333) * (x * x)), x, x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.32)
		tmp = Float64(Float64(2.0 / Float64(fma(-1.3333333333333333, x, 2.0) * Float64(x * x))) - 1.0);
	else
		tmp = fma(Float64(fma(Float64(x * x), 0.13333333333333333, -0.3333333333333333) * Float64(x * x)), x, x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1.32], N[(N[(2.0 / N[(N[(-1.3333333333333333 * x + 2.0), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(N[(x * x), $MachinePrecision] * 0.13333333333333333 + -0.3333333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * x + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.32000000000000006

   1. Initial program 100.0%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{2}{\color{blue}{2 + x \cdot \left(x \cdot \left(2 + \frac{-4}{3} \cdot x\right) - 2\right)}} - 1 \]
   4. Step-by-step derivation

      1. fp-cancel-sign-sub-inv N/A

         \[\leadsto \frac{2}{\color{blue}{2 - \left(\mathsf{neg}\left(x\right)\right) \cdot \left(x \cdot \left(2 + \frac{-4}{3} \cdot x\right) - 2\right)}} - 1 \]

      2. fp-cancel-sub-sign-inv N/A

         \[\leadsto \frac{2}{\color{blue}{2 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \cdot \left(x \cdot \left(2 + \frac{-4}{3} \cdot x\right) - 2\right)}} - 1 \]

      3. +-commutative N/A

         \[\leadsto \frac{2}{\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \cdot \left(x \cdot \left(2 + \frac{-4}{3} \cdot x\right) - 2\right) + 2}} - 1 \]

      4. remove-double-neg N/A

         \[\leadsto \frac{2}{\color{blue}{x} \cdot \left(x \cdot \left(2 + \frac{-4}{3} \cdot x\right) - 2\right) + 2} - 1 \]

      5. *-commutative N/A

         \[\leadsto \frac{2}{\color{blue}{\left(x \cdot \left(2 + \frac{-4}{3} \cdot x\right) - 2\right) \cdot x} + 2} - 1 \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(x \cdot \left(2 + \frac{-4}{3} \cdot x\right) - 2, x, 2\right)}} - 1 \]

      7. metadata-eval N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(x \cdot \left(2 + \frac{-4}{3} \cdot x\right) - \color{blue}{2 \cdot 1}, x, 2\right)} - 1 \]

      8. fp-cancel-sub-sign-inv N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(\color{blue}{x \cdot \left(2 + \frac{-4}{3} \cdot x\right) + \left(\mathsf{neg}\left(2\right)\right) \cdot 1}, x, 2\right)} - 1 \]

      9. *-commutative N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(\color{blue}{\left(2 + \frac{-4}{3} \cdot x\right) \cdot x} + \left(\mathsf{neg}\left(2\right)\right) \cdot 1, x, 2\right)} - 1 \]

      10. metadata-eval N/A

          \[\leadsto \frac{2}{\mathsf{fma}\left(\left(2 + \frac{-4}{3} \cdot x\right) \cdot x + \color{blue}{-2} \cdot 1, x, 2\right)} - 1 \]

      11. metadata-eval N/A

          \[\leadsto \frac{2}{\mathsf{fma}\left(\left(2 + \frac{-4}{3} \cdot x\right) \cdot x + \color{blue}{-2}, x, 2\right)} - 1 \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{2}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(2 + \frac{-4}{3} \cdot x, x, -2\right)}, x, 2\right)} - 1 \]

      13. +-commutative N/A

          \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{-4}{3} \cdot x + 2}, x, -2\right), x, 2\right)} - 1 \]

      14. lower-fma.f64 98.3

          \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(-1.3333333333333333, x, 2\right)}, x, -2\right), x, 2\right)} - 1 \]

   5. Applied rewrites 98.3%

      \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-1.3333333333333333, x, 2\right), x, -2\right), x, 2\right)}} - 1 \]

   6. Taylor expanded in x around inf

      \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{-4}{3} \cdot x, x, -2\right), x, 2\right)} - 1 \]
   7. Step-by-step derivation

   8. Applied rewrites 98.3%

      \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(-1.3333333333333333 \cdot x, x, -2\right), x, 2\right)} - 1 \]

   9. Taylor expanded in x around inf

      \[\leadsto \frac{2}{{x}^{3} \cdot \color{blue}{\left(2 \cdot \frac{1}{x} - \frac{4}{3}\right)}} - 1 \]
   10. Step-by-step derivation

   11. Applied rewrites 98.3%

       \[\leadsto \frac{2}{\mathsf{fma}\left(-1.3333333333333333, x, 2\right) \cdot \color{blue}{\left(x \cdot x\right)}} - 1 \]

   if -1.32000000000000006 < x

   1. Initial program 37.4%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right) + 1\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{x \cdot \left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)\right) + x \cdot 1} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(x \cdot {x}^{2}\right) \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)} + x \cdot 1 \]

      4. *-rgt-identity N/A

         \[\leadsto \left(x \cdot {x}^{2}\right) \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right) + \color{blue}{x} \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot {x}^{2}, \frac{2}{15} \cdot {x}^{2} - \frac{1}{3}, x\right)} \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{2} \cdot x}, \frac{2}{15} \cdot {x}^{2} - \frac{1}{3}, x\right) \]

      7. pow-plus N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{\left(2 + 1\right)}}, \frac{2}{15} \cdot {x}^{2} - \frac{1}{3}, x\right) \]

      8. lower-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{\left(2 + 1\right)}}, \frac{2}{15} \cdot {x}^{2} - \frac{1}{3}, x\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left({x}^{\color{blue}{3}}, \frac{2}{15} \cdot {x}^{2} - \frac{1}{3}, x\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, \frac{2}{15} \cdot {x}^{2} - \color{blue}{\frac{1}{3} \cdot 1}, x\right) \]

      11. fp-cancel-sub-sign-inv N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{\frac{2}{15} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot 1}, x\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, \frac{2}{15} \cdot {x}^{2} + \color{blue}{\frac{-1}{3}} \cdot 1, x\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, \frac{2}{15} \cdot {x}^{2} + \color{blue}{\frac{-1}{3}}, x\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{\mathsf{fma}\left(\frac{2}{15}, {x}^{2}, \frac{-1}{3}\right)}, x\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\frac{2}{15}, \color{blue}{x \cdot x}, \frac{-1}{3}\right), x\right) \]

      16. lower-*.f64 69.5

          \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(0.13333333333333333, \color{blue}{x \cdot x}, -0.3333333333333333\right), x\right) \]

   5. Applied rewrites 69.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(0.13333333333333333, x \cdot x, -0.3333333333333333\right), x\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 69.5%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.13333333333333333, -0.3333333333333333\right) \cdot \left(x \cdot x\right), \color{blue}{x}, x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 75.2% accurate, 3.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.2:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(2, x, -2\right), x, 2\right)} - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.13333333333333333, -0.3333333333333333\right) \cdot \left(x \cdot x\right), x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.2)
   (- (/ 2.0 (fma (fma 2.0 x -2.0) x 2.0)) 1.0)
   (fma
    (* (fma (* x x) 0.13333333333333333 -0.3333333333333333) (* x x))
    x
    x)))

C

double code(double x) {
	double tmp;
	if (x <= -1.2) {
		tmp = (2.0 / fma(fma(2.0, x, -2.0), x, 2.0)) - 1.0;
	} else {
		tmp = fma((fma((x * x), 0.13333333333333333, -0.3333333333333333) * (x * x)), x, x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.2)
		tmp = Float64(Float64(2.0 / fma(fma(2.0, x, -2.0), x, 2.0)) - 1.0);
	else
		tmp = fma(Float64(fma(Float64(x * x), 0.13333333333333333, -0.3333333333333333) * Float64(x * x)), x, x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1.2], N[(N[(2.0 / N[(N[(2.0 * x + -2.0), $MachinePrecision] * x + 2.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(N[(x * x), $MachinePrecision] * 0.13333333333333333 + -0.3333333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * x + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.19999999999999996

   1. Initial program 100.0%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{2}{\color{blue}{2 + x \cdot \left(2 \cdot x - 2\right)}} - 1 \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2}{\color{blue}{x \cdot \left(2 \cdot x - 2\right) + 2}} - 1 \]

      2. *-commutative N/A

         \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot x - 2\right) \cdot x} + 2} - 1 \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(2 \cdot x - 2, x, 2\right)}} - 1 \]

      4. metadata-eval N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(2 \cdot x - \color{blue}{2 \cdot 1}, x, 2\right)} - 1 \]

      5. fp-cancel-sub-sign-inv N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(\color{blue}{2 \cdot x + \left(\mathsf{neg}\left(2\right)\right) \cdot 1}, x, 2\right)} - 1 \]

      6. metadata-eval N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(2 \cdot x + \color{blue}{-2} \cdot 1, x, 2\right)} - 1 \]

      7. metadata-eval N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(2 \cdot x + \color{blue}{-2}, x, 2\right)} - 1 \]

      8. lower-fma.f64 98.0

         \[\leadsto \frac{2}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(2, x, -2\right)}, x, 2\right)} - 1 \]

   5. Applied rewrites 98.0%

      \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(2, x, -2\right), x, 2\right)}} - 1 \]

   if -1.19999999999999996 < x

   1. Initial program 37.4%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right) + 1\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{x \cdot \left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)\right) + x \cdot 1} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(x \cdot {x}^{2}\right) \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)} + x \cdot 1 \]

      4. *-rgt-identity N/A

         \[\leadsto \left(x \cdot {x}^{2}\right) \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right) + \color{blue}{x} \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot {x}^{2}, \frac{2}{15} \cdot {x}^{2} - \frac{1}{3}, x\right)} \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{2} \cdot x}, \frac{2}{15} \cdot {x}^{2} - \frac{1}{3}, x\right) \]

      7. pow-plus N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{\left(2 + 1\right)}}, \frac{2}{15} \cdot {x}^{2} - \frac{1}{3}, x\right) \]

      8. lower-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{\left(2 + 1\right)}}, \frac{2}{15} \cdot {x}^{2} - \frac{1}{3}, x\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left({x}^{\color{blue}{3}}, \frac{2}{15} \cdot {x}^{2} - \frac{1}{3}, x\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, \frac{2}{15} \cdot {x}^{2} - \color{blue}{\frac{1}{3} \cdot 1}, x\right) \]

      11. fp-cancel-sub-sign-inv N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{\frac{2}{15} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot 1}, x\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, \frac{2}{15} \cdot {x}^{2} + \color{blue}{\frac{-1}{3}} \cdot 1, x\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, \frac{2}{15} \cdot {x}^{2} + \color{blue}{\frac{-1}{3}}, x\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{\mathsf{fma}\left(\frac{2}{15}, {x}^{2}, \frac{-1}{3}\right)}, x\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\frac{2}{15}, \color{blue}{x \cdot x}, \frac{-1}{3}\right), x\right) \]

      16. lower-*.f64 69.5

          \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(0.13333333333333333, \color{blue}{x \cdot x}, -0.3333333333333333\right), x\right) \]

   5. Applied rewrites 69.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(0.13333333333333333, x \cdot x, -0.3333333333333333\right), x\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 69.5%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.13333333333333333, -0.3333333333333333\right) \cdot \left(x \cdot x\right), \color{blue}{x}, x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 74.4% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(2, x, -2\right), x, 2\right)} - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333 \cdot \left(x \cdot x\right), x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.0)
   (- (/ 2.0 (fma (fma 2.0 x -2.0) x 2.0)) 1.0)
   (fma (* -0.3333333333333333 (* x x)) x x)))

C

double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = (2.0 / fma(fma(2.0, x, -2.0), x, 2.0)) - 1.0;
	} else {
		tmp = fma((-0.3333333333333333 * (x * x)), x, x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(Float64(2.0 / fma(fma(2.0, x, -2.0), x, 2.0)) - 1.0);
	else
		tmp = fma(Float64(-0.3333333333333333 * Float64(x * x)), x, x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1.0], N[(N[(2.0 / N[(N[(2.0 * x + -2.0), $MachinePrecision] * x + 2.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(-0.3333333333333333 * N[(x * x), $MachinePrecision]), $MachinePrecision] * x + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1

   1. Initial program 100.0%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{2}{\color{blue}{2 + x \cdot \left(2 \cdot x - 2\right)}} - 1 \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2}{\color{blue}{x \cdot \left(2 \cdot x - 2\right) + 2}} - 1 \]

      2. *-commutative N/A

         \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot x - 2\right) \cdot x} + 2} - 1 \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(2 \cdot x - 2, x, 2\right)}} - 1 \]

      4. metadata-eval N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(2 \cdot x - \color{blue}{2 \cdot 1}, x, 2\right)} - 1 \]

      5. fp-cancel-sub-sign-inv N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(\color{blue}{2 \cdot x + \left(\mathsf{neg}\left(2\right)\right) \cdot 1}, x, 2\right)} - 1 \]

      6. metadata-eval N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(2 \cdot x + \color{blue}{-2} \cdot 1, x, 2\right)} - 1 \]

      7. metadata-eval N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(2 \cdot x + \color{blue}{-2}, x, 2\right)} - 1 \]

      8. lower-fma.f64 98.0

         \[\leadsto \frac{2}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(2, x, -2\right)}, x, 2\right)} - 1 \]

   5. Applied rewrites 98.0%

      \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(2, x, -2\right), x, 2\right)}} - 1 \]

   if -1 < x

   1. Initial program 37.4%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right) + 1\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{x \cdot \left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)\right) + x \cdot 1} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(x \cdot {x}^{2}\right) \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)} + x \cdot 1 \]

      4. *-rgt-identity N/A

         \[\leadsto \left(x \cdot {x}^{2}\right) \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right) + \color{blue}{x} \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot {x}^{2}, \frac{2}{15} \cdot {x}^{2} - \frac{1}{3}, x\right)} \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{2} \cdot x}, \frac{2}{15} \cdot {x}^{2} - \frac{1}{3}, x\right) \]

      7. pow-plus N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{\left(2 + 1\right)}}, \frac{2}{15} \cdot {x}^{2} - \frac{1}{3}, x\right) \]

      8. lower-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{\left(2 + 1\right)}}, \frac{2}{15} \cdot {x}^{2} - \frac{1}{3}, x\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left({x}^{\color{blue}{3}}, \frac{2}{15} \cdot {x}^{2} - \frac{1}{3}, x\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, \frac{2}{15} \cdot {x}^{2} - \color{blue}{\frac{1}{3} \cdot 1}, x\right) \]

      11. fp-cancel-sub-sign-inv N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{\frac{2}{15} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot 1}, x\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, \frac{2}{15} \cdot {x}^{2} + \color{blue}{\frac{-1}{3}} \cdot 1, x\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, \frac{2}{15} \cdot {x}^{2} + \color{blue}{\frac{-1}{3}}, x\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{\mathsf{fma}\left(\frac{2}{15}, {x}^{2}, \frac{-1}{3}\right)}, x\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\frac{2}{15}, \color{blue}{x \cdot x}, \frac{-1}{3}\right), x\right) \]

      16. lower-*.f64 69.5

          \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(0.13333333333333333, \color{blue}{x \cdot x}, -0.3333333333333333\right), x\right) \]

   5. Applied rewrites 69.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(0.13333333333333333, x \cdot x, -0.3333333333333333\right), x\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 69.5%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.13333333333333333, -0.3333333333333333\right) \cdot \left(x \cdot x\right), \color{blue}{x}, x\right) \]

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(\frac{-1}{3} \cdot \left(x \cdot x\right), x, x\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 68.4%

       \[\leadsto \mathsf{fma}\left(-0.3333333333333333 \cdot \left(x \cdot x\right), x, x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 74.4% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.2:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(2, x, -2\right) \cdot x} - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333 \cdot \left(x \cdot x\right), x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.2)
   (- (/ 2.0 (* (fma 2.0 x -2.0) x)) 1.0)
   (fma (* -0.3333333333333333 (* x x)) x x)))

C

double code(double x) {
	double tmp;
	if (x <= -1.2) {
		tmp = (2.0 / (fma(2.0, x, -2.0) * x)) - 1.0;
	} else {
		tmp = fma((-0.3333333333333333 * (x * x)), x, x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.2)
		tmp = Float64(Float64(2.0 / Float64(fma(2.0, x, -2.0) * x)) - 1.0);
	else
		tmp = fma(Float64(-0.3333333333333333 * Float64(x * x)), x, x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1.2], N[(N[(2.0 / N[(N[(2.0 * x + -2.0), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(-0.3333333333333333 * N[(x * x), $MachinePrecision]), $MachinePrecision] * x + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.19999999999999996

   1. Initial program 100.0%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{2}{\color{blue}{2 + -2 \cdot x}} - 1 \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2}{\color{blue}{-2 \cdot x + 2}} - 1 \]

      2. lower-fma.f64 96.2

         \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(-2, x, 2\right)}} - 1 \]

   5. Applied rewrites 96.2%

      \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(-2, x, 2\right)}} - 1 \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{2}{\color{blue}{2 + x \cdot \left(2 \cdot x - 2\right)}} - 1 \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2}{\color{blue}{x \cdot \left(2 \cdot x - 2\right) + 2}} - 1 \]

      2. *-commutative N/A

         \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot x - 2\right) \cdot x} + 2} - 1 \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(2 \cdot x - 2, x, 2\right)}} - 1 \]

      4. metadata-eval N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(2 \cdot x - \color{blue}{2 \cdot 1}, x, 2\right)} - 1 \]

      5. fp-cancel-sub-sign-inv N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(\color{blue}{2 \cdot x + \left(\mathsf{neg}\left(2\right)\right) \cdot 1}, x, 2\right)} - 1 \]

      6. *-commutative N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(\color{blue}{x \cdot 2} + \left(\mathsf{neg}\left(2\right)\right) \cdot 1, x, 2\right)} - 1 \]

      7. metadata-eval N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(x \cdot 2 + \color{blue}{-2} \cdot 1, x, 2\right)} - 1 \]

      8. metadata-eval N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(x \cdot 2 + \color{blue}{-2}, x, 2\right)} - 1 \]

      9. lower-fma.f64 98.0

         \[\leadsto \frac{2}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(x, 2, -2\right)}, x, 2\right)} - 1 \]

   8. Applied rewrites 98.0%

      \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x, 2, -2\right), x, 2\right)}} - 1 \]

   9. Taylor expanded in x around inf

      \[\leadsto \frac{2}{{x}^{2} \cdot \color{blue}{\left(2 - 2 \cdot \frac{1}{x}\right)}} - 1 \]

   11. Applied rewrites 98.0%

       \[\leadsto \frac{2}{\mathsf{fma}\left(2, x, -2\right) \cdot \color{blue}{x}} - 1 \]

   if -1.19999999999999996 < x

   1. Initial program 37.4%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right) + 1\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{x \cdot \left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)\right) + x \cdot 1} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(x \cdot {x}^{2}\right) \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)} + x \cdot 1 \]

      4. *-rgt-identity N/A

         \[\leadsto \left(x \cdot {x}^{2}\right) \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right) + \color{blue}{x} \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot {x}^{2}, \frac{2}{15} \cdot {x}^{2} - \frac{1}{3}, x\right)} \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{2} \cdot x}, \frac{2}{15} \cdot {x}^{2} - \frac{1}{3}, x\right) \]

      7. pow-plus N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{\left(2 + 1\right)}}, \frac{2}{15} \cdot {x}^{2} - \frac{1}{3}, x\right) \]

      8. lower-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{\left(2 + 1\right)}}, \frac{2}{15} \cdot {x}^{2} - \frac{1}{3}, x\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left({x}^{\color{blue}{3}}, \frac{2}{15} \cdot {x}^{2} - \frac{1}{3}, x\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, \frac{2}{15} \cdot {x}^{2} - \color{blue}{\frac{1}{3} \cdot 1}, x\right) \]

      11. fp-cancel-sub-sign-inv N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{\frac{2}{15} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot 1}, x\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, \frac{2}{15} \cdot {x}^{2} + \color{blue}{\frac{-1}{3}} \cdot 1, x\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, \frac{2}{15} \cdot {x}^{2} + \color{blue}{\frac{-1}{3}}, x\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{\mathsf{fma}\left(\frac{2}{15}, {x}^{2}, \frac{-1}{3}\right)}, x\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\frac{2}{15}, \color{blue}{x \cdot x}, \frac{-1}{3}\right), x\right) \]

      16. lower-*.f64 69.5

          \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(0.13333333333333333, \color{blue}{x \cdot x}, -0.3333333333333333\right), x\right) \]

   5. Applied rewrites 69.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(0.13333333333333333, x \cdot x, -0.3333333333333333\right), x\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 69.5%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.13333333333333333, -0.3333333333333333\right) \cdot \left(x \cdot x\right), \color{blue}{x}, x\right) \]

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(\frac{-1}{3} \cdot \left(x \cdot x\right), x, x\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 68.4%

       \[\leadsto \mathsf{fma}\left(-0.3333333333333333 \cdot \left(x \cdot x\right), x, x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 74.4% accurate, 4.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.4:\\ \;\;\;\;\frac{2}{\left(2 \cdot x\right) \cdot x} - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333 \cdot \left(x \cdot x\right), x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.4)
   (- (/ 2.0 (* (* 2.0 x) x)) 1.0)
   (fma (* -0.3333333333333333 (* x x)) x x)))

C

double code(double x) {
	double tmp;
	if (x <= -1.4) {
		tmp = (2.0 / ((2.0 * x) * x)) - 1.0;
	} else {
		tmp = fma((-0.3333333333333333 * (x * x)), x, x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.4)
		tmp = Float64(Float64(2.0 / Float64(Float64(2.0 * x) * x)) - 1.0);
	else
		tmp = fma(Float64(-0.3333333333333333 * Float64(x * x)), x, x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1.4], N[(N[(2.0 / N[(N[(2.0 * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(-0.3333333333333333 * N[(x * x), $MachinePrecision]), $MachinePrecision] * x + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.3999999999999999

   1. Initial program 100.0%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \frac{2}{\color{blue}{2 + -2 \cdot x}} - 1 \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2}{\color{blue}{-2 \cdot x + 2}} - 1 \]

      2. lower-fma.f64 96.2

         \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(-2, x, 2\right)}} - 1 \]

   5. Applied rewrites 96.2%

      \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(-2, x, 2\right)}} - 1 \]

   6. Taylor expanded in x around 0

      \[\leadsto \frac{2}{\color{blue}{2 + x \cdot \left(2 \cdot x - 2\right)}} - 1 \]
   7. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2}{\color{blue}{x \cdot \left(2 \cdot x - 2\right) + 2}} - 1 \]

      2. *-commutative N/A

         \[\leadsto \frac{2}{\color{blue}{\left(2 \cdot x - 2\right) \cdot x} + 2} - 1 \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(2 \cdot x - 2, x, 2\right)}} - 1 \]

      4. metadata-eval N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(2 \cdot x - \color{blue}{2 \cdot 1}, x, 2\right)} - 1 \]

      5. fp-cancel-sub-sign-inv N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(\color{blue}{2 \cdot x + \left(\mathsf{neg}\left(2\right)\right) \cdot 1}, x, 2\right)} - 1 \]

      6. *-commutative N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(\color{blue}{x \cdot 2} + \left(\mathsf{neg}\left(2\right)\right) \cdot 1, x, 2\right)} - 1 \]

      7. metadata-eval N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(x \cdot 2 + \color{blue}{-2} \cdot 1, x, 2\right)} - 1 \]

      8. metadata-eval N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(x \cdot 2 + \color{blue}{-2}, x, 2\right)} - 1 \]

      9. lower-fma.f64 98.0

         \[\leadsto \frac{2}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(x, 2, -2\right)}, x, 2\right)} - 1 \]

   8. Applied rewrites 98.0%

      \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x, 2, -2\right), x, 2\right)}} - 1 \]

   9. Taylor expanded in x around inf

      \[\leadsto \frac{2}{2 \cdot \color{blue}{{x}^{2}}} - 1 \]
   10. Step-by-step derivation

   11. Applied rewrites 98.0%

       \[\leadsto \frac{2}{\left(2 \cdot x\right) \cdot \color{blue}{x}} - 1 \]

   if -1.3999999999999999 < x

   1. Initial program 37.4%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right) + 1\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{x \cdot \left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)\right) + x \cdot 1} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(x \cdot {x}^{2}\right) \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)} + x \cdot 1 \]

      4. *-rgt-identity N/A

         \[\leadsto \left(x \cdot {x}^{2}\right) \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right) + \color{blue}{x} \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot {x}^{2}, \frac{2}{15} \cdot {x}^{2} - \frac{1}{3}, x\right)} \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{2} \cdot x}, \frac{2}{15} \cdot {x}^{2} - \frac{1}{3}, x\right) \]

      7. pow-plus N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{\left(2 + 1\right)}}, \frac{2}{15} \cdot {x}^{2} - \frac{1}{3}, x\right) \]

      8. lower-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{\left(2 + 1\right)}}, \frac{2}{15} \cdot {x}^{2} - \frac{1}{3}, x\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left({x}^{\color{blue}{3}}, \frac{2}{15} \cdot {x}^{2} - \frac{1}{3}, x\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, \frac{2}{15} \cdot {x}^{2} - \color{blue}{\frac{1}{3} \cdot 1}, x\right) \]

      11. fp-cancel-sub-sign-inv N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{\frac{2}{15} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot 1}, x\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, \frac{2}{15} \cdot {x}^{2} + \color{blue}{\frac{-1}{3}} \cdot 1, x\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, \frac{2}{15} \cdot {x}^{2} + \color{blue}{\frac{-1}{3}}, x\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{\mathsf{fma}\left(\frac{2}{15}, {x}^{2}, \frac{-1}{3}\right)}, x\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\frac{2}{15}, \color{blue}{x \cdot x}, \frac{-1}{3}\right), x\right) \]

      16. lower-*.f64 69.5

          \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(0.13333333333333333, \color{blue}{x \cdot x}, -0.3333333333333333\right), x\right) \]

   5. Applied rewrites 69.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(0.13333333333333333, x \cdot x, -0.3333333333333333\right), x\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 69.5%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.13333333333333333, -0.3333333333333333\right) \cdot \left(x \cdot x\right), \color{blue}{x}, x\right) \]

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(\frac{-1}{3} \cdot \left(x \cdot x\right), x, x\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 68.4%

       \[\leadsto \mathsf{fma}\left(-0.3333333333333333 \cdot \left(x \cdot x\right), x, x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 74.1% accurate, 5.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.3:\\ \;\;\;\;\frac{-1}{-1 + x} - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333 \cdot \left(x \cdot x\right), x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.3)
   (- (/ -1.0 (+ -1.0 x)) 1.0)
   (fma (* -0.3333333333333333 (* x x)) x x)))

C

double code(double x) {
	double tmp;
	if (x <= -1.3) {
		tmp = (-1.0 / (-1.0 + x)) - 1.0;
	} else {
		tmp = fma((-0.3333333333333333 * (x * x)), x, x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.3)
		tmp = Float64(Float64(-1.0 / Float64(-1.0 + x)) - 1.0);
	else
		tmp = fma(Float64(-0.3333333333333333 * Float64(x * x)), x, x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1.3], N[(N[(-1.0 / N[(-1.0 + x), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(-0.3333333333333333 * N[(x * x), $MachinePrecision]), $MachinePrecision] * x + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.30000000000000004

   1. Initial program 100.0%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(1 + x\right)} - 1 \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\left(x + 1\right)} - 1 \]

      2. metadata-eval N/A

         \[\leadsto \left(x + \color{blue}{1 \cdot 1}\right) - 1 \]

      3. fp-cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{\left(x - \left(\mathsf{neg}\left(1\right)\right) \cdot 1\right)} - 1 \]

      4. *-rgt-identity N/A

         \[\leadsto \left(x - \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right) - 1 \]

      5. lower--.f64 N/A

         \[\leadsto \color{blue}{\left(x - \left(\mathsf{neg}\left(1\right)\right)\right)} - 1 \]

      6. metadata-eval 5.5

         \[\leadsto \left(x - \color{blue}{-1}\right) - 1 \]

   5. Applied rewrites 5.5%

      \[\leadsto \color{blue}{\left(x - -1\right)} - 1 \]
   6. Step-by-step derivation

   7. Applied rewrites 5.1%

      \[\leadsto \frac{\mathsf{fma}\left(x, x, -1\right)}{\color{blue}{-1 + x}} - 1 \]

   8. Taylor expanded in x around 0

      \[\leadsto \frac{-1}{\color{blue}{-1} + x} - 1 \]
   9. Step-by-step derivation

   10. Applied rewrites 96.2%

       \[\leadsto \frac{-1}{\color{blue}{-1} + x} - 1 \]

   if -1.30000000000000004 < x

   1. Initial program 37.4%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right) + 1\right)} \]

      2. distribute-lft-in N/A

         \[\leadsto \color{blue}{x \cdot \left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)\right) + x \cdot 1} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(x \cdot {x}^{2}\right) \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)} + x \cdot 1 \]

      4. *-rgt-identity N/A

         \[\leadsto \left(x \cdot {x}^{2}\right) \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right) + \color{blue}{x} \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot {x}^{2}, \frac{2}{15} \cdot {x}^{2} - \frac{1}{3}, x\right)} \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{2} \cdot x}, \frac{2}{15} \cdot {x}^{2} - \frac{1}{3}, x\right) \]

      7. pow-plus N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{\left(2 + 1\right)}}, \frac{2}{15} \cdot {x}^{2} - \frac{1}{3}, x\right) \]

      8. lower-pow.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{\left(2 + 1\right)}}, \frac{2}{15} \cdot {x}^{2} - \frac{1}{3}, x\right) \]

      9. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left({x}^{\color{blue}{3}}, \frac{2}{15} \cdot {x}^{2} - \frac{1}{3}, x\right) \]

      10. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, \frac{2}{15} \cdot {x}^{2} - \color{blue}{\frac{1}{3} \cdot 1}, x\right) \]

      11. fp-cancel-sub-sign-inv N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{\frac{2}{15} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot 1}, x\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, \frac{2}{15} \cdot {x}^{2} + \color{blue}{\frac{-1}{3}} \cdot 1, x\right) \]

      13. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, \frac{2}{15} \cdot {x}^{2} + \color{blue}{\frac{-1}{3}}, x\right) \]

      14. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{\mathsf{fma}\left(\frac{2}{15}, {x}^{2}, \frac{-1}{3}\right)}, x\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\frac{2}{15}, \color{blue}{x \cdot x}, \frac{-1}{3}\right), x\right) \]

      16. lower-*.f64 69.5

          \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(0.13333333333333333, \color{blue}{x \cdot x}, -0.3333333333333333\right), x\right) \]

   5. Applied rewrites 69.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(0.13333333333333333, x \cdot x, -0.3333333333333333\right), x\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 69.5%

      \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.13333333333333333, -0.3333333333333333\right) \cdot \left(x \cdot x\right), \color{blue}{x}, x\right) \]

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(\frac{-1}{3} \cdot \left(x \cdot x\right), x, x\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 68.4%

       \[\leadsto \mathsf{fma}\left(-0.3333333333333333 \cdot \left(x \cdot x\right), x, x\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 50.2% accurate, 7.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-0.3333333333333333 \cdot \left(x \cdot x\right), x, x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (fma (* -0.3333333333333333 (* x x)) x x))

C

double code(double x) {
	return fma((-0.3333333333333333 * (x * x)), x, x);
}

Julia

function code(x)
	return fma(Float64(-0.3333333333333333 * Float64(x * x)), x, x)
end

Wolfram

code[x_] := N[(N[(-0.3333333333333333 * N[(x * x), $MachinePrecision]), $MachinePrecision] * x + x), $MachinePrecision]

Derivation

1. Initial program 54.8%

   \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right) + 1\right)} \]

   2. distribute-lft-in N/A

      \[\leadsto \color{blue}{x \cdot \left({x}^{2} \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)\right) + x \cdot 1} \]

   3. associate-*r* N/A

      \[\leadsto \color{blue}{\left(x \cdot {x}^{2}\right) \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right)} + x \cdot 1 \]

   4. *-rgt-identity N/A

      \[\leadsto \left(x \cdot {x}^{2}\right) \cdot \left(\frac{2}{15} \cdot {x}^{2} - \frac{1}{3}\right) + \color{blue}{x} \]

   5. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot {x}^{2}, \frac{2}{15} \cdot {x}^{2} - \frac{1}{3}, x\right)} \]

   6. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{2} \cdot x}, \frac{2}{15} \cdot {x}^{2} - \frac{1}{3}, x\right) \]

   7. pow-plus N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{\left(2 + 1\right)}}, \frac{2}{15} \cdot {x}^{2} - \frac{1}{3}, x\right) \]

   8. lower-pow.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{\left(2 + 1\right)}}, \frac{2}{15} \cdot {x}^{2} - \frac{1}{3}, x\right) \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left({x}^{\color{blue}{3}}, \frac{2}{15} \cdot {x}^{2} - \frac{1}{3}, x\right) \]

   10. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left({x}^{3}, \frac{2}{15} \cdot {x}^{2} - \color{blue}{\frac{1}{3} \cdot 1}, x\right) \]

   11. fp-cancel-sub-sign-inv N/A

       \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{\frac{2}{15} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{3}\right)\right) \cdot 1}, x\right) \]

   12. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left({x}^{3}, \frac{2}{15} \cdot {x}^{2} + \color{blue}{\frac{-1}{3}} \cdot 1, x\right) \]

   13. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left({x}^{3}, \frac{2}{15} \cdot {x}^{2} + \color{blue}{\frac{-1}{3}}, x\right) \]

   14. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left({x}^{3}, \color{blue}{\mathsf{fma}\left(\frac{2}{15}, {x}^{2}, \frac{-1}{3}\right)}, x\right) \]

   15. unpow2 N/A

       \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(\frac{2}{15}, \color{blue}{x \cdot x}, \frac{-1}{3}\right), x\right) \]

   16. lower-*.f64 51.3

       \[\leadsto \mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(0.13333333333333333, \color{blue}{x \cdot x}, -0.3333333333333333\right), x\right) \]

5. Applied rewrites 51.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, \mathsf{fma}\left(0.13333333333333333, x \cdot x, -0.3333333333333333\right), x\right)} \]
6. Step-by-step derivation

7. Applied rewrites 51.3%

   \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.13333333333333333, -0.3333333333333333\right) \cdot \left(x \cdot x\right), \color{blue}{x}, x\right) \]

8. Taylor expanded in x around 0

   \[\leadsto \mathsf{fma}\left(\frac{-1}{3} \cdot \left(x \cdot x\right), x, x\right) \]
9. Step-by-step derivation

10. Applied rewrites 49.7%

    \[\leadsto \mathsf{fma}\left(-0.3333333333333333 \cdot \left(x \cdot x\right), x, x\right) \]
11. Add Preprocessing

Alternative 14: 6.6% accurate, 17.6× speedup

Math

\[\begin{array}{l} \\ \left(x - -1\right) - 1 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- (- x -1.0) 1.0))

C

double code(double x) {
	return (x - -1.0) - 1.0;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

def code(x):
	return (x - -1.0) - 1.0

Julia

function code(x)
	return Float64(Float64(x - -1.0) - 1.0)
end

MATLAB

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

Wolfram

code[x_] := N[(N[(x - -1.0), $MachinePrecision] - 1.0), $MachinePrecision]

Derivation

1. Initial program 54.8%

   \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
2. Add Preprocessing

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\left(1 + x\right)} - 1 \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{\left(x + 1\right)} - 1 \]

   2. metadata-eval N/A

      \[\leadsto \left(x + \color{blue}{1 \cdot 1}\right) - 1 \]

   3. fp-cancel-sign-sub-inv N/A

      \[\leadsto \color{blue}{\left(x - \left(\mathsf{neg}\left(1\right)\right) \cdot 1\right)} - 1 \]

   4. *-rgt-identity N/A

      \[\leadsto \left(x - \color{blue}{\left(\mathsf{neg}\left(1\right)\right)}\right) - 1 \]

   5. lower--.f64 N/A

      \[\leadsto \color{blue}{\left(x - \left(\mathsf{neg}\left(1\right)\right)\right)} - 1 \]

   6. metadata-eval 6.6

      \[\leadsto \left(x - \color{blue}{-1}\right) - 1 \]

5. Applied rewrites 6.6%

   \[\leadsto \color{blue}{\left(x - -1\right)} - 1 \]
6. Add Preprocessing

Alternative 15: 4.2% accurate, 30.8× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (- 1.0 1.0))

C

double code(double x) {
	return 1.0 - 1.0;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

def code(x):
	return 1.0 - 1.0

Julia

function code(x)
	return Float64(1.0 - 1.0)
end

MATLAB

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

Wolfram

code[x_] := N[(1.0 - 1.0), $MachinePrecision]

Derivation

1. Initial program 54.8%

   \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

5. Applied rewrites 4.1%

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

Reproduce

herbie shell --seed 2024360 
(FPCore (x)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))