Logistic function from Lakshay Garg

Percentage Accurate: 54.4% → 99.9%

Time: 3.5s

Alternatives: 14

Speedup: 4.7×

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: 54.4% 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: 99.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.3:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(x, x, \left(-x\right) \cdot x\right)}{x - x} \cdot x} - 1\\ \mathbf{elif}\;x \leq 0.025:\\ \;\;\;\;\mathsf{fma}\left(\left(x \cdot x\right) \cdot x, \mathsf{fma}\left(\mathsf{fma}\left(-0.05396825396825397, x \cdot x, 0.13333333333333333\right), x \cdot x, -0.3333333333333333\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.3)
   (- (/ 2.0 (* (/ (fma x x (* (- x) x)) (- x x)) x)) 1.0)
   (if (<= x 0.025)
     (fma
      (* (* x x) x)
      (fma
       (fma -0.05396825396825397 (* x x) 0.13333333333333333)
       (* x x)
       -0.3333333333333333)
      x)
     (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))))

C

double code(double x) {
	double tmp;
	if (x <= -1.3) {
		tmp = (2.0 / ((fma(x, x, (-x * x)) / (x - x)) * x)) - 1.0;
	} else if (x <= 0.025) {
		tmp = fma(((x * x) * x), fma(fma(-0.05396825396825397, (x * x), 0.13333333333333333), (x * x), -0.3333333333333333), x);
	} else {
		tmp = (2.0 / (1.0 + exp((-2.0 * x)))) - 1.0;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.3)
		tmp = Float64(Float64(2.0 / Float64(Float64(fma(x, x, Float64(Float64(-x) * x)) / Float64(x - x)) * x)) - 1.0);
	elseif (x <= 0.025)
		tmp = fma(Float64(Float64(x * x) * x), fma(fma(-0.05396825396825397, Float64(x * x), 0.13333333333333333), Float64(x * x), -0.3333333333333333), x);
	else
		tmp = Float64(Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x)))) - 1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1.3], N[(N[(2.0 / N[(N[(N[(x * x + N[((-x) * x), $MachinePrecision]), $MachinePrecision] / N[(x - x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], If[LessEqual[x, 0.025], N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * N[(N[(-0.05396825396825397 * N[(x * x), $MachinePrecision] + 0.13333333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] + x), $MachinePrecision], N[(N[(2.0 / N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.30000000000000004

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. metadata-eval N/A

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

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

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 99.1

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in x around inf

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

      1. pow2 N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 99.1

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

   7. Applied rewrites 99.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 99.1

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

   9. Applied rewrites 99.1%

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

       1. lift-+.f64 N/A

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

       2. flip-+ N/A

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

       3. lower-/.f64 N/A

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

       4. pow2 N/A

          \[\leadsto \frac{2}{\frac{{x}^{2} - x \cdot x}{x - x} \cdot x} - 1 \]

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

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

       6. pow2 N/A

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

       7. distribute-lft-neg-in N/A

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

       8. pow2 N/A

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

       9. lower-fma.f64 N/A

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

       10. pow2 N/A

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

       11. distribute-lft-neg-in N/A

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

       12. lower-*.f64 N/A

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

       13. lower-neg.f64 N/A

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

       14. lower--.f64 99.9

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

   11. Applied rewrites 99.9%

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

   if -1.30000000000000004 < x < 0.025000000000000001

   1. Initial program 8.1%

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

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x} \]
   3. Step-by-step derivation

   4. Applied rewrites 99.2%

      \[\leadsto \color{blue}{x} \]

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. pow2 N/A

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

      5. cube-mult N/A

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

      6. *-rgt-identity N/A

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

      7. lower-fma.f64 N/A

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

   7. Applied rewrites 99.8%

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

      1. lift-pow.f64 N/A

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

      2. pow3 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 99.8

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

   9. Applied rewrites 99.8%

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

   if 0.025000000000000001 < x

   1. Initial program 100.0%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 99.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{1 + e^{-2 \cdot x}} - 1\\ \mathbf{if}\;t\_0 \leq -1:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-1.3333333333333333, x, 2\right), x, -2\right), x, 2\right)} - 1\\ \mathbf{elif}\;t\_0 \leq 0.1:\\ \;\;\;\;\mathsf{fma}\left(\left(x \cdot x\right) \cdot x, \mathsf{fma}\left(\mathsf{fma}\left(-0.05396825396825397, x \cdot x, 0.13333333333333333\right), x \cdot x, -0.3333333333333333\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, 1, -1\right)}{x - -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0)))
   (if (<= t_0 -1.0)
     (- (/ 2.0 (fma (fma (fma -1.3333333333333333 x 2.0) x -2.0) x 2.0)) 1.0)
     (if (<= t_0 0.1)
       (fma
        (* (* x x) x)
        (fma
         (fma -0.05396825396825397 (* x x) 0.13333333333333333)
         (* x x)
         -0.3333333333333333)
        x)
       (/ (fma x 1.0 -1.0) (- x -1.0))))))

C

double code(double x) {
	double t_0 = (2.0 / (1.0 + exp((-2.0 * x)))) - 1.0;
	double tmp;
	if (t_0 <= -1.0) {
		tmp = (2.0 / fma(fma(fma(-1.3333333333333333, x, 2.0), x, -2.0), x, 2.0)) - 1.0;
	} else if (t_0 <= 0.1) {
		tmp = fma(((x * x) * x), fma(fma(-0.05396825396825397, (x * x), 0.13333333333333333), (x * x), -0.3333333333333333), x);
	} else {
		tmp = fma(x, 1.0, -1.0) / (x - -1.0);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x)))) - 1.0)
	tmp = 0.0
	if (t_0 <= -1.0)
		tmp = Float64(Float64(2.0 / fma(fma(fma(-1.3333333333333333, x, 2.0), x, -2.0), x, 2.0)) - 1.0);
	elseif (t_0 <= 0.1)
		tmp = fma(Float64(Float64(x * x) * x), fma(fma(-0.05396825396825397, Float64(x * x), 0.13333333333333333), Float64(x * x), -0.3333333333333333), x);
	else
		tmp = Float64(fma(x, 1.0, -1.0) / Float64(x - -1.0));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(2.0 / N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]}, If[LessEqual[t$95$0, -1.0], 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], If[LessEqual[t$95$0, 0.1], N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * N[(N[(-0.05396825396825397 * N[(x * x), $MachinePrecision] + 0.13333333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] + x), $MachinePrecision], N[(N[(x * 1.0 + -1.0), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (/.f64 #s(literal 2 binary64) (+.f64 #s(literal 1 binary64) (exp.f64 (*.f64 #s(literal -2 binary64) x)))) #s(literal 1 binary64)) < -1

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. metadata-eval N/A

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

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

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 99.5

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

   4. Applied rewrites 99.5%

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

   if -1 < (-.f64 (/.f64 #s(literal 2 binary64) (+.f64 #s(literal 1 binary64) (exp.f64 (*.f64 #s(literal -2 binary64) x)))) #s(literal 1 binary64)) < 0.10000000000000001

   1. Initial program 8.3%

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

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x} \]
   3. Step-by-step derivation

   4. Applied rewrites 99.1%

      \[\leadsto \color{blue}{x} \]

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. pow2 N/A

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

      5. cube-mult N/A

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

      6. *-rgt-identity N/A

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

      7. lower-fma.f64 N/A

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

   7. Applied rewrites 99.7%

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

      1. lift-pow.f64 N/A

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

      2. pow3 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 99.7

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

   9. Applied rewrites 99.7%

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

   if 0.10000000000000001 < (-.f64 (/.f64 #s(literal 2 binary64) (+.f64 #s(literal 1 binary64) (exp.f64 (*.f64 #s(literal -2 binary64) x)))) #s(literal 1 binary64))

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. metadata-eval N/A

         \[\leadsto \left(x - -1 \cdot 1\right) - 1 \]

      5. metadata-eval N/A

         \[\leadsto \left(x - -1\right) - 1 \]

      6. lower--.f64 5.4

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

   4. Applied rewrites 5.4%

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

   5. Taylor expanded in x around inf

      \[\leadsto x - 1 \]
   6. Step-by-step derivation

   7. Applied rewrites 5.4%

      \[\leadsto x - 1 \]
   8. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

   9. Applied rewrites 5.1%

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

   10. Taylor expanded in x around 0

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

   12. Applied rewrites 97.9%

       \[\leadsto \frac{\mathsf{fma}\left(x, 1, -1\right)}{x - -1} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 99.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{1 + e^{-2 \cdot x}} - 1\\ \mathbf{if}\;t\_0 \leq -1:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-1.3333333333333333, x, 2\right), x, -2\right), x, 2\right)} - 1\\ \mathbf{elif}\;t\_0 \leq 0.1:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.13333333333333333, -0.3333333333333333\right) \cdot x, x \cdot x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, 1, -1\right)}{x - -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0)))
   (if (<= t_0 -1.0)
     (- (/ 2.0 (fma (fma (fma -1.3333333333333333 x 2.0) x -2.0) x 2.0)) 1.0)
     (if (<= t_0 0.1)
       (fma
        (* (fma (* x x) 0.13333333333333333 -0.3333333333333333) x)
        (* x x)
        x)
       (/ (fma x 1.0 -1.0) (- x -1.0))))))

C

double code(double x) {
	double t_0 = (2.0 / (1.0 + exp((-2.0 * x)))) - 1.0;
	double tmp;
	if (t_0 <= -1.0) {
		tmp = (2.0 / fma(fma(fma(-1.3333333333333333, x, 2.0), x, -2.0), x, 2.0)) - 1.0;
	} else if (t_0 <= 0.1) {
		tmp = fma((fma((x * x), 0.13333333333333333, -0.3333333333333333) * x), (x * x), x);
	} else {
		tmp = fma(x, 1.0, -1.0) / (x - -1.0);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x)))) - 1.0)
	tmp = 0.0
	if (t_0 <= -1.0)
		tmp = Float64(Float64(2.0 / fma(fma(fma(-1.3333333333333333, x, 2.0), x, -2.0), x, 2.0)) - 1.0);
	elseif (t_0 <= 0.1)
		tmp = fma(Float64(fma(Float64(x * x), 0.13333333333333333, -0.3333333333333333) * x), Float64(x * x), x);
	else
		tmp = Float64(fma(x, 1.0, -1.0) / Float64(x - -1.0));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(2.0 / N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]}, If[LessEqual[t$95$0, -1.0], 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], If[LessEqual[t$95$0, 0.1], N[(N[(N[(N[(x * x), $MachinePrecision] * 0.13333333333333333 + -0.3333333333333333), $MachinePrecision] * x), $MachinePrecision] * N[(x * x), $MachinePrecision] + x), $MachinePrecision], N[(N[(x * 1.0 + -1.0), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (/.f64 #s(literal 2 binary64) (+.f64 #s(literal 1 binary64) (exp.f64 (*.f64 #s(literal -2 binary64) x)))) #s(literal 1 binary64)) < -1

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. metadata-eval N/A

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

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

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 99.5

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

   4. Applied rewrites 99.5%

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

   if -1 < (-.f64 (/.f64 #s(literal 2 binary64) (+.f64 #s(literal 1 binary64) (exp.f64 (*.f64 #s(literal -2 binary64) x)))) #s(literal 1 binary64)) < 0.10000000000000001

   1. Initial program 8.3%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. cube-mult N/A

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

      6. *-rgt-identity N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 99.7

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

   4. Applied rewrites 99.7%

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

      1. lift-pow.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. cube-mult N/A

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

      8. pow2 N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

   6. Applied rewrites 99.7%

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

   if 0.10000000000000001 < (-.f64 (/.f64 #s(literal 2 binary64) (+.f64 #s(literal 1 binary64) (exp.f64 (*.f64 #s(literal -2 binary64) x)))) #s(literal 1 binary64))

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. metadata-eval N/A

         \[\leadsto \left(x - -1 \cdot 1\right) - 1 \]

      5. metadata-eval N/A

         \[\leadsto \left(x - -1\right) - 1 \]

      6. lower--.f64 5.4

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

   4. Applied rewrites 5.4%

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

   5. Taylor expanded in x around inf

      \[\leadsto x - 1 \]
   6. Step-by-step derivation

   7. Applied rewrites 5.4%

      \[\leadsto x - 1 \]
   8. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

   9. Applied rewrites 5.1%

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

   10. Taylor expanded in x around 0

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

   12. Applied rewrites 97.9%

       \[\leadsto \frac{\mathsf{fma}\left(x, 1, -1\right)}{x - -1} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 99.3% accurate, 2.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.3:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(x, x, \left(-x\right) \cdot x\right)}{x - x} \cdot x} - 1\\ \mathbf{elif}\;x \leq 1.6:\\ \;\;\;\;\mathsf{fma}\left(\left(x \cdot x\right) \cdot x, \mathsf{fma}\left(\mathsf{fma}\left(-0.05396825396825397, x \cdot x, 0.13333333333333333\right), x \cdot x, -0.3333333333333333\right), x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, 1, -1\right)}{x - -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.3)
   (- (/ 2.0 (* (/ (fma x x (* (- x) x)) (- x x)) x)) 1.0)
   (if (<= x 1.6)
     (fma
      (* (* x x) x)
      (fma
       (fma -0.05396825396825397 (* x x) 0.13333333333333333)
       (* x x)
       -0.3333333333333333)
      x)
     (/ (fma x 1.0 -1.0) (- x -1.0)))))

C

double code(double x) {
	double tmp;
	if (x <= -1.3) {
		tmp = (2.0 / ((fma(x, x, (-x * x)) / (x - x)) * x)) - 1.0;
	} else if (x <= 1.6) {
		tmp = fma(((x * x) * x), fma(fma(-0.05396825396825397, (x * x), 0.13333333333333333), (x * x), -0.3333333333333333), x);
	} else {
		tmp = fma(x, 1.0, -1.0) / (x - -1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.3)
		tmp = Float64(Float64(2.0 / Float64(Float64(fma(x, x, Float64(Float64(-x) * x)) / Float64(x - x)) * x)) - 1.0);
	elseif (x <= 1.6)
		tmp = fma(Float64(Float64(x * x) * x), fma(fma(-0.05396825396825397, Float64(x * x), 0.13333333333333333), Float64(x * x), -0.3333333333333333), x);
	else
		tmp = Float64(fma(x, 1.0, -1.0) / Float64(x - -1.0));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1.3], N[(N[(2.0 / N[(N[(N[(x * x + N[((-x) * x), $MachinePrecision]), $MachinePrecision] / N[(x - x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], If[LessEqual[x, 1.6], N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * N[(N[(-0.05396825396825397 * N[(x * x), $MachinePrecision] + 0.13333333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] + x), $MachinePrecision], N[(N[(x * 1.0 + -1.0), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.30000000000000004

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. metadata-eval N/A

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

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

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 99.1

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in x around inf

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

      1. pow2 N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 99.1

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

   7. Applied rewrites 99.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 99.1

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

   9. Applied rewrites 99.1%

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

       1. lift-+.f64 N/A

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

       2. flip-+ N/A

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

       3. lower-/.f64 N/A

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

       4. pow2 N/A

          \[\leadsto \frac{2}{\frac{{x}^{2} - x \cdot x}{x - x} \cdot x} - 1 \]

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

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

       6. pow2 N/A

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

       7. distribute-lft-neg-in N/A

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

       8. pow2 N/A

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

       9. lower-fma.f64 N/A

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

       10. pow2 N/A

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

       11. distribute-lft-neg-in N/A

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

       12. lower-*.f64 N/A

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

       13. lower-neg.f64 N/A

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

       14. lower--.f64 99.9

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

   11. Applied rewrites 99.9%

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

   if -1.30000000000000004 < x < 1.6000000000000001

   1. Initial program 8.3%

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

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x} \]
   3. Step-by-step derivation

   4. Applied rewrites 99.0%

      \[\leadsto \color{blue}{x} \]

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. pow2 N/A

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

      5. cube-mult N/A

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

      6. *-rgt-identity N/A

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

      7. lower-fma.f64 N/A

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

   7. Applied rewrites 99.7%

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

      1. lift-pow.f64 N/A

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

      2. pow3 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 99.7

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

   9. Applied rewrites 99.7%

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

   if 1.6000000000000001 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. metadata-eval N/A

         \[\leadsto \left(x - -1 \cdot 1\right) - 1 \]

      5. metadata-eval N/A

         \[\leadsto \left(x - -1\right) - 1 \]

      6. lower--.f64 5.4

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

   4. Applied rewrites 5.4%

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

   5. Taylor expanded in x around inf

      \[\leadsto x - 1 \]
   6. Step-by-step derivation

   7. Applied rewrites 5.4%

      \[\leadsto x - 1 \]
   8. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

   9. Applied rewrites 5.1%

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

   10. Taylor expanded in x around 0

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

   12. Applied rewrites 98.0%

       \[\leadsto \frac{\mathsf{fma}\left(x, 1, -1\right)}{x - -1} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 99.2% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.2:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(-1.3333333333333333 \cdot x, x, -2\right), x, 2\right)} - 1\\ \mathbf{elif}\;x \leq 1.92:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.13333333333333333, -0.3333333333333333\right) \cdot x, x \cdot x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, 1, -1\right)}{x - -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.2)
   (- (/ 2.0 (fma (fma (* -1.3333333333333333 x) x -2.0) x 2.0)) 1.0)
   (if (<= x 1.92)
     (fma
      (* (fma (* x x) 0.13333333333333333 -0.3333333333333333) x)
      (* x x)
      x)
     (/ (fma x 1.0 -1.0) (- x -1.0)))))

C

double code(double x) {
	double tmp;
	if (x <= -1.2) {
		tmp = (2.0 / fma(fma((-1.3333333333333333 * x), x, -2.0), x, 2.0)) - 1.0;
	} else if (x <= 1.92) {
		tmp = fma((fma((x * x), 0.13333333333333333, -0.3333333333333333) * x), (x * x), x);
	} else {
		tmp = fma(x, 1.0, -1.0) / (x - -1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.2)
		tmp = Float64(Float64(2.0 / fma(fma(Float64(-1.3333333333333333 * x), x, -2.0), x, 2.0)) - 1.0);
	elseif (x <= 1.92)
		tmp = fma(Float64(fma(Float64(x * x), 0.13333333333333333, -0.3333333333333333) * x), Float64(x * x), x);
	else
		tmp = Float64(fma(x, 1.0, -1.0) / Float64(x - -1.0));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1.2], N[(N[(2.0 / N[(N[(N[(-1.3333333333333333 * x), $MachinePrecision] * x + -2.0), $MachinePrecision] * x + 2.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], If[LessEqual[x, 1.92], N[(N[(N[(N[(x * x), $MachinePrecision] * 0.13333333333333333 + -0.3333333333333333), $MachinePrecision] * x), $MachinePrecision] * N[(x * x), $MachinePrecision] + x), $MachinePrecision], N[(N[(x * 1.0 + -1.0), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.19999999999999996

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. metadata-eval N/A

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

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

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 99.5

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

   7. Applied rewrites 99.5%

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

   if -1.19999999999999996 < x < 1.9199999999999999

   1. Initial program 8.3%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. cube-mult N/A

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

      6. *-rgt-identity N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 99.6

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

   4. Applied rewrites 99.6%

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

      1. lift-pow.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. cube-mult N/A

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

      8. pow2 N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

   6. Applied rewrites 99.6%

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

   if 1.9199999999999999 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. metadata-eval N/A

         \[\leadsto \left(x - -1 \cdot 1\right) - 1 \]

      5. metadata-eval N/A

         \[\leadsto \left(x - -1\right) - 1 \]

      6. lower--.f64 5.4

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

   4. Applied rewrites 5.4%

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

   5. Taylor expanded in x around inf

      \[\leadsto x - 1 \]
   6. Step-by-step derivation

   7. Applied rewrites 5.4%

      \[\leadsto x - 1 \]
   8. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

   9. Applied rewrites 5.1%

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

   10. Taylor expanded in x around 0

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

   12. Applied rewrites 98.0%

       \[\leadsto \frac{\mathsf{fma}\left(x, 1, -1\right)}{x - -1} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 99.2% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.4:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(\left(-1.3333333333333333 \cdot x\right) \cdot x, x, 2\right)} - 1\\ \mathbf{elif}\;x \leq 1.92:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.13333333333333333, -0.3333333333333333\right) \cdot x, x \cdot x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, 1, -1\right)}{x - -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.4)
   (- (/ 2.0 (fma (* (* -1.3333333333333333 x) x) x 2.0)) 1.0)
   (if (<= x 1.92)
     (fma
      (* (fma (* x x) 0.13333333333333333 -0.3333333333333333) x)
      (* x x)
      x)
     (/ (fma x 1.0 -1.0) (- x -1.0)))))

C

double code(double x) {
	double tmp;
	if (x <= -1.4) {
		tmp = (2.0 / fma(((-1.3333333333333333 * x) * x), x, 2.0)) - 1.0;
	} else if (x <= 1.92) {
		tmp = fma((fma((x * x), 0.13333333333333333, -0.3333333333333333) * x), (x * x), x);
	} else {
		tmp = fma(x, 1.0, -1.0) / (x - -1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.4)
		tmp = Float64(Float64(2.0 / fma(Float64(Float64(-1.3333333333333333 * x) * x), x, 2.0)) - 1.0);
	elseif (x <= 1.92)
		tmp = fma(Float64(fma(Float64(x * x), 0.13333333333333333, -0.3333333333333333) * x), Float64(x * x), x);
	else
		tmp = Float64(fma(x, 1.0, -1.0) / Float64(x - -1.0));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1.4], N[(N[(2.0 / N[(N[(N[(-1.3333333333333333 * x), $MachinePrecision] * x), $MachinePrecision] * x + 2.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], If[LessEqual[x, 1.92], N[(N[(N[(N[(x * x), $MachinePrecision] * 0.13333333333333333 + -0.3333333333333333), $MachinePrecision] * x), $MachinePrecision] * N[(x * x), $MachinePrecision] + x), $MachinePrecision], N[(N[(x * 1.0 + -1.0), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.3999999999999999

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. metadata-eval N/A

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

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

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in x around inf

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

      1. pow2 N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-*.f64 99.5

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

   7. Applied rewrites 99.5%

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

   if -1.3999999999999999 < x < 1.9199999999999999

   1. Initial program 8.3%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. cube-mult N/A

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

      6. *-rgt-identity N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 99.6

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

   4. Applied rewrites 99.6%

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

      1. lift-pow.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. cube-mult N/A

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

      8. pow2 N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

   6. Applied rewrites 99.6%

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

   if 1.9199999999999999 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. metadata-eval N/A

         \[\leadsto \left(x - -1 \cdot 1\right) - 1 \]

      5. metadata-eval N/A

         \[\leadsto \left(x - -1\right) - 1 \]

      6. lower--.f64 5.4

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

   4. Applied rewrites 5.4%

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

   5. Taylor expanded in x around inf

      \[\leadsto x - 1 \]
   6. Step-by-step derivation

   7. Applied rewrites 5.4%

      \[\leadsto x - 1 \]
   8. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

   9. Applied rewrites 5.1%

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

   10. Taylor expanded in x around 0

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

   12. Applied rewrites 98.0%

       \[\leadsto \frac{\mathsf{fma}\left(x, 1, -1\right)}{x - -1} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 99.1% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.2:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(x, 2, -2\right), x, 2\right)} - 1\\ \mathbf{elif}\;x \leq 1.92:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.13333333333333333, -0.3333333333333333\right) \cdot x, x \cdot x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, 1, -1\right)}{x - -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.2)
   (- (/ 2.0 (fma (fma x 2.0 -2.0) x 2.0)) 1.0)
   (if (<= x 1.92)
     (fma
      (* (fma (* x x) 0.13333333333333333 -0.3333333333333333) x)
      (* x x)
      x)
     (/ (fma x 1.0 -1.0) (- x -1.0)))))

C

double code(double x) {
	double tmp;
	if (x <= -1.2) {
		tmp = (2.0 / fma(fma(x, 2.0, -2.0), x, 2.0)) - 1.0;
	} else if (x <= 1.92) {
		tmp = fma((fma((x * x), 0.13333333333333333, -0.3333333333333333) * x), (x * x), x);
	} else {
		tmp = fma(x, 1.0, -1.0) / (x - -1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.2)
		tmp = Float64(Float64(2.0 / fma(fma(x, 2.0, -2.0), x, 2.0)) - 1.0);
	elseif (x <= 1.92)
		tmp = fma(Float64(fma(Float64(x * x), 0.13333333333333333, -0.3333333333333333) * x), Float64(x * x), x);
	else
		tmp = Float64(fma(x, 1.0, -1.0) / Float64(x - -1.0));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1.2], N[(N[(2.0 / N[(N[(x * 2.0 + -2.0), $MachinePrecision] * x + 2.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], If[LessEqual[x, 1.92], N[(N[(N[(N[(x * x), $MachinePrecision] * 0.13333333333333333 + -0.3333333333333333), $MachinePrecision] * x), $MachinePrecision] * N[(x * x), $MachinePrecision] + x), $MachinePrecision], N[(N[(x * 1.0 + -1.0), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.19999999999999996

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. metadata-eval N/A

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

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

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 99.1

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

   4. Applied rewrites 99.1%

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

   if -1.19999999999999996 < x < 1.9199999999999999

   1. Initial program 8.3%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. cube-mult N/A

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

      6. *-rgt-identity N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 99.6

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

   4. Applied rewrites 99.6%

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

      1. lift-pow.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. cube-mult N/A

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

      8. pow2 N/A

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

      9. associate-*r* N/A

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

      10. lower-fma.f64 N/A

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

   6. Applied rewrites 99.6%

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

   if 1.9199999999999999 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. metadata-eval N/A

         \[\leadsto \left(x - -1 \cdot 1\right) - 1 \]

      5. metadata-eval N/A

         \[\leadsto \left(x - -1\right) - 1 \]

      6. lower--.f64 5.4

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

   4. Applied rewrites 5.4%

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

   5. Taylor expanded in x around inf

      \[\leadsto x - 1 \]
   6. Step-by-step derivation

   7. Applied rewrites 5.4%

      \[\leadsto x - 1 \]
   8. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

   9. Applied rewrites 5.1%

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

   10. Taylor expanded in x around 0

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

   12. Applied rewrites 98.0%

       \[\leadsto \frac{\mathsf{fma}\left(x, 1, -1\right)}{x - -1} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 99.0% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(x, 2, -2\right), x, 2\right)} - 1\\ \mathbf{elif}\;x \leq 1.6:\\ \;\;\;\;\mathsf{fma}\left(\left(x \cdot x\right) \cdot x, -0.3333333333333333, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, 1, -1\right)}{x - -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.0)
   (- (/ 2.0 (fma (fma x 2.0 -2.0) x 2.0)) 1.0)
   (if (<= x 1.6)
     (fma (* (* x x) x) -0.3333333333333333 x)
     (/ (fma x 1.0 -1.0) (- x -1.0)))))

C

double code(double x) {
	double tmp;
	if (x <= -1.0) {
		tmp = (2.0 / fma(fma(x, 2.0, -2.0), x, 2.0)) - 1.0;
	} else if (x <= 1.6) {
		tmp = fma(((x * x) * x), -0.3333333333333333, x);
	} else {
		tmp = fma(x, 1.0, -1.0) / (x - -1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(Float64(2.0 / fma(fma(x, 2.0, -2.0), x, 2.0)) - 1.0);
	elseif (x <= 1.6)
		tmp = fma(Float64(Float64(x * x) * x), -0.3333333333333333, x);
	else
		tmp = Float64(fma(x, 1.0, -1.0) / Float64(x - -1.0));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1.0], N[(N[(2.0 / N[(N[(x * 2.0 + -2.0), $MachinePrecision] * x + 2.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], If[LessEqual[x, 1.6], N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * -0.3333333333333333 + x), $MachinePrecision], N[(N[(x * 1.0 + -1.0), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. metadata-eval N/A

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

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

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 99.0

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

   4. Applied rewrites 99.0%

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

   if -1 < x < 1.6000000000000001

   1. Initial program 8.3%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. unpow2 N/A

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

      6. cube-mult N/A

         \[\leadsto {x}^{3} \cdot \frac{-1}{3} + x \cdot 1 \]

      7. *-rgt-identity N/A

         \[\leadsto {x}^{3} \cdot \frac{-1}{3} + x \]

      8. lower-fma.f64 N/A

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

      9. lower-pow.f64 99.6

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

   4. Applied rewrites 99.6%

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

      1. lift-pow.f64 N/A

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

      2. unpow3 N/A

         \[\leadsto \mathsf{fma}\left(\left(x \cdot x\right) \cdot x, \frac{-1}{3}, x\right) \]

      3. pow2 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\left(x \cdot x\right) \cdot x, \frac{-1}{3}, x\right) \]

      6. lift-*.f64 99.6

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

   6. Applied rewrites 99.6%

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

   if 1.6000000000000001 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. metadata-eval N/A

         \[\leadsto \left(x - -1 \cdot 1\right) - 1 \]

      5. metadata-eval N/A

         \[\leadsto \left(x - -1\right) - 1 \]

      6. lower--.f64 5.4

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

   4. Applied rewrites 5.4%

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

   5. Taylor expanded in x around inf

      \[\leadsto x - 1 \]
   6. Step-by-step derivation

   7. Applied rewrites 5.4%

      \[\leadsto x - 1 \]
   8. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

   9. Applied rewrites 5.1%

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

   10. Taylor expanded in x around 0

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

   12. Applied rewrites 98.0%

       \[\leadsto \frac{\mathsf{fma}\left(x, 1, -1\right)}{x - -1} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 99.0% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(x \cdot 2, x, 2\right)} - 1\\ \mathbf{elif}\;x \leq 1.6:\\ \;\;\;\;\mathsf{fma}\left(\left(x \cdot x\right) \cdot x, -0.3333333333333333, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, 1, -1\right)}{x - -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.25)
   (- (/ 2.0 (fma (* x 2.0) x 2.0)) 1.0)
   (if (<= x 1.6)
     (fma (* (* x x) x) -0.3333333333333333 x)
     (/ (fma x 1.0 -1.0) (- x -1.0)))))

C

double code(double x) {
	double tmp;
	if (x <= -1.25) {
		tmp = (2.0 / fma((x * 2.0), x, 2.0)) - 1.0;
	} else if (x <= 1.6) {
		tmp = fma(((x * x) * x), -0.3333333333333333, x);
	} else {
		tmp = fma(x, 1.0, -1.0) / (x - -1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.25)
		tmp = Float64(Float64(2.0 / fma(Float64(x * 2.0), x, 2.0)) - 1.0);
	elseif (x <= 1.6)
		tmp = fma(Float64(Float64(x * x) * x), -0.3333333333333333, x);
	else
		tmp = Float64(fma(x, 1.0, -1.0) / Float64(x - -1.0));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1.25], N[(N[(2.0 / N[(N[(x * 2.0), $MachinePrecision] * x + 2.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], If[LessEqual[x, 1.6], N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * -0.3333333333333333 + x), $MachinePrecision], N[(N[(x * 1.0 + -1.0), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.25

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. metadata-eval N/A

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

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

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 99.1

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 99.1

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

   7. Applied rewrites 99.1%

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

   if -1.25 < x < 1.6000000000000001

   1. Initial program 8.3%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. unpow2 N/A

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

      6. cube-mult N/A

         \[\leadsto {x}^{3} \cdot \frac{-1}{3} + x \cdot 1 \]

      7. *-rgt-identity N/A

         \[\leadsto {x}^{3} \cdot \frac{-1}{3} + x \]

      8. lower-fma.f64 N/A

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

      9. lower-pow.f64 99.5

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

   4. Applied rewrites 99.5%

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

      1. lift-pow.f64 N/A

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

      2. unpow3 N/A

         \[\leadsto \mathsf{fma}\left(\left(x \cdot x\right) \cdot x, \frac{-1}{3}, x\right) \]

      3. pow2 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\left(x \cdot x\right) \cdot x, \frac{-1}{3}, x\right) \]

      6. lift-*.f64 99.5

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

   6. Applied rewrites 99.5%

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

   if 1.6000000000000001 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. metadata-eval N/A

         \[\leadsto \left(x - -1 \cdot 1\right) - 1 \]

      5. metadata-eval N/A

         \[\leadsto \left(x - -1\right) - 1 \]

      6. lower--.f64 5.4

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

   4. Applied rewrites 5.4%

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

   5. Taylor expanded in x around inf

      \[\leadsto x - 1 \]
   6. Step-by-step derivation

   7. Applied rewrites 5.4%

      \[\leadsto x - 1 \]
   8. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

   9. Applied rewrites 5.1%

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

   10. Taylor expanded in x around 0

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

   12. Applied rewrites 98.0%

       \[\leadsto \frac{\mathsf{fma}\left(x, 1, -1\right)}{x - -1} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 99.0% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.4:\\ \;\;\;\;\frac{2}{\left(x + x\right) \cdot x} - 1\\ \mathbf{elif}\;x \leq 1.6:\\ \;\;\;\;\mathsf{fma}\left(\left(x \cdot x\right) \cdot x, -0.3333333333333333, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, 1, -1\right)}{x - -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.4)
   (- (/ 2.0 (* (+ x x) x)) 1.0)
   (if (<= x 1.6)
     (fma (* (* x x) x) -0.3333333333333333 x)
     (/ (fma x 1.0 -1.0) (- x -1.0)))))

C

double code(double x) {
	double tmp;
	if (x <= -1.4) {
		tmp = (2.0 / ((x + x) * x)) - 1.0;
	} else if (x <= 1.6) {
		tmp = fma(((x * x) * x), -0.3333333333333333, x);
	} else {
		tmp = fma(x, 1.0, -1.0) / (x - -1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.4)
		tmp = Float64(Float64(2.0 / Float64(Float64(x + x) * x)) - 1.0);
	elseif (x <= 1.6)
		tmp = fma(Float64(Float64(x * x) * x), -0.3333333333333333, x);
	else
		tmp = Float64(fma(x, 1.0, -1.0) / Float64(x - -1.0));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1.4], N[(N[(2.0 / N[(N[(x + x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], If[LessEqual[x, 1.6], N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * -0.3333333333333333 + x), $MachinePrecision], N[(N[(x * 1.0 + -1.0), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.3999999999999999

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. metadata-eval N/A

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

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

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

      6. *-commutative N/A

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. lower-fma.f64 99.1

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

   4. Applied rewrites 99.1%

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

   5. Taylor expanded in x around inf

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

      1. pow2 N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 99.1

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

   7. Applied rewrites 99.1%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. count-2-rev N/A

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

      4. lower-+.f64 99.1

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

   9. Applied rewrites 99.1%

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

   if -1.3999999999999999 < x < 1.6000000000000001

   1. Initial program 8.3%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. unpow2 N/A

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

      6. cube-mult N/A

         \[\leadsto {x}^{3} \cdot \frac{-1}{3} + x \cdot 1 \]

      7. *-rgt-identity N/A

         \[\leadsto {x}^{3} \cdot \frac{-1}{3} + x \]

      8. lower-fma.f64 N/A

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

      9. lower-pow.f64 99.5

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

   4. Applied rewrites 99.5%

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

      1. lift-pow.f64 N/A

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

      2. unpow3 N/A

         \[\leadsto \mathsf{fma}\left(\left(x \cdot x\right) \cdot x, \frac{-1}{3}, x\right) \]

      3. pow2 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\left(x \cdot x\right) \cdot x, \frac{-1}{3}, x\right) \]

      6. lift-*.f64 99.5

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

   6. Applied rewrites 99.5%

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

   if 1.6000000000000001 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. metadata-eval N/A

         \[\leadsto \left(x - -1 \cdot 1\right) - 1 \]

      5. metadata-eval N/A

         \[\leadsto \left(x - -1\right) - 1 \]

      6. lower--.f64 5.4

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

   4. Applied rewrites 5.4%

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

   5. Taylor expanded in x around inf

      \[\leadsto x - 1 \]
   6. Step-by-step derivation

   7. Applied rewrites 5.4%

      \[\leadsto x - 1 \]
   8. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

   9. Applied rewrites 5.1%

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

   10. Taylor expanded in x around 0

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

   12. Applied rewrites 98.0%

       \[\leadsto \frac{\mathsf{fma}\left(x, 1, -1\right)}{x - -1} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 98.7% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.3:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(x, -2, 2\right)} - 1\\ \mathbf{elif}\;x \leq 1.6:\\ \;\;\;\;\mathsf{fma}\left(\left(x \cdot x\right) \cdot x, -0.3333333333333333, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, 1, -1\right)}{x - -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.3)
   (- (/ 2.0 (fma x -2.0 2.0)) 1.0)
   (if (<= x 1.6)
     (fma (* (* x x) x) -0.3333333333333333 x)
     (/ (fma x 1.0 -1.0) (- x -1.0)))))

C

double code(double x) {
	double tmp;
	if (x <= -1.3) {
		tmp = (2.0 / fma(x, -2.0, 2.0)) - 1.0;
	} else if (x <= 1.6) {
		tmp = fma(((x * x) * x), -0.3333333333333333, x);
	} else {
		tmp = fma(x, 1.0, -1.0) / (x - -1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.3)
		tmp = Float64(Float64(2.0 / fma(x, -2.0, 2.0)) - 1.0);
	elseif (x <= 1.6)
		tmp = fma(Float64(Float64(x * x) * x), -0.3333333333333333, x);
	else
		tmp = Float64(fma(x, 1.0, -1.0) / Float64(x - -1.0));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1.3], N[(N[(2.0 / N[(x * -2.0 + 2.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], If[LessEqual[x, 1.6], N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * -0.3333333333333333 + x), $MachinePrecision], N[(N[(x * 1.0 + -1.0), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.30000000000000004

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 97.8

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

   4. Applied rewrites 97.8%

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

   if -1.30000000000000004 < x < 1.6000000000000001

   1. Initial program 8.3%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. unpow2 N/A

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

      6. cube-mult N/A

         \[\leadsto {x}^{3} \cdot \frac{-1}{3} + x \cdot 1 \]

      7. *-rgt-identity N/A

         \[\leadsto {x}^{3} \cdot \frac{-1}{3} + x \]

      8. lower-fma.f64 N/A

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

      9. lower-pow.f64 99.5

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

   4. Applied rewrites 99.5%

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

      1. lift-pow.f64 N/A

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

      2. unpow3 N/A

         \[\leadsto \mathsf{fma}\left(\left(x \cdot x\right) \cdot x, \frac{-1}{3}, x\right) \]

      3. pow2 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\left(x \cdot x\right) \cdot x, \frac{-1}{3}, x\right) \]

      6. lift-*.f64 99.5

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

   6. Applied rewrites 99.5%

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

   if 1.6000000000000001 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. metadata-eval N/A

         \[\leadsto \left(x - -1 \cdot 1\right) - 1 \]

      5. metadata-eval N/A

         \[\leadsto \left(x - -1\right) - 1 \]

      6. lower--.f64 5.4

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

   4. Applied rewrites 5.4%

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

   5. Taylor expanded in x around inf

      \[\leadsto x - 1 \]
   6. Step-by-step derivation

   7. Applied rewrites 5.4%

      \[\leadsto x - 1 \]
   8. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

   9. Applied rewrites 5.1%

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

   10. Taylor expanded in x around 0

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

   12. Applied rewrites 98.0%

       \[\leadsto \frac{\mathsf{fma}\left(x, 1, -1\right)}{x - -1} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 78.7% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.3:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(x, -2, 2\right)} - 1\\ \mathbf{elif}\;x \leq 1.2:\\ \;\;\;\;\mathsf{fma}\left(\left(x \cdot x\right) \cdot x, -0.3333333333333333, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 2}{x - -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.3)
   (- (/ 2.0 (fma x -2.0 2.0)) 1.0)
   (if (<= x 1.2)
     (fma (* (* x x) x) -0.3333333333333333 x)
     (/ (* x 2.0) (- x -1.0)))))

C

double code(double x) {
	double tmp;
	if (x <= -1.3) {
		tmp = (2.0 / fma(x, -2.0, 2.0)) - 1.0;
	} else if (x <= 1.2) {
		tmp = fma(((x * x) * x), -0.3333333333333333, x);
	} else {
		tmp = (x * 2.0) / (x - -1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.3)
		tmp = Float64(Float64(2.0 / fma(x, -2.0, 2.0)) - 1.0);
	elseif (x <= 1.2)
		tmp = fma(Float64(Float64(x * x) * x), -0.3333333333333333, x);
	else
		tmp = Float64(Float64(x * 2.0) / Float64(x - -1.0));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1.3], N[(N[(2.0 / N[(x * -2.0 + 2.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], If[LessEqual[x, 1.2], N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * -0.3333333333333333 + x), $MachinePrecision], N[(N[(x * 2.0), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.30000000000000004

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 97.8

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

   4. Applied rewrites 97.8%

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

   if -1.30000000000000004 < x < 1.19999999999999996

   1. Initial program 8.3%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. unpow2 N/A

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

      6. cube-mult N/A

         \[\leadsto {x}^{3} \cdot \frac{-1}{3} + x \cdot 1 \]

      7. *-rgt-identity N/A

         \[\leadsto {x}^{3} \cdot \frac{-1}{3} + x \]

      8. lower-fma.f64 N/A

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

      9. lower-pow.f64 99.5

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

   4. Applied rewrites 99.5%

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

      1. lift-pow.f64 N/A

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

      2. unpow3 N/A

         \[\leadsto \mathsf{fma}\left(\left(x \cdot x\right) \cdot x, \frac{-1}{3}, x\right) \]

      3. pow2 N/A

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

      4. lower-*.f64 N/A

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

      5. pow2 N/A

         \[\leadsto \mathsf{fma}\left(\left(x \cdot x\right) \cdot x, \frac{-1}{3}, x\right) \]

      6. lift-*.f64 99.5

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

   6. Applied rewrites 99.5%

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

   if 1.19999999999999996 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. metadata-eval N/A

         \[\leadsto \left(x - -1 \cdot 1\right) - 1 \]

      5. metadata-eval N/A

         \[\leadsto \left(x - -1\right) - 1 \]

      6. lower--.f64 5.4

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

   4. Applied rewrites 5.4%

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

   5. Taylor expanded in x around inf

      \[\leadsto x - 1 \]
   6. Step-by-step derivation

   7. Applied rewrites 5.4%

      \[\leadsto x - 1 \]
   8. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

   9. Applied rewrites 5.1%

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

   10. Taylor expanded in x around 0

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

       1. *-commutative N/A

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

       2. lift-*.f64 18.7

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

   12. Applied rewrites 18.7%

       \[\leadsto \frac{\color{blue}{x \cdot 2}}{x - -1} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 13: 55.4% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot 2}{x - -1}\\ \end{array} \end{array} \]

FPCore

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

C

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

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
    real(8) :: tmp
    if (x <= 1.0d0) then
        tmp = x
    else
        tmp = (x * 2.0d0) / (x - (-1.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.0) {
		tmp = x;
	} else {
		tmp = (x * 2.0) / (x - -1.0);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.0:
		tmp = x
	else:
		tmp = (x * 2.0) / (x - -1.0)
	return tmp

Julia

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

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.0)
		tmp = x;
	else
		tmp = (x * 2.0) / (x - -1.0);
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 1

   1. Initial program 38.9%

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

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x} \]
   3. Step-by-step derivation

   4. Applied rewrites 67.8%

      \[\leadsto \color{blue}{x} \]

   if 1 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. metadata-eval N/A

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

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

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

      4. metadata-eval N/A

         \[\leadsto \left(x - -1 \cdot 1\right) - 1 \]

      5. metadata-eval N/A

         \[\leadsto \left(x - -1\right) - 1 \]

      6. lower--.f64 5.4

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

   4. Applied rewrites 5.4%

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

   5. Taylor expanded in x around inf

      \[\leadsto x - 1 \]
   6. Step-by-step derivation

   7. Applied rewrites 5.4%

      \[\leadsto x - 1 \]
   8. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. lower-fma.f64 N/A

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

      8. metadata-eval N/A

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

      9. metadata-eval N/A

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

   9. Applied rewrites 5.1%

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

   10. Taylor expanded in x around 0

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

       1. *-commutative N/A

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

       2. lift-*.f64 18.7

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

   12. Applied rewrites 18.7%

       \[\leadsto \frac{\color{blue}{x \cdot 2}}{x - -1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 52.0% accurate, 123.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 x)

C

double code(double x) {
	return x;
}

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
end function

Java

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

Python

def code(x):
	return x

Julia

function code(x)
	return x
end

MATLAB

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

Wolfram

code[x_] := x

Derivation

1. Initial program 54.4%

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

2. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x} \]
3. Step-by-step derivation

4. Applied rewrites 52.0%

   \[\leadsto \color{blue}{x} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025107 
(FPCore (x)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))