Logistic function from Lakshay Garg

Percentage Accurate: 54.2% → 100.0%

Time: 4.2s

Alternatives: 13

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: 54.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.007 \lor \neg \left(x \leq 0.0077\right):\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.13333333333333333, -0.3333333333333333\right) \cdot x, x \cdot x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -0.007) (not (<= x 0.0077)))
   (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0)
   (fma
    (* (fma (* x x) 0.13333333333333333 -0.3333333333333333) x)
    (* x x)
    x)))

C

double code(double x) {
	double tmp;
	if ((x <= -0.007) || !(x <= 0.0077)) {
		tmp = (2.0 / (1.0 + exp((-2.0 * x)))) - 1.0;
	} else {
		tmp = fma((fma((x * x), 0.13333333333333333, -0.3333333333333333) * x), (x * x), x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if ((x <= -0.007) || !(x <= 0.0077))
		tmp = Float64(Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x)))) - 1.0);
	else
		tmp = fma(Float64(fma(Float64(x * x), 0.13333333333333333, -0.3333333333333333) * x), Float64(x * x), x);
	end
	return tmp
end

Wolfram

code[x_] := If[Or[LessEqual[x, -0.007], N[Not[LessEqual[x, 0.0077]], $MachinePrecision]], N[(N[(2.0 / N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(N[(x * x), $MachinePrecision] * 0.13333333333333333 + -0.3333333333333333), $MachinePrecision] * x), $MachinePrecision] * N[(x * x), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -0.00700000000000000015 or 0.0077000000000000002 < x

   1. Initial program 100.0%

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

   if -0.00700000000000000015 < x < 0.0077000000000000002

   1. Initial program 8.3%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      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 100.0

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

   5. Applied rewrites 100.0%

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

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

   7. Applied rewrites 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.007 \lor \neg \left(x \leq 0.0077\right):\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.13333333333333333, -0.3333333333333333\right) \cdot x, x \cdot x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + e^{-2 \cdot x}\\ \mathbf{if}\;t\_0 \leq 1.5:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, 1, -1\right)}{x - -1}\\ \mathbf{elif}\;t\_0 \leq 4:\\ \;\;\;\;\mathsf{fma}\left(\left(x \cdot x\right) \cdot x, -0.3333333333333333, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(x + x\right) \cdot x} - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ 1.0 (exp (* -2.0 x)))))
   (if (<= t_0 1.5)
     (/ (fma x 1.0 -1.0) (- x -1.0))
     (if (<= t_0 4.0)
       (fma (* (* x x) x) -0.3333333333333333 x)
       (- (/ 2.0 (* (+ x x) x)) 1.0)))))

C

double code(double x) {
	double t_0 = 1.0 + exp((-2.0 * x));
	double tmp;
	if (t_0 <= 1.5) {
		tmp = fma(x, 1.0, -1.0) / (x - -1.0);
	} else if (t_0 <= 4.0) {
		tmp = fma(((x * x) * x), -0.3333333333333333, x);
	} else {
		tmp = (2.0 / ((x + x) * x)) - 1.0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(1.0 + exp(Float64(-2.0 * x)))
	tmp = 0.0
	if (t_0 <= 1.5)
		tmp = Float64(fma(x, 1.0, -1.0) / Float64(x - -1.0));
	elseif (t_0 <= 4.0)
		tmp = fma(Float64(Float64(x * x) * x), -0.3333333333333333, x);
	else
		tmp = Float64(Float64(2.0 / Float64(Float64(x + x) * x)) - 1.0);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 1.5], N[(N[(x * 1.0 + -1.0), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 4.0], N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * -0.3333333333333333 + x), $MachinePrecision], N[(N[(2.0 / N[(N[(x + x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 #s(literal 1 binary64) (exp.f64 (*.f64 #s(literal -2 binary64) x))) < 1.5

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      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 5.2

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

   5. Applied rewrites 5.2%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 5.2%

      \[\leadsto x - 1 \]
   9. 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

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

   10. Applied rewrites 4.8%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 98.1%

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

   if 1.5 < (+.f64 #s(literal 1 binary64) (exp.f64 (*.f64 #s(literal -2 binary64) x))) < 4

   1. Initial program 8.3%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + \frac{-1}{3} \cdot {x}^{2}\right)} \]
   4. 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 99.8

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

   5. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, -0.3333333333333333, x\right)} \]
   6. 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 99.8

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

   7. Applied rewrites 99.8%

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

   if 4 < (+.f64 #s(literal 1 binary64) (exp.f64 (*.f64 #s(literal -2 binary64) x)))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      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 98.0

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

   5. Applied rewrites 98.0%

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

   6. Taylor expanded in x around inf

      \[\leadsto \frac{2}{2 \cdot \color{blue}{{x}^{2}}} - 1 \]
   7. 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 98.0

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

   8. Applied rewrites 98.0%

      \[\leadsto \frac{2}{\left(x \cdot 2\right) \cdot \color{blue}{x}} - 1 \]
   9. 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 98.0

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

   10. Applied rewrites 98.0%

       \[\leadsto \frac{2}{\left(x + x\right) \cdot x} - 1 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 98.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + e^{-2 \cdot x}\\ \mathbf{if}\;t\_0 \leq 1.5:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, 1, -1\right)}{x - -1}\\ \mathbf{elif}\;t\_0 \leq 4:\\ \;\;\;\;\mathsf{fma}\left(\left(x \cdot x\right) \cdot x, -0.3333333333333333, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(x, -2, 2\right)} - 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ 1.0 (exp (* -2.0 x)))))
   (if (<= t_0 1.5)
     (/ (fma x 1.0 -1.0) (- x -1.0))
     (if (<= t_0 4.0)
       (fma (* (* x x) x) -0.3333333333333333 x)
       (- (/ 2.0 (fma x -2.0 2.0)) 1.0)))))

C

double code(double x) {
	double t_0 = 1.0 + exp((-2.0 * x));
	double tmp;
	if (t_0 <= 1.5) {
		tmp = fma(x, 1.0, -1.0) / (x - -1.0);
	} else if (t_0 <= 4.0) {
		tmp = fma(((x * x) * x), -0.3333333333333333, x);
	} else {
		tmp = (2.0 / fma(x, -2.0, 2.0)) - 1.0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(1.0 + exp(Float64(-2.0 * x)))
	tmp = 0.0
	if (t_0 <= 1.5)
		tmp = Float64(fma(x, 1.0, -1.0) / Float64(x - -1.0));
	elseif (t_0 <= 4.0)
		tmp = fma(Float64(Float64(x * x) * x), -0.3333333333333333, x);
	else
		tmp = Float64(Float64(2.0 / fma(x, -2.0, 2.0)) - 1.0);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 1.5], N[(N[(x * 1.0 + -1.0), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 4.0], N[(N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision] * -0.3333333333333333 + x), $MachinePrecision], N[(N[(2.0 / N[(x * -2.0 + 2.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 #s(literal 1 binary64) (exp.f64 (*.f64 #s(literal -2 binary64) x))) < 1.5

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      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 5.2

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

   5. Applied rewrites 5.2%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 5.2%

      \[\leadsto x - 1 \]
   9. 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

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

   10. Applied rewrites 4.8%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 98.1%

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

   if 1.5 < (+.f64 #s(literal 1 binary64) (exp.f64 (*.f64 #s(literal -2 binary64) x))) < 4

   1. Initial program 8.3%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + \frac{-1}{3} \cdot {x}^{2}\right)} \]
   4. 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 99.8

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

   5. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, -0.3333333333333333, x\right)} \]
   6. 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 99.8

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

   7. Applied rewrites 99.8%

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

   if 4 < (+.f64 #s(literal 1 binary64) (exp.f64 (*.f64 #s(literal -2 binary64) x)))

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 96.3

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

   5. Applied rewrites 96.3%

      \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(x, -2, 2\right)}} - 1 \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 99.2% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(x, x, \left(-x\right) \cdot x\right)}{0} \cdot x} - 1\\ \mathbf{elif}\;x \leq 1.92:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.13333333333333333, -0.3333333333333333\right) \cdot x, x \cdot x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, 1, -1\right)}{x - -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.25)
   (- (/ 2.0 (* (/ (fma x x (* (- x) x)) 0.0) x)) 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 <= -1.25) {
		tmp = (2.0 / ((fma(x, x, (-x * x)) / 0.0) * x)) - 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 <= -1.25)
		tmp = Float64(Float64(2.0 / Float64(Float64(fma(x, x, Float64(Float64(-x) * x)) / 0.0) * x)) - 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, -1.25], N[(N[(2.0 / N[(N[(N[(x * x + N[((-x) * x), $MachinePrecision]), $MachinePrecision] / 0.0), $MachinePrecision] * x), $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 < -1.25

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      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 98.0

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

   5. Applied rewrites 98.0%

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

   6. Taylor expanded in x around inf

      \[\leadsto \frac{2}{2 \cdot \color{blue}{{x}^{2}}} - 1 \]
   7. 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 98.0

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

   8. Applied rewrites 98.0%

      \[\leadsto \frac{2}{\left(x \cdot 2\right) \cdot \color{blue}{x}} - 1 \]
   9. 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 98.0

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

   10. Applied rewrites 98.0%

       \[\leadsto \frac{2}{\left(x + x\right) \cdot x} - 1 \]
   11. 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 99.2

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

   12. Applied rewrites 99.2%

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

   if -1.25 < x < 1.9199999999999999

   1. Initial program 8.3%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      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 100.0

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

   5. Applied rewrites 100.0%

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

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

   7. Applied rewrites 100.0%

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

   if 1.9199999999999999 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      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 5.2

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

   5. Applied rewrites 5.2%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 5.2%

      \[\leadsto x - 1 \]
   9. 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

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

   10. Applied rewrites 4.8%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 98.1%

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

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;\frac{2}{\frac{\mathsf{fma}\left(x, x, \left(-x\right) \cdot x\right)}{0} \cdot x} - 1\\ \mathbf{elif}\;x \leq 1.92:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.13333333333333333, -0.3333333333333333\right) \cdot x, x \cdot x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, 1, -1\right)}{x - -1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 99.1% accurate, 3.1× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.0)
   (- (/ 2.0 (fma (fma (fma -1.3333333333333333 x 2.0) 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 <= -1.0) {
		tmp = (2.0 / fma(fma(fma(-1.3333333333333333, x, 2.0), 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 <= -1.0)
		tmp = Float64(Float64(2.0 / fma(fma(fma(-1.3333333333333333, x, 2.0), 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, -1.0], 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[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

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-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 98.6

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

   5. Applied rewrites 98.6%

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

   if -1 < x < 1.9199999999999999

   1. Initial program 8.3%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      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 100.0

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

   5. Applied rewrites 100.0%

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

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

   7. Applied rewrites 100.0%

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

   if 1.9199999999999999 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      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 5.2

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

   5. Applied rewrites 5.2%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 5.2%

      \[\leadsto x - 1 \]
   9. 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

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

   10. Applied rewrites 4.8%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 98.1%

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

Alternative 6: 99.1% accurate, 3.1× speedup

Math

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

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. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. +-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 98.6

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

   5. Applied rewrites 98.6%

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

   6. Taylor expanded in x around inf

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

      1. lower-*.f64 98.6

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

   8. Applied rewrites 98.6%

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

   if -1.19999999999999996 < x < 1.9199999999999999

   1. Initial program 8.3%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      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 100.0

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

   5. Applied rewrites 100.0%

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

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

   7. Applied rewrites 100.0%

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

   if 1.9199999999999999 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      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 5.2

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

   5. Applied rewrites 5.2%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 5.2%

      \[\leadsto x - 1 \]
   9. 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

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

   10. Applied rewrites 4.8%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 98.1%

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

Alternative 7: 99.1% accurate, 3.1× speedup

Math

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

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. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. +-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 98.6

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

   5. Applied rewrites 98.6%

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

   6. Taylor expanded in x around inf

      \[\leadsto \frac{2}{\mathsf{fma}\left(\frac{-4}{3} \cdot {x}^{2}, x, 2\right)} - 1 \]
   7. 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 98.6

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

   8. Applied rewrites 98.6%

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

   if -1.3999999999999999 < x < 1.9199999999999999

   1. Initial program 8.3%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      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 100.0

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

   5. Applied rewrites 100.0%

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

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

   7. Applied rewrites 100.0%

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

   if 1.9199999999999999 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      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 5.2

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

   5. Applied rewrites 5.2%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 5.2%

      \[\leadsto x - 1 \]
   9. 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

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

   10. Applied rewrites 4.8%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 98.1%

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

Alternative 8: 99.0% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.15:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(x, 2, -2\right), x, 2\right)} - 1\\ \mathbf{elif}\;x \leq 1.92:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.13333333333333333, -0.3333333333333333\right) \cdot x, x \cdot x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, 1, -1\right)}{x - -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.15)
   (- (/ 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 <= -1.15) {
		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 <= -1.15)
		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, -1.15], 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 < -1.1499999999999999

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      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 98.0

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

   5. Applied rewrites 98.0%

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

   if -1.1499999999999999 < x < 1.9199999999999999

   1. Initial program 8.3%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      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 100.0

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

   5. Applied rewrites 100.0%

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

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

   7. Applied rewrites 100.0%

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

   if 1.9199999999999999 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      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 5.2

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

   5. Applied rewrites 5.2%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 5.2%

      \[\leadsto x - 1 \]
   9. 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

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

   10. Applied rewrites 4.8%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 98.1%

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

Alternative 9: 98.9% 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. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      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 98.0

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

   5. Applied rewrites 98.0%

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

   if -1 < x < 1.6000000000000001

   1. Initial program 8.3%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + \frac{-1}{3} \cdot {x}^{2}\right)} \]
   4. 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 99.8

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

   5. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, -0.3333333333333333, x\right)} \]
   6. 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 99.8

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

   7. Applied rewrites 99.8%

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

   if 1.6000000000000001 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      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 5.2

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

   5. Applied rewrites 5.2%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 5.2%

      \[\leadsto x - 1 \]
   9. 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

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

   10. Applied rewrites 4.8%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 98.1%

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

Alternative 10: 98.9% 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. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      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 98.0

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

   5. Applied rewrites 98.0%

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

   6. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 98.0

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

   8. Applied rewrites 98.0%

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

   if -1.25 < x < 1.6000000000000001

   1. Initial program 8.3%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + \frac{-1}{3} \cdot {x}^{2}\right)} \]
   4. 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 99.8

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

   5. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, -0.3333333333333333, x\right)} \]
   6. 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 99.8

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

   7. Applied rewrites 99.8%

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

   if 1.6000000000000001 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      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 5.2

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

   5. Applied rewrites 5.2%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 5.2%

      \[\leadsto x - 1 \]
   9. 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

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

   10. Applied rewrites 4.8%

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

   11. Taylor expanded in x around 0

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

   13. Applied rewrites 98.1%

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

Alternative 11: 79.3% 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. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 96.3

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

   5. Applied rewrites 96.3%

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

   if -1.30000000000000004 < x < 1.19999999999999996

   1. Initial program 8.3%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + \frac{-1}{3} \cdot {x}^{2}\right)} \]
   4. 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 99.8

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

   5. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, -0.3333333333333333, x\right)} \]
   6. 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 99.8

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

   7. Applied rewrites 99.8%

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

   if 1.19999999999999996 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      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 5.2

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

   5. Applied rewrites 5.2%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 5.2%

      \[\leadsto x - 1 \]
   9. 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

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

   10. Applied rewrites 4.8%

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

   11. Taylor expanded in x around 0

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

       1. *-commutative N/A

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

       2. lift-*.f64 18.8

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

   13. Applied rewrites 18.8%

       \[\leadsto \frac{\color{blue}{x \cdot 2}}{x - -1} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 55.5% 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 40.3%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 66.3%

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

   if 1 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      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 5.2

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

   5. Applied rewrites 5.2%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 5.2%

      \[\leadsto x - 1 \]
   9. 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

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

   10. Applied rewrites 4.8%

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

   11. Taylor expanded in x around 0

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

       1. *-commutative N/A

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

       2. lift-*.f64 18.8

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

   13. Applied rewrites 18.8%

       \[\leadsto \frac{\color{blue}{x \cdot 2}}{x - -1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 52.2% 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 57.4%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 48.9%

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

Reproduce

herbie shell --seed 2025076 
(FPCore (x)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))