Logistic function from Lakshay Garg

Percentage Accurate: 54.3% → 100.0%

Time: 4.3s

Alternatives: 12

Speedup: 3.8×

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: 54.3% 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, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-2 \cdot x}\\ t_1 := \frac{-1}{e^{x}}\\ \mathbf{if}\;x \leq -0.0072:\\ \;\;\;\;\frac{2}{1 + t\_0} - 1\\ \mathbf{elif}\;x \leq 0.0074:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.13333333333333333, -0.3333333333333333\right) \cdot x, x \cdot x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{4 - \left(t\_0 - -1\right) \cdot 2}{\mathsf{fma}\left(t\_1, t\_1, 1\right) \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (* -2.0 x))) (t_1 (/ -1.0 (exp x))))
   (if (<= x -0.0072)
     (- (/ 2.0 (+ 1.0 t_0)) 1.0)
     (if (<= x 0.0074)
       (fma
        (* (fma (* x x) 0.13333333333333333 -0.3333333333333333) x)
        (* x x)
        x)
       (/ (- 4.0 (* (- t_0 -1.0) 2.0)) (* (fma t_1 t_1 1.0) 2.0))))))

C

double code(double x) {
	double t_0 = exp((-2.0 * x));
	double t_1 = -1.0 / exp(x);
	double tmp;
	if (x <= -0.0072) {
		tmp = (2.0 / (1.0 + t_0)) - 1.0;
	} else if (x <= 0.0074) {
		tmp = fma((fma((x * x), 0.13333333333333333, -0.3333333333333333) * x), (x * x), x);
	} else {
		tmp = (4.0 - ((t_0 - -1.0) * 2.0)) / (fma(t_1, t_1, 1.0) * 2.0);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = exp(Float64(-2.0 * x))
	t_1 = Float64(-1.0 / exp(x))
	tmp = 0.0
	if (x <= -0.0072)
		tmp = Float64(Float64(2.0 / Float64(1.0 + t_0)) - 1.0);
	elseif (x <= 0.0074)
		tmp = fma(Float64(fma(Float64(x * x), 0.13333333333333333, -0.3333333333333333) * x), Float64(x * x), x);
	else
		tmp = Float64(Float64(4.0 - Float64(Float64(t_0 - -1.0) * 2.0)) / Float64(fma(t_1, t_1, 1.0) * 2.0));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(-1.0 / N[Exp[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.0072], N[(N[(2.0 / N[(1.0 + t$95$0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], If[LessEqual[x, 0.0074], N[(N[(N[(N[(x * x), $MachinePrecision] * 0.13333333333333333 + -0.3333333333333333), $MachinePrecision] * x), $MachinePrecision] * N[(x * x), $MachinePrecision] + x), $MachinePrecision], N[(N[(4.0 - N[(N[(t$95$0 - -1.0), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(t$95$1 * t$95$1 + 1.0), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.0071999999999999998

   1. Initial program 100.0%

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

   if -0.0071999999999999998 < x < 0.0074000000000000003

   1. Initial program 7.6%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   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 0.0074000000000000003 < x

   1. Initial program 100.0%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. metadata-eval N/A

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

      7. frac-sub N/A

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

      8. lower-/.f64 N/A

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

   4. Applied rewrites 100.0%

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

      1. lift-exp.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. lower--.f64 N/A

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

      4. metadata-eval N/A

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

      5. metadata-eval N/A

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

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

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

      7. metadata-eval N/A

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

      8. metadata-eval N/A

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

      9. pow-flip N/A

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

      10. metadata-eval N/A

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

      11. unpow2 N/A

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

      12. times-frac N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lift-exp.f64 N/A

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

      16. lower-/.f64 N/A

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

      17. lift-exp.f64 100.0

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

   6. Applied rewrites 100.0%

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

      1. lift-exp.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow-exp N/A

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

      4. *-commutative N/A

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

      5. lower-exp.f64 N/A

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

      6. lower-*.f64 100.0

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

   8. Applied rewrites 100.0%

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

Alternative 2: 100.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.0072:\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} - 1\\ \mathbf{elif}\;x \leq 0.0075:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.13333333333333333, -0.3333333333333333\right) \cdot x, x \cdot x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{4 - \left({\left(e^{x}\right)}^{-2} - -1\right) \cdot 2}{\left(e^{x \cdot -2} - -1\right) \cdot 2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -0.0072)
   (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0)
   (if (<= x 0.0075)
     (fma
      (* (fma (* x x) 0.13333333333333333 -0.3333333333333333) x)
      (* x x)
      x)
     (/
      (- 4.0 (* (- (pow (exp x) -2.0) -1.0) 2.0))
      (* (- (exp (* x -2.0)) -1.0) 2.0)))))

C

double code(double x) {
	double tmp;
	if (x <= -0.0072) {
		tmp = (2.0 / (1.0 + exp((-2.0 * x)))) - 1.0;
	} else if (x <= 0.0075) {
		tmp = fma((fma((x * x), 0.13333333333333333, -0.3333333333333333) * x), (x * x), x);
	} else {
		tmp = (4.0 - ((pow(exp(x), -2.0) - -1.0) * 2.0)) / ((exp((x * -2.0)) - -1.0) * 2.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -0.0072)
		tmp = Float64(Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x)))) - 1.0);
	elseif (x <= 0.0075)
		tmp = fma(Float64(fma(Float64(x * x), 0.13333333333333333, -0.3333333333333333) * x), Float64(x * x), x);
	else
		tmp = Float64(Float64(4.0 - Float64(Float64((exp(x) ^ -2.0) - -1.0) * 2.0)) / Float64(Float64(exp(Float64(x * -2.0)) - -1.0) * 2.0));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -0.0072], N[(N[(2.0 / N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], If[LessEqual[x, 0.0075], N[(N[(N[(N[(x * x), $MachinePrecision] * 0.13333333333333333 + -0.3333333333333333), $MachinePrecision] * x), $MachinePrecision] * N[(x * x), $MachinePrecision] + x), $MachinePrecision], N[(N[(4.0 - N[(N[(N[Power[N[Exp[x], $MachinePrecision], -2.0], $MachinePrecision] - -1.0), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Exp[N[(x * -2.0), $MachinePrecision]], $MachinePrecision] - -1.0), $MachinePrecision] * 2.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.0071999999999999998

   1. Initial program 100.0%

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

   if -0.0071999999999999998 < x < 0.0074999999999999997

   1. Initial program 7.6%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   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 0.0074999999999999997 < x

   1. Initial program 100.0%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift-exp.f64 N/A

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

      6. metadata-eval N/A

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

      7. frac-sub N/A

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

      8. lower-/.f64 N/A

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

   4. Applied rewrites 100.0%

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

      1. lift-exp.f64 N/A

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

      2. lift-pow.f64 N/A

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

      3. pow-exp N/A

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

      4. *-commutative N/A

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

      5. lower-exp.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 100.0

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

   6. Applied rewrites 100.0%

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

Alternative 3: 100.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{2}{1 + e^{-2 \cdot x}} - 1\\ \mathbf{if}\;x \leq -0.0072:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 0.01:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.13333333333333333, -0.3333333333333333\right) \cdot x, x \cdot x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0)))
   (if (<= x -0.0072)
     t_0
     (if (<= x 0.01)
       (fma
        (* (fma (* x x) 0.13333333333333333 -0.3333333333333333) x)
        (* x x)
        x)
       t_0))))

C

double code(double x) {
	double t_0 = (2.0 / (1.0 + exp((-2.0 * x)))) - 1.0;
	double tmp;
	if (x <= -0.0072) {
		tmp = t_0;
	} else if (x <= 0.01) {
		tmp = fma((fma((x * x), 0.13333333333333333, -0.3333333333333333) * x), (x * x), x);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x)))) - 1.0)
	tmp = 0.0
	if (x <= -0.0072)
		tmp = t_0;
	elseif (x <= 0.01)
		tmp = fma(Float64(fma(Float64(x * x), 0.13333333333333333, -0.3333333333333333) * x), Float64(x * x), x);
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(2.0 / N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision]}, If[LessEqual[x, -0.0072], t$95$0, If[LessEqual[x, 0.01], N[(N[(N[(N[(x * x), $MachinePrecision] * 0.13333333333333333 + -0.3333333333333333), $MachinePrecision] * x), $MachinePrecision] * N[(x * x), $MachinePrecision] + x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -0.0071999999999999998 or 0.0100000000000000002 < x

   1. Initial program 100.0%

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

   if -0.0071999999999999998 < x < 0.0100000000000000002

   1. Initial program 7.6%

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

Alternative 4: 55.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;1 + e^{-2 \cdot x} \leq 1.5:\\ \;\;\;\;\frac{2 \cdot x}{x - -1}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (+ 1.0 (exp (* -2.0 x))) 1.5) (/ (* 2.0 x) (- x -1.0)) x))

C

double code(double x) {
	double tmp;
	if ((1.0 + exp((-2.0 * x))) <= 1.5) {
		tmp = (2.0 * x) / (x - -1.0);
	} else {
		tmp = x;
	}
	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 ((1.0d0 + exp(((-2.0d0) * x))) <= 1.5d0) then
        tmp = (2.0d0 * x) / (x - (-1.0d0))
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((1.0 + Math.exp((-2.0 * x))) <= 1.5) {
		tmp = (2.0 * x) / (x - -1.0);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (1.0 + math.exp((-2.0 * x))) <= 1.5:
		tmp = (2.0 * x) / (x - -1.0)
	else:
		tmp = x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(1.0 + exp(Float64(-2.0 * x))) <= 1.5)
		tmp = Float64(Float64(2.0 * x) / Float64(x - -1.0));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((1.0 + exp((-2.0 * x))) <= 1.5)
		tmp = (2.0 * x) / (x - -1.0);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1.5], N[(N[(2.0 * x), $MachinePrecision] / N[(x - -1.0), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 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 5.4

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

   5. Applied rewrites 5.4%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 5.4%

      \[\leadsto x - 1 \]
   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. fp-cancel-sub-sign-inv N/A

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

      5. metadata-eval N/A

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

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

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

   10. Applied rewrites 5.0%

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

   11. Taylor expanded in x around 0

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

       1. lower-*.f64 18.8

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

   13. Applied rewrites 18.8%

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

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

   1. Initial program 39.6%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 67.0%

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

Alternative 5: 99.1% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.4:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(\left(x \cdot x\right) \cdot -1.3333333333333333, x, 2\right)} - 1\\ \mathbf{elif}\;x \leq 1.6:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, -0.05396825396825397, 0.13333333333333333\right), x \cdot x, -0.3333333333333333\right) \cdot x, x, 1\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, 1, -1\right)}{x - -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.4)
   (- (/ 2.0 (fma (* (* x x) -1.3333333333333333) x 2.0)) 1.0)
   (if (<= x 1.6)
     (*
      (fma
       (*
        (fma
         (fma (* x x) -0.05396825396825397 0.13333333333333333)
         (* x x)
         -0.3333333333333333)
        x)
       x
       1.0)
      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(((x * x) * -1.3333333333333333), x, 2.0)) - 1.0;
	} else if (x <= 1.6) {
		tmp = fma((fma(fma((x * x), -0.05396825396825397, 0.13333333333333333), (x * x), -0.3333333333333333) * x), x, 1.0) * 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(x * x) * -1.3333333333333333), x, 2.0)) - 1.0);
	elseif (x <= 1.6)
		tmp = Float64(fma(Float64(fma(fma(Float64(x * x), -0.05396825396825397, 0.13333333333333333), Float64(x * x), -0.3333333333333333) * x), x, 1.0) * 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[(x * x), $MachinePrecision] * -1.3333333333333333), $MachinePrecision] * x + 2.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], If[LessEqual[x, 1.6], N[(N[(N[(N[(N[(N[(x * x), $MachinePrecision] * -0.05396825396825397 + 0.13333333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision] + -0.3333333333333333), $MachinePrecision] * x), $MachinePrecision] * x + 1.0), $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

      \[\leadsto \frac{2}{\color{blue}{2 + x \cdot \left(2 \cdot x - 2\right)}} - 1 \]
   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 98.9

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

   5. Applied rewrites 98.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 98.0%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. pow-exp N/A

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

       4. metadata-eval N/A

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

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

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

       6. metadata-eval N/A

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

       7. metadata-eval N/A

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

       8. +-commutative N/A

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

       9. *-commutative N/A

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

       10. lower-fma.f64 N/A

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

       11. metadata-eval N/A

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

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

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

       13. metadata-eval N/A

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

       14. metadata-eval N/A

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

       15. *-commutative N/A

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

       16. lower-fma.f64 N/A

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

       17. +-commutative N/A

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

       18. lower-fma.f64 99.2

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

   11. Applied rewrites 99.2%

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

   12. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 99.2

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

   14. Applied rewrites 99.2%

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

   if -1.3999999999999999 < x < 1.6000000000000001

   1. Initial program 8.4%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. cube-mult N/A

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

      6. *-rgt-identity N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 99.6

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

   5. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, 0.13333333333333333 \cdot \left(x \cdot x\right) - 0.3333333333333333, x\right)} \]
   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 99.6%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 99.7%

       \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.05396825396825397, x \cdot x, 0.13333333333333333\right), x \cdot x, -0.3333333333333333\right), x \cdot x, 1\right) \cdot x} \]
   11. Step-by-step derivation

       1. lift-*.f64 N/A

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

       2. lift-fma.f64 N/A

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

       3. lift-*.f64 N/A

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

       4. lift-fma.f64 N/A

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

       5. lift-*.f64 N/A

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

       6. lift-fma.f64 N/A

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

       7. associate-*r* N/A

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

       8. lower-fma.f64 N/A

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

   12. Applied rewrites 99.7%

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, -0.05396825396825397, 0.13333333333333333\right), x \cdot x, -0.3333333333333333\right) \cdot x, x, 1\right) \cdot x \]

   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 5.4

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

   5. Applied rewrites 5.4%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 5.4%

      \[\leadsto x - 1 \]
   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. fp-cancel-sub-sign-inv N/A

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

      5. metadata-eval N/A

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

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

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

   10. Applied rewrites 5.0%

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

   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 97.9%

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

Alternative 6: 99.1% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.45:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(\left(x \cdot x\right) \cdot -1.3333333333333333, x, 2\right)} - 1\\ \mathbf{elif}\;x \leq 1.95:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.13333333333333333, -0.3333333333333333\right) \cdot x, x \cdot x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, 1, -1\right)}{x - -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.45)
   (- (/ 2.0 (fma (* (* x x) -1.3333333333333333) x 2.0)) 1.0)
   (if (<= x 1.95)
     (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 <= -1.45) {
		tmp = (2.0 / fma(((x * x) * -1.3333333333333333), x, 2.0)) - 1.0;
	} else if (x <= 1.95) {
		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 <= -1.45)
		tmp = Float64(Float64(2.0 / fma(Float64(Float64(x * x) * -1.3333333333333333), x, 2.0)) - 1.0);
	elseif (x <= 1.95)
		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, -1.45], N[(N[(2.0 / N[(N[(N[(x * x), $MachinePrecision] * -1.3333333333333333), $MachinePrecision] * x + 2.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], If[LessEqual[x, 1.95], 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 < -1.44999999999999996

   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 98.9

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

   5. Applied rewrites 98.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 98.0%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. pow-exp N/A

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

       4. metadata-eval N/A

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

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

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

       6. metadata-eval N/A

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

       7. metadata-eval N/A

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

       8. +-commutative N/A

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

       9. *-commutative N/A

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

       10. lower-fma.f64 N/A

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

       11. metadata-eval N/A

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

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

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

       13. metadata-eval N/A

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

       14. metadata-eval N/A

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

       15. *-commutative N/A

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

       16. lower-fma.f64 N/A

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

       17. +-commutative N/A

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

       18. lower-fma.f64 99.2

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

   11. Applied rewrites 99.2%

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

   12. Taylor expanded in x around inf

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

       1. *-commutative N/A

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

       2. lower-*.f64 N/A

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

       3. pow2 N/A

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

       4. lift-*.f64 99.2

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

   14. Applied rewrites 99.2%

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

   if -1.44999999999999996 < x < 1.94999999999999996

   1. Initial program 8.4%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. cube-mult N/A

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

      6. *-rgt-identity N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 99.5

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

   5. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, 0.13333333333333333 \cdot \left(x \cdot x\right) - 0.3333333333333333, x\right)} \]
   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 99.5%

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

   if 1.94999999999999996 < 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 5.4

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

   5. Applied rewrites 5.4%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 5.4%

      \[\leadsto x - 1 \]
   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. fp-cancel-sub-sign-inv N/A

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

      5. metadata-eval N/A

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

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

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

   10. Applied rewrites 5.0%

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

   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 98.0%

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

Alternative 7: 99.0% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.2:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(x, 2, -2\right), x, 2\right)} - 1\\ \mathbf{elif}\;x \leq 1.95:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.13333333333333333, -0.3333333333333333\right) \cdot x, x \cdot x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(x, 1, -1\right)}{x - -1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.2)
   (- (/ 2.0 (fma (fma x 2.0 -2.0) x 2.0)) 1.0)
   (if (<= x 1.95)
     (fma
      (* (fma (* x x) 0.13333333333333333 -0.3333333333333333) x)
      (* x x)
      x)
     (/ (fma x 1.0 -1.0) (- x -1.0)))))

C

double code(double x) {
	double tmp;
	if (x <= -1.2) {
		tmp = (2.0 / fma(fma(x, 2.0, -2.0), x, 2.0)) - 1.0;
	} else if (x <= 1.95) {
		tmp = fma((fma((x * x), 0.13333333333333333, -0.3333333333333333) * x), (x * x), x);
	} else {
		tmp = fma(x, 1.0, -1.0) / (x - -1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.2)
		tmp = Float64(Float64(2.0 / fma(fma(x, 2.0, -2.0), x, 2.0)) - 1.0);
	elseif (x <= 1.95)
		tmp = fma(Float64(fma(Float64(x * x), 0.13333333333333333, -0.3333333333333333) * x), Float64(x * x), x);
	else
		tmp = Float64(fma(x, 1.0, -1.0) / Float64(x - -1.0));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1.2], N[(N[(2.0 / N[(N[(x * 2.0 + -2.0), $MachinePrecision] * x + 2.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], If[LessEqual[x, 1.95], 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

      \[\leadsto \frac{2}{\color{blue}{2 + x \cdot \left(2 \cdot x - 2\right)}} - 1 \]
   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 98.9

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

   5. Applied rewrites 98.9%

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

   if -1.19999999999999996 < x < 1.94999999999999996

   1. Initial program 8.4%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. unpow2 N/A

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

      5. cube-mult N/A

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

      6. *-rgt-identity N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-pow.f64 N/A

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

      9. lower--.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. unpow2 N/A

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

      12. lower-*.f64 99.6

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

   5. Applied rewrites 99.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{3}, 0.13333333333333333 \cdot \left(x \cdot x\right) - 0.3333333333333333, x\right)} \]
   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 99.6%

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

   if 1.94999999999999996 < 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 5.4

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

   5. Applied rewrites 5.4%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 5.4%

      \[\leadsto x - 1 \]
   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. fp-cancel-sub-sign-inv N/A

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

      5. metadata-eval N/A

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

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

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

   10. Applied rewrites 5.0%

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

   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 98.0%

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

Alternative 8: 98.9% accurate, 3.7× speedup

Math

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

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) -0.3333333333333333 1.0) 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), -0.3333333333333333, 1.0) * 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 = Float64(fma(Float64(x * x), -0.3333333333333333, 1.0) * 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] * -0.3333333333333333 + 1.0), $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

      \[\leadsto \frac{2}{\color{blue}{2 + x \cdot \left(2 \cdot x - 2\right)}} - 1 \]
   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 98.9

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

   5. Applied rewrites 98.9%

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

   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 99.5

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

   5. Applied rewrites 99.5%

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

      1. lift-pow.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. cube-mult N/A

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

      4. pow2 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. *-rgt-identity N/A

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

      8. distribute-lft-in N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 99.4

          \[\leadsto \mathsf{fma}\left(x \cdot x, -0.3333333333333333, 1\right) \cdot x \]

   7. Applied rewrites 99.4%

      \[\leadsto \mathsf{fma}\left(x \cdot x, -0.3333333333333333, 1\right) \cdot \color{blue}{x} \]

   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 5.4

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

   5. Applied rewrites 5.4%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 5.4%

      \[\leadsto x - 1 \]
   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. fp-cancel-sub-sign-inv N/A

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

      5. metadata-eval N/A

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

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

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

   10. Applied rewrites 5.0%

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

   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 97.9%

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

Alternative 9: 98.9% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

      \[\leadsto \frac{2}{\color{blue}{2 + x \cdot \left(2 \cdot x - 2\right)}} - 1 \]
   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 98.9

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

   5. Applied rewrites 98.9%

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

   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 98.9

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

   8. Applied rewrites 98.9%

      \[\leadsto \frac{2}{\left(x \cdot 2\right) \cdot \color{blue}{x}} - 1 \]
   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 98.9

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

   10. Applied rewrites 98.9%

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

   if -1.3999999999999999 < x < 1.6000000000000001

   1. Initial program 8.4%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + \frac{-1}{3} \cdot {x}^{2}\right)} \]
   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 99.4

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

   5. Applied rewrites 99.4%

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

      1. lift-pow.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. cube-mult N/A

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

      4. pow2 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. *-rgt-identity N/A

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

      8. distribute-lft-in N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 99.4

          \[\leadsto \mathsf{fma}\left(x \cdot x, -0.3333333333333333, 1\right) \cdot x \]

   7. Applied rewrites 99.4%

      \[\leadsto \mathsf{fma}\left(x \cdot x, -0.3333333333333333, 1\right) \cdot \color{blue}{x} \]

   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 5.4

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

   5. Applied rewrites 5.4%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 5.4%

      \[\leadsto x - 1 \]
   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. fp-cancel-sub-sign-inv N/A

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

      5. metadata-eval N/A

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

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

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

   10. Applied rewrites 5.0%

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

   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 97.9%

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

Alternative 10: 98.7% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if x < -1.30000000000000004

   1. Initial program 100.0%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   2. 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 98.0

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

   5. Applied rewrites 98.0%

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

   if -1.30000000000000004 < x < 1.6000000000000001

   1. Initial program 8.3%

      \[\frac{2}{1 + e^{-2 \cdot x}} - 1 \]
   2. 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 99.4

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

   5. Applied rewrites 99.4%

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

      1. lift-pow.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. cube-mult N/A

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

      4. pow2 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. *-rgt-identity N/A

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

      8. distribute-lft-in N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 99.4

          \[\leadsto \mathsf{fma}\left(x \cdot x, -0.3333333333333333, 1\right) \cdot x \]

   7. Applied rewrites 99.4%

      \[\leadsto \mathsf{fma}\left(x \cdot x, -0.3333333333333333, 1\right) \cdot \color{blue}{x} \]

   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 5.4

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

   5. Applied rewrites 5.4%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 5.4%

      \[\leadsto x - 1 \]
   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. fp-cancel-sub-sign-inv N/A

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

      5. metadata-eval N/A

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

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

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

   10. Applied rewrites 5.0%

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

   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 97.9%

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

Alternative 11: 79.5% accurate, 3.8× speedup

Math

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

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) -0.3333333333333333 1.0) x)
     (/ (* 2.0 x) (- 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), -0.3333333333333333, 1.0) * x;
	} else {
		tmp = (2.0 * x) / (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 = Float64(fma(Float64(x * x), -0.3333333333333333, 1.0) * x);
	else
		tmp = Float64(Float64(2.0 * x) / 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] * -0.3333333333333333 + 1.0), $MachinePrecision] * x), $MachinePrecision], N[(N[(2.0 * x), $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 98.0

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

   5. Applied rewrites 98.0%

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

   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 99.5

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

   5. Applied rewrites 99.5%

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

      1. lift-pow.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. cube-mult N/A

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

      4. pow2 N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. *-rgt-identity N/A

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

      8. distribute-lft-in N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

      11. lower-*.f64 N/A

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

      12. +-commutative N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. pow2 N/A

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

      16. lift-*.f64 99.5

          \[\leadsto \mathsf{fma}\left(x \cdot x, -0.3333333333333333, 1\right) \cdot x \]

   7. Applied rewrites 99.5%

      \[\leadsto \mathsf{fma}\left(x \cdot x, -0.3333333333333333, 1\right) \cdot \color{blue}{x} \]

   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 5.4

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

   5. Applied rewrites 5.4%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 5.4%

      \[\leadsto x - 1 \]
   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. fp-cancel-sub-sign-inv N/A

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

      5. metadata-eval N/A

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

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

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

   10. Applied rewrites 5.0%

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

   11. Taylor expanded in x around 0

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

       1. lower-*.f64 18.8

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

   13. Applied rewrites 18.8%

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

Alternative 12: 52.0% accurate, 123.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 x)

C

double code(double x) {
	return x;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

def code(x):
	return x

Julia

function code(x)
	return x
end

MATLAB

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

Wolfram

code[x_] := x

Derivation

1. Initial program 54.3%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 52.0%

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

Reproduce

herbie shell --seed 2025088 
(FPCore (x)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))