Logistic function from Lakshay Garg

Percentage Accurate: 53.6% → 100.0%

Time: 8.0s

Alternatives: 16

Speedup: 4.7×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.6% 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, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(e^{x}\right)}^{-2}\\ t_1 := t\_0 + 1\\ t_2 := \left(\frac{4}{{t\_1}^{2}} + 1\right) + \frac{2}{t\_1}\\ \mathbf{if}\;x \leq -0.00068:\\ \;\;\;\;\mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left(t\_0\right)\right)\\ \mathbf{elif}\;x \leq 0.00072:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333 \cdot \left(x \cdot x\right), x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{8}{{t\_1}^{3}}}{t\_2} - {t\_2}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (pow (exp x) -2.0))
        (t_1 (+ t_0 1.0))
        (t_2 (+ (+ (/ 4.0 (pow t_1 2.0)) 1.0) (/ 2.0 t_1))))
   (if (<= x -0.00068)
     (expm1 (- (log 2.0) (log1p t_0)))
     (if (<= x 0.00072)
       (fma (* -0.3333333333333333 (* x x)) x x)
       (- (/ (/ 8.0 (pow t_1 3.0)) t_2) (pow t_2 -1.0))))))

C

double code(double x) {
	double t_0 = pow(exp(x), -2.0);
	double t_1 = t_0 + 1.0;
	double t_2 = ((4.0 / pow(t_1, 2.0)) + 1.0) + (2.0 / t_1);
	double tmp;
	if (x <= -0.00068) {
		tmp = expm1((log(2.0) - log1p(t_0)));
	} else if (x <= 0.00072) {
		tmp = fma((-0.3333333333333333 * (x * x)), x, x);
	} else {
		tmp = ((8.0 / pow(t_1, 3.0)) / t_2) - pow(t_2, -1.0);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = exp(x) ^ -2.0
	t_1 = Float64(t_0 + 1.0)
	t_2 = Float64(Float64(Float64(4.0 / (t_1 ^ 2.0)) + 1.0) + Float64(2.0 / t_1))
	tmp = 0.0
	if (x <= -0.00068)
		tmp = expm1(Float64(log(2.0) - log1p(t_0)));
	elseif (x <= 0.00072)
		tmp = fma(Float64(-0.3333333333333333 * Float64(x * x)), x, x);
	else
		tmp = Float64(Float64(Float64(8.0 / (t_1 ^ 3.0)) / t_2) - (t_2 ^ -1.0));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[Exp[x], $MachinePrecision], -2.0], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(4.0 / N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision] + 1.0), $MachinePrecision] + N[(2.0 / t$95$1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.00068], N[(Exp[N[(N[Log[2.0], $MachinePrecision] - N[Log[1 + t$95$0], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision], If[LessEqual[x, 0.00072], N[(N[(-0.3333333333333333 * N[(x * x), $MachinePrecision]), $MachinePrecision] * x + x), $MachinePrecision], N[(N[(N[(8.0 / N[Power[t$95$1, 3.0], $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision] - N[Power[t$95$2, -1.0], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.8e-4

   1. Initial program 99.9%

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

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left({\left(e^{x}\right)}^{-2}\right)\right)} \]

   if -6.8e-4 < x < 7.20000000000000045e-4

   1. Initial program 6.4%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. pow-plus N/A

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

      8. lower-pow.f64 N/A

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

      9. metadata-eval N/A

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

      10. lower--.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 100.0

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

   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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 100.0%

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

   if 7.20000000000000045e-4 < x

   1. Initial program 99.9%

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

      1. lift-exp.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. exp-prod N/A

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

      4. sqr-pow N/A

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

      5. fabs-sqr N/A

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

      6. sqr-pow N/A

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

      7. exp-prod N/A

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

      8. lift-*.f64 N/A

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

      9. lift-exp.f64 N/A

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

      10. rem-sqrt-square-rev N/A

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

      11. lower-sqrt.f64 N/A

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

      12. pow2 N/A

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

      13. lift-exp.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. exp-prod N/A

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

      17. pow-pow N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. lower-pow.f64 N/A

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

      21. lower-exp.f64 N/A

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

      22. metadata-eval 99.9

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

   4. Applied rewrites 99.9%

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

   6. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\frac{\frac{8}{{\left({\left(e^{x}\right)}^{-2} + 1\right)}^{3}}}{\left(\frac{4}{{\left({\left(e^{x}\right)}^{-2} + 1\right)}^{2}} + 1\right) + \frac{2}{{\left(e^{x}\right)}^{-2} + 1}} - {\left(\left(\frac{4}{{\left({\left(e^{x}\right)}^{-2} + 1\right)}^{2}} + 1\right) + \frac{2}{{\left(e^{x}\right)}^{-2} + 1}\right)}^{-1}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 100.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := {\left(e^{x}\right)}^{-2}\\ t_1 := t\_0 + 1\\ t_2 := \frac{2}{t\_1} + 1\\ \mathbf{if}\;x \leq -0.00068:\\ \;\;\;\;\mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left(t\_0\right)\right)\\ \mathbf{elif}\;x \leq 0.00085:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333 \cdot \left(x \cdot x\right), x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{4}{t\_2}}{{t\_1}^{2}} - {t\_2}^{-1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (pow (exp x) -2.0)) (t_1 (+ t_0 1.0)) (t_2 (+ (/ 2.0 t_1) 1.0)))
   (if (<= x -0.00068)
     (expm1 (- (log 2.0) (log1p t_0)))
     (if (<= x 0.00085)
       (fma (* -0.3333333333333333 (* x x)) x x)
       (- (/ (/ 4.0 t_2) (pow t_1 2.0)) (pow t_2 -1.0))))))

C

double code(double x) {
	double t_0 = pow(exp(x), -2.0);
	double t_1 = t_0 + 1.0;
	double t_2 = (2.0 / t_1) + 1.0;
	double tmp;
	if (x <= -0.00068) {
		tmp = expm1((log(2.0) - log1p(t_0)));
	} else if (x <= 0.00085) {
		tmp = fma((-0.3333333333333333 * (x * x)), x, x);
	} else {
		tmp = ((4.0 / t_2) / pow(t_1, 2.0)) - pow(t_2, -1.0);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = exp(x) ^ -2.0
	t_1 = Float64(t_0 + 1.0)
	t_2 = Float64(Float64(2.0 / t_1) + 1.0)
	tmp = 0.0
	if (x <= -0.00068)
		tmp = expm1(Float64(log(2.0) - log1p(t_0)));
	elseif (x <= 0.00085)
		tmp = fma(Float64(-0.3333333333333333 * Float64(x * x)), x, x);
	else
		tmp = Float64(Float64(Float64(4.0 / t_2) / (t_1 ^ 2.0)) - (t_2 ^ -1.0));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[Exp[x], $MachinePrecision], -2.0], $MachinePrecision]}, Block[{t$95$1 = N[(t$95$0 + 1.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 / t$95$1), $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[x, -0.00068], N[(Exp[N[(N[Log[2.0], $MachinePrecision] - N[Log[1 + t$95$0], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision], If[LessEqual[x, 0.00085], N[(N[(-0.3333333333333333 * N[(x * x), $MachinePrecision]), $MachinePrecision] * x + x), $MachinePrecision], N[(N[(N[(4.0 / t$95$2), $MachinePrecision] / N[Power[t$95$1, 2.0], $MachinePrecision]), $MachinePrecision] - N[Power[t$95$2, -1.0], $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -6.8e-4

   1. Initial program 99.9%

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

   4. Applied rewrites 100.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left({\left(e^{x}\right)}^{-2}\right)\right)} \]

   if -6.8e-4 < x < 8.49999999999999953e-4

   1. Initial program 6.4%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. pow-plus N/A

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

      8. lower-pow.f64 N/A

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

      9. metadata-eval N/A

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

      10. lower--.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 100.0

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

   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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 100.0%

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

   if 8.49999999999999953e-4 < x

   1. Initial program 99.9%

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

      1. lift-exp.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. exp-prod N/A

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

      4. sqr-pow N/A

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

      5. fabs-sqr N/A

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

      6. sqr-pow N/A

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

      7. exp-prod N/A

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

      8. lift-*.f64 N/A

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

      9. lift-exp.f64 N/A

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

      10. rem-sqrt-square-rev N/A

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

      11. lower-sqrt.f64 N/A

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

      12. pow2 N/A

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

      13. lift-exp.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. exp-prod N/A

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

      17. pow-pow N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. lower-pow.f64 N/A

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

      21. lower-exp.f64 N/A

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

      22. metadata-eval 99.9

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

   4. Applied rewrites 99.9%

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in x around inf

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

      1. lower--.f64 N/A

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

   9. Applied rewrites 99.9%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00068:\\ \;\;\;\;\mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left({\left(e^{x}\right)}^{-2}\right)\right)\\ \mathbf{elif}\;x \leq 0.00085:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333 \cdot \left(x \cdot x\right), x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{4}{\frac{2}{{\left(e^{x}\right)}^{-2} + 1} + 1}}{{\left({\left(e^{x}\right)}^{-2} + 1\right)}^{2}} - {\left(\frac{2}{{\left(e^{x}\right)}^{-2} + 1} + 1\right)}^{-1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 100.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.00068 \lor \neg \left(x \leq 0.00085\right):\\ \;\;\;\;\mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left({\left(e^{x}\right)}^{-2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333 \cdot \left(x \cdot x\right), x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -0.00068) (not (<= x 0.00085)))
   (expm1 (- (log 2.0) (log1p (pow (exp x) -2.0))))
   (fma (* -0.3333333333333333 (* x x)) x x)))

C

double code(double x) {
	double tmp;
	if ((x <= -0.00068) || !(x <= 0.00085)) {
		tmp = expm1((log(2.0) - log1p(pow(exp(x), -2.0))));
	} else {
		tmp = fma((-0.3333333333333333 * (x * x)), x, x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if ((x <= -0.00068) || !(x <= 0.00085))
		tmp = expm1(Float64(log(2.0) - log1p((exp(x) ^ -2.0))));
	else
		tmp = fma(Float64(-0.3333333333333333 * Float64(x * x)), x, x);
	end
	return tmp
end

Wolfram

code[x_] := If[Or[LessEqual[x, -0.00068], N[Not[LessEqual[x, 0.00085]], $MachinePrecision]], N[(Exp[N[(N[Log[2.0], $MachinePrecision] - N[Log[1 + N[Power[N[Exp[x], $MachinePrecision], -2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]] - 1), $MachinePrecision], N[(N[(-0.3333333333333333 * N[(x * x), $MachinePrecision]), $MachinePrecision] * x + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -6.8e-4 or 8.49999999999999953e-4 < x

   1. Initial program 99.9%

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

   4. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left({\left(e^{x}\right)}^{-2}\right)\right)} \]

   if -6.8e-4 < x < 8.49999999999999953e-4

   1. Initial program 6.4%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. pow-plus N/A

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

      8. lower-pow.f64 N/A

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

      9. metadata-eval N/A

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

      10. lower--.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 100.0

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

   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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00068 \lor \neg \left(x \leq 0.00085\right):\\ \;\;\;\;\mathsf{expm1}\left(\log 2 - \mathsf{log1p}\left({\left(e^{x}\right)}^{-2}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333 \cdot \left(x \cdot x\right), x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 100.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.00085 \lor \neg \left(x \leq 0.0009\right):\\ \;\;\;\;\frac{2}{1 + \sqrt{e^{-4 \cdot x}}} - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333 \cdot \left(x \cdot x\right), x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -0.00085) (not (<= x 0.0009)))
   (- (/ 2.0 (+ 1.0 (sqrt (exp (* -4.0 x))))) 1.0)
   (fma (* -0.3333333333333333 (* x x)) x x)))

C

double code(double x) {
	double tmp;
	if ((x <= -0.00085) || !(x <= 0.0009)) {
		tmp = (2.0 / (1.0 + sqrt(exp((-4.0 * x))))) - 1.0;
	} else {
		tmp = fma((-0.3333333333333333 * (x * x)), x, x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if ((x <= -0.00085) || !(x <= 0.0009))
		tmp = Float64(Float64(2.0 / Float64(1.0 + sqrt(exp(Float64(-4.0 * x))))) - 1.0);
	else
		tmp = fma(Float64(-0.3333333333333333 * Float64(x * x)), x, x);
	end
	return tmp
end

Wolfram

code[x_] := If[Or[LessEqual[x, -0.00085], N[Not[LessEqual[x, 0.0009]], $MachinePrecision]], N[(N[(2.0 / N[(1.0 + N[Sqrt[N[Exp[N[(-4.0 * x), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(-0.3333333333333333 * N[(x * x), $MachinePrecision]), $MachinePrecision] * x + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -8.49999999999999953e-4 or 8.9999999999999998e-4 < x

   1. Initial program 99.9%

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

      1. lift-exp.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. exp-prod N/A

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

      4. sqr-pow N/A

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

      5. fabs-sqr N/A

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

      6. sqr-pow N/A

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

      7. exp-prod N/A

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

      8. lift-*.f64 N/A

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

      9. lift-exp.f64 N/A

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

      10. rem-sqrt-square-rev N/A

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

      11. lower-sqrt.f64 N/A

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

      12. pow2 N/A

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

      13. lift-exp.f64 N/A

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

      14. lift-*.f64 N/A

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

      15. *-commutative N/A

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

      16. exp-prod N/A

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

      17. pow-pow N/A

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

      18. metadata-eval N/A

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

      19. metadata-eval N/A

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

      20. lower-pow.f64 N/A

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

      21. lower-exp.f64 N/A

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

      22. metadata-eval 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift-pow.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. pow-exp N/A

         \[\leadsto \frac{2}{1 + \sqrt{\color{blue}{e^{x \cdot -4}}}} - 1 \]

      4. lower-exp.f64 N/A

         \[\leadsto \frac{2}{1 + \sqrt{\color{blue}{e^{x \cdot -4}}}} - 1 \]

      5. *-commutative N/A

         \[\leadsto \frac{2}{1 + \sqrt{e^{\color{blue}{-4 \cdot x}}}} - 1 \]

      6. lower-*.f64 99.9

         \[\leadsto \frac{2}{1 + \sqrt{e^{\color{blue}{-4 \cdot x}}}} - 1 \]

   6. Applied rewrites 99.9%

      \[\leadsto \frac{2}{1 + \sqrt{\color{blue}{e^{-4 \cdot x}}}} - 1 \]

   if -8.49999999999999953e-4 < x < 8.9999999999999998e-4

   1. Initial program 6.4%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. pow-plus N/A

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

      8. lower-pow.f64 N/A

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

      9. metadata-eval N/A

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

      10. lower--.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 100.0

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

   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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00085 \lor \neg \left(x \leq 0.0009\right):\\ \;\;\;\;\frac{2}{1 + \sqrt{e^{-4 \cdot x}}} - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333 \cdot \left(x \cdot x\right), x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 100.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.00085 \lor \neg \left(x \leq 0.0009\right):\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333 \cdot \left(x \cdot x\right), x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -0.00085) (not (<= x 0.0009)))
   (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0)
   (fma (* -0.3333333333333333 (* x x)) x x)))

C

double code(double x) {
	double tmp;
	if ((x <= -0.00085) || !(x <= 0.0009)) {
		tmp = (2.0 / (1.0 + exp((-2.0 * x)))) - 1.0;
	} else {
		tmp = fma((-0.3333333333333333 * (x * x)), x, x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if ((x <= -0.00085) || !(x <= 0.0009))
		tmp = Float64(Float64(2.0 / Float64(1.0 + exp(Float64(-2.0 * x)))) - 1.0);
	else
		tmp = fma(Float64(-0.3333333333333333 * Float64(x * x)), x, x);
	end
	return tmp
end

Wolfram

code[x_] := If[Or[LessEqual[x, -0.00085], N[Not[LessEqual[x, 0.0009]], $MachinePrecision]], N[(N[(2.0 / N[(1.0 + N[Exp[N[(-2.0 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(-0.3333333333333333 * N[(x * x), $MachinePrecision]), $MachinePrecision] * x + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -8.49999999999999953e-4 or 8.9999999999999998e-4 < x

   1. Initial program 99.9%

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

   if -8.49999999999999953e-4 < x < 8.9999999999999998e-4

   1. Initial program 6.4%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. pow-plus N/A

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

      8. lower-pow.f64 N/A

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

      9. metadata-eval N/A

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

      10. lower--.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 100.0

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

   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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.00085 \lor \neg \left(x \leq 0.0009\right):\\ \;\;\;\;\frac{2}{1 + e^{-2 \cdot x}} - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333 \cdot \left(x \cdot x\right), x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.5% accurate, 3.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

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

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

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

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

      3. +-commutative N/A

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

      4. remove-double-neg N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 98.8

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

   5. Applied rewrites 98.8%

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

   if -1 < x

   1. Initial program 34.5%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. pow-plus N/A

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

      8. lower-pow.f64 N/A

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

      9. metadata-eval N/A

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

      10. lower--.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 71.5

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

   5. Applied rewrites 71.5%

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

   7. Applied rewrites 71.5%

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

Alternative 7: 75.5% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.15:\\ \;\;\;\;\frac{2}{\left(\mathsf{fma}\left(-1.3333333333333333, x, 2\right) \cdot x - 2\right) \cdot x} - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot 0.13333333333333333 - 0.3333333333333333\right) \cdot \left(x \cdot x\right), x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.15)
   (- (/ 2.0 (* (- (* (fma -1.3333333333333333 x 2.0) x) 2.0) x)) 1.0)
   (fma
    (* (- (* (* x x) 0.13333333333333333) 0.3333333333333333) (* x x))
    x
    x)))

C

double code(double x) {
	double tmp;
	if (x <= -1.15) {
		tmp = (2.0 / (((fma(-1.3333333333333333, x, 2.0) * x) - 2.0) * x)) - 1.0;
	} else {
		tmp = fma(((((x * x) * 0.13333333333333333) - 0.3333333333333333) * (x * x)), x, x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.15)
		tmp = Float64(Float64(2.0 / Float64(Float64(Float64(fma(-1.3333333333333333, x, 2.0) * x) - 2.0) * x)) - 1.0);
	else
		tmp = fma(Float64(Float64(Float64(Float64(x * x) * 0.13333333333333333) - 0.3333333333333333) * Float64(x * x)), x, x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1.15], N[(N[(2.0 / N[(N[(N[(N[(-1.3333333333333333 * x + 2.0), $MachinePrecision] * x), $MachinePrecision] - 2.0), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(N[(N[(x * x), $MachinePrecision] * 0.13333333333333333), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * x + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.1499999999999999

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

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

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

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

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

      3. +-commutative N/A

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

      4. remove-double-neg N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 98.8

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

   5. Applied rewrites 98.8%

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

   6. Taylor expanded in x around -inf

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

   8. Applied rewrites 98.8%

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

   if -1.1499999999999999 < x

   1. Initial program 34.5%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. pow-plus N/A

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

      8. lower-pow.f64 N/A

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

      9. metadata-eval N/A

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

      10. lower--.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 71.5

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

   5. Applied rewrites 71.5%

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

   7. Applied rewrites 71.5%

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

Alternative 8: 75.5% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.3:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(-1.3333333333333333, x, 2\right) \cdot \left(x \cdot x\right)} - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot 0.13333333333333333 - 0.3333333333333333\right) \cdot \left(x \cdot x\right), x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.3)
   (- (/ 2.0 (* (fma -1.3333333333333333 x 2.0) (* x x))) 1.0)
   (fma
    (* (- (* (* x x) 0.13333333333333333) 0.3333333333333333) (* x x))
    x
    x)))

C

double code(double x) {
	double tmp;
	if (x <= -1.3) {
		tmp = (2.0 / (fma(-1.3333333333333333, x, 2.0) * (x * x))) - 1.0;
	} else {
		tmp = fma(((((x * x) * 0.13333333333333333) - 0.3333333333333333) * (x * x)), x, x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.3)
		tmp = Float64(Float64(2.0 / Float64(fma(-1.3333333333333333, x, 2.0) * Float64(x * x))) - 1.0);
	else
		tmp = fma(Float64(Float64(Float64(Float64(x * x) * 0.13333333333333333) - 0.3333333333333333) * Float64(x * x)), x, x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1.3], N[(N[(2.0 / N[(N[(-1.3333333333333333 * x + 2.0), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(N[(N[(x * x), $MachinePrecision] * 0.13333333333333333), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * x + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.30000000000000004

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

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

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

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

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

      3. +-commutative N/A

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

      4. remove-double-neg N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 98.8

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

   5. Applied rewrites 98.8%

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

   6. Taylor expanded in x around -inf

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

   8. Applied rewrites 98.8%

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

   if -1.30000000000000004 < x

   1. Initial program 34.5%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. pow-plus N/A

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

      8. lower-pow.f64 N/A

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

      9. metadata-eval N/A

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

      10. lower--.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 71.5

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

   5. Applied rewrites 71.5%

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

   7. Applied rewrites 71.5%

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

Alternative 9: 75.5% accurate, 3.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.55:\\ \;\;\;\;\frac{2}{\left(-1.3333333333333333 \cdot x\right) \cdot \left(x \cdot x\right)} - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot 0.13333333333333333 - 0.3333333333333333\right) \cdot \left(x \cdot x\right), x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.55)
   (- (/ 2.0 (* (* -1.3333333333333333 x) (* x x))) 1.0)
   (fma
    (* (- (* (* x x) 0.13333333333333333) 0.3333333333333333) (* x x))
    x
    x)))

C

double code(double x) {
	double tmp;
	if (x <= -1.55) {
		tmp = (2.0 / ((-1.3333333333333333 * x) * (x * x))) - 1.0;
	} else {
		tmp = fma(((((x * x) * 0.13333333333333333) - 0.3333333333333333) * (x * x)), x, x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.55)
		tmp = Float64(Float64(2.0 / Float64(Float64(-1.3333333333333333 * x) * Float64(x * x))) - 1.0);
	else
		tmp = fma(Float64(Float64(Float64(Float64(x * x) * 0.13333333333333333) - 0.3333333333333333) * Float64(x * x)), x, x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1.55], N[(N[(2.0 / N[(N[(-1.3333333333333333 * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(N[(N[(x * x), $MachinePrecision] * 0.13333333333333333), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * x + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.55000000000000004

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

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

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

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

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

      3. +-commutative N/A

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

      4. remove-double-neg N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. +-commutative N/A

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

      11. lower-fma.f64 98.8

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

   5. Applied rewrites 98.8%

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

   6. Taylor expanded in x around -inf

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

   8. Applied rewrites 98.8%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 98.8%

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

   if -1.55000000000000004 < x

   1. Initial program 34.5%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. pow-plus N/A

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

      8. lower-pow.f64 N/A

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

      9. metadata-eval N/A

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

      10. lower--.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 71.5

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

   5. Applied rewrites 71.5%

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

   7. Applied rewrites 71.5%

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

Alternative 10: 75.4% accurate, 3.4× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.15)
   (- (/ 2.0 (fma (fma x 2.0 -2.0) x 2.0)) 1.0)
   (fma
    (* (- (* (* x x) 0.13333333333333333) 0.3333333333333333) (* x x))
    x
    x)))

C

double code(double x) {
	double tmp;
	if (x <= -1.15) {
		tmp = (2.0 / fma(fma(x, 2.0, -2.0), x, 2.0)) - 1.0;
	} else {
		tmp = fma(((((x * x) * 0.13333333333333333) - 0.3333333333333333) * (x * x)), x, x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.15)
		tmp = Float64(Float64(2.0 / fma(fma(x, 2.0, -2.0), x, 2.0)) - 1.0);
	else
		tmp = fma(Float64(Float64(Float64(Float64(x * x) * 0.13333333333333333) - 0.3333333333333333) * Float64(x * x)), x, x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1.15], N[(N[(2.0 / N[(N[(x * 2.0 + -2.0), $MachinePrecision] * x + 2.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(N[(N[(N[(x * x), $MachinePrecision] * 0.13333333333333333), $MachinePrecision] - 0.3333333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision] * x + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.1499999999999999

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 97.8

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

   5. Applied rewrites 97.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 98.4%

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

   if -1.1499999999999999 < x

   1. Initial program 34.5%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. pow-plus N/A

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

      8. lower-pow.f64 N/A

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

      9. metadata-eval N/A

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

      10. lower--.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 71.5

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

   5. Applied rewrites 71.5%

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

   7. Applied rewrites 71.5%

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

Alternative 11: 74.6% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 97.8

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

   5. Applied rewrites 97.8%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 98.4%

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

   if -1 < x

   1. Initial program 34.5%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. pow-plus N/A

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

      8. lower-pow.f64 N/A

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

      9. metadata-eval N/A

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

      10. lower--.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 71.5

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

   5. Applied rewrites 71.5%

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

   7. Applied rewrites 71.5%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 70.6%

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

Alternative 12: 74.4% accurate, 4.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1.30000000000000004

   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

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

      2. lower-fma.f64 97.8

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

   5. Applied rewrites 97.8%

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

   if -1.30000000000000004 < x

   1. Initial program 34.5%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. pow-plus N/A

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

      8. lower-pow.f64 N/A

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

      9. metadata-eval N/A

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

      10. lower--.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 71.5

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

   5. Applied rewrites 71.5%

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

   7. Applied rewrites 71.5%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 70.6%

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

Alternative 13: 74.4% accurate, 4.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.55:\\ \;\;\;\;\frac{2}{x \cdot -2} - 1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.3333333333333333 \cdot \left(x \cdot x\right), x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -1.55)
   (- (/ 2.0 (* x -2.0)) 1.0)
   (fma (* -0.3333333333333333 (* x x)) x x)))

C

double code(double x) {
	double tmp;
	if (x <= -1.55) {
		tmp = (2.0 / (x * -2.0)) - 1.0;
	} else {
		tmp = fma((-0.3333333333333333 * (x * x)), x, x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -1.55)
		tmp = Float64(Float64(2.0 / Float64(x * -2.0)) - 1.0);
	else
		tmp = fma(Float64(-0.3333333333333333 * Float64(x * x)), x, x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -1.55], N[(N[(2.0 / N[(x * -2.0), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision], N[(N[(-0.3333333333333333 * N[(x * x), $MachinePrecision]), $MachinePrecision] * x + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.55000000000000004

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 97.8

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

   5. Applied rewrites 97.8%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 97.8%

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

   if -1.55000000000000004 < x

   1. Initial program 34.5%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. associate-*r* N/A

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

      4. *-rgt-identity N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. pow-plus N/A

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

      8. lower-pow.f64 N/A

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

      9. metadata-eval N/A

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

      10. lower--.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. unpow2 N/A

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

      13. lower-*.f64 71.5

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

   5. Applied rewrites 71.5%

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

   7. Applied rewrites 71.5%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 70.6%

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

Alternative 14: 50.6% accurate, 7.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 49.9%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. associate-*r* N/A

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

   4. *-rgt-identity N/A

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

   5. lower-fma.f64 N/A

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

   6. *-commutative N/A

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

   7. pow-plus N/A

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

   8. lower-pow.f64 N/A

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

   9. metadata-eval N/A

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

   10. lower--.f64 N/A

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

   11. lower-*.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 55.6

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

5. Applied rewrites 55.6%

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

7. Applied rewrites 55.6%

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

8. Taylor expanded in x around 0

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

10. Applied rewrites 54.2%

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

Alternative 15: 6.4% accurate, 17.6× speedup

Math

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

FPCore

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

C

double code(double x) {
	return (1.0 + 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 = (1.0d0 + x) - 1.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 49.9%

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

3. Taylor expanded in x around 0

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

   1. lower-+.f64 6.1

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

5. Applied rewrites 6.1%

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

Alternative 16: 4.3% accurate, 30.8× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

def code(x):
	return 1.0 - 1.0

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 49.9%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 4.4%

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

Reproduce

herbie shell --seed 2024353 
(FPCore (x)
  :name "Logistic function from Lakshay Garg"
  :precision binary64
  (- (/ 2.0 (+ 1.0 (exp (* -2.0 x)))) 1.0))