Hyperbolic secant

Percentage Accurate: 100.0% → 100.0%

Time: 3.1s

Alternatives: 15

Speedup: 1.9×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \frac{1}{\cosh x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 1.0 (cosh x)))

C

double code(double x) {
	return 1.0 / cosh(x);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

public static double code(double x) {
	return 1.0 / Math.cosh(x);
}

Python

def code(x):
	return 1.0 / math.cosh(x)

Julia

function code(x)
	return Float64(1.0 / cosh(x))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 / cosh(x);
end

Wolfram

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

Derivation

1. Initial program 100.0%

   \[\frac{2}{e^{x} + e^{-x}} \]
2. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift-+.f64 N/A

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

   3. lift-exp.f64 N/A

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

   4. lift-neg.f64 N/A

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

   5. lift-exp.f64 N/A

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

   6. cosh-undef N/A

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

   7. associate-/r* N/A

      \[\leadsto \color{blue}{\frac{\frac{2}{2}}{\cosh x}} \]

   8. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{1}}{\cosh x} \]

   9. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{1}{\cosh x}} \]

   10. lower-cosh.f64 100.0

       \[\leadsto \frac{1}{\color{blue}{\cosh x}} \]

3. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\frac{1}{\cosh x}} \]
4. Add Preprocessing

Alternative 2: 92.2% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot x\right)\\ \frac{2}{\frac{\mathsf{fma}\left(t\_0, t\_0, 8\right)}{4}} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* x (* x x)))) (/ 2.0 (/ (fma t_0 t_0 8.0) 4.0))))

C

double code(double x) {
	double t_0 = x * (x * x);
	return 2.0 / (fma(t_0, t_0, 8.0) / 4.0);
}

Julia

function code(x)
	t_0 = Float64(x * Float64(x * x))
	return Float64(2.0 / Float64(fma(t_0, t_0, 8.0) / 4.0))
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]}, N[(2.0 / N[(N[(t$95$0 * t$95$0 + 8.0), $MachinePrecision] / 4.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in x around 0

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

   1. +-commutative N/A

      \[\leadsto \frac{2}{{x}^{2} + \color{blue}{2}} \]

   2. unpow2 N/A

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

   3. lower-fma.f64 76.2

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

4. Applied rewrites 76.2%

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

   1. lift-fma.f64 N/A

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

   2. pow2 N/A

      \[\leadsto \frac{2}{{x}^{2} + 2} \]

   3. +-commutative N/A

      \[\leadsto \frac{2}{2 + \color{blue}{{x}^{2}}} \]

   4. flip3-+ N/A

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

   5. lower-/.f64 N/A

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

   6. +-commutative N/A

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

   7. pow-pow N/A

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

   8. metadata-eval N/A

      \[\leadsto \frac{2}{\frac{{x}^{6} + {2}^{3}}{2 \cdot 2 + \left({x}^{2} \cdot {x}^{2} - 2 \cdot {x}^{2}\right)}} \]

   9. metadata-eval N/A

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

   10. pow-prod-up N/A

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

   11. pow2 N/A

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

   12. metadata-eval N/A

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

   13. pow-prod-up N/A

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

   14. unswap-sqr N/A

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

   15. lower-fma.f64 N/A

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

   16. lower-*.f64 N/A

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

   17. pow2 N/A

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

   18. lift-*.f64 N/A

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

   19. lower-*.f64 N/A

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

   20. pow2 N/A

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

   21. lift-*.f64 N/A

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

   22. metadata-eval N/A

       \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), x \cdot \left(x \cdot x\right), 8\right)}{2 \cdot 2 + \left({x}^{2} \cdot {x}^{2} - 2 \cdot {x}^{2}\right)}} \]

   23. metadata-eval N/A

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

6. Applied rewrites 54.2%

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

7. Taylor expanded in x around 0

   \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), x \cdot \left(x \cdot x\right), 8\right)}{4}} \]
8. Step-by-step derivation

9. Applied rewrites 91.7%

   \[\leadsto \frac{2}{\frac{\mathsf{fma}\left(x \cdot \left(x \cdot x\right), x \cdot \left(x \cdot x\right), 8\right)}{4}} \]
10. Add Preprocessing

Alternative 3: 92.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{2}{e^{x} + e^{-x}} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{2}{\left(\mathsf{fma}\left(x \cdot x, 0.002777777777777778, 0.08333333333333333\right) \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, -0.5, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (/ 2.0 (+ (exp x) (exp (- x)))) 5e-6)
   (/
    2.0
    (*
     (* (fma (* x x) 0.002777777777777778 0.08333333333333333) x)
     (* (* x x) x)))
   (fma (* x x) -0.5 1.0)))

C

double code(double x) {
	double tmp;
	if ((2.0 / (exp(x) + exp(-x))) <= 5e-6) {
		tmp = 2.0 / ((fma((x * x), 0.002777777777777778, 0.08333333333333333) * x) * ((x * x) * x));
	} else {
		tmp = fma((x * x), -0.5, 1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(2.0 / Float64(exp(x) + exp(Float64(-x)))) <= 5e-6)
		tmp = Float64(2.0 / Float64(Float64(fma(Float64(x * x), 0.002777777777777778, 0.08333333333333333) * x) * Float64(Float64(x * x) * x)));
	else
		tmp = fma(Float64(x * x), -0.5, 1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(2.0 / N[(N[Exp[x], $MachinePrecision] + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e-6], N[(2.0 / N[(N[(N[(N[(x * x), $MachinePrecision] * 0.002777777777777778 + 0.08333333333333333), $MachinePrecision] * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 2 binary64) (+.f64 (exp.f64 x) (exp.f64 (neg.f64 x)))) < 5.00000000000000041e-6

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

      \[\leadsto \frac{2}{\color{blue}{2 + {x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right)}} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2}{{x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right) + \color{blue}{2}} \]

      2. *-commutative N/A

         \[\leadsto \frac{2}{\left(1 + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right) \cdot {x}^{2} + 2} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(1 + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right), \color{blue}{{x}^{2}}, 2\right)} \]

      4. +-commutative N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left({x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right) + 1, {\color{blue}{x}}^{2}, 2\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(\left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right) \cdot {x}^{2} + 1, {x}^{2}, 2\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}, {x}^{2}, 1\right), {\color{blue}{x}}^{2}, 2\right)} \]

      7. +-commutative N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360} \cdot {x}^{2} + \frac{1}{12}, {x}^{2}, 1\right), {x}^{2}, 2\right)} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, {x}^{2}, \frac{1}{12}\right), {x}^{2}, 1\right), {x}^{2}, 2\right)} \]

      9. unpow2 N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, x \cdot x, \frac{1}{12}\right), {x}^{2}, 1\right), {x}^{2}, 2\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, x \cdot x, \frac{1}{12}\right), {x}^{2}, 1\right), {x}^{2}, 2\right)} \]

      11. unpow2 N/A

          \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, x \cdot x, \frac{1}{12}\right), x \cdot x, 1\right), {x}^{2}, 2\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, x \cdot x, \frac{1}{12}\right), x \cdot x, 1\right), {x}^{2}, 2\right)} \]

      13. unpow2 N/A

          \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, x \cdot x, \frac{1}{12}\right), x \cdot x, 1\right), x \cdot \color{blue}{x}, 2\right)} \]

      14. lower-*.f64 84.8

          \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, x \cdot x, 0.08333333333333333\right), x \cdot x, 1\right), x \cdot \color{blue}{x}, 2\right)} \]

   4. Applied rewrites 84.8%

      \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, x \cdot x, 0.08333333333333333\right), x \cdot x, 1\right), x \cdot x, 2\right)}} \]

   5. Taylor expanded in x around inf

      \[\leadsto \frac{2}{{x}^{6} \cdot \color{blue}{\left(\frac{1}{360} + \frac{1}{12} \cdot \frac{1}{{x}^{2}}\right)}} \]
   6. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \frac{2}{\left(\frac{1}{360} + \frac{1}{12} \cdot \frac{1}{{x}^{2}}\right) \cdot {x}^{\color{blue}{6}}} \]

      2. metadata-eval N/A

         \[\leadsto \frac{2}{\left(\frac{1}{360} + \frac{1}{12} \cdot \frac{1}{{x}^{2}}\right) \cdot {x}^{\left(2 + 4\right)}} \]

      3. pow-prod-up N/A

         \[\leadsto \frac{2}{\left(\frac{1}{360} + \frac{1}{12} \cdot \frac{1}{{x}^{2}}\right) \cdot \left({x}^{2} \cdot {x}^{\color{blue}{4}}\right)} \]

      4. pow2 N/A

         \[\leadsto \frac{2}{\left(\frac{1}{360} + \frac{1}{12} \cdot \frac{1}{{x}^{2}}\right) \cdot \left(\left(x \cdot x\right) \cdot {x}^{4}\right)} \]

      5. associate-*r* N/A

         \[\leadsto \frac{2}{\left(\left(\frac{1}{360} + \frac{1}{12} \cdot \frac{1}{{x}^{2}}\right) \cdot \left(x \cdot x\right)\right) \cdot {x}^{\color{blue}{4}}} \]

      6. *-commutative N/A

         \[\leadsto \frac{2}{\left(\left(x \cdot x\right) \cdot \left(\frac{1}{360} + \frac{1}{12} \cdot \frac{1}{{x}^{2}}\right)\right) \cdot {x}^{4}} \]

      7. distribute-rgt-in N/A

         \[\leadsto \frac{2}{\left(\frac{1}{360} \cdot \left(x \cdot x\right) + \left(\frac{1}{12} \cdot \frac{1}{{x}^{2}}\right) \cdot \left(x \cdot x\right)\right) \cdot {x}^{4}} \]

      8. associate-*l* N/A

         \[\leadsto \frac{2}{\left(\frac{1}{360} \cdot \left(x \cdot x\right) + \frac{1}{12} \cdot \left(\frac{1}{{x}^{2}} \cdot \left(x \cdot x\right)\right)\right) \cdot {x}^{4}} \]

      9. pow2 N/A

         \[\leadsto \frac{2}{\left(\frac{1}{360} \cdot \left(x \cdot x\right) + \frac{1}{12} \cdot \left(\frac{1}{x \cdot x} \cdot \left(x \cdot x\right)\right)\right) \cdot {x}^{4}} \]

      10. lft-mult-inverse N/A

          \[\leadsto \frac{2}{\left(\frac{1}{360} \cdot \left(x \cdot x\right) + \frac{1}{12} \cdot 1\right) \cdot {x}^{4}} \]

      11. metadata-eval N/A

          \[\leadsto \frac{2}{\left(\frac{1}{360} \cdot \left(x \cdot x\right) + \frac{1}{12}\right) \cdot {x}^{4}} \]

      12. metadata-eval N/A

          \[\leadsto \frac{2}{\left(\frac{1}{360} \cdot \left(x \cdot x\right) + \frac{1}{12}\right) \cdot {x}^{\left(2 \cdot 2\right)}} \]

   7. Applied rewrites 84.8%

      \[\leadsto \frac{2}{\left(\mathsf{fma}\left(x \cdot x, 0.002777777777777778, 0.08333333333333333\right) \cdot x\right) \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot x\right)}} \]

   if 5.00000000000000041e-6 < (/.f64 #s(literal 2 binary64) (+.f64 (exp.f64 x) (exp.f64 (neg.f64 x))))

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 99.1

         \[\leadsto \mathsf{fma}\left(x \cdot x, -0.5, 1\right) \]

   4. Applied rewrites 99.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, -0.5, 1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 91.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(x \cdot x\right) \cdot x\\ \mathbf{if}\;\frac{2}{e^{x} + e^{-x}} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{2}{\left(0.002777777777777778 \cdot t\_0\right) \cdot t\_0}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, -0.5, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (* x x) x)))
   (if (<= (/ 2.0 (+ (exp x) (exp (- x)))) 5e-6)
     (/ 2.0 (* (* 0.002777777777777778 t_0) t_0))
     (fma (* x x) -0.5 1.0))))

C

double code(double x) {
	double t_0 = (x * x) * x;
	double tmp;
	if ((2.0 / (exp(x) + exp(-x))) <= 5e-6) {
		tmp = 2.0 / ((0.002777777777777778 * t_0) * t_0);
	} else {
		tmp = fma((x * x), -0.5, 1.0);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(Float64(x * x) * x)
	tmp = 0.0
	if (Float64(2.0 / Float64(exp(x) + exp(Float64(-x)))) <= 5e-6)
		tmp = Float64(2.0 / Float64(Float64(0.002777777777777778 * t_0) * t_0));
	else
		tmp = fma(Float64(x * x), -0.5, 1.0);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[N[(2.0 / N[(N[Exp[x], $MachinePrecision] + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e-6], N[(2.0 / N[(N[(0.002777777777777778 * t$95$0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 2 binary64) (+.f64 (exp.f64 x) (exp.f64 (neg.f64 x)))) < 5.00000000000000041e-6

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

      \[\leadsto \frac{2}{\color{blue}{2 + {x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right)}} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2}{{x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right) + \color{blue}{2}} \]

      2. *-commutative N/A

         \[\leadsto \frac{2}{\left(1 + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right) \cdot {x}^{2} + 2} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(1 + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right), \color{blue}{{x}^{2}}, 2\right)} \]

      4. +-commutative N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left({x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right) + 1, {\color{blue}{x}}^{2}, 2\right)} \]

      5. *-commutative N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(\left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right) \cdot {x}^{2} + 1, {x}^{2}, 2\right)} \]

      6. lower-fma.f64 N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}, {x}^{2}, 1\right), {\color{blue}{x}}^{2}, 2\right)} \]

      7. +-commutative N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360} \cdot {x}^{2} + \frac{1}{12}, {x}^{2}, 1\right), {x}^{2}, 2\right)} \]

      8. lower-fma.f64 N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, {x}^{2}, \frac{1}{12}\right), {x}^{2}, 1\right), {x}^{2}, 2\right)} \]

      9. unpow2 N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, x \cdot x, \frac{1}{12}\right), {x}^{2}, 1\right), {x}^{2}, 2\right)} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, x \cdot x, \frac{1}{12}\right), {x}^{2}, 1\right), {x}^{2}, 2\right)} \]

      11. unpow2 N/A

          \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, x \cdot x, \frac{1}{12}\right), x \cdot x, 1\right), {x}^{2}, 2\right)} \]

      12. lower-*.f64 N/A

          \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, x \cdot x, \frac{1}{12}\right), x \cdot x, 1\right), {x}^{2}, 2\right)} \]

      13. unpow2 N/A

          \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, x \cdot x, \frac{1}{12}\right), x \cdot x, 1\right), x \cdot \color{blue}{x}, 2\right)} \]

      14. lower-*.f64 84.8

          \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, x \cdot x, 0.08333333333333333\right), x \cdot x, 1\right), x \cdot \color{blue}{x}, 2\right)} \]

   4. Applied rewrites 84.8%

      \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, x \cdot x, 0.08333333333333333\right), x \cdot x, 1\right), x \cdot x, 2\right)}} \]

   5. Taylor expanded in x around inf

      \[\leadsto \frac{2}{\frac{1}{360} \cdot \color{blue}{{x}^{6}}} \]
   6. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{2}{\frac{1}{360} \cdot {x}^{\left(3 \cdot 2\right)}} \]

      2. pow-pow N/A

         \[\leadsto \frac{2}{\frac{1}{360} \cdot {\left({x}^{3}\right)}^{2}} \]

      3. cube-unmult N/A

         \[\leadsto \frac{2}{\frac{1}{360} \cdot {\left(x \cdot \left(x \cdot x\right)\right)}^{2}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{1}{360} \cdot {\left(x \cdot \left(x \cdot x\right)\right)}^{2}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{2}{\frac{1}{360} \cdot {\left(x \cdot \left(x \cdot x\right)\right)}^{2}} \]

      6. pow2 N/A

         \[\leadsto \frac{2}{\frac{1}{360} \cdot \left(\left(x \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)\right)} \]

      7. associate-*r* N/A

         \[\leadsto \frac{2}{\left(\frac{1}{360} \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\frac{1}{360} \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(x \cdot \color{blue}{\left(x \cdot x\right)}\right)} \]

      9. lower-*.f64 84.8

         \[\leadsto \frac{2}{\left(0.002777777777777778 \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(x \cdot \left(\color{blue}{x} \cdot x\right)\right)} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{1}{360} \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{1}{360} \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)} \]

      12. associate-*r* N/A

          \[\leadsto \frac{2}{\left(\frac{1}{360} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{1}{360} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)} \]

      14. lift-*.f64 84.8

          \[\leadsto \frac{2}{\left(0.002777777777777778 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)} \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{1}{360} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot x\right)\right)} \]

      16. lift-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{1}{360} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(x \cdot \left(x \cdot \color{blue}{x}\right)\right)} \]

      17. associate-*r* N/A

          \[\leadsto \frac{2}{\left(\frac{1}{360} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)} \]

      18. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\frac{1}{360} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)} \]

      19. lift-*.f64 84.8

          \[\leadsto \frac{2}{\left(0.002777777777777778 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)} \]

   7. Applied rewrites 84.8%

      \[\leadsto \frac{2}{\left(0.002777777777777778 \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot x\right)}} \]

   if 5.00000000000000041e-6 < (/.f64 #s(literal 2 binary64) (+.f64 (exp.f64 x) (exp.f64 (neg.f64 x))))

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 99.1

         \[\leadsto \mathsf{fma}\left(x \cdot x, -0.5, 1\right) \]

   4. Applied rewrites 99.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, -0.5, 1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 91.9% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.002777777777777778, 0.08333333333333333\right), x \cdot x, 1\right) \cdot x, x, 2\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  2.0
  (fma
   (*
    (fma (fma (* x x) 0.002777777777777778 0.08333333333333333) (* x x) 1.0)
    x)
   x
   2.0)))

C

double code(double x) {
	return 2.0 / fma((fma(fma((x * x), 0.002777777777777778, 0.08333333333333333), (x * x), 1.0) * x), x, 2.0);
}

Julia

function code(x)
	return Float64(2.0 / fma(Float64(fma(fma(Float64(x * x), 0.002777777777777778, 0.08333333333333333), Float64(x * x), 1.0) * x), x, 2.0))
end

Wolfram

code[x_] := N[(2.0 / N[(N[(N[(N[(N[(x * x), $MachinePrecision] * 0.002777777777777778 + 0.08333333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * x), $MachinePrecision] * x + 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in x around 0

   \[\leadsto \frac{2}{\color{blue}{2 + {x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right)}} \]
3. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{2}{{x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right) + \color{blue}{2}} \]

   2. *-commutative N/A

      \[\leadsto \frac{2}{\left(1 + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right) \cdot {x}^{2} + 2} \]

   3. lower-fma.f64 N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left(1 + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right), \color{blue}{{x}^{2}}, 2\right)} \]

   4. +-commutative N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left({x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right) + 1, {\color{blue}{x}}^{2}, 2\right)} \]

   5. *-commutative N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left(\left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right) \cdot {x}^{2} + 1, {x}^{2}, 2\right)} \]

   6. lower-fma.f64 N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}, {x}^{2}, 1\right), {\color{blue}{x}}^{2}, 2\right)} \]

   7. +-commutative N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360} \cdot {x}^{2} + \frac{1}{12}, {x}^{2}, 1\right), {x}^{2}, 2\right)} \]

   8. lower-fma.f64 N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, {x}^{2}, \frac{1}{12}\right), {x}^{2}, 1\right), {x}^{2}, 2\right)} \]

   9. unpow2 N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, x \cdot x, \frac{1}{12}\right), {x}^{2}, 1\right), {x}^{2}, 2\right)} \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, x \cdot x, \frac{1}{12}\right), {x}^{2}, 1\right), {x}^{2}, 2\right)} \]

   11. unpow2 N/A

       \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, x \cdot x, \frac{1}{12}\right), x \cdot x, 1\right), {x}^{2}, 2\right)} \]

   12. lower-*.f64 N/A

       \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, x \cdot x, \frac{1}{12}\right), x \cdot x, 1\right), {x}^{2}, 2\right)} \]

   13. unpow2 N/A

       \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, x \cdot x, \frac{1}{12}\right), x \cdot x, 1\right), x \cdot \color{blue}{x}, 2\right)} \]

   14. lower-*.f64 92.2

       \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, x \cdot x, 0.08333333333333333\right), x \cdot x, 1\right), x \cdot \color{blue}{x}, 2\right)} \]

4. Applied rewrites 92.2%

   \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, x \cdot x, 0.08333333333333333\right), x \cdot x, 1\right), x \cdot x, 2\right)}} \]
5. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, x \cdot x, \frac{1}{12}\right), x \cdot x, 1\right), x \cdot \color{blue}{x}, 2\right)} \]

   2. lift-fma.f64 N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, x \cdot x, \frac{1}{12}\right), x \cdot x, 1\right) \cdot \left(x \cdot x\right) + \color{blue}{2}} \]

   3. lift-*.f64 N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, x \cdot x, \frac{1}{12}\right), x \cdot x, 1\right) \cdot \left(x \cdot x\right) + 2} \]

   4. lift-fma.f64 N/A

      \[\leadsto \frac{2}{\left(\mathsf{fma}\left(\frac{1}{360}, x \cdot x, \frac{1}{12}\right) \cdot \left(x \cdot x\right) + 1\right) \cdot \left(x \cdot x\right) + 2} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{2}{\left(\mathsf{fma}\left(\frac{1}{360}, x \cdot x, \frac{1}{12}\right) \cdot \left(x \cdot x\right) + 1\right) \cdot \left(x \cdot x\right) + 2} \]

   6. lift-fma.f64 N/A

      \[\leadsto \frac{2}{\left(\left(\frac{1}{360} \cdot \left(x \cdot x\right) + \frac{1}{12}\right) \cdot \left(x \cdot x\right) + 1\right) \cdot \left(x \cdot x\right) + 2} \]

   7. associate-*r* N/A

      \[\leadsto \frac{2}{\left(\left(\left(\frac{1}{360} \cdot \left(x \cdot x\right) + \frac{1}{12}\right) \cdot \left(x \cdot x\right) + 1\right) \cdot x\right) \cdot x + 2} \]

   8. lower-fma.f64 N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left(\left(\left(\frac{1}{360} \cdot \left(x \cdot x\right) + \frac{1}{12}\right) \cdot \left(x \cdot x\right) + 1\right) \cdot x, \color{blue}{x}, 2\right)} \]

   9. lower-*.f64 N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left(\left(\left(\frac{1}{360} \cdot \left(x \cdot x\right) + \frac{1}{12}\right) \cdot \left(x \cdot x\right) + 1\right) \cdot x, x, 2\right)} \]

   10. lower-fma.f64 N/A

       \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360} \cdot \left(x \cdot x\right) + \frac{1}{12}, x \cdot x, 1\right) \cdot x, x, 2\right)} \]

   11. *-commutative N/A

       \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\left(x \cdot x\right) \cdot \frac{1}{360} + \frac{1}{12}, x \cdot x, 1\right) \cdot x, x, 2\right)} \]

   12. lower-fma.f64 N/A

       \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{1}{360}, \frac{1}{12}\right), x \cdot x, 1\right) \cdot x, x, 2\right)} \]

   13. lift-*.f64 N/A

       \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{1}{360}, \frac{1}{12}\right), x \cdot x, 1\right) \cdot x, x, 2\right)} \]

   14. lift-*.f64 92.2

       \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.002777777777777778, 0.08333333333333333\right), x \cdot x, 1\right) \cdot x, x, 2\right)} \]

6. Applied rewrites 92.2%

   \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.002777777777777778, 0.08333333333333333\right), x \cdot x, 1\right) \cdot x, \color{blue}{x}, 2\right)} \]
7. Add Preprocessing

Alternative 6: 91.7% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\left(x \cdot x\right) \cdot 0.002777777777777778, x \cdot x, 1\right), x \cdot x, 2\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ 2.0 (fma (fma (* (* x x) 0.002777777777777778) (* x x) 1.0) (* x x) 2.0)))

C

double code(double x) {
	return 2.0 / fma(fma(((x * x) * 0.002777777777777778), (x * x), 1.0), (x * x), 2.0);
}

Julia

function code(x)
	return Float64(2.0 / fma(fma(Float64(Float64(x * x) * 0.002777777777777778), Float64(x * x), 1.0), Float64(x * x), 2.0))
end

Wolfram

code[x_] := N[(2.0 / N[(N[(N[(N[(x * x), $MachinePrecision] * 0.002777777777777778), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in x around 0

   \[\leadsto \frac{2}{\color{blue}{2 + {x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right)}} \]
3. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{2}{{x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right) + \color{blue}{2}} \]

   2. *-commutative N/A

      \[\leadsto \frac{2}{\left(1 + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right) \cdot {x}^{2} + 2} \]

   3. lower-fma.f64 N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left(1 + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right), \color{blue}{{x}^{2}}, 2\right)} \]

   4. +-commutative N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left({x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right) + 1, {\color{blue}{x}}^{2}, 2\right)} \]

   5. *-commutative N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left(\left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right) \cdot {x}^{2} + 1, {x}^{2}, 2\right)} \]

   6. lower-fma.f64 N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}, {x}^{2}, 1\right), {\color{blue}{x}}^{2}, 2\right)} \]

   7. +-commutative N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360} \cdot {x}^{2} + \frac{1}{12}, {x}^{2}, 1\right), {x}^{2}, 2\right)} \]

   8. lower-fma.f64 N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, {x}^{2}, \frac{1}{12}\right), {x}^{2}, 1\right), {x}^{2}, 2\right)} \]

   9. unpow2 N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, x \cdot x, \frac{1}{12}\right), {x}^{2}, 1\right), {x}^{2}, 2\right)} \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, x \cdot x, \frac{1}{12}\right), {x}^{2}, 1\right), {x}^{2}, 2\right)} \]

   11. unpow2 N/A

       \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, x \cdot x, \frac{1}{12}\right), x \cdot x, 1\right), {x}^{2}, 2\right)} \]

   12. lower-*.f64 N/A

       \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, x \cdot x, \frac{1}{12}\right), x \cdot x, 1\right), {x}^{2}, 2\right)} \]

   13. unpow2 N/A

       \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, x \cdot x, \frac{1}{12}\right), x \cdot x, 1\right), x \cdot \color{blue}{x}, 2\right)} \]

   14. lower-*.f64 92.2

       \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, x \cdot x, 0.08333333333333333\right), x \cdot x, 1\right), x \cdot \color{blue}{x}, 2\right)} \]

4. Applied rewrites 92.2%

   \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, x \cdot x, 0.08333333333333333\right), x \cdot x, 1\right), x \cdot x, 2\right)}} \]

5. Taylor expanded in x around inf

   \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360} \cdot {x}^{2}, x \cdot x, 1\right), x \cdot x, 2\right)} \]
6. Step-by-step derivation

   1. pow2 N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360} \cdot \left(x \cdot x\right), x \cdot x, 1\right), x \cdot x, 2\right)} \]

   2. *-commutative N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\left(x \cdot x\right) \cdot \frac{1}{360}, x \cdot x, 1\right), x \cdot x, 2\right)} \]

   3. lower-*.f64 N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\left(x \cdot x\right) \cdot \frac{1}{360}, x \cdot x, 1\right), x \cdot x, 2\right)} \]

   4. lift-*.f64 92.0

      \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\left(x \cdot x\right) \cdot 0.002777777777777778, x \cdot x, 1\right), x \cdot x, 2\right)} \]

7. Applied rewrites 92.0%

   \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\left(x \cdot x\right) \cdot 0.002777777777777778, x \cdot x, 1\right), x \cdot x, 2\right)} \]
8. Add Preprocessing

Alternative 7: 91.7% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\mathsf{fma}\left(\left(\mathsf{fma}\left(0.002777777777777778 \cdot x, x, 0.08333333333333333\right) \cdot x\right) \cdot x, x \cdot x, 2\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  2.0
  (fma
   (* (* (fma (* 0.002777777777777778 x) x 0.08333333333333333) x) x)
   (* x x)
   2.0)))

C

double code(double x) {
	return 2.0 / fma(((fma((0.002777777777777778 * x), x, 0.08333333333333333) * x) * x), (x * x), 2.0);
}

Julia

function code(x)
	return Float64(2.0 / fma(Float64(Float64(fma(Float64(0.002777777777777778 * x), x, 0.08333333333333333) * x) * x), Float64(x * x), 2.0))
end

Wolfram

code[x_] := N[(2.0 / N[(N[(N[(N[(N[(0.002777777777777778 * x), $MachinePrecision] * x + 0.08333333333333333), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in x around 0

   \[\leadsto \frac{2}{\color{blue}{2 + {x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right)}} \]
3. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{2}{{x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right) + \color{blue}{2}} \]

   2. *-commutative N/A

      \[\leadsto \frac{2}{\left(1 + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right) \cdot {x}^{2} + 2} \]

   3. lower-fma.f64 N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left(1 + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right), \color{blue}{{x}^{2}}, 2\right)} \]

   4. +-commutative N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left({x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right) + 1, {\color{blue}{x}}^{2}, 2\right)} \]

   5. *-commutative N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left(\left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right) \cdot {x}^{2} + 1, {x}^{2}, 2\right)} \]

   6. lower-fma.f64 N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}, {x}^{2}, 1\right), {\color{blue}{x}}^{2}, 2\right)} \]

   7. +-commutative N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360} \cdot {x}^{2} + \frac{1}{12}, {x}^{2}, 1\right), {x}^{2}, 2\right)} \]

   8. lower-fma.f64 N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, {x}^{2}, \frac{1}{12}\right), {x}^{2}, 1\right), {x}^{2}, 2\right)} \]

   9. unpow2 N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, x \cdot x, \frac{1}{12}\right), {x}^{2}, 1\right), {x}^{2}, 2\right)} \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, x \cdot x, \frac{1}{12}\right), {x}^{2}, 1\right), {x}^{2}, 2\right)} \]

   11. unpow2 N/A

       \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, x \cdot x, \frac{1}{12}\right), x \cdot x, 1\right), {x}^{2}, 2\right)} \]

   12. lower-*.f64 N/A

       \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, x \cdot x, \frac{1}{12}\right), x \cdot x, 1\right), {x}^{2}, 2\right)} \]

   13. unpow2 N/A

       \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, x \cdot x, \frac{1}{12}\right), x \cdot x, 1\right), x \cdot \color{blue}{x}, 2\right)} \]

   14. lower-*.f64 92.2

       \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, x \cdot x, 0.08333333333333333\right), x \cdot x, 1\right), x \cdot \color{blue}{x}, 2\right)} \]

4. Applied rewrites 92.2%

   \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, x \cdot x, 0.08333333333333333\right), x \cdot x, 1\right), x \cdot x, 2\right)}} \]

5. Taylor expanded in x around inf

   \[\leadsto \frac{2}{\mathsf{fma}\left({x}^{4} \cdot \left(\frac{1}{360} + \frac{1}{12} \cdot \frac{1}{{x}^{2}}\right), \color{blue}{x} \cdot x, 2\right)} \]
6. Step-by-step derivation

   1. metadata-eval N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left({x}^{\left(2 \cdot 2\right)} \cdot \left(\frac{1}{360} + \frac{1}{12} \cdot \frac{1}{{x}^{2}}\right), x \cdot x, 2\right)} \]

   2. pow-unpow N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left({\left({x}^{2}\right)}^{2} \cdot \left(\frac{1}{360} + \frac{1}{12} \cdot \frac{1}{{x}^{2}}\right), x \cdot x, 2\right)} \]

   3. pow2 N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left({\left(x \cdot x\right)}^{2} \cdot \left(\frac{1}{360} + \frac{1}{12} \cdot \frac{1}{{x}^{2}}\right), x \cdot x, 2\right)} \]

   4. pow2 N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{1}{360} + \frac{1}{12} \cdot \frac{1}{{x}^{2}}\right), x \cdot x, 2\right)} \]

   5. associate-*l* N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{360} + \frac{1}{12} \cdot \frac{1}{{x}^{2}}\right)\right), x \cdot x, 2\right)} \]

   6. distribute-rgt-in N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left(\left(x \cdot x\right) \cdot \left(\frac{1}{360} \cdot \left(x \cdot x\right) + \left(\frac{1}{12} \cdot \frac{1}{{x}^{2}}\right) \cdot \left(x \cdot x\right)\right), x \cdot x, 2\right)} \]

   7. associate-*l* N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left(\left(x \cdot x\right) \cdot \left(\frac{1}{360} \cdot \left(x \cdot x\right) + \frac{1}{12} \cdot \left(\frac{1}{{x}^{2}} \cdot \left(x \cdot x\right)\right)\right), x \cdot x, 2\right)} \]

   8. pow2 N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left(\left(x \cdot x\right) \cdot \left(\frac{1}{360} \cdot \left(x \cdot x\right) + \frac{1}{12} \cdot \left(\frac{1}{x \cdot x} \cdot \left(x \cdot x\right)\right)\right), x \cdot x, 2\right)} \]

   9. lft-mult-inverse N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left(\left(x \cdot x\right) \cdot \left(\frac{1}{360} \cdot \left(x \cdot x\right) + \frac{1}{12} \cdot 1\right), x \cdot x, 2\right)} \]

   10. metadata-eval N/A

       \[\leadsto \frac{2}{\mathsf{fma}\left(\left(x \cdot x\right) \cdot \left(\frac{1}{360} \cdot \left(x \cdot x\right) + \frac{1}{12}\right), x \cdot x, 2\right)} \]

   11. *-commutative N/A

       \[\leadsto \frac{2}{\mathsf{fma}\left(\left(\frac{1}{360} \cdot \left(x \cdot x\right) + \frac{1}{12}\right) \cdot \left(x \cdot x\right), x \cdot x, 2\right)} \]

   12. associate-*r* N/A

       \[\leadsto \frac{2}{\mathsf{fma}\left(\left(\left(\frac{1}{360} \cdot \left(x \cdot x\right) + \frac{1}{12}\right) \cdot x\right) \cdot x, x \cdot x, 2\right)} \]

   13. lower-*.f64 N/A

       \[\leadsto \frac{2}{\mathsf{fma}\left(\left(\left(\frac{1}{360} \cdot \left(x \cdot x\right) + \frac{1}{12}\right) \cdot x\right) \cdot x, x \cdot x, 2\right)} \]

7. Applied rewrites 91.7%

   \[\leadsto \frac{2}{\mathsf{fma}\left(\left(\mathsf{fma}\left(x \cdot x, 0.002777777777777778, 0.08333333333333333\right) \cdot x\right) \cdot x, \color{blue}{x} \cdot x, 2\right)} \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left(\left(\mathsf{fma}\left(x \cdot x, \frac{1}{360}, \frac{1}{12}\right) \cdot x\right) \cdot x, x \cdot x, 2\right)} \]

   2. lift-fma.f64 N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left(\left(\left(\left(x \cdot x\right) \cdot \frac{1}{360} + \frac{1}{12}\right) \cdot x\right) \cdot x, x \cdot x, 2\right)} \]

   3. *-commutative N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left(\left(\left(\frac{1}{360} \cdot \left(x \cdot x\right) + \frac{1}{12}\right) \cdot x\right) \cdot x, x \cdot x, 2\right)} \]

   4. associate-*r* N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left(\left(\left(\left(\frac{1}{360} \cdot x\right) \cdot x + \frac{1}{12}\right) \cdot x\right) \cdot x, x \cdot x, 2\right)} \]

   5. lower-fma.f64 N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left(\left(\mathsf{fma}\left(\frac{1}{360} \cdot x, x, \frac{1}{12}\right) \cdot x\right) \cdot x, x \cdot x, 2\right)} \]

   6. lower-*.f64 91.7

      \[\leadsto \frac{2}{\mathsf{fma}\left(\left(\mathsf{fma}\left(0.002777777777777778 \cdot x, x, 0.08333333333333333\right) \cdot x\right) \cdot x, x \cdot x, 2\right)} \]

9. Applied rewrites 91.7%

   \[\leadsto \frac{2}{\mathsf{fma}\left(\left(\mathsf{fma}\left(0.002777777777777778 \cdot x, x, 0.08333333333333333\right) \cdot x\right) \cdot x, x \cdot x, 2\right)} \]
10. Add Preprocessing

Alternative 8: 91.7% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\mathsf{fma}\left(\left(\left(0.002777777777777778 \cdot \left(x \cdot x\right)\right) \cdot x\right) \cdot x, x \cdot x, 2\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ 2.0 (fma (* (* (* 0.002777777777777778 (* x x)) x) x) (* x x) 2.0)))

C

double code(double x) {
	return 2.0 / fma((((0.002777777777777778 * (x * x)) * x) * x), (x * x), 2.0);
}

Julia

function code(x)
	return Float64(2.0 / fma(Float64(Float64(Float64(0.002777777777777778 * Float64(x * x)) * x) * x), Float64(x * x), 2.0))
end

Wolfram

code[x_] := N[(2.0 / N[(N[(N[(N[(0.002777777777777778 * N[(x * x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in x around 0

   \[\leadsto \frac{2}{\color{blue}{2 + {x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right)}} \]
3. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{2}{{x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right) + \color{blue}{2}} \]

   2. *-commutative N/A

      \[\leadsto \frac{2}{\left(1 + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right)\right) \cdot {x}^{2} + 2} \]

   3. lower-fma.f64 N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left(1 + {x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right), \color{blue}{{x}^{2}}, 2\right)} \]

   4. +-commutative N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left({x}^{2} \cdot \left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right) + 1, {\color{blue}{x}}^{2}, 2\right)} \]

   5. *-commutative N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left(\left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}\right) \cdot {x}^{2} + 1, {x}^{2}, 2\right)} \]

   6. lower-fma.f64 N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{12} + \frac{1}{360} \cdot {x}^{2}, {x}^{2}, 1\right), {\color{blue}{x}}^{2}, 2\right)} \]

   7. +-commutative N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360} \cdot {x}^{2} + \frac{1}{12}, {x}^{2}, 1\right), {x}^{2}, 2\right)} \]

   8. lower-fma.f64 N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, {x}^{2}, \frac{1}{12}\right), {x}^{2}, 1\right), {x}^{2}, 2\right)} \]

   9. unpow2 N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, x \cdot x, \frac{1}{12}\right), {x}^{2}, 1\right), {x}^{2}, 2\right)} \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, x \cdot x, \frac{1}{12}\right), {x}^{2}, 1\right), {x}^{2}, 2\right)} \]

   11. unpow2 N/A

       \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, x \cdot x, \frac{1}{12}\right), x \cdot x, 1\right), {x}^{2}, 2\right)} \]

   12. lower-*.f64 N/A

       \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, x \cdot x, \frac{1}{12}\right), x \cdot x, 1\right), {x}^{2}, 2\right)} \]

   13. unpow2 N/A

       \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{360}, x \cdot x, \frac{1}{12}\right), x \cdot x, 1\right), x \cdot \color{blue}{x}, 2\right)} \]

   14. lower-*.f64 92.2

       \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, x \cdot x, 0.08333333333333333\right), x \cdot x, 1\right), x \cdot \color{blue}{x}, 2\right)} \]

4. Applied rewrites 92.2%

   \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.002777777777777778, x \cdot x, 0.08333333333333333\right), x \cdot x, 1\right), x \cdot x, 2\right)}} \]

5. Taylor expanded in x around inf

   \[\leadsto \frac{2}{\mathsf{fma}\left({x}^{4} \cdot \left(\frac{1}{360} + \frac{1}{12} \cdot \frac{1}{{x}^{2}}\right), \color{blue}{x} \cdot x, 2\right)} \]
6. Step-by-step derivation

   1. metadata-eval N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left({x}^{\left(2 \cdot 2\right)} \cdot \left(\frac{1}{360} + \frac{1}{12} \cdot \frac{1}{{x}^{2}}\right), x \cdot x, 2\right)} \]

   2. pow-unpow N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left({\left({x}^{2}\right)}^{2} \cdot \left(\frac{1}{360} + \frac{1}{12} \cdot \frac{1}{{x}^{2}}\right), x \cdot x, 2\right)} \]

   3. pow2 N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left({\left(x \cdot x\right)}^{2} \cdot \left(\frac{1}{360} + \frac{1}{12} \cdot \frac{1}{{x}^{2}}\right), x \cdot x, 2\right)} \]

   4. pow2 N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(\frac{1}{360} + \frac{1}{12} \cdot \frac{1}{{x}^{2}}\right), x \cdot x, 2\right)} \]

   5. associate-*l* N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left(\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \left(\frac{1}{360} + \frac{1}{12} \cdot \frac{1}{{x}^{2}}\right)\right), x \cdot x, 2\right)} \]

   6. distribute-rgt-in N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left(\left(x \cdot x\right) \cdot \left(\frac{1}{360} \cdot \left(x \cdot x\right) + \left(\frac{1}{12} \cdot \frac{1}{{x}^{2}}\right) \cdot \left(x \cdot x\right)\right), x \cdot x, 2\right)} \]

   7. associate-*l* N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left(\left(x \cdot x\right) \cdot \left(\frac{1}{360} \cdot \left(x \cdot x\right) + \frac{1}{12} \cdot \left(\frac{1}{{x}^{2}} \cdot \left(x \cdot x\right)\right)\right), x \cdot x, 2\right)} \]

   8. pow2 N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left(\left(x \cdot x\right) \cdot \left(\frac{1}{360} \cdot \left(x \cdot x\right) + \frac{1}{12} \cdot \left(\frac{1}{x \cdot x} \cdot \left(x \cdot x\right)\right)\right), x \cdot x, 2\right)} \]

   9. lft-mult-inverse N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left(\left(x \cdot x\right) \cdot \left(\frac{1}{360} \cdot \left(x \cdot x\right) + \frac{1}{12} \cdot 1\right), x \cdot x, 2\right)} \]

   10. metadata-eval N/A

       \[\leadsto \frac{2}{\mathsf{fma}\left(\left(x \cdot x\right) \cdot \left(\frac{1}{360} \cdot \left(x \cdot x\right) + \frac{1}{12}\right), x \cdot x, 2\right)} \]

   11. *-commutative N/A

       \[\leadsto \frac{2}{\mathsf{fma}\left(\left(\frac{1}{360} \cdot \left(x \cdot x\right) + \frac{1}{12}\right) \cdot \left(x \cdot x\right), x \cdot x, 2\right)} \]

   12. associate-*r* N/A

       \[\leadsto \frac{2}{\mathsf{fma}\left(\left(\left(\frac{1}{360} \cdot \left(x \cdot x\right) + \frac{1}{12}\right) \cdot x\right) \cdot x, x \cdot x, 2\right)} \]

   13. lower-*.f64 N/A

       \[\leadsto \frac{2}{\mathsf{fma}\left(\left(\left(\frac{1}{360} \cdot \left(x \cdot x\right) + \frac{1}{12}\right) \cdot x\right) \cdot x, x \cdot x, 2\right)} \]

7. Applied rewrites 91.7%

   \[\leadsto \frac{2}{\mathsf{fma}\left(\left(\mathsf{fma}\left(x \cdot x, 0.002777777777777778, 0.08333333333333333\right) \cdot x\right) \cdot x, \color{blue}{x} \cdot x, 2\right)} \]

8. Taylor expanded in x around inf

   \[\leadsto \frac{2}{\mathsf{fma}\left(\frac{1}{360} \cdot {x}^{4}, \color{blue}{x} \cdot x, 2\right)} \]
9. Step-by-step derivation

   1. metadata-eval N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left(\frac{1}{360} \cdot {x}^{\left(2 + 2\right)}, x \cdot x, 2\right)} \]

   2. pow-prod-up N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left(\frac{1}{360} \cdot \left({x}^{2} \cdot {x}^{2}\right), x \cdot x, 2\right)} \]

   3. pow2 N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left(\frac{1}{360} \cdot \left(\left(x \cdot x\right) \cdot {x}^{2}\right), x \cdot x, 2\right)} \]

   4. pow2 N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left(\frac{1}{360} \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right), x \cdot x, 2\right)} \]

   5. associate-*r* N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left(\left(\frac{1}{360} \cdot \left(x \cdot x\right)\right) \cdot \left(x \cdot x\right), x \cdot x, 2\right)} \]

   6. associate-*r* N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left(\left(\left(\frac{1}{360} \cdot \left(x \cdot x\right)\right) \cdot x\right) \cdot x, x \cdot x, 2\right)} \]

   7. associate-*r* N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left(\left(\frac{1}{360} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot x, x \cdot x, 2\right)} \]

   8. unpow3 N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left(\left(\frac{1}{360} \cdot {x}^{3}\right) \cdot x, x \cdot x, 2\right)} \]

   9. cube-unmult N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left(\left(\frac{1}{360} \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot x, x \cdot x, 2\right)} \]

   10. lift-*.f64 N/A

       \[\leadsto \frac{2}{\mathsf{fma}\left(\left(\frac{1}{360} \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot x, x \cdot x, 2\right)} \]

   11. lift-*.f64 N/A

       \[\leadsto \frac{2}{\mathsf{fma}\left(\left(\frac{1}{360} \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot x, x \cdot x, 2\right)} \]

   12. lower-*.f64 N/A

       \[\leadsto \frac{2}{\mathsf{fma}\left(\left(\frac{1}{360} \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot x, x \cdot x, 2\right)} \]

   13. lift-*.f64 N/A

       \[\leadsto \frac{2}{\mathsf{fma}\left(\left(\frac{1}{360} \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot x, x \cdot x, 2\right)} \]

   14. lift-*.f64 N/A

       \[\leadsto \frac{2}{\mathsf{fma}\left(\left(\frac{1}{360} \cdot \left(x \cdot \left(x \cdot x\right)\right)\right) \cdot x, x \cdot x, 2\right)} \]

   15. cube-unmult N/A

       \[\leadsto \frac{2}{\mathsf{fma}\left(\left(\frac{1}{360} \cdot {x}^{3}\right) \cdot x, x \cdot x, 2\right)} \]

   16. unpow3 N/A

       \[\leadsto \frac{2}{\mathsf{fma}\left(\left(\frac{1}{360} \cdot \left(\left(x \cdot x\right) \cdot x\right)\right) \cdot x, x \cdot x, 2\right)} \]

   17. associate-*r* N/A

       \[\leadsto \frac{2}{\mathsf{fma}\left(\left(\left(\frac{1}{360} \cdot \left(x \cdot x\right)\right) \cdot x\right) \cdot x, x \cdot x, 2\right)} \]

   18. lower-*.f64 N/A

       \[\leadsto \frac{2}{\mathsf{fma}\left(\left(\left(\frac{1}{360} \cdot \left(x \cdot x\right)\right) \cdot x\right) \cdot x, x \cdot x, 2\right)} \]

   19. lower-*.f64 N/A

       \[\leadsto \frac{2}{\mathsf{fma}\left(\left(\left(\frac{1}{360} \cdot \left(x \cdot x\right)\right) \cdot x\right) \cdot x, x \cdot x, 2\right)} \]

   20. lift-*.f64 91.7

       \[\leadsto \frac{2}{\mathsf{fma}\left(\left(\left(0.002777777777777778 \cdot \left(x \cdot x\right)\right) \cdot x\right) \cdot x, x \cdot x, 2\right)} \]

10. Applied rewrites 91.7%

    \[\leadsto \frac{2}{\mathsf{fma}\left(\left(\left(0.002777777777777778 \cdot \left(x \cdot x\right)\right) \cdot x\right) \cdot x, \color{blue}{x} \cdot x, 2\right)} \]
11. Add Preprocessing

Alternative 9: 88.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{2}{e^{x} + e^{-x}} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{2}{\left(\mathsf{fma}\left(0.08333333333333333 \cdot x, x, 1\right) \cdot x\right) \cdot x}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, -0.5, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (/ 2.0 (+ (exp x) (exp (- x)))) 5e-6)
   (/ 2.0 (* (* (fma (* 0.08333333333333333 x) x 1.0) x) x))
   (fma (* x x) -0.5 1.0)))

C

double code(double x) {
	double tmp;
	if ((2.0 / (exp(x) + exp(-x))) <= 5e-6) {
		tmp = 2.0 / ((fma((0.08333333333333333 * x), x, 1.0) * x) * x);
	} else {
		tmp = fma((x * x), -0.5, 1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(2.0 / Float64(exp(x) + exp(Float64(-x)))) <= 5e-6)
		tmp = Float64(2.0 / Float64(Float64(fma(Float64(0.08333333333333333 * x), x, 1.0) * x) * x));
	else
		tmp = fma(Float64(x * x), -0.5, 1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(2.0 / N[(N[Exp[x], $MachinePrecision] + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e-6], N[(2.0 / N[(N[(N[(N[(0.08333333333333333 * x), $MachinePrecision] * x + 1.0), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 2 binary64) (+.f64 (exp.f64 x) (exp.f64 (neg.f64 x)))) < 5.00000000000000041e-6

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

      \[\leadsto \frac{2}{\color{blue}{2 + {x}^{2} \cdot \left(1 + \frac{1}{12} \cdot {x}^{2}\right)}} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2}{{x}^{2} \cdot \left(1 + \frac{1}{12} \cdot {x}^{2}\right) + \color{blue}{2}} \]

      2. *-commutative N/A

         \[\leadsto \frac{2}{\left(1 + \frac{1}{12} \cdot {x}^{2}\right) \cdot {x}^{2} + 2} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(1 + \frac{1}{12} \cdot {x}^{2}, \color{blue}{{x}^{2}}, 2\right)} \]

      4. +-commutative N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(\frac{1}{12} \cdot {x}^{2} + 1, {\color{blue}{x}}^{2}, 2\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{12}, {x}^{2}, 1\right), {\color{blue}{x}}^{2}, 2\right)} \]

      6. unpow2 N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{12}, x \cdot x, 1\right), {x}^{2}, 2\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{12}, x \cdot x, 1\right), {x}^{2}, 2\right)} \]

      8. unpow2 N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{12}, x \cdot x, 1\right), x \cdot \color{blue}{x}, 2\right)} \]

      9. lower-*.f64 76.5

         \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(0.08333333333333333, x \cdot x, 1\right), x \cdot \color{blue}{x}, 2\right)} \]

   4. Applied rewrites 76.5%

      \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.08333333333333333, x \cdot x, 1\right), x \cdot x, 2\right)}} \]

   5. Taylor expanded in x around inf

      \[\leadsto \frac{2}{{x}^{4} \cdot \color{blue}{\left(\frac{1}{12} + \frac{1}{{x}^{2}}\right)}} \]
   6. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{2}{{x}^{\left(2 + 2\right)} \cdot \left(\frac{1}{12} + \frac{1}{{x}^{2}}\right)} \]

      2. pow-prod-up N/A

         \[\leadsto \frac{2}{\left({x}^{2} \cdot {x}^{2}\right) \cdot \left(\frac{1}{12} + \frac{\color{blue}{1}}{{x}^{2}}\right)} \]

      3. associate-*l* N/A

         \[\leadsto \frac{2}{{x}^{2} \cdot \left({x}^{2} \cdot \color{blue}{\left(\frac{1}{12} + \frac{1}{{x}^{2}}\right)}\right)} \]

      4. distribute-rgt-in N/A

         \[\leadsto \frac{2}{{x}^{2} \cdot \left(\frac{1}{12} \cdot {x}^{2} + \frac{1}{{x}^{2}} \cdot \color{blue}{{x}^{2}}\right)} \]

      5. lft-mult-inverse N/A

         \[\leadsto \frac{2}{{x}^{2} \cdot \left(\frac{1}{12} \cdot {x}^{2} + 1\right)} \]

      6. pow2 N/A

         \[\leadsto \frac{2}{{x}^{2} \cdot \left(\frac{1}{12} \cdot \left(x \cdot x\right) + 1\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{2}{\left(\frac{1}{12} \cdot \left(x \cdot x\right) + 1\right) \cdot {x}^{\color{blue}{2}}} \]

      8. pow2 N/A

         \[\leadsto \frac{2}{\left(\frac{1}{12} \cdot \left(x \cdot x\right) + 1\right) \cdot \left(x \cdot x\right)} \]

      9. associate-*r* N/A

         \[\leadsto \frac{2}{\left(\left(\frac{1}{12} \cdot \left(x \cdot x\right) + 1\right) \cdot x\right) \cdot x} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{1}{12} \cdot \left(x \cdot x\right) + 1\right) \cdot x\right) \cdot x} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{1}{12} \cdot \left(x \cdot x\right) + 1\right) \cdot x\right) \cdot x} \]

      12. pow2 N/A

          \[\leadsto \frac{2}{\left(\left(\frac{1}{12} \cdot {x}^{2} + 1\right) \cdot x\right) \cdot x} \]

      13. *-commutative N/A

          \[\leadsto \frac{2}{\left(\left({x}^{2} \cdot \frac{1}{12} + 1\right) \cdot x\right) \cdot x} \]

      14. lower-fma.f64 N/A

          \[\leadsto \frac{2}{\left(\mathsf{fma}\left({x}^{2}, \frac{1}{12}, 1\right) \cdot x\right) \cdot x} \]

      15. pow2 N/A

          \[\leadsto \frac{2}{\left(\mathsf{fma}\left(x \cdot x, \frac{1}{12}, 1\right) \cdot x\right) \cdot x} \]

      16. lift-*.f64 76.5

          \[\leadsto \frac{2}{\left(\mathsf{fma}\left(x \cdot x, 0.08333333333333333, 1\right) \cdot x\right) \cdot x} \]

   7. Applied rewrites 76.5%

      \[\leadsto \frac{2}{\left(\mathsf{fma}\left(x \cdot x, 0.08333333333333333, 1\right) \cdot x\right) \cdot \color{blue}{x}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\mathsf{fma}\left(x \cdot x, \frac{1}{12}, 1\right) \cdot x\right) \cdot x} \]

      2. lift-fma.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\left(x \cdot x\right) \cdot \frac{1}{12} + 1\right) \cdot x\right) \cdot x} \]

      3. *-commutative N/A

         \[\leadsto \frac{2}{\left(\left(\frac{1}{12} \cdot \left(x \cdot x\right) + 1\right) \cdot x\right) \cdot x} \]

      4. associate-*r* N/A

         \[\leadsto \frac{2}{\left(\left(\left(\frac{1}{12} \cdot x\right) \cdot x + 1\right) \cdot x\right) \cdot x} \]

      5. lift-fma.f64 N/A

         \[\leadsto \frac{2}{\left(\mathsf{fma}\left(\frac{1}{12} \cdot x, x, 1\right) \cdot x\right) \cdot x} \]

      6. lift-*.f64 76.5

         \[\leadsto \frac{2}{\left(\mathsf{fma}\left(0.08333333333333333 \cdot x, x, 1\right) \cdot x\right) \cdot x} \]

   9. Applied rewrites 76.5%

      \[\leadsto \frac{2}{\left(\mathsf{fma}\left(0.08333333333333333 \cdot x, x, 1\right) \cdot x\right) \cdot x} \]

   if 5.00000000000000041e-6 < (/.f64 #s(literal 2 binary64) (+.f64 (exp.f64 x) (exp.f64 (neg.f64 x))))

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 99.1

         \[\leadsto \mathsf{fma}\left(x \cdot x, -0.5, 1\right) \]

   4. Applied rewrites 99.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, -0.5, 1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 87.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{2}{e^{x} + e^{-x}} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{2}{\left(x \cdot x\right) \cdot \left(0.08333333333333333 \cdot \left(x \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, -0.5, 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (/ 2.0 (+ (exp x) (exp (- x)))) 5e-6)
   (/ 2.0 (* (* x x) (* 0.08333333333333333 (* x x))))
   (fma (* x x) -0.5 1.0)))

C

double code(double x) {
	double tmp;
	if ((2.0 / (exp(x) + exp(-x))) <= 5e-6) {
		tmp = 2.0 / ((x * x) * (0.08333333333333333 * (x * x)));
	} else {
		tmp = fma((x * x), -0.5, 1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(2.0 / Float64(exp(x) + exp(Float64(-x)))) <= 5e-6)
		tmp = Float64(2.0 / Float64(Float64(x * x) * Float64(0.08333333333333333 * Float64(x * x))));
	else
		tmp = fma(Float64(x * x), -0.5, 1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(2.0 / N[(N[Exp[x], $MachinePrecision] + N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 5e-6], N[(2.0 / N[(N[(x * x), $MachinePrecision] * N[(0.08333333333333333 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * x), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 #s(literal 2 binary64) (+.f64 (exp.f64 x) (exp.f64 (neg.f64 x)))) < 5.00000000000000041e-6

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

      \[\leadsto \frac{2}{\color{blue}{2 + {x}^{2} \cdot \left(1 + \frac{1}{12} \cdot {x}^{2}\right)}} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{2}{{x}^{2} \cdot \left(1 + \frac{1}{12} \cdot {x}^{2}\right) + \color{blue}{2}} \]

      2. *-commutative N/A

         \[\leadsto \frac{2}{\left(1 + \frac{1}{12} \cdot {x}^{2}\right) \cdot {x}^{2} + 2} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(1 + \frac{1}{12} \cdot {x}^{2}, \color{blue}{{x}^{2}}, 2\right)} \]

      4. +-commutative N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(\frac{1}{12} \cdot {x}^{2} + 1, {\color{blue}{x}}^{2}, 2\right)} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{12}, {x}^{2}, 1\right), {\color{blue}{x}}^{2}, 2\right)} \]

      6. unpow2 N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{12}, x \cdot x, 1\right), {x}^{2}, 2\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{12}, x \cdot x, 1\right), {x}^{2}, 2\right)} \]

      8. unpow2 N/A

         \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{12}, x \cdot x, 1\right), x \cdot \color{blue}{x}, 2\right)} \]

      9. lower-*.f64 76.5

         \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(0.08333333333333333, x \cdot x, 1\right), x \cdot \color{blue}{x}, 2\right)} \]

   4. Applied rewrites 76.5%

      \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.08333333333333333, x \cdot x, 1\right), x \cdot x, 2\right)}} \]

   5. Taylor expanded in x around inf

      \[\leadsto \frac{2}{\frac{1}{12} \cdot \color{blue}{{x}^{4}}} \]
   6. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \frac{2}{\frac{1}{12} \cdot {x}^{\left(2 + 2\right)}} \]

      2. pow-prod-up N/A

         \[\leadsto \frac{2}{\frac{1}{12} \cdot \left({x}^{2} \cdot {x}^{\color{blue}{2}}\right)} \]

      3. associate-*r* N/A

         \[\leadsto \frac{2}{\left(\frac{1}{12} \cdot {x}^{2}\right) \cdot {x}^{\color{blue}{2}}} \]

      4. pow2 N/A

         \[\leadsto \frac{2}{\left(\frac{1}{12} \cdot {x}^{2}\right) \cdot \left(x \cdot x\right)} \]

      5. associate-*r* N/A

         \[\leadsto \frac{2}{\left(\left(\frac{1}{12} \cdot {x}^{2}\right) \cdot x\right) \cdot x} \]

      6. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{1}{12} \cdot {x}^{2}\right) \cdot x\right) \cdot x} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\frac{1}{12} \cdot {x}^{2}\right) \cdot x\right) \cdot x} \]

      8. *-commutative N/A

         \[\leadsto \frac{2}{\left(\left({x}^{2} \cdot \frac{1}{12}\right) \cdot x\right) \cdot x} \]

      9. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left({x}^{2} \cdot \frac{1}{12}\right) \cdot x\right) \cdot x} \]

      10. pow2 N/A

          \[\leadsto \frac{2}{\left(\left(\left(x \cdot x\right) \cdot \frac{1}{12}\right) \cdot x\right) \cdot x} \]

      11. lift-*.f64 76.5

          \[\leadsto \frac{2}{\left(\left(\left(x \cdot x\right) \cdot 0.08333333333333333\right) \cdot x\right) \cdot x} \]

   7. Applied rewrites 76.5%

      \[\leadsto \frac{2}{\left(\left(\left(x \cdot x\right) \cdot 0.08333333333333333\right) \cdot x\right) \cdot \color{blue}{x}} \]
   8. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\left(x \cdot x\right) \cdot \frac{1}{12}\right) \cdot x\right) \cdot x} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\left(x \cdot x\right) \cdot \frac{1}{12}\right) \cdot x\right) \cdot x} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\left(x \cdot x\right) \cdot \frac{1}{12}\right) \cdot x\right) \cdot x} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(\left(\left(x \cdot x\right) \cdot \frac{1}{12}\right) \cdot x\right) \cdot x} \]

      5. *-commutative N/A

         \[\leadsto \frac{2}{x \cdot \left(\left(\left(x \cdot x\right) \cdot \frac{1}{12}\right) \cdot \color{blue}{x}\right)} \]

      6. *-commutative N/A

         \[\leadsto \frac{2}{x \cdot \left(x \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{\frac{1}{12}}\right)\right)} \]

      7. associate-*r* N/A

         \[\leadsto \frac{2}{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{\frac{1}{12}}\right)} \]

      8. lower-*.f64 N/A

         \[\leadsto \frac{2}{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{\frac{1}{12}}\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{2}{\left(x \cdot x\right) \cdot \left(\left(x \cdot x\right) \cdot \frac{1}{12}\right)} \]

      10. *-commutative N/A

          \[\leadsto \frac{2}{\left(x \cdot x\right) \cdot \left(\frac{1}{12} \cdot \left(x \cdot \color{blue}{x}\right)\right)} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{2}{\left(x \cdot x\right) \cdot \left(\frac{1}{12} \cdot \left(x \cdot \color{blue}{x}\right)\right)} \]

      12. lift-*.f64 76.5

          \[\leadsto \frac{2}{\left(x \cdot x\right) \cdot \left(0.08333333333333333 \cdot \left(x \cdot x\right)\right)} \]

   9. Applied rewrites 76.5%

      \[\leadsto \frac{2}{\left(x \cdot x\right) \cdot \left(0.08333333333333333 \cdot \color{blue}{\left(x \cdot x\right)}\right)} \]

   if 5.00000000000000041e-6 < (/.f64 #s(literal 2 binary64) (+.f64 (exp.f64 x) (exp.f64 (neg.f64 x))))

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 99.1

         \[\leadsto \mathsf{fma}\left(x \cdot x, -0.5, 1\right) \]

   4. Applied rewrites 99.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, -0.5, 1\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 87.8% accurate, 6.4× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(0.08333333333333333, x \cdot x, 1\right), x \cdot x, 2\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ 2.0 (fma (fma 0.08333333333333333 (* x x) 1.0) (* x x) 2.0)))

C

double code(double x) {
	return 2.0 / fma(fma(0.08333333333333333, (x * x), 1.0), (x * x), 2.0);
}

Julia

function code(x)
	return Float64(2.0 / fma(fma(0.08333333333333333, Float64(x * x), 1.0), Float64(x * x), 2.0))
end

Wolfram

code[x_] := N[(2.0 / N[(N[(0.08333333333333333 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in x around 0

   \[\leadsto \frac{2}{\color{blue}{2 + {x}^{2} \cdot \left(1 + \frac{1}{12} \cdot {x}^{2}\right)}} \]
3. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{2}{{x}^{2} \cdot \left(1 + \frac{1}{12} \cdot {x}^{2}\right) + \color{blue}{2}} \]

   2. *-commutative N/A

      \[\leadsto \frac{2}{\left(1 + \frac{1}{12} \cdot {x}^{2}\right) \cdot {x}^{2} + 2} \]

   3. lower-fma.f64 N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left(1 + \frac{1}{12} \cdot {x}^{2}, \color{blue}{{x}^{2}}, 2\right)} \]

   4. +-commutative N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left(\frac{1}{12} \cdot {x}^{2} + 1, {\color{blue}{x}}^{2}, 2\right)} \]

   5. lower-fma.f64 N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{12}, {x}^{2}, 1\right), {\color{blue}{x}}^{2}, 2\right)} \]

   6. unpow2 N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{12}, x \cdot x, 1\right), {x}^{2}, 2\right)} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{12}, x \cdot x, 1\right), {x}^{2}, 2\right)} \]

   8. unpow2 N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{12}, x \cdot x, 1\right), x \cdot \color{blue}{x}, 2\right)} \]

   9. lower-*.f64 88.0

      \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(0.08333333333333333, x \cdot x, 1\right), x \cdot \color{blue}{x}, 2\right)} \]

4. Applied rewrites 88.0%

   \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.08333333333333333, x \cdot x, 1\right), x \cdot x, 2\right)}} \]
5. Add Preprocessing

Alternative 12: 87.5% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\mathsf{fma}\left(\left(x \cdot x\right) \cdot 0.08333333333333333, x \cdot x, 2\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ 2.0 (fma (* (* x x) 0.08333333333333333) (* x x) 2.0)))

C

double code(double x) {
	return 2.0 / fma(((x * x) * 0.08333333333333333), (x * x), 2.0);
}

Julia

function code(x)
	return Float64(2.0 / fma(Float64(Float64(x * x) * 0.08333333333333333), Float64(x * x), 2.0))
end

Wolfram

code[x_] := N[(2.0 / N[(N[(N[(x * x), $MachinePrecision] * 0.08333333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in x around 0

   \[\leadsto \frac{2}{\color{blue}{2 + {x}^{2} \cdot \left(1 + \frac{1}{12} \cdot {x}^{2}\right)}} \]
3. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{2}{{x}^{2} \cdot \left(1 + \frac{1}{12} \cdot {x}^{2}\right) + \color{blue}{2}} \]

   2. *-commutative N/A

      \[\leadsto \frac{2}{\left(1 + \frac{1}{12} \cdot {x}^{2}\right) \cdot {x}^{2} + 2} \]

   3. lower-fma.f64 N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left(1 + \frac{1}{12} \cdot {x}^{2}, \color{blue}{{x}^{2}}, 2\right)} \]

   4. +-commutative N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left(\frac{1}{12} \cdot {x}^{2} + 1, {\color{blue}{x}}^{2}, 2\right)} \]

   5. lower-fma.f64 N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{12}, {x}^{2}, 1\right), {\color{blue}{x}}^{2}, 2\right)} \]

   6. unpow2 N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{12}, x \cdot x, 1\right), {x}^{2}, 2\right)} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{12}, x \cdot x, 1\right), {x}^{2}, 2\right)} \]

   8. unpow2 N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{12}, x \cdot x, 1\right), x \cdot \color{blue}{x}, 2\right)} \]

   9. lower-*.f64 88.0

      \[\leadsto \frac{2}{\mathsf{fma}\left(\mathsf{fma}\left(0.08333333333333333, x \cdot x, 1\right), x \cdot \color{blue}{x}, 2\right)} \]

4. Applied rewrites 88.0%

   \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.08333333333333333, x \cdot x, 1\right), x \cdot x, 2\right)}} \]

5. Taylor expanded in x around inf

   \[\leadsto \frac{2}{\mathsf{fma}\left(\frac{1}{12} \cdot {x}^{2}, \color{blue}{x} \cdot x, 2\right)} \]
6. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left({x}^{2} \cdot \frac{1}{12}, x \cdot x, 2\right)} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left({x}^{2} \cdot \frac{1}{12}, x \cdot x, 2\right)} \]

   3. pow2 N/A

      \[\leadsto \frac{2}{\mathsf{fma}\left(\left(x \cdot x\right) \cdot \frac{1}{12}, x \cdot x, 2\right)} \]

   4. lift-*.f64 87.5

      \[\leadsto \frac{2}{\mathsf{fma}\left(\left(x \cdot x\right) \cdot 0.08333333333333333, x \cdot x, 2\right)} \]

7. Applied rewrites 87.5%

   \[\leadsto \frac{2}{\mathsf{fma}\left(\left(x \cdot x\right) \cdot 0.08333333333333333, \color{blue}{x} \cdot x, 2\right)} \]
8. Add Preprocessing

Alternative 13: 76.2% accurate, 9.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.25:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, -0.5, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.25) (fma (* x x) -0.5 1.0) (/ 2.0 (* x x))))

C

double code(double x) {
	double tmp;
	if (x <= 1.25) {
		tmp = fma((x * x), -0.5, 1.0);
	} else {
		tmp = 2.0 / (x * x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.25)
		tmp = fma(Float64(x * x), -0.5, 1.0);
	else
		tmp = Float64(2.0 / Float64(x * x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 1.25], N[(N[(x * x), $MachinePrecision] * -0.5 + 1.0), $MachinePrecision], N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.25

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 66.3

         \[\leadsto \mathsf{fma}\left(x \cdot x, -0.5, 1\right) \]

   4. Applied rewrites 66.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, -0.5, 1\right)} \]

   if 1.25 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

         \[\leadsto \frac{2}{{x}^{2} + \color{blue}{2}} \]

      2. unpow2 N/A

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

      3. lower-fma.f64 53.4

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

   4. Applied rewrites 53.4%

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

   5. Taylor expanded in x around inf

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

      1. pow2 N/A

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

      2. lift-*.f64 53.4

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

   7. Applied rewrites 53.4%

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

Alternative 14: 63.1% accurate, 12.1× speedup

Math

\[\begin{array}{l} \\ \frac{2}{\mathsf{fma}\left(x, x, 2\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 2.0 (fma x x 2.0)))

C

double code(double x) {
	return 2.0 / fma(x, x, 2.0);
}

Julia

function code(x)
	return Float64(2.0 / fma(x, x, 2.0))
end

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in x around 0

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

   1. +-commutative N/A

      \[\leadsto \frac{2}{{x}^{2} + \color{blue}{2}} \]

   2. unpow2 N/A

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

   3. lower-fma.f64 76.2

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

4. Applied rewrites 76.2%

   \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(x, x, 2\right)}} \]
5. Add Preprocessing

Alternative 15: 50.8% accurate, 217.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 1.0)

C

double code(double x) {
	return 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
end function

Java

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

Python

def code(x):
	return 1.0

Julia

function code(x)
	return 1.0
end

MATLAB

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

Wolfram

code[x_] := 1.0

Derivation

1. Initial program 100.0%

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

2. Taylor expanded in x around 0

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

4. Applied rewrites 50.8%

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

Reproduce

herbie shell --seed 2025101 
(FPCore (x)
  :name "Hyperbolic secant"
  :precision binary64
  (/ 2.0 (+ (exp x) (exp (- x)))))