Hyperbolic secant

Percentage Accurate: 100.0% → 100.0%

Time: 6.1s

Alternatives: 10

Speedup: 1.1×

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, 1.1× speedup

Math

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

FPCore

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

C

double code(double x) {
	return pow(cosh(x), -1.0);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

public static double code(double x) {
	return Math.pow(Math.cosh(x), -1.0);
}

Python

def code(x):
	return math.pow(math.cosh(x), -1.0)

Julia

function code(x)
	return cosh(x) ^ -1.0
end

MATLAB

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

Wolfram

code[x_] := N[Power[N[Cosh[x], $MachinePrecision], -1.0], $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\frac{2}{e^{x} + e^{-x}} \]
2. Add Preprocessing
3. 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-exp.f64 N/A

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

   5. lift-neg.f64 N/A

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

   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

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

4. Applied rewrites 100.0%

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

5. Final simplification 100.0%

   \[\leadsto {\cosh x}^{-1} \]
6. Add Preprocessing

Alternative 2: 92.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ {\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right) \cdot x\right) \cdot x + 1\right)}^{-1} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (pow
  (+
   (*
    (*
     (fma (fma 0.001388888888888889 (* x x) 0.041666666666666664) (* x x) 0.5)
     x)
    x)
   1.0)
  -1.0))

C

double code(double x) {
	return pow((((fma(fma(0.001388888888888889, (x * x), 0.041666666666666664), (x * x), 0.5) * x) * x) + 1.0), -1.0);
}

Julia

function code(x)
	return Float64(Float64(Float64(fma(fma(0.001388888888888889, Float64(x * x), 0.041666666666666664), Float64(x * x), 0.5) * x) * x) + 1.0) ^ -1.0
end

Wolfram

code[x_] := N[Power[N[(N[(N[(N[(N[(0.001388888888888889 * N[(x * x), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] + 1.0), $MachinePrecision], -1.0], $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\frac{2}{e^{x} + e^{-x}} \]
2. Add Preprocessing
3. 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-exp.f64 N/A

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

   5. lift-neg.f64 N/A

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

   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

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in x around 0

   \[\leadsto \frac{1}{\color{blue}{1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)}} \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{1}{\color{blue}{{x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) + 1}} \]

   2. *-commutative N/A

      \[\leadsto \frac{1}{\color{blue}{\left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) \cdot {x}^{2}} + 1} \]

   3. lower-fma.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right), {x}^{2}, 1\right)}} \]

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

      \[\leadsto \frac{1}{\mathsf{fma}\left(\color{blue}{\frac{1}{2} - \left(\mathsf{neg}\left({x}^{2}\right)\right) \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)}, {x}^{2}, 1\right)} \]

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

      \[\leadsto \frac{1}{\mathsf{fma}\left(\color{blue}{\frac{1}{2} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left({x}^{2}\right)\right)\right)\right) \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)}, {x}^{2}, 1\right)} \]

   6. +-commutative N/A

      \[\leadsto \frac{1}{\mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left({x}^{2}\right)\right)\right)\right) \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right) + \frac{1}{2}}, {x}^{2}, 1\right)} \]

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

      \[\leadsto \frac{1}{\mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left({x}^{2}\right)\right) \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)} + \frac{1}{2}, {x}^{2}, 1\right)} \]

   8. distribute-lft-neg-out N/A

      \[\leadsto \frac{1}{\mathsf{fma}\left(\left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left({x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)}\right)\right) + \frac{1}{2}, {x}^{2}, 1\right)} \]

   9. remove-double-neg N/A

      \[\leadsto \frac{1}{\mathsf{fma}\left(\color{blue}{{x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)} + \frac{1}{2}, {x}^{2}, 1\right)} \]

   10. *-commutative N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left(\color{blue}{\left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right) \cdot {x}^{2}} + \frac{1}{2}, {x}^{2}, 1\right)} \]

   11. lower-fma.f64 N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}, {x}^{2}, \frac{1}{2}\right)}, {x}^{2}, 1\right)} \]

   12. +-commutative N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{720} \cdot {x}^{2} + \frac{1}{24}}, {x}^{2}, \frac{1}{2}\right), {x}^{2}, 1\right)} \]

   13. lower-fma.f64 N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{720}, {x}^{2}, \frac{1}{24}\right)}, {x}^{2}, \frac{1}{2}\right), {x}^{2}, 1\right)} \]

   14. unpow2 N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, \color{blue}{x \cdot x}, \frac{1}{24}\right), {x}^{2}, \frac{1}{2}\right), {x}^{2}, 1\right)} \]

   15. lower-*.f64 N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, \color{blue}{x \cdot x}, \frac{1}{24}\right), {x}^{2}, \frac{1}{2}\right), {x}^{2}, 1\right)} \]

   16. unpow2 N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, x \cdot x, \frac{1}{24}\right), \color{blue}{x \cdot x}, \frac{1}{2}\right), {x}^{2}, 1\right)} \]

   17. lower-*.f64 N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, x \cdot x, \frac{1}{24}\right), \color{blue}{x \cdot x}, \frac{1}{2}\right), {x}^{2}, 1\right)} \]

   18. unpow2 N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, x \cdot x, \frac{1}{24}\right), x \cdot x, \frac{1}{2}\right), \color{blue}{x \cdot x}, 1\right)} \]

   19. lower-*.f64 91.8

       \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right), \color{blue}{x \cdot x}, 1\right)} \]

7. Applied rewrites 91.8%

   \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right)}} \]
8. Step-by-step derivation

9. Applied rewrites 91.8%

   \[\leadsto \frac{1}{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right) \cdot x\right) \cdot x + \color{blue}{1}} \]

10. Final simplification 91.8%

    \[\leadsto {\left(\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right) \cdot x\right) \cdot x + 1\right)}^{-1} \]
11. Add Preprocessing

Alternative 3: 92.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ {\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right) \cdot x, x, 0.5\right), x \cdot x, 1\right)\right)}^{-1} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (pow
  (fma
   (fma (* (fma 0.001388888888888889 (* x x) 0.041666666666666664) x) x 0.5)
   (* x x)
   1.0)
  -1.0))

C

double code(double x) {
	return pow(fma(fma((fma(0.001388888888888889, (x * x), 0.041666666666666664) * x), x, 0.5), (x * x), 1.0), -1.0);
}

Julia

function code(x)
	return fma(fma(Float64(fma(0.001388888888888889, Float64(x * x), 0.041666666666666664) * x), x, 0.5), Float64(x * x), 1.0) ^ -1.0
end

Wolfram

code[x_] := N[Power[N[(N[(N[(N[(0.001388888888888889 * N[(x * x), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * x), $MachinePrecision] * x + 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision], -1.0], $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\frac{2}{e^{x} + e^{-x}} \]
2. Add Preprocessing
3. 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-exp.f64 N/A

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

   5. lift-neg.f64 N/A

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

   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

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in x around 0

   \[\leadsto \frac{1}{\color{blue}{1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)}} \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{1}{\color{blue}{{x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) + 1}} \]

   2. *-commutative N/A

      \[\leadsto \frac{1}{\color{blue}{\left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) \cdot {x}^{2}} + 1} \]

   3. lower-fma.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right), {x}^{2}, 1\right)}} \]

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

      \[\leadsto \frac{1}{\mathsf{fma}\left(\color{blue}{\frac{1}{2} - \left(\mathsf{neg}\left({x}^{2}\right)\right) \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)}, {x}^{2}, 1\right)} \]

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

      \[\leadsto \frac{1}{\mathsf{fma}\left(\color{blue}{\frac{1}{2} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left({x}^{2}\right)\right)\right)\right) \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)}, {x}^{2}, 1\right)} \]

   6. +-commutative N/A

      \[\leadsto \frac{1}{\mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left({x}^{2}\right)\right)\right)\right) \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right) + \frac{1}{2}}, {x}^{2}, 1\right)} \]

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

      \[\leadsto \frac{1}{\mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left({x}^{2}\right)\right) \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)} + \frac{1}{2}, {x}^{2}, 1\right)} \]

   8. distribute-lft-neg-out N/A

      \[\leadsto \frac{1}{\mathsf{fma}\left(\left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left({x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)}\right)\right) + \frac{1}{2}, {x}^{2}, 1\right)} \]

   9. remove-double-neg N/A

      \[\leadsto \frac{1}{\mathsf{fma}\left(\color{blue}{{x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)} + \frac{1}{2}, {x}^{2}, 1\right)} \]

   10. *-commutative N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left(\color{blue}{\left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right) \cdot {x}^{2}} + \frac{1}{2}, {x}^{2}, 1\right)} \]

   11. lower-fma.f64 N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}, {x}^{2}, \frac{1}{2}\right)}, {x}^{2}, 1\right)} \]

   12. +-commutative N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{720} \cdot {x}^{2} + \frac{1}{24}}, {x}^{2}, \frac{1}{2}\right), {x}^{2}, 1\right)} \]

   13. lower-fma.f64 N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{720}, {x}^{2}, \frac{1}{24}\right)}, {x}^{2}, \frac{1}{2}\right), {x}^{2}, 1\right)} \]

   14. unpow2 N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, \color{blue}{x \cdot x}, \frac{1}{24}\right), {x}^{2}, \frac{1}{2}\right), {x}^{2}, 1\right)} \]

   15. lower-*.f64 N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, \color{blue}{x \cdot x}, \frac{1}{24}\right), {x}^{2}, \frac{1}{2}\right), {x}^{2}, 1\right)} \]

   16. unpow2 N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, x \cdot x, \frac{1}{24}\right), \color{blue}{x \cdot x}, \frac{1}{2}\right), {x}^{2}, 1\right)} \]

   17. lower-*.f64 N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, x \cdot x, \frac{1}{24}\right), \color{blue}{x \cdot x}, \frac{1}{2}\right), {x}^{2}, 1\right)} \]

   18. unpow2 N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, x \cdot x, \frac{1}{24}\right), x \cdot x, \frac{1}{2}\right), \color{blue}{x \cdot x}, 1\right)} \]

   19. lower-*.f64 91.8

       \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right), \color{blue}{x \cdot x}, 1\right)} \]

7. Applied rewrites 91.8%

   \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right)}} \]
8. Step-by-step derivation

9. Applied rewrites 91.8%

   \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right) \cdot x, x, 0.5\right), \color{blue}{x} \cdot x, 1\right)} \]

10. Final simplification 91.8%

    \[\leadsto {\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right) \cdot x, x, 0.5\right), x \cdot x, 1\right)\right)}^{-1} \]
11. Add Preprocessing

Alternative 4: 92.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ {\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right) \cdot x, x, 1\right)\right)}^{-1} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (pow
  (fma
   (*
    (fma (fma 0.001388888888888889 (* x x) 0.041666666666666664) (* x x) 0.5)
    x)
   x
   1.0)
  -1.0))

C

double code(double x) {
	return pow(fma((fma(fma(0.001388888888888889, (x * x), 0.041666666666666664), (x * x), 0.5) * x), x, 1.0), -1.0);
}

Julia

function code(x)
	return fma(Float64(fma(fma(0.001388888888888889, Float64(x * x), 0.041666666666666664), Float64(x * x), 0.5) * x), x, 1.0) ^ -1.0
end

Wolfram

code[x_] := N[Power[N[(N[(N[(N[(0.001388888888888889 * N[(x * x), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision] * x), $MachinePrecision] * x + 1.0), $MachinePrecision], -1.0], $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\frac{2}{e^{x} + e^{-x}} \]
2. Add Preprocessing
3. 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-exp.f64 N/A

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

   5. lift-neg.f64 N/A

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

   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

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in x around 0

   \[\leadsto \frac{1}{\color{blue}{1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)}} \]
6. Step-by-step derivation

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

      \[\leadsto \frac{1}{\color{blue}{1 - \left(\mathsf{neg}\left({x}^{2}\right)\right) \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)}} \]

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

      \[\leadsto \frac{1}{\color{blue}{1 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left({x}^{2}\right)\right)\right)\right) \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)}} \]

   3. +-commutative N/A

      \[\leadsto \frac{1}{\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left({x}^{2}\right)\right)\right)\right) \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) + 1}} \]

   4. distribute-lft-neg-out N/A

      \[\leadsto \frac{1}{\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left({x}^{2}\right)\right) \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)\right)} + 1} \]

   5. distribute-lft-neg-out N/A

      \[\leadsto \frac{1}{\left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left({x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)\right)}\right)\right) + 1} \]

   6. remove-double-neg N/A

      \[\leadsto \frac{1}{\color{blue}{{x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)} + 1} \]

   7. *-commutative N/A

      \[\leadsto \frac{1}{\color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) \cdot {x}^{2}} + 1} \]

   8. lower-fma.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, {x}^{2}, 1\right)}} \]

   9. +-commutative N/A

      \[\leadsto \frac{1}{\mathsf{fma}\left(\color{blue}{\frac{1}{24} \cdot {x}^{2} + \frac{1}{2}}, {x}^{2}, 1\right)} \]

   10. lower-fma.f64 N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24}, {x}^{2}, \frac{1}{2}\right)}, {x}^{2}, 1\right)} \]

   11. unpow2 N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{x \cdot x}, \frac{1}{2}\right), {x}^{2}, 1\right)} \]

   12. lower-*.f64 N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{x \cdot x}, \frac{1}{2}\right), {x}^{2}, 1\right)} \]

   13. unpow2 N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), \color{blue}{x \cdot x}, 1\right)} \]

   14. lower-*.f64 87.0

       \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), \color{blue}{x \cdot x}, 1\right)} \]

7. Applied rewrites 87.0%

   \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right)}} \]

8. Taylor expanded in x around 0

   \[\leadsto \frac{1}{\color{blue}{1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)}} \]
9. Step-by-step derivation

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

      \[\leadsto \frac{1}{\color{blue}{1 - \left(\mathsf{neg}\left({x}^{2}\right)\right) \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)}} \]

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

      \[\leadsto \frac{1}{\color{blue}{1 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left({x}^{2}\right)\right)\right)\right) \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)}} \]

   3. +-commutative N/A

      \[\leadsto \frac{1}{\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left({x}^{2}\right)\right)\right)\right) \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) + 1}} \]

   4. distribute-lft-neg-out N/A

      \[\leadsto \frac{1}{\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left({x}^{2}\right)\right) \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)} + 1} \]

   5. distribute-lft-neg-out N/A

      \[\leadsto \frac{1}{\left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left({x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)\right)}\right)\right) + 1} \]

   6. remove-double-neg N/A

      \[\leadsto \frac{1}{\color{blue}{{x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)} + 1} \]

   7. unpow2 N/A

      \[\leadsto \frac{1}{\color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) + 1} \]

   8. associate-*l* N/A

      \[\leadsto \frac{1}{\color{blue}{x \cdot \left(x \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)} + 1} \]

   9. *-commutative N/A

      \[\leadsto \frac{1}{\color{blue}{\left(x \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right) \cdot x} + 1} \]

   10. lower-fma.f64 N/A

       \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(x \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right), x, 1\right)}} \]

10. Applied rewrites 91.8%

    \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right) \cdot x, x, 1\right)}} \]

11. Final simplification 91.8%

    \[\leadsto {\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right) \cdot x, x, 1\right)\right)}^{-1} \]
12. Add Preprocessing

Alternative 5: 92.1% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ {\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(x \cdot x\right) \cdot 0.001388888888888889, x \cdot x, 0.5\right) \cdot x, x, 1\right)\right)}^{-1} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (pow
  (fma (* (fma (* (* x x) 0.001388888888888889) (* x x) 0.5) x) x 1.0)
  -1.0))

C

double code(double x) {
	return pow(fma((fma(((x * x) * 0.001388888888888889), (x * x), 0.5) * x), x, 1.0), -1.0);
}

Julia

function code(x)
	return fma(Float64(fma(Float64(Float64(x * x) * 0.001388888888888889), Float64(x * x), 0.5) * x), x, 1.0) ^ -1.0
end

Wolfram

code[x_] := N[Power[N[(N[(N[(N[(N[(x * x), $MachinePrecision] * 0.001388888888888889), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision] * x), $MachinePrecision] * x + 1.0), $MachinePrecision], -1.0], $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\frac{2}{e^{x} + e^{-x}} \]
2. Add Preprocessing
3. 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-exp.f64 N/A

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

   5. lift-neg.f64 N/A

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

   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

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in x around 0

   \[\leadsto \frac{1}{\color{blue}{1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)}} \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{1}{\color{blue}{{x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) + 1}} \]

   2. *-commutative N/A

      \[\leadsto \frac{1}{\color{blue}{\left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) \cdot {x}^{2}} + 1} \]

   3. lower-fma.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right), {x}^{2}, 1\right)}} \]

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

      \[\leadsto \frac{1}{\mathsf{fma}\left(\color{blue}{\frac{1}{2} - \left(\mathsf{neg}\left({x}^{2}\right)\right) \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)}, {x}^{2}, 1\right)} \]

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

      \[\leadsto \frac{1}{\mathsf{fma}\left(\color{blue}{\frac{1}{2} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left({x}^{2}\right)\right)\right)\right) \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)}, {x}^{2}, 1\right)} \]

   6. +-commutative N/A

      \[\leadsto \frac{1}{\mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left({x}^{2}\right)\right)\right)\right) \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right) + \frac{1}{2}}, {x}^{2}, 1\right)} \]

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

      \[\leadsto \frac{1}{\mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left({x}^{2}\right)\right) \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)} + \frac{1}{2}, {x}^{2}, 1\right)} \]

   8. distribute-lft-neg-out N/A

      \[\leadsto \frac{1}{\mathsf{fma}\left(\left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left({x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)}\right)\right) + \frac{1}{2}, {x}^{2}, 1\right)} \]

   9. remove-double-neg N/A

      \[\leadsto \frac{1}{\mathsf{fma}\left(\color{blue}{{x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)} + \frac{1}{2}, {x}^{2}, 1\right)} \]

   10. *-commutative N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left(\color{blue}{\left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right) \cdot {x}^{2}} + \frac{1}{2}, {x}^{2}, 1\right)} \]

   11. lower-fma.f64 N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}, {x}^{2}, \frac{1}{2}\right)}, {x}^{2}, 1\right)} \]

   12. +-commutative N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{720} \cdot {x}^{2} + \frac{1}{24}}, {x}^{2}, \frac{1}{2}\right), {x}^{2}, 1\right)} \]

   13. lower-fma.f64 N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{720}, {x}^{2}, \frac{1}{24}\right)}, {x}^{2}, \frac{1}{2}\right), {x}^{2}, 1\right)} \]

   14. unpow2 N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, \color{blue}{x \cdot x}, \frac{1}{24}\right), {x}^{2}, \frac{1}{2}\right), {x}^{2}, 1\right)} \]

   15. lower-*.f64 N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, \color{blue}{x \cdot x}, \frac{1}{24}\right), {x}^{2}, \frac{1}{2}\right), {x}^{2}, 1\right)} \]

   16. unpow2 N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, x \cdot x, \frac{1}{24}\right), \color{blue}{x \cdot x}, \frac{1}{2}\right), {x}^{2}, 1\right)} \]

   17. lower-*.f64 N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, x \cdot x, \frac{1}{24}\right), \color{blue}{x \cdot x}, \frac{1}{2}\right), {x}^{2}, 1\right)} \]

   18. unpow2 N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, x \cdot x, \frac{1}{24}\right), x \cdot x, \frac{1}{2}\right), \color{blue}{x \cdot x}, 1\right)} \]

   19. lower-*.f64 91.8

       \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right), \color{blue}{x \cdot x}, 1\right)} \]

7. Applied rewrites 91.8%

   \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right)}} \]

8. Taylor expanded in x around inf

   \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720} \cdot {x}^{2}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right)} \]
9. Step-by-step derivation

10. Applied rewrites 91.5%

    \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889 \cdot \left(x \cdot x\right), x \cdot x, 0.5\right), x \cdot x, 1\right)} \]
11. Step-by-step derivation

12. Applied rewrites 91.5%

    \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\left(x \cdot x\right) \cdot 0.001388888888888889, x \cdot x, 0.5\right) \cdot x, \color{blue}{x}, 1\right)} \]

13. Final simplification 91.5%

    \[\leadsto {\left(\mathsf{fma}\left(\mathsf{fma}\left(\left(x \cdot x\right) \cdot 0.001388888888888889, x \cdot x, 0.5\right) \cdot x, x, 1\right)\right)}^{-1} \]
14. Add Preprocessing

Alternative 6: 91.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ {\left(\mathsf{fma}\left(\left(\mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right) \cdot x\right) \cdot x, x \cdot x, 1\right)\right)}^{-1} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (pow
  (fma
   (* (* (fma (* x x) 0.001388888888888889 0.041666666666666664) x) x)
   (* x x)
   1.0)
  -1.0))

C

double code(double x) {
	return pow(fma(((fma((x * x), 0.001388888888888889, 0.041666666666666664) * x) * x), (x * x), 1.0), -1.0);
}

Julia

function code(x)
	return fma(Float64(Float64(fma(Float64(x * x), 0.001388888888888889, 0.041666666666666664) * x) * x), Float64(x * x), 1.0) ^ -1.0
end

Wolfram

code[x_] := N[Power[N[(N[(N[(N[(N[(x * x), $MachinePrecision] * 0.001388888888888889 + 0.041666666666666664), $MachinePrecision] * x), $MachinePrecision] * x), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision], -1.0], $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\frac{2}{e^{x} + e^{-x}} \]
2. Add Preprocessing
3. 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-exp.f64 N/A

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

   5. lift-neg.f64 N/A

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

   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

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in x around 0

   \[\leadsto \frac{1}{\color{blue}{1 + {x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)}} \]
6. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{1}{\color{blue}{{x}^{2} \cdot \left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) + 1}} \]

   2. *-commutative N/A

      \[\leadsto \frac{1}{\color{blue}{\left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right) \cdot {x}^{2}} + 1} \]

   3. lower-fma.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right), {x}^{2}, 1\right)}} \]

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

      \[\leadsto \frac{1}{\mathsf{fma}\left(\color{blue}{\frac{1}{2} - \left(\mathsf{neg}\left({x}^{2}\right)\right) \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)}, {x}^{2}, 1\right)} \]

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

      \[\leadsto \frac{1}{\mathsf{fma}\left(\color{blue}{\frac{1}{2} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left({x}^{2}\right)\right)\right)\right) \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)}, {x}^{2}, 1\right)} \]

   6. +-commutative N/A

      \[\leadsto \frac{1}{\mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left({x}^{2}\right)\right)\right)\right) \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right) + \frac{1}{2}}, {x}^{2}, 1\right)} \]

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

      \[\leadsto \frac{1}{\mathsf{fma}\left(\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left({x}^{2}\right)\right) \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)} + \frac{1}{2}, {x}^{2}, 1\right)} \]

   8. distribute-lft-neg-out N/A

      \[\leadsto \frac{1}{\mathsf{fma}\left(\left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left({x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)\right)\right)}\right)\right) + \frac{1}{2}, {x}^{2}, 1\right)} \]

   9. remove-double-neg N/A

      \[\leadsto \frac{1}{\mathsf{fma}\left(\color{blue}{{x}^{2} \cdot \left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right)} + \frac{1}{2}, {x}^{2}, 1\right)} \]

   10. *-commutative N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left(\color{blue}{\left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}\right) \cdot {x}^{2}} + \frac{1}{2}, {x}^{2}, 1\right)} \]

   11. lower-fma.f64 N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24} + \frac{1}{720} \cdot {x}^{2}, {x}^{2}, \frac{1}{2}\right)}, {x}^{2}, 1\right)} \]

   12. +-commutative N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{720} \cdot {x}^{2} + \frac{1}{24}}, {x}^{2}, \frac{1}{2}\right), {x}^{2}, 1\right)} \]

   13. lower-fma.f64 N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{720}, {x}^{2}, \frac{1}{24}\right)}, {x}^{2}, \frac{1}{2}\right), {x}^{2}, 1\right)} \]

   14. unpow2 N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, \color{blue}{x \cdot x}, \frac{1}{24}\right), {x}^{2}, \frac{1}{2}\right), {x}^{2}, 1\right)} \]

   15. lower-*.f64 N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, \color{blue}{x \cdot x}, \frac{1}{24}\right), {x}^{2}, \frac{1}{2}\right), {x}^{2}, 1\right)} \]

   16. unpow2 N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, x \cdot x, \frac{1}{24}\right), \color{blue}{x \cdot x}, \frac{1}{2}\right), {x}^{2}, 1\right)} \]

   17. lower-*.f64 N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, x \cdot x, \frac{1}{24}\right), \color{blue}{x \cdot x}, \frac{1}{2}\right), {x}^{2}, 1\right)} \]

   18. unpow2 N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, x \cdot x, \frac{1}{24}\right), x \cdot x, \frac{1}{2}\right), \color{blue}{x \cdot x}, 1\right)} \]

   19. lower-*.f64 91.8

       \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right), \color{blue}{x \cdot x}, 1\right)} \]

7. Applied rewrites 91.8%

   \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, 0.041666666666666664\right), x \cdot x, 0.5\right), x \cdot x, 1\right)}} \]

8. Taylor expanded in x around inf

   \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720} \cdot {x}^{2}, x \cdot x, \frac{1}{2}\right), x \cdot x, 1\right)} \]
9. Step-by-step derivation

10. Applied rewrites 91.5%

    \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889 \cdot \left(x \cdot x\right), x \cdot x, 0.5\right), x \cdot x, 1\right)} \]

11. Taylor expanded in x around inf

    \[\leadsto \frac{1}{\mathsf{fma}\left({x}^{4} \cdot \left(\frac{1}{720} + \frac{1}{24} \cdot \frac{1}{{x}^{2}}\right), \color{blue}{x} \cdot x, 1\right)} \]

13. Applied rewrites 91.0%

    \[\leadsto \frac{1}{\mathsf{fma}\left(\left(\mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right) \cdot x\right) \cdot x, \color{blue}{x} \cdot x, 1\right)} \]

14. Final simplification 91.0%

    \[\leadsto {\left(\mathsf{fma}\left(\left(\mathsf{fma}\left(x \cdot x, 0.001388888888888889, 0.041666666666666664\right) \cdot x\right) \cdot x, x \cdot x, 1\right)\right)}^{-1} \]
15. Add Preprocessing

Alternative 7: 88.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ {\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right) \cdot x, x, 1\right)\right)}^{-1} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (pow (fma (* (fma 0.041666666666666664 (* x x) 0.5) x) x 1.0) -1.0))

C

double code(double x) {
	return pow(fma((fma(0.041666666666666664, (x * x), 0.5) * x), x, 1.0), -1.0);
}

Julia

function code(x)
	return fma(Float64(fma(0.041666666666666664, Float64(x * x), 0.5) * x), x, 1.0) ^ -1.0
end

Wolfram

code[x_] := N[Power[N[(N[(N[(0.041666666666666664 * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision] * x), $MachinePrecision] * x + 1.0), $MachinePrecision], -1.0], $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\frac{2}{e^{x} + e^{-x}} \]
2. Add Preprocessing
3. 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-exp.f64 N/A

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

   5. lift-neg.f64 N/A

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

   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

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in x around 0

   \[\leadsto \frac{1}{\color{blue}{1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)}} \]
6. Step-by-step derivation

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

      \[\leadsto \frac{1}{\color{blue}{1 - \left(\mathsf{neg}\left({x}^{2}\right)\right) \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)}} \]

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

      \[\leadsto \frac{1}{\color{blue}{1 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left({x}^{2}\right)\right)\right)\right) \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)}} \]

   3. +-commutative N/A

      \[\leadsto \frac{1}{\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left({x}^{2}\right)\right)\right)\right) \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) + 1}} \]

   4. distribute-lft-neg-out N/A

      \[\leadsto \frac{1}{\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left({x}^{2}\right)\right) \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)\right)} + 1} \]

   5. distribute-lft-neg-out N/A

      \[\leadsto \frac{1}{\left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left({x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)\right)}\right)\right) + 1} \]

   6. remove-double-neg N/A

      \[\leadsto \frac{1}{\color{blue}{{x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)} + 1} \]

   7. *-commutative N/A

      \[\leadsto \frac{1}{\color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) \cdot {x}^{2}} + 1} \]

   8. lower-fma.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, {x}^{2}, 1\right)}} \]

   9. +-commutative N/A

      \[\leadsto \frac{1}{\mathsf{fma}\left(\color{blue}{\frac{1}{24} \cdot {x}^{2} + \frac{1}{2}}, {x}^{2}, 1\right)} \]

   10. lower-fma.f64 N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24}, {x}^{2}, \frac{1}{2}\right)}, {x}^{2}, 1\right)} \]

   11. unpow2 N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{x \cdot x}, \frac{1}{2}\right), {x}^{2}, 1\right)} \]

   12. lower-*.f64 N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{x \cdot x}, \frac{1}{2}\right), {x}^{2}, 1\right)} \]

   13. unpow2 N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), \color{blue}{x \cdot x}, 1\right)} \]

   14. lower-*.f64 87.0

       \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), \color{blue}{x \cdot x}, 1\right)} \]

7. Applied rewrites 87.0%

   \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right)}} \]
8. Step-by-step derivation

9. Applied rewrites 87.0%

   \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right) \cdot x, \color{blue}{x}, 1\right)} \]

10. Final simplification 87.0%

    \[\leadsto {\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right) \cdot x, x, 1\right)\right)}^{-1} \]
11. Add Preprocessing

Alternative 8: 87.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ {\left(\mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right), x \cdot x, 1\right)\right)}^{-1} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (pow (fma (* 0.041666666666666664 (* x x)) (* x x) 1.0) -1.0))

C

double code(double x) {
	return pow(fma((0.041666666666666664 * (x * x)), (x * x), 1.0), -1.0);
}

Julia

function code(x)
	return fma(Float64(0.041666666666666664 * Float64(x * x)), Float64(x * x), 1.0) ^ -1.0
end

Wolfram

code[x_] := N[Power[N[(N[(0.041666666666666664 * N[(x * x), $MachinePrecision]), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision], -1.0], $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\frac{2}{e^{x} + e^{-x}} \]
2. Add Preprocessing
3. 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-exp.f64 N/A

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

   5. lift-neg.f64 N/A

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

   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

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

4. Applied rewrites 100.0%

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

5. Taylor expanded in x around 0

   \[\leadsto \frac{1}{\color{blue}{1 + {x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)}} \]
6. Step-by-step derivation

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

      \[\leadsto \frac{1}{\color{blue}{1 - \left(\mathsf{neg}\left({x}^{2}\right)\right) \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)}} \]

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

      \[\leadsto \frac{1}{\color{blue}{1 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left({x}^{2}\right)\right)\right)\right) \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)}} \]

   3. +-commutative N/A

      \[\leadsto \frac{1}{\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left({x}^{2}\right)\right)\right)\right) \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) + 1}} \]

   4. distribute-lft-neg-out N/A

      \[\leadsto \frac{1}{\color{blue}{\left(\mathsf{neg}\left(\left(\mathsf{neg}\left({x}^{2}\right)\right) \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)\right)} + 1} \]

   5. distribute-lft-neg-out N/A

      \[\leadsto \frac{1}{\left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left({x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)\right)\right)}\right)\right) + 1} \]

   6. remove-double-neg N/A

      \[\leadsto \frac{1}{\color{blue}{{x}^{2} \cdot \left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right)} + 1} \]

   7. *-commutative N/A

      \[\leadsto \frac{1}{\color{blue}{\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}\right) \cdot {x}^{2}} + 1} \]

   8. lower-fma.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\frac{1}{2} + \frac{1}{24} \cdot {x}^{2}, {x}^{2}, 1\right)}} \]

   9. +-commutative N/A

      \[\leadsto \frac{1}{\mathsf{fma}\left(\color{blue}{\frac{1}{24} \cdot {x}^{2} + \frac{1}{2}}, {x}^{2}, 1\right)} \]

   10. lower-fma.f64 N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{24}, {x}^{2}, \frac{1}{2}\right)}, {x}^{2}, 1\right)} \]

   11. unpow2 N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{x \cdot x}, \frac{1}{2}\right), {x}^{2}, 1\right)} \]

   12. lower-*.f64 N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, \color{blue}{x \cdot x}, \frac{1}{2}\right), {x}^{2}, 1\right)} \]

   13. unpow2 N/A

       \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{24}, x \cdot x, \frac{1}{2}\right), \color{blue}{x \cdot x}, 1\right)} \]

   14. lower-*.f64 87.0

       \[\leadsto \frac{1}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), \color{blue}{x \cdot x}, 1\right)} \]

7. Applied rewrites 87.0%

   \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, x \cdot x, 0.5\right), x \cdot x, 1\right)}} \]

8. Taylor expanded in x around inf

   \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{1}{24} \cdot {x}^{2}, \color{blue}{x} \cdot x, 1\right)} \]
9. Step-by-step derivation

10. Applied rewrites 86.3%

    \[\leadsto \frac{1}{\mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right), \color{blue}{x} \cdot x, 1\right)} \]

11. Final simplification 86.3%

    \[\leadsto {\left(\mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right), x \cdot x, 1\right)\right)}^{-1} \]
12. Add Preprocessing

Alternative 9: 75.3% 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. Add Preprocessing

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. lower-fma.f64 70.9

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

5. Applied rewrites 70.9%

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

Alternative 10: 50.5% 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. Add Preprocessing

3. Taylor expanded in x around 0

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

5. Applied rewrites 51.1%

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

Reproduce

herbie shell --seed 2024351 
(FPCore (x)
  :name "Hyperbolic secant"
  :precision binary64
  (/ 2.0 (+ (exp x) (exp (- x)))))