exp2 (problem 3.3.7)

Percentage Accurate: 53.4% → 99.3%

Time: 4.5s

Alternatives: 4

Speedup: 34.8×

Specification

\[\left|x\right| \leq 710\]

Math

\[\begin{array}{l} \\ \left(e^{x} - 2\right) + e^{-x} \end{array} \]

FPCore

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

C

double code(double x) {
	return (exp(x) - 2.0) + 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 = (exp(x) - 2.0d0) + exp(-x)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(e^{x} - 2\right) + e^{-x} \end{array} \]

FPCore

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

C

double code(double x) {
	return (exp(x) - 2.0) + 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 = (exp(x) - 2.0d0) + exp(-x)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.3% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.96031746031746 \cdot 10^{-5}, x \cdot x, 0.002777777777777778\right), x \cdot x, 0.08333333333333333\right), x \cdot x, 1\right) \cdot \left(x \cdot x\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (fma
   (fma
    (fma 4.96031746031746e-5 (* x x) 0.002777777777777778)
    (* x x)
    0.08333333333333333)
   (* x x)
   1.0)
  (* x x)))

C

double code(double x) {
	return fma(fma(fma(4.96031746031746e-5, (x * x), 0.002777777777777778), (x * x), 0.08333333333333333), (x * x), 1.0) * (x * x);
}

Julia

function code(x)
	return Float64(fma(fma(fma(4.96031746031746e-5, Float64(x * x), 0.002777777777777778), Float64(x * x), 0.08333333333333333), Float64(x * x), 1.0) * Float64(x * x))
end

Wolfram

code[x_] := N[(N[(N[(N[(4.96031746031746e-5 * N[(x * x), $MachinePrecision] + 0.002777777777777778), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.08333333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 53.4%

   \[\left(e^{x} - 2\right) + e^{-x} \]
2. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. lift--.f64 N/A

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

   3. flip-- N/A

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

   4. lift-exp.f64 N/A

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

   5. lift-neg.f64 N/A

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

   6. exp-neg N/A

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

   7. lift-exp.f64 N/A

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

   8. frac-add N/A

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

   9. lower-/.f64 N/A

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

   10. lower-fma.f64 N/A

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

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

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

   12. lower-fma.f64 N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. lower-*.f64 N/A

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

   16. +-commutative N/A

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

   17. lower-+.f64 N/A

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

   18. lower-*.f64 N/A

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

3. Applied rewrites 53.6%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-fma.f64 N/A

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

   4. pow2 N/A

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

   5. lift-pow.f64 N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

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

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

   9. lift-pow.f64 N/A

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

   10. pow2 N/A

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

   11. lift-*.f64 N/A

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

   12. *-rgt-identity N/A

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

   13. lower-fma.f64 N/A

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

   14. *-rgt-identity N/A

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

5. Applied rewrites 54.7%

   \[\leadsto \color{blue}{\frac{{\left(\mathsf{expm1}\left(x\right)\right)}^{2} - \left(-\mathsf{expm1}\left(-x\right)\right) \cdot \left(-\mathsf{expm1}\left(-x\right)\right)}{\mathsf{expm1}\left(x\right) + \left(-\mathsf{expm1}\left(-x\right)\right)}} \]
6. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \frac{{\left(\mathsf{expm1}\left(x\right)\right)}^{2} - \left(-\mathsf{expm1}\left(-x\right)\right) \cdot \left(-\mathsf{expm1}\left(-x\right)\right)}{\color{blue}{\mathsf{expm1}\left(x\right) + \left(-\mathsf{expm1}\left(-x\right)\right)}} \]

   2. lift-expm1.f64 N/A

      \[\leadsto \frac{{\left(\mathsf{expm1}\left(x\right)\right)}^{2} - \left(-\mathsf{expm1}\left(-x\right)\right) \cdot \left(-\mathsf{expm1}\left(-x\right)\right)}{\color{blue}{\left(e^{x} - 1\right)} + \left(-\mathsf{expm1}\left(-x\right)\right)} \]

   3. associate-+l- N/A

      \[\leadsto \frac{{\left(\mathsf{expm1}\left(x\right)\right)}^{2} - \left(-\mathsf{expm1}\left(-x\right)\right) \cdot \left(-\mathsf{expm1}\left(-x\right)\right)}{\color{blue}{e^{x} - \left(1 - \left(-\mathsf{expm1}\left(-x\right)\right)\right)}} \]

   4. sinh-+-cosh-rev N/A

      \[\leadsto \frac{{\left(\mathsf{expm1}\left(x\right)\right)}^{2} - \left(-\mathsf{expm1}\left(-x\right)\right) \cdot \left(-\mathsf{expm1}\left(-x\right)\right)}{\color{blue}{\left(\cosh x + \sinh x\right)} - \left(1 - \left(-\mathsf{expm1}\left(-x\right)\right)\right)} \]

   5. lift-cosh.f64 N/A

      \[\leadsto \frac{{\left(\mathsf{expm1}\left(x\right)\right)}^{2} - \left(-\mathsf{expm1}\left(-x\right)\right) \cdot \left(-\mathsf{expm1}\left(-x\right)\right)}{\left(\color{blue}{\cosh x} + \sinh x\right) - \left(1 - \left(-\mathsf{expm1}\left(-x\right)\right)\right)} \]

   6. lift-sinh.f64 N/A

      \[\leadsto \frac{{\left(\mathsf{expm1}\left(x\right)\right)}^{2} - \left(-\mathsf{expm1}\left(-x\right)\right) \cdot \left(-\mathsf{expm1}\left(-x\right)\right)}{\left(\cosh x + \color{blue}{\sinh x}\right) - \left(1 - \left(-\mathsf{expm1}\left(-x\right)\right)\right)} \]

   7. associate--l+ N/A

      \[\leadsto \frac{{\left(\mathsf{expm1}\left(x\right)\right)}^{2} - \left(-\mathsf{expm1}\left(-x\right)\right) \cdot \left(-\mathsf{expm1}\left(-x\right)\right)}{\color{blue}{\cosh x + \left(\sinh x - \left(1 - \left(-\mathsf{expm1}\left(-x\right)\right)\right)\right)}} \]

   8. lower-+.f64 N/A

      \[\leadsto \frac{{\left(\mathsf{expm1}\left(x\right)\right)}^{2} - \left(-\mathsf{expm1}\left(-x\right)\right) \cdot \left(-\mathsf{expm1}\left(-x\right)\right)}{\color{blue}{\cosh x + \left(\sinh x - \left(1 - \left(-\mathsf{expm1}\left(-x\right)\right)\right)\right)}} \]

   9. lower--.f64 N/A

      \[\leadsto \frac{{\left(\mathsf{expm1}\left(x\right)\right)}^{2} - \left(-\mathsf{expm1}\left(-x\right)\right) \cdot \left(-\mathsf{expm1}\left(-x\right)\right)}{\cosh x + \color{blue}{\left(\sinh x - \left(1 - \left(-\mathsf{expm1}\left(-x\right)\right)\right)\right)}} \]

   10. sub-negate1 N/A

       \[\leadsto \frac{{\left(\mathsf{expm1}\left(x\right)\right)}^{2} - \left(-\mathsf{expm1}\left(-x\right)\right) \cdot \left(-\mathsf{expm1}\left(-x\right)\right)}{\cosh x + \left(\sinh x - \color{blue}{\left(1 + \left(\mathsf{neg}\left(\left(-\mathsf{expm1}\left(-x\right)\right)\right)\right)\right)}\right)} \]

   11. lift-neg.f64 N/A

       \[\leadsto \frac{{\left(\mathsf{expm1}\left(x\right)\right)}^{2} - \left(-\mathsf{expm1}\left(-x\right)\right) \cdot \left(-\mathsf{expm1}\left(-x\right)\right)}{\cosh x + \left(\sinh x - \left(1 + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\mathsf{expm1}\left(-x\right)\right)\right)}\right)\right)\right)\right)} \]

   12. lift-expm1.f64 N/A

       \[\leadsto \frac{{\left(\mathsf{expm1}\left(x\right)\right)}^{2} - \left(-\mathsf{expm1}\left(-x\right)\right) \cdot \left(-\mathsf{expm1}\left(-x\right)\right)}{\cosh x + \left(\sinh x - \left(1 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\left(e^{-x} - 1\right)}\right)\right)\right)\right)\right)\right)} \]

   13. lift-exp.f64 N/A

       \[\leadsto \frac{{\left(\mathsf{expm1}\left(x\right)\right)}^{2} - \left(-\mathsf{expm1}\left(-x\right)\right) \cdot \left(-\mathsf{expm1}\left(-x\right)\right)}{\cosh x + \left(\sinh x - \left(1 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\color{blue}{e^{-x}} - 1\right)\right)\right)\right)\right)\right)\right)} \]

   14. sub-negate1 N/A

       \[\leadsto \frac{{\left(\mathsf{expm1}\left(x\right)\right)}^{2} - \left(-\mathsf{expm1}\left(-x\right)\right) \cdot \left(-\mathsf{expm1}\left(-x\right)\right)}{\cosh x + \left(\sinh x - \left(1 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\left(e^{-x} + \left(\mathsf{neg}\left(1\right)\right)\right)}\right)\right)\right)\right)\right)\right)} \]

   15. metadata-eval N/A

       \[\leadsto \frac{{\left(\mathsf{expm1}\left(x\right)\right)}^{2} - \left(-\mathsf{expm1}\left(-x\right)\right) \cdot \left(-\mathsf{expm1}\left(-x\right)\right)}{\cosh x + \left(\sinh x - \left(1 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(e^{-x} + \color{blue}{-1}\right)\right)\right)\right)\right)\right)\right)} \]

   16. +-commutative N/A

       \[\leadsto \frac{{\left(\mathsf{expm1}\left(x\right)\right)}^{2} - \left(-\mathsf{expm1}\left(-x\right)\right) \cdot \left(-\mathsf{expm1}\left(-x\right)\right)}{\cosh x + \left(\sinh x - \left(1 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\left(-1 + e^{-x}\right)}\right)\right)\right)\right)\right)\right)} \]

   17. lift-+.f64 N/A

       \[\leadsto \frac{{\left(\mathsf{expm1}\left(x\right)\right)}^{2} - \left(-\mathsf{expm1}\left(-x\right)\right) \cdot \left(-\mathsf{expm1}\left(-x\right)\right)}{\cosh x + \left(\sinh x - \left(1 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\left(-1 + e^{-x}\right)}\right)\right)\right)\right)\right)\right)} \]

   18. remove-double-neg N/A

       \[\leadsto \frac{{\left(\mathsf{expm1}\left(x\right)\right)}^{2} - \left(-\mathsf{expm1}\left(-x\right)\right) \cdot \left(-\mathsf{expm1}\left(-x\right)\right)}{\cosh x + \left(\sinh x - \left(1 + \color{blue}{\left(-1 + e^{-x}\right)}\right)\right)} \]

   19. lower-+.f64 3.7

       \[\leadsto \frac{{\left(\mathsf{expm1}\left(x\right)\right)}^{2} - \left(-\mathsf{expm1}\left(-x\right)\right) \cdot \left(-\mathsf{expm1}\left(-x\right)\right)}{\cosh x + \left(\sinh x - \color{blue}{\left(1 + \left(-1 + e^{-x}\right)\right)}\right)} \]

   20. lift-+.f64 N/A

       \[\leadsto \frac{{\left(\mathsf{expm1}\left(x\right)\right)}^{2} - \left(-\mathsf{expm1}\left(-x\right)\right) \cdot \left(-\mathsf{expm1}\left(-x\right)\right)}{\cosh x + \left(\sinh x - \left(1 + \color{blue}{\left(-1 + e^{-x}\right)}\right)\right)} \]

7. Applied rewrites 3.7%

   \[\leadsto \frac{{\left(\mathsf{expm1}\left(x\right)\right)}^{2} - \left(-\mathsf{expm1}\left(-x\right)\right) \cdot \left(-\mathsf{expm1}\left(-x\right)\right)}{\color{blue}{\cosh x + \left(\sinh x - \left(1 + \mathsf{expm1}\left(-x\right)\right)\right)}} \]

8. Taylor expanded in x around 0

   \[\leadsto \color{blue}{{x}^{2} \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)\right)} \]
9. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)\right) \cdot \color{blue}{{x}^{2}} \]

   2. lower-*.f64 N/A

      \[\leadsto \left(1 + {x}^{2} \cdot \left(\frac{1}{12} + {x}^{2} \cdot \left(\frac{1}{360} + \frac{1}{20160} \cdot {x}^{2}\right)\right)\right) \cdot \color{blue}{{x}^{2}} \]

10. Applied rewrites 99.3%

    \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(4.96031746031746 \cdot 10^{-5}, x \cdot x, 0.002777777777777778\right), x \cdot x, 0.08333333333333333\right), x \cdot x, 1\right) \cdot \left(x \cdot x\right)} \]
11. Add Preprocessing

Alternative 2: 99.2% accurate, 6.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 53.4%

   \[\left(e^{x} - 2\right) + e^{-x} \]
2. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. lift--.f64 N/A

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

   3. flip-- N/A

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

   4. lift-exp.f64 N/A

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

   5. lift-neg.f64 N/A

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

   6. exp-neg N/A

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

   7. lift-exp.f64 N/A

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

   8. frac-add N/A

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

   9. lower-/.f64 N/A

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

   10. lower-fma.f64 N/A

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

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

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

   12. lower-fma.f64 N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. lower-*.f64 N/A

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

   16. +-commutative N/A

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

   17. lower-+.f64 N/A

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

   18. lower-*.f64 N/A

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

3. Applied rewrites 53.6%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-fma.f64 N/A

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

   4. pow2 N/A

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

   5. lift-pow.f64 N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

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

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

   9. lift-pow.f64 N/A

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

   10. pow2 N/A

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

   11. lift-*.f64 N/A

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

   12. *-rgt-identity N/A

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

   13. lower-fma.f64 N/A

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

   14. *-rgt-identity N/A

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

5. Applied rewrites 54.7%

   \[\leadsto \color{blue}{\frac{{\left(\mathsf{expm1}\left(x\right)\right)}^{2} - \left(-\mathsf{expm1}\left(-x\right)\right) \cdot \left(-\mathsf{expm1}\left(-x\right)\right)}{\mathsf{expm1}\left(x\right) + \left(-\mathsf{expm1}\left(-x\right)\right)}} \]
6. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \frac{{\left(\mathsf{expm1}\left(x\right)\right)}^{2} - \left(-\mathsf{expm1}\left(-x\right)\right) \cdot \left(-\mathsf{expm1}\left(-x\right)\right)}{\color{blue}{\mathsf{expm1}\left(x\right) + \left(-\mathsf{expm1}\left(-x\right)\right)}} \]

   2. lift-expm1.f64 N/A

      \[\leadsto \frac{{\left(\mathsf{expm1}\left(x\right)\right)}^{2} - \left(-\mathsf{expm1}\left(-x\right)\right) \cdot \left(-\mathsf{expm1}\left(-x\right)\right)}{\color{blue}{\left(e^{x} - 1\right)} + \left(-\mathsf{expm1}\left(-x\right)\right)} \]

   3. associate-+l- N/A

      \[\leadsto \frac{{\left(\mathsf{expm1}\left(x\right)\right)}^{2} - \left(-\mathsf{expm1}\left(-x\right)\right) \cdot \left(-\mathsf{expm1}\left(-x\right)\right)}{\color{blue}{e^{x} - \left(1 - \left(-\mathsf{expm1}\left(-x\right)\right)\right)}} \]

   4. sinh-+-cosh-rev N/A

      \[\leadsto \frac{{\left(\mathsf{expm1}\left(x\right)\right)}^{2} - \left(-\mathsf{expm1}\left(-x\right)\right) \cdot \left(-\mathsf{expm1}\left(-x\right)\right)}{\color{blue}{\left(\cosh x + \sinh x\right)} - \left(1 - \left(-\mathsf{expm1}\left(-x\right)\right)\right)} \]

   5. lift-cosh.f64 N/A

      \[\leadsto \frac{{\left(\mathsf{expm1}\left(x\right)\right)}^{2} - \left(-\mathsf{expm1}\left(-x\right)\right) \cdot \left(-\mathsf{expm1}\left(-x\right)\right)}{\left(\color{blue}{\cosh x} + \sinh x\right) - \left(1 - \left(-\mathsf{expm1}\left(-x\right)\right)\right)} \]

   6. lift-sinh.f64 N/A

      \[\leadsto \frac{{\left(\mathsf{expm1}\left(x\right)\right)}^{2} - \left(-\mathsf{expm1}\left(-x\right)\right) \cdot \left(-\mathsf{expm1}\left(-x\right)\right)}{\left(\cosh x + \color{blue}{\sinh x}\right) - \left(1 - \left(-\mathsf{expm1}\left(-x\right)\right)\right)} \]

   7. associate--l+ N/A

      \[\leadsto \frac{{\left(\mathsf{expm1}\left(x\right)\right)}^{2} - \left(-\mathsf{expm1}\left(-x\right)\right) \cdot \left(-\mathsf{expm1}\left(-x\right)\right)}{\color{blue}{\cosh x + \left(\sinh x - \left(1 - \left(-\mathsf{expm1}\left(-x\right)\right)\right)\right)}} \]

   8. lower-+.f64 N/A

      \[\leadsto \frac{{\left(\mathsf{expm1}\left(x\right)\right)}^{2} - \left(-\mathsf{expm1}\left(-x\right)\right) \cdot \left(-\mathsf{expm1}\left(-x\right)\right)}{\color{blue}{\cosh x + \left(\sinh x - \left(1 - \left(-\mathsf{expm1}\left(-x\right)\right)\right)\right)}} \]

   9. lower--.f64 N/A

      \[\leadsto \frac{{\left(\mathsf{expm1}\left(x\right)\right)}^{2} - \left(-\mathsf{expm1}\left(-x\right)\right) \cdot \left(-\mathsf{expm1}\left(-x\right)\right)}{\cosh x + \color{blue}{\left(\sinh x - \left(1 - \left(-\mathsf{expm1}\left(-x\right)\right)\right)\right)}} \]

   10. sub-negate1 N/A

       \[\leadsto \frac{{\left(\mathsf{expm1}\left(x\right)\right)}^{2} - \left(-\mathsf{expm1}\left(-x\right)\right) \cdot \left(-\mathsf{expm1}\left(-x\right)\right)}{\cosh x + \left(\sinh x - \color{blue}{\left(1 + \left(\mathsf{neg}\left(\left(-\mathsf{expm1}\left(-x\right)\right)\right)\right)\right)}\right)} \]

   11. lift-neg.f64 N/A

       \[\leadsto \frac{{\left(\mathsf{expm1}\left(x\right)\right)}^{2} - \left(-\mathsf{expm1}\left(-x\right)\right) \cdot \left(-\mathsf{expm1}\left(-x\right)\right)}{\cosh x + \left(\sinh x - \left(1 + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\mathsf{expm1}\left(-x\right)\right)\right)}\right)\right)\right)\right)} \]

   12. lift-expm1.f64 N/A

       \[\leadsto \frac{{\left(\mathsf{expm1}\left(x\right)\right)}^{2} - \left(-\mathsf{expm1}\left(-x\right)\right) \cdot \left(-\mathsf{expm1}\left(-x\right)\right)}{\cosh x + \left(\sinh x - \left(1 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\left(e^{-x} - 1\right)}\right)\right)\right)\right)\right)\right)} \]

   13. lift-exp.f64 N/A

       \[\leadsto \frac{{\left(\mathsf{expm1}\left(x\right)\right)}^{2} - \left(-\mathsf{expm1}\left(-x\right)\right) \cdot \left(-\mathsf{expm1}\left(-x\right)\right)}{\cosh x + \left(\sinh x - \left(1 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\color{blue}{e^{-x}} - 1\right)\right)\right)\right)\right)\right)\right)} \]

   14. sub-negate1 N/A

       \[\leadsto \frac{{\left(\mathsf{expm1}\left(x\right)\right)}^{2} - \left(-\mathsf{expm1}\left(-x\right)\right) \cdot \left(-\mathsf{expm1}\left(-x\right)\right)}{\cosh x + \left(\sinh x - \left(1 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\left(e^{-x} + \left(\mathsf{neg}\left(1\right)\right)\right)}\right)\right)\right)\right)\right)\right)} \]

   15. metadata-eval N/A

       \[\leadsto \frac{{\left(\mathsf{expm1}\left(x\right)\right)}^{2} - \left(-\mathsf{expm1}\left(-x\right)\right) \cdot \left(-\mathsf{expm1}\left(-x\right)\right)}{\cosh x + \left(\sinh x - \left(1 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(e^{-x} + \color{blue}{-1}\right)\right)\right)\right)\right)\right)\right)} \]

   16. +-commutative N/A

       \[\leadsto \frac{{\left(\mathsf{expm1}\left(x\right)\right)}^{2} - \left(-\mathsf{expm1}\left(-x\right)\right) \cdot \left(-\mathsf{expm1}\left(-x\right)\right)}{\cosh x + \left(\sinh x - \left(1 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\left(-1 + e^{-x}\right)}\right)\right)\right)\right)\right)\right)} \]

   17. lift-+.f64 N/A

       \[\leadsto \frac{{\left(\mathsf{expm1}\left(x\right)\right)}^{2} - \left(-\mathsf{expm1}\left(-x\right)\right) \cdot \left(-\mathsf{expm1}\left(-x\right)\right)}{\cosh x + \left(\sinh x - \left(1 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\left(-1 + e^{-x}\right)}\right)\right)\right)\right)\right)\right)} \]

   18. remove-double-neg N/A

       \[\leadsto \frac{{\left(\mathsf{expm1}\left(x\right)\right)}^{2} - \left(-\mathsf{expm1}\left(-x\right)\right) \cdot \left(-\mathsf{expm1}\left(-x\right)\right)}{\cosh x + \left(\sinh x - \left(1 + \color{blue}{\left(-1 + e^{-x}\right)}\right)\right)} \]

   19. lower-+.f64 3.7

       \[\leadsto \frac{{\left(\mathsf{expm1}\left(x\right)\right)}^{2} - \left(-\mathsf{expm1}\left(-x\right)\right) \cdot \left(-\mathsf{expm1}\left(-x\right)\right)}{\cosh x + \left(\sinh x - \color{blue}{\left(1 + \left(-1 + e^{-x}\right)\right)}\right)} \]

   20. lift-+.f64 N/A

       \[\leadsto \frac{{\left(\mathsf{expm1}\left(x\right)\right)}^{2} - \left(-\mathsf{expm1}\left(-x\right)\right) \cdot \left(-\mathsf{expm1}\left(-x\right)\right)}{\cosh x + \left(\sinh x - \left(1 + \color{blue}{\left(-1 + e^{-x}\right)}\right)\right)} \]

7. Applied rewrites 3.7%

   \[\leadsto \frac{{\left(\mathsf{expm1}\left(x\right)\right)}^{2} - \left(-\mathsf{expm1}\left(-x\right)\right) \cdot \left(-\mathsf{expm1}\left(-x\right)\right)}{\color{blue}{\cosh x + \left(\sinh x - \left(1 + \mathsf{expm1}\left(-x\right)\right)\right)}} \]

8. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. +-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 99.2

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

10. Applied rewrites 99.2%

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

Alternative 3: 99.0% accurate, 9.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 53.4%

   \[\left(e^{x} - 2\right) + e^{-x} \]
2. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. lift--.f64 N/A

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

   3. flip-- N/A

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

   4. lift-exp.f64 N/A

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

   5. lift-neg.f64 N/A

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

   6. exp-neg N/A

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

   7. lift-exp.f64 N/A

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

   8. frac-add N/A

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

   9. lower-/.f64 N/A

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

   10. lower-fma.f64 N/A

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

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

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

   12. lower-fma.f64 N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. lower-*.f64 N/A

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

   16. +-commutative N/A

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

   17. lower-+.f64 N/A

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

   18. lower-*.f64 N/A

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

3. Applied rewrites 53.6%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-fma.f64 N/A

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

   4. pow2 N/A

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

   5. lift-pow.f64 N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

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

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

   9. lift-pow.f64 N/A

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

   10. pow2 N/A

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

   11. lift-*.f64 N/A

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

   12. *-rgt-identity N/A

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

   13. lower-fma.f64 N/A

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

   14. *-rgt-identity N/A

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

5. Applied rewrites 54.7%

   \[\leadsto \color{blue}{\frac{{\left(\mathsf{expm1}\left(x\right)\right)}^{2} - \left(-\mathsf{expm1}\left(-x\right)\right) \cdot \left(-\mathsf{expm1}\left(-x\right)\right)}{\mathsf{expm1}\left(x\right) + \left(-\mathsf{expm1}\left(-x\right)\right)}} \]
6. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \frac{{\left(\mathsf{expm1}\left(x\right)\right)}^{2} - \left(-\mathsf{expm1}\left(-x\right)\right) \cdot \left(-\mathsf{expm1}\left(-x\right)\right)}{\color{blue}{\mathsf{expm1}\left(x\right) + \left(-\mathsf{expm1}\left(-x\right)\right)}} \]

   2. lift-expm1.f64 N/A

      \[\leadsto \frac{{\left(\mathsf{expm1}\left(x\right)\right)}^{2} - \left(-\mathsf{expm1}\left(-x\right)\right) \cdot \left(-\mathsf{expm1}\left(-x\right)\right)}{\color{blue}{\left(e^{x} - 1\right)} + \left(-\mathsf{expm1}\left(-x\right)\right)} \]

   3. associate-+l- N/A

      \[\leadsto \frac{{\left(\mathsf{expm1}\left(x\right)\right)}^{2} - \left(-\mathsf{expm1}\left(-x\right)\right) \cdot \left(-\mathsf{expm1}\left(-x\right)\right)}{\color{blue}{e^{x} - \left(1 - \left(-\mathsf{expm1}\left(-x\right)\right)\right)}} \]

   4. sinh-+-cosh-rev N/A

      \[\leadsto \frac{{\left(\mathsf{expm1}\left(x\right)\right)}^{2} - \left(-\mathsf{expm1}\left(-x\right)\right) \cdot \left(-\mathsf{expm1}\left(-x\right)\right)}{\color{blue}{\left(\cosh x + \sinh x\right)} - \left(1 - \left(-\mathsf{expm1}\left(-x\right)\right)\right)} \]

   5. lift-cosh.f64 N/A

      \[\leadsto \frac{{\left(\mathsf{expm1}\left(x\right)\right)}^{2} - \left(-\mathsf{expm1}\left(-x\right)\right) \cdot \left(-\mathsf{expm1}\left(-x\right)\right)}{\left(\color{blue}{\cosh x} + \sinh x\right) - \left(1 - \left(-\mathsf{expm1}\left(-x\right)\right)\right)} \]

   6. lift-sinh.f64 N/A

      \[\leadsto \frac{{\left(\mathsf{expm1}\left(x\right)\right)}^{2} - \left(-\mathsf{expm1}\left(-x\right)\right) \cdot \left(-\mathsf{expm1}\left(-x\right)\right)}{\left(\cosh x + \color{blue}{\sinh x}\right) - \left(1 - \left(-\mathsf{expm1}\left(-x\right)\right)\right)} \]

   7. associate--l+ N/A

      \[\leadsto \frac{{\left(\mathsf{expm1}\left(x\right)\right)}^{2} - \left(-\mathsf{expm1}\left(-x\right)\right) \cdot \left(-\mathsf{expm1}\left(-x\right)\right)}{\color{blue}{\cosh x + \left(\sinh x - \left(1 - \left(-\mathsf{expm1}\left(-x\right)\right)\right)\right)}} \]

   8. lower-+.f64 N/A

      \[\leadsto \frac{{\left(\mathsf{expm1}\left(x\right)\right)}^{2} - \left(-\mathsf{expm1}\left(-x\right)\right) \cdot \left(-\mathsf{expm1}\left(-x\right)\right)}{\color{blue}{\cosh x + \left(\sinh x - \left(1 - \left(-\mathsf{expm1}\left(-x\right)\right)\right)\right)}} \]

   9. lower--.f64 N/A

      \[\leadsto \frac{{\left(\mathsf{expm1}\left(x\right)\right)}^{2} - \left(-\mathsf{expm1}\left(-x\right)\right) \cdot \left(-\mathsf{expm1}\left(-x\right)\right)}{\cosh x + \color{blue}{\left(\sinh x - \left(1 - \left(-\mathsf{expm1}\left(-x\right)\right)\right)\right)}} \]

   10. sub-negate1 N/A

       \[\leadsto \frac{{\left(\mathsf{expm1}\left(x\right)\right)}^{2} - \left(-\mathsf{expm1}\left(-x\right)\right) \cdot \left(-\mathsf{expm1}\left(-x\right)\right)}{\cosh x + \left(\sinh x - \color{blue}{\left(1 + \left(\mathsf{neg}\left(\left(-\mathsf{expm1}\left(-x\right)\right)\right)\right)\right)}\right)} \]

   11. lift-neg.f64 N/A

       \[\leadsto \frac{{\left(\mathsf{expm1}\left(x\right)\right)}^{2} - \left(-\mathsf{expm1}\left(-x\right)\right) \cdot \left(-\mathsf{expm1}\left(-x\right)\right)}{\cosh x + \left(\sinh x - \left(1 + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\mathsf{expm1}\left(-x\right)\right)\right)}\right)\right)\right)\right)} \]

   12. lift-expm1.f64 N/A

       \[\leadsto \frac{{\left(\mathsf{expm1}\left(x\right)\right)}^{2} - \left(-\mathsf{expm1}\left(-x\right)\right) \cdot \left(-\mathsf{expm1}\left(-x\right)\right)}{\cosh x + \left(\sinh x - \left(1 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\left(e^{-x} - 1\right)}\right)\right)\right)\right)\right)\right)} \]

   13. lift-exp.f64 N/A

       \[\leadsto \frac{{\left(\mathsf{expm1}\left(x\right)\right)}^{2} - \left(-\mathsf{expm1}\left(-x\right)\right) \cdot \left(-\mathsf{expm1}\left(-x\right)\right)}{\cosh x + \left(\sinh x - \left(1 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\color{blue}{e^{-x}} - 1\right)\right)\right)\right)\right)\right)\right)} \]

   14. sub-negate1 N/A

       \[\leadsto \frac{{\left(\mathsf{expm1}\left(x\right)\right)}^{2} - \left(-\mathsf{expm1}\left(-x\right)\right) \cdot \left(-\mathsf{expm1}\left(-x\right)\right)}{\cosh x + \left(\sinh x - \left(1 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\left(e^{-x} + \left(\mathsf{neg}\left(1\right)\right)\right)}\right)\right)\right)\right)\right)\right)} \]

   15. metadata-eval N/A

       \[\leadsto \frac{{\left(\mathsf{expm1}\left(x\right)\right)}^{2} - \left(-\mathsf{expm1}\left(-x\right)\right) \cdot \left(-\mathsf{expm1}\left(-x\right)\right)}{\cosh x + \left(\sinh x - \left(1 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(e^{-x} + \color{blue}{-1}\right)\right)\right)\right)\right)\right)\right)} \]

   16. +-commutative N/A

       \[\leadsto \frac{{\left(\mathsf{expm1}\left(x\right)\right)}^{2} - \left(-\mathsf{expm1}\left(-x\right)\right) \cdot \left(-\mathsf{expm1}\left(-x\right)\right)}{\cosh x + \left(\sinh x - \left(1 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\left(-1 + e^{-x}\right)}\right)\right)\right)\right)\right)\right)} \]

   17. lift-+.f64 N/A

       \[\leadsto \frac{{\left(\mathsf{expm1}\left(x\right)\right)}^{2} - \left(-\mathsf{expm1}\left(-x\right)\right) \cdot \left(-\mathsf{expm1}\left(-x\right)\right)}{\cosh x + \left(\sinh x - \left(1 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\left(-1 + e^{-x}\right)}\right)\right)\right)\right)\right)\right)} \]

   18. remove-double-neg N/A

       \[\leadsto \frac{{\left(\mathsf{expm1}\left(x\right)\right)}^{2} - \left(-\mathsf{expm1}\left(-x\right)\right) \cdot \left(-\mathsf{expm1}\left(-x\right)\right)}{\cosh x + \left(\sinh x - \left(1 + \color{blue}{\left(-1 + e^{-x}\right)}\right)\right)} \]

   19. lower-+.f64 3.7

       \[\leadsto \frac{{\left(\mathsf{expm1}\left(x\right)\right)}^{2} - \left(-\mathsf{expm1}\left(-x\right)\right) \cdot \left(-\mathsf{expm1}\left(-x\right)\right)}{\cosh x + \left(\sinh x - \color{blue}{\left(1 + \left(-1 + e^{-x}\right)\right)}\right)} \]

   20. lift-+.f64 N/A

       \[\leadsto \frac{{\left(\mathsf{expm1}\left(x\right)\right)}^{2} - \left(-\mathsf{expm1}\left(-x\right)\right) \cdot \left(-\mathsf{expm1}\left(-x\right)\right)}{\cosh x + \left(\sinh x - \left(1 + \color{blue}{\left(-1 + e^{-x}\right)}\right)\right)} \]

7. Applied rewrites 3.7%

   \[\leadsto \frac{{\left(\mathsf{expm1}\left(x\right)\right)}^{2} - \left(-\mathsf{expm1}\left(-x\right)\right) \cdot \left(-\mathsf{expm1}\left(-x\right)\right)}{\color{blue}{\cosh x + \left(\sinh x - \left(1 + \mathsf{expm1}\left(-x\right)\right)\right)}} \]

8. Taylor expanded in x around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 99.0

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

10. Applied rewrites 99.0%

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

Alternative 4: 98.4% accurate, 34.8× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (* x x))

C

double code(double x) {
	return x * x;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

def code(x):
	return x * x

Julia

function code(x)
	return Float64(x * x)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.4%

   \[\left(e^{x} - 2\right) + e^{-x} \]
2. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. lift--.f64 N/A

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

   3. flip-- N/A

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

   4. lift-exp.f64 N/A

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

   5. lift-neg.f64 N/A

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

   6. exp-neg N/A

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

   7. lift-exp.f64 N/A

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

   8. frac-add N/A

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

   9. lower-/.f64 N/A

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

   10. lower-fma.f64 N/A

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

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

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

   12. lower-fma.f64 N/A

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. lower-*.f64 N/A

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

   16. +-commutative N/A

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

   17. lower-+.f64 N/A

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

   18. lower-*.f64 N/A

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

3. Applied rewrites 53.6%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-fma.f64 N/A

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

   4. pow2 N/A

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

   5. lift-pow.f64 N/A

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

   6. metadata-eval N/A

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

   7. metadata-eval N/A

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

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

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

   9. lift-pow.f64 N/A

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

   10. pow2 N/A

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

   11. lift-*.f64 N/A

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

   12. *-rgt-identity N/A

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

   13. lower-fma.f64 N/A

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

   14. *-rgt-identity N/A

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

5. Applied rewrites 54.7%

   \[\leadsto \color{blue}{\frac{{\left(\mathsf{expm1}\left(x\right)\right)}^{2} - \left(-\mathsf{expm1}\left(-x\right)\right) \cdot \left(-\mathsf{expm1}\left(-x\right)\right)}{\mathsf{expm1}\left(x\right) + \left(-\mathsf{expm1}\left(-x\right)\right)}} \]
6. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \frac{{\left(\mathsf{expm1}\left(x\right)\right)}^{2} - \left(-\mathsf{expm1}\left(-x\right)\right) \cdot \left(-\mathsf{expm1}\left(-x\right)\right)}{\color{blue}{\mathsf{expm1}\left(x\right) + \left(-\mathsf{expm1}\left(-x\right)\right)}} \]

   2. lift-expm1.f64 N/A

      \[\leadsto \frac{{\left(\mathsf{expm1}\left(x\right)\right)}^{2} - \left(-\mathsf{expm1}\left(-x\right)\right) \cdot \left(-\mathsf{expm1}\left(-x\right)\right)}{\color{blue}{\left(e^{x} - 1\right)} + \left(-\mathsf{expm1}\left(-x\right)\right)} \]

   3. associate-+l- N/A

      \[\leadsto \frac{{\left(\mathsf{expm1}\left(x\right)\right)}^{2} - \left(-\mathsf{expm1}\left(-x\right)\right) \cdot \left(-\mathsf{expm1}\left(-x\right)\right)}{\color{blue}{e^{x} - \left(1 - \left(-\mathsf{expm1}\left(-x\right)\right)\right)}} \]

   4. sinh-+-cosh-rev N/A

      \[\leadsto \frac{{\left(\mathsf{expm1}\left(x\right)\right)}^{2} - \left(-\mathsf{expm1}\left(-x\right)\right) \cdot \left(-\mathsf{expm1}\left(-x\right)\right)}{\color{blue}{\left(\cosh x + \sinh x\right)} - \left(1 - \left(-\mathsf{expm1}\left(-x\right)\right)\right)} \]

   5. lift-cosh.f64 N/A

      \[\leadsto \frac{{\left(\mathsf{expm1}\left(x\right)\right)}^{2} - \left(-\mathsf{expm1}\left(-x\right)\right) \cdot \left(-\mathsf{expm1}\left(-x\right)\right)}{\left(\color{blue}{\cosh x} + \sinh x\right) - \left(1 - \left(-\mathsf{expm1}\left(-x\right)\right)\right)} \]

   6. lift-sinh.f64 N/A

      \[\leadsto \frac{{\left(\mathsf{expm1}\left(x\right)\right)}^{2} - \left(-\mathsf{expm1}\left(-x\right)\right) \cdot \left(-\mathsf{expm1}\left(-x\right)\right)}{\left(\cosh x + \color{blue}{\sinh x}\right) - \left(1 - \left(-\mathsf{expm1}\left(-x\right)\right)\right)} \]

   7. associate--l+ N/A

      \[\leadsto \frac{{\left(\mathsf{expm1}\left(x\right)\right)}^{2} - \left(-\mathsf{expm1}\left(-x\right)\right) \cdot \left(-\mathsf{expm1}\left(-x\right)\right)}{\color{blue}{\cosh x + \left(\sinh x - \left(1 - \left(-\mathsf{expm1}\left(-x\right)\right)\right)\right)}} \]

   8. lower-+.f64 N/A

      \[\leadsto \frac{{\left(\mathsf{expm1}\left(x\right)\right)}^{2} - \left(-\mathsf{expm1}\left(-x\right)\right) \cdot \left(-\mathsf{expm1}\left(-x\right)\right)}{\color{blue}{\cosh x + \left(\sinh x - \left(1 - \left(-\mathsf{expm1}\left(-x\right)\right)\right)\right)}} \]

   9. lower--.f64 N/A

      \[\leadsto \frac{{\left(\mathsf{expm1}\left(x\right)\right)}^{2} - \left(-\mathsf{expm1}\left(-x\right)\right) \cdot \left(-\mathsf{expm1}\left(-x\right)\right)}{\cosh x + \color{blue}{\left(\sinh x - \left(1 - \left(-\mathsf{expm1}\left(-x\right)\right)\right)\right)}} \]

   10. sub-negate1 N/A

       \[\leadsto \frac{{\left(\mathsf{expm1}\left(x\right)\right)}^{2} - \left(-\mathsf{expm1}\left(-x\right)\right) \cdot \left(-\mathsf{expm1}\left(-x\right)\right)}{\cosh x + \left(\sinh x - \color{blue}{\left(1 + \left(\mathsf{neg}\left(\left(-\mathsf{expm1}\left(-x\right)\right)\right)\right)\right)}\right)} \]

   11. lift-neg.f64 N/A

       \[\leadsto \frac{{\left(\mathsf{expm1}\left(x\right)\right)}^{2} - \left(-\mathsf{expm1}\left(-x\right)\right) \cdot \left(-\mathsf{expm1}\left(-x\right)\right)}{\cosh x + \left(\sinh x - \left(1 + \left(\mathsf{neg}\left(\color{blue}{\left(\mathsf{neg}\left(\mathsf{expm1}\left(-x\right)\right)\right)}\right)\right)\right)\right)} \]

   12. lift-expm1.f64 N/A

       \[\leadsto \frac{{\left(\mathsf{expm1}\left(x\right)\right)}^{2} - \left(-\mathsf{expm1}\left(-x\right)\right) \cdot \left(-\mathsf{expm1}\left(-x\right)\right)}{\cosh x + \left(\sinh x - \left(1 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\left(e^{-x} - 1\right)}\right)\right)\right)\right)\right)\right)} \]

   13. lift-exp.f64 N/A

       \[\leadsto \frac{{\left(\mathsf{expm1}\left(x\right)\right)}^{2} - \left(-\mathsf{expm1}\left(-x\right)\right) \cdot \left(-\mathsf{expm1}\left(-x\right)\right)}{\cosh x + \left(\sinh x - \left(1 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(\color{blue}{e^{-x}} - 1\right)\right)\right)\right)\right)\right)\right)} \]

   14. sub-negate1 N/A

       \[\leadsto \frac{{\left(\mathsf{expm1}\left(x\right)\right)}^{2} - \left(-\mathsf{expm1}\left(-x\right)\right) \cdot \left(-\mathsf{expm1}\left(-x\right)\right)}{\cosh x + \left(\sinh x - \left(1 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\left(e^{-x} + \left(\mathsf{neg}\left(1\right)\right)\right)}\right)\right)\right)\right)\right)\right)} \]

   15. metadata-eval N/A

       \[\leadsto \frac{{\left(\mathsf{expm1}\left(x\right)\right)}^{2} - \left(-\mathsf{expm1}\left(-x\right)\right) \cdot \left(-\mathsf{expm1}\left(-x\right)\right)}{\cosh x + \left(\sinh x - \left(1 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\left(e^{-x} + \color{blue}{-1}\right)\right)\right)\right)\right)\right)\right)} \]

   16. +-commutative N/A

       \[\leadsto \frac{{\left(\mathsf{expm1}\left(x\right)\right)}^{2} - \left(-\mathsf{expm1}\left(-x\right)\right) \cdot \left(-\mathsf{expm1}\left(-x\right)\right)}{\cosh x + \left(\sinh x - \left(1 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\left(-1 + e^{-x}\right)}\right)\right)\right)\right)\right)\right)} \]

   17. lift-+.f64 N/A

       \[\leadsto \frac{{\left(\mathsf{expm1}\left(x\right)\right)}^{2} - \left(-\mathsf{expm1}\left(-x\right)\right) \cdot \left(-\mathsf{expm1}\left(-x\right)\right)}{\cosh x + \left(\sinh x - \left(1 + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\color{blue}{\left(-1 + e^{-x}\right)}\right)\right)\right)\right)\right)\right)} \]

   18. remove-double-neg N/A

       \[\leadsto \frac{{\left(\mathsf{expm1}\left(x\right)\right)}^{2} - \left(-\mathsf{expm1}\left(-x\right)\right) \cdot \left(-\mathsf{expm1}\left(-x\right)\right)}{\cosh x + \left(\sinh x - \left(1 + \color{blue}{\left(-1 + e^{-x}\right)}\right)\right)} \]

   19. lower-+.f64 3.7

       \[\leadsto \frac{{\left(\mathsf{expm1}\left(x\right)\right)}^{2} - \left(-\mathsf{expm1}\left(-x\right)\right) \cdot \left(-\mathsf{expm1}\left(-x\right)\right)}{\cosh x + \left(\sinh x - \color{blue}{\left(1 + \left(-1 + e^{-x}\right)\right)}\right)} \]

   20. lift-+.f64 N/A

       \[\leadsto \frac{{\left(\mathsf{expm1}\left(x\right)\right)}^{2} - \left(-\mathsf{expm1}\left(-x\right)\right) \cdot \left(-\mathsf{expm1}\left(-x\right)\right)}{\cosh x + \left(\sinh x - \left(1 + \color{blue}{\left(-1 + e^{-x}\right)}\right)\right)} \]

7. Applied rewrites 3.7%

   \[\leadsto \frac{{\left(\mathsf{expm1}\left(x\right)\right)}^{2} - \left(-\mathsf{expm1}\left(-x\right)\right) \cdot \left(-\mathsf{expm1}\left(-x\right)\right)}{\color{blue}{\cosh x + \left(\sinh x - \left(1 + \mathsf{expm1}\left(-x\right)\right)\right)}} \]

8. Taylor expanded in x around 0

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

   1. unpow2 N/A

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

   2. lower-*.f64 98.4

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

10. Applied rewrites 98.4%

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

Developer Target 1: 99.9% accurate, 0.9× speedup

Math

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

FPCore

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

C

double code(double x) {
	double t_0 = sinh((x / 2.0));
	return 4.0 * (t_0 * t_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
    real(8) :: t_0
    t_0 = sinh((x / 2.0d0))
    code = 4.0d0 * (t_0 * t_0)
end function

Java

public static double code(double x) {
	double t_0 = Math.sinh((x / 2.0));
	return 4.0 * (t_0 * t_0);
}

Python

def code(x):
	t_0 = math.sinh((x / 2.0))
	return 4.0 * (t_0 * t_0)

Julia

function code(x)
	t_0 = sinh(Float64(x / 2.0))
	return Float64(4.0 * Float64(t_0 * t_0))
end

MATLAB

function tmp = code(x)
	t_0 = sinh((x / 2.0));
	tmp = 4.0 * (t_0 * t_0);
end

Wolfram

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

Reproduce

herbie shell --seed 2025107 
(FPCore (x)
  :name "exp2 (problem 3.3.7)"
  :precision binary64
  :pre (<= (fabs x) 710.0)

  :alt
  (! :herbie-platform default (* 4 (* (sinh (/ x 2)) (sinh (/ x 2)))))

  (+ (- (exp x) 2.0) (exp (- x))))