expfmod (used to be hard to sample)

Percentage Accurate: 8.8% → 39.9%

Time: 17.5s

Alternatives: 15

Speedup: 3.8×

• could not determine a ground truth

  1. x = -3.0405563741636946e-34

Specification

Math

\[\begin{array}{l} \\ \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))

C

double code(double x) {
	return fmod(exp(x), sqrt(cos(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 = mod(exp(x), sqrt(cos(x))) * exp(-x)
end function

Python

def code(x):
	return math.fmod(math.exp(x), math.sqrt(math.cos(x))) * math.exp(-x)

Julia

function code(x)
	return Float64(rem(exp(x), sqrt(cos(x))) * exp(Float64(-x)))
end

Wolfram

code[x_] := N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]

Initial Program: 8.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))

C

double code(double x) {
	return fmod(exp(x), sqrt(cos(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 = mod(exp(x), sqrt(cos(x))) * exp(-x)
end function

Python

def code(x):
	return math.fmod(math.exp(x), math.sqrt(math.cos(x))) * math.exp(-x)

Julia

function code(x)
	return Float64(rem(exp(x), sqrt(cos(x))) * exp(Float64(-x)))
end

Wolfram

code[x_] := N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]

Alternative 1: 39.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ t_1 := \frac{\cosh x \cdot \cosh x - \sinh x \cdot \sinh x}{t\_0}\\ \mathbf{if}\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot t\_0 \leq 2:\\ \;\;\;\;\frac{\left(t\_1 \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.003298611111111111 \cdot \left(x \cdot x\right) - 0.010416666666666666, x \cdot x, -0.25\right) \cdot x, x, 1\right)\right)\right) \cdot 1}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod \left(\sqrt{1}\right)\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x)))
        (t_1 (/ (- (* (cosh x) (cosh x)) (* (sinh x) (sinh x))) t_0)))
   (if (<= (* (fmod (exp x) (sqrt (cos x))) t_0) 2.0)
     (/
      (*
       (fmod
        t_1
        (fma
         (*
          (fma
           (- (* -0.003298611111111111 (* x x)) 0.010416666666666666)
           (* x x)
           -0.25)
          x)
         x
         1.0))
       1.0)
      t_1)
     (* (fmod 1.0 (sqrt 1.0)) 1.0))))

C

double code(double x) {
	double t_0 = exp(-x);
	double t_1 = ((cosh(x) * cosh(x)) - (sinh(x) * sinh(x))) / t_0;
	double tmp;
	if ((fmod(exp(x), sqrt(cos(x))) * t_0) <= 2.0) {
		tmp = (fmod(t_1, fma((fma(((-0.003298611111111111 * (x * x)) - 0.010416666666666666), (x * x), -0.25) * x), x, 1.0)) * 1.0) / t_1;
	} else {
		tmp = fmod(1.0, sqrt(1.0)) * 1.0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = exp(Float64(-x))
	t_1 = Float64(Float64(Float64(cosh(x) * cosh(x)) - Float64(sinh(x) * sinh(x))) / t_0)
	tmp = 0.0
	if (Float64(rem(exp(x), sqrt(cos(x))) * t_0) <= 2.0)
		tmp = Float64(Float64(rem(t_1, fma(Float64(fma(Float64(Float64(-0.003298611111111111 * Float64(x * x)) - 0.010416666666666666), Float64(x * x), -0.25) * x), x, 1.0)) * 1.0) / t_1);
	else
		tmp = Float64(rem(1.0, sqrt(1.0)) * 1.0);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, Block[{t$95$1 = N[(N[(N[(N[Cosh[x], $MachinePrecision] * N[Cosh[x], $MachinePrecision]), $MachinePrecision] - N[(N[Sinh[x], $MachinePrecision] * N[Sinh[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]}, If[LessEqual[N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision], 2.0], N[(N[(N[With[{TMP1 = t$95$1, TMP2 = N[(N[(N[(N[(N[(-0.003298611111111111 * N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.010416666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision] + -0.25), $MachinePrecision] * x), $MachinePrecision] * x + 1.0), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * 1.0), $MachinePrecision] / t$95$1), $MachinePrecision], N[(N[With[{TMP1 = 1.0, TMP2 = N[Sqrt[1.0], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * 1.0), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x))) < 2

   1. Initial program 12.9%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x}} \]

      2. lift-exp.f64 N/A

         \[\leadsto \left(\color{blue}{\left(e^{x}\right)} \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

      3. lift-fmod.f64 N/A

         \[\leadsto \color{blue}{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)} \cdot e^{-x} \]

      4. lift-cos.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\color{blue}{\cos x}}\right)\right) \cdot e^{-x} \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(\sqrt{\cos x}\right)}\right) \cdot e^{-x} \]

      6. lift-exp.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot \color{blue}{e^{-x}} \]

      7. lift-neg.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{\color{blue}{\mathsf{neg}\left(x\right)}} \]

      8. exp-neg N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot \color{blue}{\frac{1}{e^{x}}} \]

      9. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot 1}{e^{x}}} \]

      10. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot 1}{e^{x}}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot 1}}{e^{x}} \]

      12. lift-sqrt.f64 N/A

          \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \color{blue}{\left(\sqrt{\cos x}\right)}\right) \cdot 1}{e^{x}} \]

      13. lift-cos.f64 N/A

          \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\color{blue}{\cos x}}\right)\right) \cdot 1}{e^{x}} \]

      14. lift-fmod.f64 N/A

          \[\leadsto \frac{\color{blue}{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)} \cdot 1}{e^{x}} \]

      15. lift-exp.f64 N/A

          \[\leadsto \frac{\left(\color{blue}{\left(e^{x}\right)} \bmod \left(\sqrt{\cos x}\right)\right) \cdot 1}{e^{x}} \]

      16. lift-exp.f64 12.9

          \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot 1}{\color{blue}{e^{x}}} \]

   3. Applied rewrites 12.9%

      \[\leadsto \color{blue}{\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot 1}{e^{x}}} \]

   4. Taylor expanded in x around 0

      \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \color{blue}{\left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-19}{5760} \cdot {x}^{2} - \frac{1}{96}\right) - \frac{1}{4}\right)\right)}\right) \cdot 1}{e^{x}} \]
   5. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-19}{5760} \cdot {x}^{2} - \frac{1}{96}\right) - \frac{1}{4}\right) + \color{blue}{1}\right)\right) \cdot 1}{e^{x}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\left({x}^{2} \cdot \left(\frac{-19}{5760} \cdot {x}^{2} - \frac{1}{96}\right) - \frac{1}{4}\right) \cdot {x}^{2} + 1\right)\right) \cdot 1}{e^{x}} \]

      3. pow2 N/A

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\left({x}^{2} \cdot \left(\frac{-19}{5760} \cdot {x}^{2} - \frac{1}{96}\right) - \frac{1}{4}\right) \cdot \left(x \cdot x\right) + 1\right)\right) \cdot 1}{e^{x}} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\left(\left({x}^{2} \cdot \left(\frac{-19}{5760} \cdot {x}^{2} - \frac{1}{96}\right) - \frac{1}{4}\right) \cdot x\right) \cdot x + 1\right)\right) \cdot 1}{e^{x}} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\left({x}^{2} \cdot \left(\frac{-19}{5760} \cdot {x}^{2} - \frac{1}{96}\right) - \frac{1}{4}\right) \cdot x, \color{blue}{x}, 1\right)\right)\right) \cdot 1}{e^{x}} \]

   6. Applied rewrites 12.6%

      \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.003298611111111111 \cdot \left(x \cdot x\right) - 0.010416666666666666, x \cdot x, -0.25\right) \cdot x, x, 1\right)\right)}\right) \cdot 1}{e^{x}} \]
   7. Step-by-step derivation

      1. lift-exp.f64 N/A

         \[\leadsto \frac{\left(\color{blue}{\left(e^{x}\right)} \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-19}{5760} \cdot \left(x \cdot x\right) - \frac{1}{96}, x \cdot x, \frac{-1}{4}\right) \cdot x, x, 1\right)\right)\right) \cdot 1}{e^{x}} \]

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

         \[\leadsto \frac{\left(\color{blue}{\left(\cosh x + \sinh x\right)} \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-19}{5760} \cdot \left(x \cdot x\right) - \frac{1}{96}, x \cdot x, \frac{-1}{4}\right) \cdot x, x, 1\right)\right)\right) \cdot 1}{e^{x}} \]

      3. flip-+ N/A

         \[\leadsto \frac{\left(\color{blue}{\left(\frac{\cosh x \cdot \cosh x - \sinh x \cdot \sinh x}{\cosh x - \sinh x}\right)} \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-19}{5760} \cdot \left(x \cdot x\right) - \frac{1}{96}, x \cdot x, \frac{-1}{4}\right) \cdot x, x, 1\right)\right)\right) \cdot 1}{e^{x}} \]

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

         \[\leadsto \frac{\left(\left(\frac{\cosh x \cdot \cosh x - \sinh x \cdot \sinh x}{\color{blue}{e^{\mathsf{neg}\left(x\right)}}}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-19}{5760} \cdot \left(x \cdot x\right) - \frac{1}{96}, x \cdot x, \frac{-1}{4}\right) \cdot x, x, 1\right)\right)\right) \cdot 1}{e^{x}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\left(\color{blue}{\left(\frac{\cosh x \cdot \cosh x - \sinh x \cdot \sinh x}{e^{\mathsf{neg}\left(x\right)}}\right)} \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-19}{5760} \cdot \left(x \cdot x\right) - \frac{1}{96}, x \cdot x, \frac{-1}{4}\right) \cdot x, x, 1\right)\right)\right) \cdot 1}{e^{x}} \]

      6. lower--.f64 N/A

         \[\leadsto \frac{\left(\left(\frac{\color{blue}{\cosh x \cdot \cosh x - \sinh x \cdot \sinh x}}{e^{\mathsf{neg}\left(x\right)}}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-19}{5760} \cdot \left(x \cdot x\right) - \frac{1}{96}, x \cdot x, \frac{-1}{4}\right) \cdot x, x, 1\right)\right)\right) \cdot 1}{e^{x}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\left(\left(\frac{\color{blue}{\cosh x \cdot \cosh x} - \sinh x \cdot \sinh x}{e^{\mathsf{neg}\left(x\right)}}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-19}{5760} \cdot \left(x \cdot x\right) - \frac{1}{96}, x \cdot x, \frac{-1}{4}\right) \cdot x, x, 1\right)\right)\right) \cdot 1}{e^{x}} \]

      8. lower-cosh.f64 N/A

         \[\leadsto \frac{\left(\left(\frac{\color{blue}{\cosh x} \cdot \cosh x - \sinh x \cdot \sinh x}{e^{\mathsf{neg}\left(x\right)}}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-19}{5760} \cdot \left(x \cdot x\right) - \frac{1}{96}, x \cdot x, \frac{-1}{4}\right) \cdot x, x, 1\right)\right)\right) \cdot 1}{e^{x}} \]

      9. lower-cosh.f64 N/A

         \[\leadsto \frac{\left(\left(\frac{\cosh x \cdot \color{blue}{\cosh x} - \sinh x \cdot \sinh x}{e^{\mathsf{neg}\left(x\right)}}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-19}{5760} \cdot \left(x \cdot x\right) - \frac{1}{96}, x \cdot x, \frac{-1}{4}\right) \cdot x, x, 1\right)\right)\right) \cdot 1}{e^{x}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\left(\left(\frac{\cosh x \cdot \cosh x - \color{blue}{\sinh x \cdot \sinh x}}{e^{\mathsf{neg}\left(x\right)}}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-19}{5760} \cdot \left(x \cdot x\right) - \frac{1}{96}, x \cdot x, \frac{-1}{4}\right) \cdot x, x, 1\right)\right)\right) \cdot 1}{e^{x}} \]

      11. lower-sinh.f64 N/A

          \[\leadsto \frac{\left(\left(\frac{\cosh x \cdot \cosh x - \color{blue}{\sinh x} \cdot \sinh x}{e^{\mathsf{neg}\left(x\right)}}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-19}{5760} \cdot \left(x \cdot x\right) - \frac{1}{96}, x \cdot x, \frac{-1}{4}\right) \cdot x, x, 1\right)\right)\right) \cdot 1}{e^{x}} \]

      12. lower-sinh.f64 N/A

          \[\leadsto \frac{\left(\left(\frac{\cosh x \cdot \cosh x - \sinh x \cdot \color{blue}{\sinh x}}{e^{\mathsf{neg}\left(x\right)}}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-19}{5760} \cdot \left(x \cdot x\right) - \frac{1}{96}, x \cdot x, \frac{-1}{4}\right) \cdot x, x, 1\right)\right)\right) \cdot 1}{e^{x}} \]

      13. lift-neg.f64 N/A

          \[\leadsto \frac{\left(\left(\frac{\cosh x \cdot \cosh x - \sinh x \cdot \sinh x}{e^{\color{blue}{-x}}}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-19}{5760} \cdot \left(x \cdot x\right) - \frac{1}{96}, x \cdot x, \frac{-1}{4}\right) \cdot x, x, 1\right)\right)\right) \cdot 1}{e^{x}} \]

      14. lift-exp.f64 12.2

          \[\leadsto \frac{\left(\left(\frac{\cosh x \cdot \cosh x - \sinh x \cdot \sinh x}{\color{blue}{e^{-x}}}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.003298611111111111 \cdot \left(x \cdot x\right) - 0.010416666666666666, x \cdot x, -0.25\right) \cdot x, x, 1\right)\right)\right) \cdot 1}{e^{x}} \]

   8. Applied rewrites 12.2%

      \[\leadsto \frac{\left(\color{blue}{\left(\frac{\cosh x \cdot \cosh x - \sinh x \cdot \sinh x}{e^{-x}}\right)} \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.003298611111111111 \cdot \left(x \cdot x\right) - 0.010416666666666666, x \cdot x, -0.25\right) \cdot x, x, 1\right)\right)\right) \cdot 1}{e^{x}} \]
   9. Step-by-step derivation

      1. lift-exp.f64 N/A

         \[\leadsto \frac{\left(\left(\frac{\cosh x \cdot \cosh x - \sinh x \cdot \sinh x}{e^{-x}}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-19}{5760} \cdot \left(x \cdot x\right) - \frac{1}{96}, x \cdot x, \frac{-1}{4}\right) \cdot x, x, 1\right)\right)\right) \cdot 1}{\color{blue}{e^{x}}} \]

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

         \[\leadsto \frac{\left(\left(\frac{\cosh x \cdot \cosh x - \sinh x \cdot \sinh x}{e^{-x}}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-19}{5760} \cdot \left(x \cdot x\right) - \frac{1}{96}, x \cdot x, \frac{-1}{4}\right) \cdot x, x, 1\right)\right)\right) \cdot 1}{\color{blue}{\cosh x + \sinh x}} \]

      3. flip-+ N/A

         \[\leadsto \frac{\left(\left(\frac{\cosh x \cdot \cosh x - \sinh x \cdot \sinh x}{e^{-x}}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-19}{5760} \cdot \left(x \cdot x\right) - \frac{1}{96}, x \cdot x, \frac{-1}{4}\right) \cdot x, x, 1\right)\right)\right) \cdot 1}{\color{blue}{\frac{\cosh x \cdot \cosh x - \sinh x \cdot \sinh x}{\cosh x - \sinh x}}} \]

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

         \[\leadsto \frac{\left(\left(\frac{\cosh x \cdot \cosh x - \sinh x \cdot \sinh x}{e^{-x}}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-19}{5760} \cdot \left(x \cdot x\right) - \frac{1}{96}, x \cdot x, \frac{-1}{4}\right) \cdot x, x, 1\right)\right)\right) \cdot 1}{\frac{\cosh x \cdot \cosh x - \sinh x \cdot \sinh x}{\color{blue}{e^{\mathsf{neg}\left(x\right)}}}} \]

      5. lower-/.f64 N/A

         \[\leadsto \frac{\left(\left(\frac{\cosh x \cdot \cosh x - \sinh x \cdot \sinh x}{e^{-x}}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-19}{5760} \cdot \left(x \cdot x\right) - \frac{1}{96}, x \cdot x, \frac{-1}{4}\right) \cdot x, x, 1\right)\right)\right) \cdot 1}{\color{blue}{\frac{\cosh x \cdot \cosh x - \sinh x \cdot \sinh x}{e^{\mathsf{neg}\left(x\right)}}}} \]

      6. lower--.f64 N/A

         \[\leadsto \frac{\left(\left(\frac{\cosh x \cdot \cosh x - \sinh x \cdot \sinh x}{e^{-x}}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-19}{5760} \cdot \left(x \cdot x\right) - \frac{1}{96}, x \cdot x, \frac{-1}{4}\right) \cdot x, x, 1\right)\right)\right) \cdot 1}{\frac{\color{blue}{\cosh x \cdot \cosh x - \sinh x \cdot \sinh x}}{e^{\mathsf{neg}\left(x\right)}}} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\left(\left(\frac{\cosh x \cdot \cosh x - \sinh x \cdot \sinh x}{e^{-x}}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-19}{5760} \cdot \left(x \cdot x\right) - \frac{1}{96}, x \cdot x, \frac{-1}{4}\right) \cdot x, x, 1\right)\right)\right) \cdot 1}{\frac{\color{blue}{\cosh x \cdot \cosh x} - \sinh x \cdot \sinh x}{e^{\mathsf{neg}\left(x\right)}}} \]

      8. lower-cosh.f64 N/A

         \[\leadsto \frac{\left(\left(\frac{\cosh x \cdot \cosh x - \sinh x \cdot \sinh x}{e^{-x}}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-19}{5760} \cdot \left(x \cdot x\right) - \frac{1}{96}, x \cdot x, \frac{-1}{4}\right) \cdot x, x, 1\right)\right)\right) \cdot 1}{\frac{\color{blue}{\cosh x} \cdot \cosh x - \sinh x \cdot \sinh x}{e^{\mathsf{neg}\left(x\right)}}} \]

      9. lower-cosh.f64 N/A

         \[\leadsto \frac{\left(\left(\frac{\cosh x \cdot \cosh x - \sinh x \cdot \sinh x}{e^{-x}}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-19}{5760} \cdot \left(x \cdot x\right) - \frac{1}{96}, x \cdot x, \frac{-1}{4}\right) \cdot x, x, 1\right)\right)\right) \cdot 1}{\frac{\cosh x \cdot \color{blue}{\cosh x} - \sinh x \cdot \sinh x}{e^{\mathsf{neg}\left(x\right)}}} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\left(\left(\frac{\cosh x \cdot \cosh x - \sinh x \cdot \sinh x}{e^{-x}}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-19}{5760} \cdot \left(x \cdot x\right) - \frac{1}{96}, x \cdot x, \frac{-1}{4}\right) \cdot x, x, 1\right)\right)\right) \cdot 1}{\frac{\cosh x \cdot \cosh x - \color{blue}{\sinh x \cdot \sinh x}}{e^{\mathsf{neg}\left(x\right)}}} \]

      11. lower-sinh.f64 N/A

          \[\leadsto \frac{\left(\left(\frac{\cosh x \cdot \cosh x - \sinh x \cdot \sinh x}{e^{-x}}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-19}{5760} \cdot \left(x \cdot x\right) - \frac{1}{96}, x \cdot x, \frac{-1}{4}\right) \cdot x, x, 1\right)\right)\right) \cdot 1}{\frac{\cosh x \cdot \cosh x - \color{blue}{\sinh x} \cdot \sinh x}{e^{\mathsf{neg}\left(x\right)}}} \]

      12. lower-sinh.f64 N/A

          \[\leadsto \frac{\left(\left(\frac{\cosh x \cdot \cosh x - \sinh x \cdot \sinh x}{e^{-x}}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-19}{5760} \cdot \left(x \cdot x\right) - \frac{1}{96}, x \cdot x, \frac{-1}{4}\right) \cdot x, x, 1\right)\right)\right) \cdot 1}{\frac{\cosh x \cdot \cosh x - \sinh x \cdot \color{blue}{\sinh x}}{e^{\mathsf{neg}\left(x\right)}}} \]

      13. lift-neg.f64 N/A

          \[\leadsto \frac{\left(\left(\frac{\cosh x \cdot \cosh x - \sinh x \cdot \sinh x}{e^{-x}}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-19}{5760} \cdot \left(x \cdot x\right) - \frac{1}{96}, x \cdot x, \frac{-1}{4}\right) \cdot x, x, 1\right)\right)\right) \cdot 1}{\frac{\cosh x \cdot \cosh x - \sinh x \cdot \sinh x}{e^{\color{blue}{-x}}}} \]

      14. lift-exp.f64 12.3

          \[\leadsto \frac{\left(\left(\frac{\cosh x \cdot \cosh x - \sinh x \cdot \sinh x}{e^{-x}}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.003298611111111111 \cdot \left(x \cdot x\right) - 0.010416666666666666, x \cdot x, -0.25\right) \cdot x, x, 1\right)\right)\right) \cdot 1}{\frac{\cosh x \cdot \cosh x - \sinh x \cdot \sinh x}{\color{blue}{e^{-x}}}} \]

   10. Applied rewrites 12.3%

       \[\leadsto \frac{\left(\left(\frac{\cosh x \cdot \cosh x - \sinh x \cdot \sinh x}{e^{-x}}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.003298611111111111 \cdot \left(x \cdot x\right) - 0.010416666666666666, x \cdot x, -0.25\right) \cdot x, x, 1\right)\right)\right) \cdot 1}{\color{blue}{\frac{\cosh x \cdot \cosh x - \sinh x \cdot \sinh x}{e^{-x}}}} \]

   if 2 < (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x)))

   1. Initial program 0.0%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

   2. Taylor expanded in x around 0

      \[\leadsto \left(\color{blue}{1} \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   3. Step-by-step derivation

   4. Applied rewrites 97.9%

      \[\leadsto \left(\color{blue}{1} \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

   5. Taylor expanded in x around 0

      \[\leadsto \left(1 \bmod \left(\sqrt{\color{blue}{1}}\right)\right) \cdot e^{-x} \]
   6. Step-by-step derivation

   7. Applied rewrites 97.8%

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

   8. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

         \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot \left(1 + \left(\mathsf{neg}\left(x\right)\right)\right) \]

      2. lift-neg.f64 N/A

         \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot \left(1 + \left(-x\right)\right) \]

      3. +-commutative N/A

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

      4. metadata-eval N/A

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

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

         \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot \left(\left(-x\right) - \color{blue}{\left(\mathsf{neg}\left(1\right)\right) \cdot 1}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot \left(\left(-x\right) - -1 \cdot 1\right) \]

      7. metadata-eval N/A

         \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot \left(\left(-x\right) - -1\right) \]

      8. lower--.f64 97.9

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

   10. Applied rewrites 97.9%

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

   11. Taylor expanded in x around 0

       \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot 1 \]
   12. Step-by-step derivation

   13. Applied rewrites 97.9%

       \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot 1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 39.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \leq 2:\\ \;\;\;\;\frac{\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, -0.003298611111111111, -0.010416666666666666\right), x \cdot x, -0.25\right) \cdot x, x, 1\right)\right)\right)}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod \left(\sqrt{1}\right)\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* (fmod (exp x) (sqrt (cos x))) (exp (- x))) 2.0)
   (/
    (fmod
     (exp x)
     (fma
      (*
       (fma
        (fma (* x x) -0.003298611111111111 -0.010416666666666666)
        (* x x)
        -0.25)
       x)
      x
      1.0))
    (exp x))
   (* (fmod 1.0 (sqrt 1.0)) 1.0)))

C

double code(double x) {
	double tmp;
	if ((fmod(exp(x), sqrt(cos(x))) * exp(-x)) <= 2.0) {
		tmp = fmod(exp(x), fma((fma(fma((x * x), -0.003298611111111111, -0.010416666666666666), (x * x), -0.25) * x), x, 1.0)) / exp(x);
	} else {
		tmp = fmod(1.0, sqrt(1.0)) * 1.0;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(rem(exp(x), sqrt(cos(x))) * exp(Float64(-x))) <= 2.0)
		tmp = Float64(rem(exp(x), fma(Float64(fma(fma(Float64(x * x), -0.003298611111111111, -0.010416666666666666), Float64(x * x), -0.25) * x), x, 1.0)) / exp(x));
	else
		tmp = Float64(rem(1.0, sqrt(1.0)) * 1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision], 2.0], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(N[(N[(N[(N[(x * x), $MachinePrecision] * -0.003298611111111111 + -0.010416666666666666), $MachinePrecision] * N[(x * x), $MachinePrecision] + -0.25), $MachinePrecision] * x), $MachinePrecision] * x + 1.0), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = 1.0, TMP2 = N[Sqrt[1.0], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x))) < 2

   1. Initial program 12.9%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x}} \]

      2. lift-exp.f64 N/A

         \[\leadsto \left(\color{blue}{\left(e^{x}\right)} \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

      3. lift-fmod.f64 N/A

         \[\leadsto \color{blue}{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)} \cdot e^{-x} \]

      4. lift-cos.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\color{blue}{\cos x}}\right)\right) \cdot e^{-x} \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(\sqrt{\cos x}\right)}\right) \cdot e^{-x} \]

      6. lift-exp.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot \color{blue}{e^{-x}} \]

      7. lift-neg.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{\color{blue}{\mathsf{neg}\left(x\right)}} \]

      8. exp-neg N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot \color{blue}{\frac{1}{e^{x}}} \]

      9. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot 1}{e^{x}}} \]

      10. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot 1}{e^{x}}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot 1}}{e^{x}} \]

      12. lift-sqrt.f64 N/A

          \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \color{blue}{\left(\sqrt{\cos x}\right)}\right) \cdot 1}{e^{x}} \]

      13. lift-cos.f64 N/A

          \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\color{blue}{\cos x}}\right)\right) \cdot 1}{e^{x}} \]

      14. lift-fmod.f64 N/A

          \[\leadsto \frac{\color{blue}{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)} \cdot 1}{e^{x}} \]

      15. lift-exp.f64 N/A

          \[\leadsto \frac{\left(\color{blue}{\left(e^{x}\right)} \bmod \left(\sqrt{\cos x}\right)\right) \cdot 1}{e^{x}} \]

      16. lift-exp.f64 12.9

          \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot 1}{\color{blue}{e^{x}}} \]

   3. Applied rewrites 12.9%

      \[\leadsto \color{blue}{\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot 1}{e^{x}}} \]

   4. Taylor expanded in x around 0

      \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \color{blue}{\left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-19}{5760} \cdot {x}^{2} - \frac{1}{96}\right) - \frac{1}{4}\right)\right)}\right) \cdot 1}{e^{x}} \]
   5. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{-19}{5760} \cdot {x}^{2} - \frac{1}{96}\right) - \frac{1}{4}\right) + \color{blue}{1}\right)\right) \cdot 1}{e^{x}} \]

      2. *-commutative N/A

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\left({x}^{2} \cdot \left(\frac{-19}{5760} \cdot {x}^{2} - \frac{1}{96}\right) - \frac{1}{4}\right) \cdot {x}^{2} + 1\right)\right) \cdot 1}{e^{x}} \]

      3. pow2 N/A

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\left({x}^{2} \cdot \left(\frac{-19}{5760} \cdot {x}^{2} - \frac{1}{96}\right) - \frac{1}{4}\right) \cdot \left(x \cdot x\right) + 1\right)\right) \cdot 1}{e^{x}} \]

      4. associate-*r* N/A

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\left(\left({x}^{2} \cdot \left(\frac{-19}{5760} \cdot {x}^{2} - \frac{1}{96}\right) - \frac{1}{4}\right) \cdot x\right) \cdot x + 1\right)\right) \cdot 1}{e^{x}} \]

      5. lower-fma.f64 N/A

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\left({x}^{2} \cdot \left(\frac{-19}{5760} \cdot {x}^{2} - \frac{1}{96}\right) - \frac{1}{4}\right) \cdot x, \color{blue}{x}, 1\right)\right)\right) \cdot 1}{e^{x}} \]

   6. Applied rewrites 12.6%

      \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.003298611111111111 \cdot \left(x \cdot x\right) - 0.010416666666666666, x \cdot x, -0.25\right) \cdot x, x, 1\right)\right)}\right) \cdot 1}{e^{x}} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-19}{5760} \cdot \left(x \cdot x\right) - \frac{1}{96}, x \cdot x, \frac{-1}{4}\right) \cdot x, x, 1\right)\right)\right) \cdot 1}}{e^{x}} \]

      2. *-rgt-identity 12.6

         \[\leadsto \frac{\color{blue}{\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.003298611111111111 \cdot \left(x \cdot x\right) - 0.010416666666666666, x \cdot x, -0.25\right) \cdot x, x, 1\right)\right)\right)}}{e^{x}} \]

      3. lift--.f64 N/A

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-19}{5760} \cdot \left(x \cdot x\right) - \frac{1}{96}, x \cdot x, \frac{-1}{4}\right) \cdot x, x, 1\right)\right)\right)}{e^{x}} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-19}{5760} \cdot \left(x \cdot x\right) - \frac{1}{96}, x \cdot x, \frac{-1}{4}\right) \cdot x, x, 1\right)\right)\right)}{e^{x}} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-19}{5760} \cdot \left(x \cdot x\right) - \frac{1}{96}, x \cdot x, \frac{-1}{4}\right) \cdot x, x, 1\right)\right)\right)}{e^{x}} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-19}{5760} \cdot \left(x \cdot x\right) - \frac{1}{96} \cdot 1, x \cdot x, \frac{-1}{4}\right) \cdot x, x, 1\right)\right)\right)}{e^{x}} \]

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

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-19}{5760} \cdot \left(x \cdot x\right) + \left(\mathsf{neg}\left(\frac{1}{96}\right)\right) \cdot 1, x \cdot x, \frac{-1}{4}\right) \cdot x, x, 1\right)\right)\right)}{e^{x}} \]

      8. metadata-eval N/A

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-19}{5760} \cdot \left(x \cdot x\right) + \frac{-1}{96} \cdot 1, x \cdot x, \frac{-1}{4}\right) \cdot x, x, 1\right)\right)\right)}{e^{x}} \]

      9. metadata-eval N/A

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-19}{5760} \cdot \left(x \cdot x\right) + \frac{-1}{96}, x \cdot x, \frac{-1}{4}\right) \cdot x, x, 1\right)\right)\right)}{e^{x}} \]

      10. pow2 N/A

          \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-19}{5760} \cdot {x}^{2} + \frac{-1}{96}, x \cdot x, \frac{-1}{4}\right) \cdot x, x, 1\right)\right)\right)}{e^{x}} \]

      11. *-commutative N/A

          \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left({x}^{2} \cdot \frac{-19}{5760} + \frac{-1}{96}, x \cdot x, \frac{-1}{4}\right) \cdot x, x, 1\right)\right)\right)}{e^{x}} \]

      12. lower-fma.f64 N/A

          \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left({x}^{2}, \frac{-19}{5760}, \frac{-1}{96}\right), x \cdot x, \frac{-1}{4}\right) \cdot x, x, 1\right)\right)\right)}{e^{x}} \]

      13. pow2 N/A

          \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{-19}{5760}, \frac{-1}{96}\right), x \cdot x, \frac{-1}{4}\right) \cdot x, x, 1\right)\right)\right)}{e^{x}} \]

      14. lift-*.f64 12.6

          \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, -0.003298611111111111, -0.010416666666666666\right), x \cdot x, -0.25\right) \cdot x, x, 1\right)\right)\right)}{e^{x}} \]

   8. Applied rewrites 12.6%

      \[\leadsto \frac{\color{blue}{\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, -0.003298611111111111, -0.010416666666666666\right), x \cdot x, -0.25\right) \cdot x, x, 1\right)\right)\right)}}{e^{x}} \]

   if 2 < (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x)))

   1. Initial program 0.0%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

   2. Taylor expanded in x around 0

      \[\leadsto \left(\color{blue}{1} \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   3. Step-by-step derivation

   4. Applied rewrites 97.9%

      \[\leadsto \left(\color{blue}{1} \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

   5. Taylor expanded in x around 0

      \[\leadsto \left(1 \bmod \left(\sqrt{\color{blue}{1}}\right)\right) \cdot e^{-x} \]
   6. Step-by-step derivation

   7. Applied rewrites 97.8%

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

   8. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

         \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot \left(1 + \left(\mathsf{neg}\left(x\right)\right)\right) \]

      2. lift-neg.f64 N/A

         \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot \left(1 + \left(-x\right)\right) \]

      3. +-commutative N/A

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

      4. metadata-eval N/A

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

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

         \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot \left(\left(-x\right) - \color{blue}{\left(\mathsf{neg}\left(1\right)\right) \cdot 1}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot \left(\left(-x\right) - -1 \cdot 1\right) \]

      7. metadata-eval N/A

         \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot \left(\left(-x\right) - -1\right) \]

      8. lower--.f64 97.9

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

   10. Applied rewrites 97.9%

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

   11. Taylor expanded in x around 0

       \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot 1 \]
   12. Step-by-step derivation

   13. Applied rewrites 97.9%

       \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot 1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 39.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot t\_0 \leq 2:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right)}\right)\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod \left(\sqrt{1}\right)\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= (* (fmod (exp x) (sqrt (cos x))) t_0) 2.0)
     (*
      (fmod
       (exp x)
       (sqrt (fma (- (* 0.041666666666666664 (* x x)) 0.5) (* x x) 1.0)))
      t_0)
     (* (fmod 1.0 (sqrt 1.0)) 1.0))))

C

double code(double x) {
	double t_0 = exp(-x);
	double tmp;
	if ((fmod(exp(x), sqrt(cos(x))) * t_0) <= 2.0) {
		tmp = fmod(exp(x), sqrt(fma(((0.041666666666666664 * (x * x)) - 0.5), (x * x), 1.0))) * t_0;
	} else {
		tmp = fmod(1.0, sqrt(1.0)) * 1.0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = exp(Float64(-x))
	tmp = 0.0
	if (Float64(rem(exp(x), sqrt(cos(x))) * t_0) <= 2.0)
		tmp = Float64(rem(exp(x), sqrt(fma(Float64(Float64(0.041666666666666664 * Float64(x * x)) - 0.5), Float64(x * x), 1.0))) * t_0);
	else
		tmp = Float64(rem(1.0, sqrt(1.0)) * 1.0);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, If[LessEqual[N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision], 2.0], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Sqrt[N[(N[(N[(0.041666666666666664 * N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[With[{TMP1 = 1.0, TMP2 = N[Sqrt[1.0], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x))) < 2

   1. Initial program 12.9%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right) \cdot {x}^{2} + 1}\right)\right) \cdot e^{-x} \]

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}, {x}^{2}, 1\right)}\right)\right) \cdot e^{-x} \]

      6. unpow2 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(\frac{1}{24} \cdot \left(x \cdot x\right) - \frac{1}{2}, {x}^{2}, 1\right)}\right)\right) \cdot e^{-x} \]

      7. lower-*.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(\frac{1}{24} \cdot \left(x \cdot x\right) - \frac{1}{2}, {x}^{2}, 1\right)}\right)\right) \cdot e^{-x} \]

      8. unpow2 N/A

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

      9. lower-*.f64 12.6

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right) - 0.5, x \cdot \color{blue}{x}, 1\right)}\right)\right) \cdot e^{-x} \]

   4. Applied rewrites 12.6%

      \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\color{blue}{\mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right)}}\right)\right) \cdot e^{-x} \]

   if 2 < (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x)))

   1. Initial program 0.0%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

   2. Taylor expanded in x around 0

      \[\leadsto \left(\color{blue}{1} \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   3. Step-by-step derivation

   4. Applied rewrites 97.9%

      \[\leadsto \left(\color{blue}{1} \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

   5. Taylor expanded in x around 0

      \[\leadsto \left(1 \bmod \left(\sqrt{\color{blue}{1}}\right)\right) \cdot e^{-x} \]
   6. Step-by-step derivation

   7. Applied rewrites 97.8%

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

   8. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

         \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot \left(1 + \left(\mathsf{neg}\left(x\right)\right)\right) \]

      2. lift-neg.f64 N/A

         \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot \left(1 + \left(-x\right)\right) \]

      3. +-commutative N/A

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

      4. metadata-eval N/A

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

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

         \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot \left(\left(-x\right) - \color{blue}{\left(\mathsf{neg}\left(1\right)\right) \cdot 1}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot \left(\left(-x\right) - -1 \cdot 1\right) \]

      7. metadata-eval N/A

         \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot \left(\left(-x\right) - -1\right) \]

      8. lower--.f64 97.9

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

   10. Applied rewrites 97.9%

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

   11. Taylor expanded in x around 0

       \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot 1 \]
   12. Step-by-step derivation

   13. Applied rewrites 97.9%

       \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot 1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 39.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot t\_0 \leq 2:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.010416666666666666, x \cdot x, -0.25\right), x \cdot x, 1\right)\right)\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod \left(\sqrt{1}\right)\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= (* (fmod (exp x) (sqrt (cos x))) t_0) 2.0)
     (*
      (fmod
       (exp x)
       (fma (fma -0.010416666666666666 (* x x) -0.25) (* x x) 1.0))
      t_0)
     (* (fmod 1.0 (sqrt 1.0)) 1.0))))

C

double code(double x) {
	double t_0 = exp(-x);
	double tmp;
	if ((fmod(exp(x), sqrt(cos(x))) * t_0) <= 2.0) {
		tmp = fmod(exp(x), fma(fma(-0.010416666666666666, (x * x), -0.25), (x * x), 1.0)) * t_0;
	} else {
		tmp = fmod(1.0, sqrt(1.0)) * 1.0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = exp(Float64(-x))
	tmp = 0.0
	if (Float64(rem(exp(x), sqrt(cos(x))) * t_0) <= 2.0)
		tmp = Float64(rem(exp(x), fma(fma(-0.010416666666666666, Float64(x * x), -0.25), Float64(x * x), 1.0)) * t_0);
	else
		tmp = Float64(rem(1.0, sqrt(1.0)) * 1.0);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, If[LessEqual[N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision], 2.0], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(N[(-0.010416666666666666 * N[(x * x), $MachinePrecision] + -0.25), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[With[{TMP1 = 1.0, TMP2 = N[Sqrt[1.0], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x))) < 2

   1. Initial program 12.9%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

   2. Taylor expanded in x around 0

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{-1}{96} \cdot {x}^{2} - \frac{1}{4}\right)\right)}\right) \cdot e^{-x} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left({x}^{2} \cdot \left(\frac{-1}{96} \cdot {x}^{2} - \frac{1}{4}\right) + \color{blue}{1}\right)\right) \cdot e^{-x} \]

      2. *-commutative N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\frac{-1}{96} \cdot {x}^{2} - \frac{1}{4}\right) \cdot {x}^{2} + 1\right)\right) \cdot e^{-x} \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot {x}^{2} - \frac{1}{4}, \color{blue}{{x}^{2}}, 1\right)\right)\right) \cdot e^{-x} \]

      4. metadata-eval N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot {x}^{2} - \frac{1}{2} \cdot \frac{1}{2}, {x}^{2}, 1\right)\right)\right) \cdot e^{-x} \]

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

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{1}{2}, {\color{blue}{x}}^{2}, 1\right)\right)\right) \cdot e^{-x} \]

      6. metadata-eval N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot {x}^{2} + \frac{-1}{2} \cdot \frac{1}{2}, {x}^{2}, 1\right)\right)\right) \cdot e^{-x} \]

      7. metadata-eval N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot {x}^{2} + \frac{-1}{4}, {x}^{2}, 1\right)\right)\right) \cdot e^{-x} \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{96}, {x}^{2}, \frac{-1}{4}\right), {\color{blue}{x}}^{2}, 1\right)\right)\right) \cdot e^{-x} \]

      9. unpow2 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{96}, x \cdot x, \frac{-1}{4}\right), {x}^{2}, 1\right)\right)\right) \cdot e^{-x} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{96}, x \cdot x, \frac{-1}{4}\right), {x}^{2}, 1\right)\right)\right) \cdot e^{-x} \]

      11. unpow2 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{96}, x \cdot x, \frac{-1}{4}\right), x \cdot \color{blue}{x}, 1\right)\right)\right) \cdot e^{-x} \]

      12. lower-*.f64 12.5

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.010416666666666666, x \cdot x, -0.25\right), x \cdot \color{blue}{x}, 1\right)\right)\right) \cdot e^{-x} \]

   4. Applied rewrites 12.5%

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.010416666666666666, x \cdot x, -0.25\right), x \cdot x, 1\right)\right)}\right) \cdot e^{-x} \]

   if 2 < (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x)))

   1. Initial program 0.0%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

   2. Taylor expanded in x around 0

      \[\leadsto \left(\color{blue}{1} \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   3. Step-by-step derivation

   4. Applied rewrites 97.9%

      \[\leadsto \left(\color{blue}{1} \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

   5. Taylor expanded in x around 0

      \[\leadsto \left(1 \bmod \left(\sqrt{\color{blue}{1}}\right)\right) \cdot e^{-x} \]
   6. Step-by-step derivation

   7. Applied rewrites 97.8%

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

   8. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

         \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot \left(1 + \left(\mathsf{neg}\left(x\right)\right)\right) \]

      2. lift-neg.f64 N/A

         \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot \left(1 + \left(-x\right)\right) \]

      3. +-commutative N/A

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

      4. metadata-eval N/A

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

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

         \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot \left(\left(-x\right) - \color{blue}{\left(\mathsf{neg}\left(1\right)\right) \cdot 1}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot \left(\left(-x\right) - -1 \cdot 1\right) \]

      7. metadata-eval N/A

         \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot \left(\left(-x\right) - -1\right) \]

      8. lower--.f64 97.9

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

   10. Applied rewrites 97.9%

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

   11. Taylor expanded in x around 0

       \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot 1 \]
   12. Step-by-step derivation

   13. Applied rewrites 97.9%

       \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot 1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 39.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \leq 2:\\ \;\;\;\;\frac{\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)\right) \cdot 1}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod \left(\sqrt{1}\right)\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* (fmod (exp x) (sqrt (cos x))) (exp (- x))) 2.0)
   (/ (* (fmod (exp x) (fma (* x x) -0.25 1.0)) 1.0) (exp x))
   (* (fmod 1.0 (sqrt 1.0)) 1.0)))

C

double code(double x) {
	double tmp;
	if ((fmod(exp(x), sqrt(cos(x))) * exp(-x)) <= 2.0) {
		tmp = (fmod(exp(x), fma((x * x), -0.25, 1.0)) * 1.0) / exp(x);
	} else {
		tmp = fmod(1.0, sqrt(1.0)) * 1.0;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(rem(exp(x), sqrt(cos(x))) * exp(Float64(-x))) <= 2.0)
		tmp = Float64(Float64(rem(exp(x), fma(Float64(x * x), -0.25, 1.0)) * 1.0) / exp(x));
	else
		tmp = Float64(rem(1.0, sqrt(1.0)) * 1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision], 2.0], N[(N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(N[(x * x), $MachinePrecision] * -0.25 + 1.0), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * 1.0), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = 1.0, TMP2 = N[Sqrt[1.0], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x))) < 2

   1. Initial program 12.9%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x}} \]

      2. lift-exp.f64 N/A

         \[\leadsto \left(\color{blue}{\left(e^{x}\right)} \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

      3. lift-fmod.f64 N/A

         \[\leadsto \color{blue}{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)} \cdot e^{-x} \]

      4. lift-cos.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\color{blue}{\cos x}}\right)\right) \cdot e^{-x} \]

      5. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(\sqrt{\cos x}\right)}\right) \cdot e^{-x} \]

      6. lift-exp.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot \color{blue}{e^{-x}} \]

      7. lift-neg.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{\color{blue}{\mathsf{neg}\left(x\right)}} \]

      8. exp-neg N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot \color{blue}{\frac{1}{e^{x}}} \]

      9. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot 1}{e^{x}}} \]

      10. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot 1}{e^{x}}} \]

      11. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot 1}}{e^{x}} \]

      12. lift-sqrt.f64 N/A

          \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \color{blue}{\left(\sqrt{\cos x}\right)}\right) \cdot 1}{e^{x}} \]

      13. lift-cos.f64 N/A

          \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\color{blue}{\cos x}}\right)\right) \cdot 1}{e^{x}} \]

      14. lift-fmod.f64 N/A

          \[\leadsto \frac{\color{blue}{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)} \cdot 1}{e^{x}} \]

      15. lift-exp.f64 N/A

          \[\leadsto \frac{\left(\color{blue}{\left(e^{x}\right)} \bmod \left(\sqrt{\cos x}\right)\right) \cdot 1}{e^{x}} \]

      16. lift-exp.f64 12.9

          \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot 1}{\color{blue}{e^{x}}} \]

   3. Applied rewrites 12.9%

      \[\leadsto \color{blue}{\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot 1}{e^{x}}} \]

   4. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. pow2 N/A

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

      5. lift-*.f64 12.4

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)\right) \cdot 1}{e^{x}} \]

   6. Applied rewrites 12.4%

      \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \color{blue}{\left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)}\right) \cdot 1}{e^{x}} \]

   if 2 < (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x)))

   1. Initial program 0.0%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

   2. Taylor expanded in x around 0

      \[\leadsto \left(\color{blue}{1} \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   3. Step-by-step derivation

   4. Applied rewrites 97.9%

      \[\leadsto \left(\color{blue}{1} \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

   5. Taylor expanded in x around 0

      \[\leadsto \left(1 \bmod \left(\sqrt{\color{blue}{1}}\right)\right) \cdot e^{-x} \]
   6. Step-by-step derivation

   7. Applied rewrites 97.8%

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

   8. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

         \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot \left(1 + \left(\mathsf{neg}\left(x\right)\right)\right) \]

      2. lift-neg.f64 N/A

         \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot \left(1 + \left(-x\right)\right) \]

      3. +-commutative N/A

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

      4. metadata-eval N/A

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

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

         \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot \left(\left(-x\right) - \color{blue}{\left(\mathsf{neg}\left(1\right)\right) \cdot 1}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot \left(\left(-x\right) - -1 \cdot 1\right) \]

      7. metadata-eval N/A

         \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot \left(\left(-x\right) - -1\right) \]

      8. lower--.f64 97.9

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

   10. Applied rewrites 97.9%

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

   11. Taylor expanded in x around 0

       \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot 1 \]
   12. Step-by-step derivation

   13. Applied rewrites 97.9%

       \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot 1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 39.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot t\_0 \leq 2:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod \left(\sqrt{1}\right)\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= (* (fmod (exp x) (sqrt (cos x))) t_0) 2.0)
     (* (fmod (exp x) (fma (* x x) -0.25 1.0)) t_0)
     (* (fmod 1.0 (sqrt 1.0)) 1.0))))

C

double code(double x) {
	double t_0 = exp(-x);
	double tmp;
	if ((fmod(exp(x), sqrt(cos(x))) * t_0) <= 2.0) {
		tmp = fmod(exp(x), fma((x * x), -0.25, 1.0)) * t_0;
	} else {
		tmp = fmod(1.0, sqrt(1.0)) * 1.0;
	}
	return tmp;
}

Julia

function code(x)
	t_0 = exp(Float64(-x))
	tmp = 0.0
	if (Float64(rem(exp(x), sqrt(cos(x))) * t_0) <= 2.0)
		tmp = Float64(rem(exp(x), fma(Float64(x * x), -0.25, 1.0)) * t_0);
	else
		tmp = Float64(rem(1.0, sqrt(1.0)) * 1.0);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, If[LessEqual[N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision], 2.0], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(N[(x * x), $MachinePrecision] * -0.25 + 1.0), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[With[{TMP1 = 1.0, TMP2 = N[Sqrt[1.0], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x))) < 2

   1. Initial program 12.9%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

   2. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 12.4

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)\right) \cdot e^{-x} \]

   4. Applied rewrites 12.4%

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(\mathsf{fma}\left(x \cdot x, -0.25, 1\right)\right)}\right) \cdot e^{-x} \]

   if 2 < (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x)))

   1. Initial program 0.0%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

   2. Taylor expanded in x around 0

      \[\leadsto \left(\color{blue}{1} \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   3. Step-by-step derivation

   4. Applied rewrites 97.9%

      \[\leadsto \left(\color{blue}{1} \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

   5. Taylor expanded in x around 0

      \[\leadsto \left(1 \bmod \left(\sqrt{\color{blue}{1}}\right)\right) \cdot e^{-x} \]
   6. Step-by-step derivation

   7. Applied rewrites 97.8%

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

   8. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

         \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot \left(1 + \left(\mathsf{neg}\left(x\right)\right)\right) \]

      2. lift-neg.f64 N/A

         \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot \left(1 + \left(-x\right)\right) \]

      3. +-commutative N/A

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

      4. metadata-eval N/A

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

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

         \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot \left(\left(-x\right) - \color{blue}{\left(\mathsf{neg}\left(1\right)\right) \cdot 1}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot \left(\left(-x\right) - -1 \cdot 1\right) \]

      7. metadata-eval N/A

         \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot \left(\left(-x\right) - -1\right) \]

      8. lower--.f64 97.9

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

   10. Applied rewrites 97.9%

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

   11. Taylor expanded in x around 0

       \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot 1 \]
   12. Step-by-step derivation

   13. Applied rewrites 97.9%

       \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot 1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 39.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \leq 2:\\ \;\;\;\;\frac{\left(\left(e^{x}\right) \bmod 1\right) \cdot 1}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod \left(\sqrt{1}\right)\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* (fmod (exp x) (sqrt (cos x))) (exp (- x))) 2.0)
   (/ (* (fmod (exp x) 1.0) 1.0) (exp x))
   (* (fmod 1.0 (sqrt 1.0)) 1.0)))

C

double code(double x) {
	double tmp;
	if ((fmod(exp(x), sqrt(cos(x))) * exp(-x)) <= 2.0) {
		tmp = (fmod(exp(x), 1.0) * 1.0) / exp(x);
	} else {
		tmp = fmod(1.0, sqrt(1.0)) * 1.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((mod(exp(x), sqrt(cos(x))) * exp(-x)) <= 2.0d0) then
        tmp = (mod(exp(x), 1.0d0) * 1.0d0) / exp(x)
    else
        tmp = mod(1.0d0, sqrt(1.0d0)) * 1.0d0
    end if
    code = tmp
end function

Python

def code(x):
	tmp = 0
	if (math.fmod(math.exp(x), math.sqrt(math.cos(x))) * math.exp(-x)) <= 2.0:
		tmp = (math.fmod(math.exp(x), 1.0) * 1.0) / math.exp(x)
	else:
		tmp = math.fmod(1.0, math.sqrt(1.0)) * 1.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(rem(exp(x), sqrt(cos(x))) * exp(Float64(-x))) <= 2.0)
		tmp = Float64(Float64(rem(exp(x), 1.0) * 1.0) / exp(x));
	else
		tmp = Float64(rem(1.0, sqrt(1.0)) * 1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision], 2.0], N[(N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * 1.0), $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = 1.0, TMP2 = N[Sqrt[1.0], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x))) < 2

   1. Initial program 12.9%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

   2. Taylor expanded in x around 0

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{1}\right) \cdot e^{-x} \]
   3. Step-by-step derivation

   4. Applied rewrites 12.0%

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

      1. lift-*.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-neg.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod 1\right) \cdot e^{\color{blue}{\mathsf{neg}\left(x\right)}} \]

      4. exp-neg N/A

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

      5. associate-*r/ N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift-exp.f64 12.0

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

   6. Applied rewrites 12.0%

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

   if 2 < (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x)))

   1. Initial program 0.0%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

   2. Taylor expanded in x around 0

      \[\leadsto \left(\color{blue}{1} \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   3. Step-by-step derivation

   4. Applied rewrites 97.9%

      \[\leadsto \left(\color{blue}{1} \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

   5. Taylor expanded in x around 0

      \[\leadsto \left(1 \bmod \left(\sqrt{\color{blue}{1}}\right)\right) \cdot e^{-x} \]
   6. Step-by-step derivation

   7. Applied rewrites 97.8%

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

   8. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

         \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot \left(1 + \left(\mathsf{neg}\left(x\right)\right)\right) \]

      2. lift-neg.f64 N/A

         \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot \left(1 + \left(-x\right)\right) \]

      3. +-commutative N/A

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

      4. metadata-eval N/A

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

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

         \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot \left(\left(-x\right) - \color{blue}{\left(\mathsf{neg}\left(1\right)\right) \cdot 1}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot \left(\left(-x\right) - -1 \cdot 1\right) \]

      7. metadata-eval N/A

         \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot \left(\left(-x\right) - -1\right) \]

      8. lower--.f64 97.9

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

   10. Applied rewrites 97.9%

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

   11. Taylor expanded in x around 0

       \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot 1 \]
   12. Step-by-step derivation

   13. Applied rewrites 97.9%

       \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot 1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 39.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{-x}\\ \mathbf{if}\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot t\_0 \leq 2:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod 1\right) \cdot t\_0\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod \left(\sqrt{1}\right)\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x))))
   (if (<= (* (fmod (exp x) (sqrt (cos x))) t_0) 2.0)
     (* (fmod (exp x) 1.0) t_0)
     (* (fmod 1.0 (sqrt 1.0)) 1.0))))

C

double code(double x) {
	double t_0 = exp(-x);
	double tmp;
	if ((fmod(exp(x), sqrt(cos(x))) * t_0) <= 2.0) {
		tmp = fmod(exp(x), 1.0) * t_0;
	} else {
		tmp = fmod(1.0, sqrt(1.0)) * 1.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = exp(-x)
    if ((mod(exp(x), sqrt(cos(x))) * t_0) <= 2.0d0) then
        tmp = mod(exp(x), 1.0d0) * t_0
    else
        tmp = mod(1.0d0, sqrt(1.0d0)) * 1.0d0
    end if
    code = tmp
end function

Python

def code(x):
	t_0 = math.exp(-x)
	tmp = 0
	if (math.fmod(math.exp(x), math.sqrt(math.cos(x))) * t_0) <= 2.0:
		tmp = math.fmod(math.exp(x), 1.0) * t_0
	else:
		tmp = math.fmod(1.0, math.sqrt(1.0)) * 1.0
	return tmp

Julia

function code(x)
	t_0 = exp(Float64(-x))
	tmp = 0.0
	if (Float64(rem(exp(x), sqrt(cos(x))) * t_0) <= 2.0)
		tmp = Float64(rem(exp(x), 1.0) * t_0);
	else
		tmp = Float64(rem(1.0, sqrt(1.0)) * 1.0);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Exp[(-x)], $MachinePrecision]}, If[LessEqual[N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision], 2.0], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision], N[(N[With[{TMP1 = 1.0, TMP2 = N[Sqrt[1.0], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x))) < 2

   1. Initial program 12.9%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

   2. Taylor expanded in x around 0

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{1}\right) \cdot e^{-x} \]
   3. Step-by-step derivation

   4. Applied rewrites 12.0%

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

   if 2 < (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x)))

   1. Initial program 0.0%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

   2. Taylor expanded in x around 0

      \[\leadsto \left(\color{blue}{1} \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   3. Step-by-step derivation

   4. Applied rewrites 97.9%

      \[\leadsto \left(\color{blue}{1} \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

   5. Taylor expanded in x around 0

      \[\leadsto \left(1 \bmod \left(\sqrt{\color{blue}{1}}\right)\right) \cdot e^{-x} \]
   6. Step-by-step derivation

   7. Applied rewrites 97.8%

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

   8. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

         \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot \left(1 + \left(\mathsf{neg}\left(x\right)\right)\right) \]

      2. lift-neg.f64 N/A

         \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot \left(1 + \left(-x\right)\right) \]

      3. +-commutative N/A

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

      4. metadata-eval N/A

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

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

         \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot \left(\left(-x\right) - \color{blue}{\left(\mathsf{neg}\left(1\right)\right) \cdot 1}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot \left(\left(-x\right) - -1 \cdot 1\right) \]

      7. metadata-eval N/A

         \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot \left(\left(-x\right) - -1\right) \]

      8. lower--.f64 97.9

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

   10. Applied rewrites 97.9%

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

   11. Taylor expanded in x around 0

       \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot 1 \]
   12. Step-by-step derivation

   13. Applied rewrites 97.9%

       \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot 1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 38.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.5:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(-0.25, x \cdot x, 1\right)\right)\right) \cdot \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod \left(\sqrt{1}\right)\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.5)
   (* (fmod (exp x) (fma -0.25 (* x x) 1.0)) (- 1.0 x))
   (* (fmod 1.0 (sqrt 1.0)) 1.0)))

C

double code(double x) {
	double tmp;
	if (x <= 0.5) {
		tmp = fmod(exp(x), fma(-0.25, (x * x), 1.0)) * (1.0 - x);
	} else {
		tmp = fmod(1.0, sqrt(1.0)) * 1.0;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.5)
		tmp = Float64(rem(exp(x), fma(-0.25, Float64(x * x), 1.0)) * Float64(1.0 - x));
	else
		tmp = Float64(rem(1.0, sqrt(1.0)) * 1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 0.5], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(-0.25 * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[(1.0 - x), $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = 1.0, TMP2 = N[Sqrt[1.0], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.5

   1. Initial program 12.5%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

   2. Taylor expanded in x around 0

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{-1}{96} \cdot {x}^{2} - \frac{1}{4}\right)\right)}\right) \cdot e^{-x} \]
   3. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left({x}^{2} \cdot \left(\frac{-1}{96} \cdot {x}^{2} - \frac{1}{4}\right) + \color{blue}{1}\right)\right) \cdot e^{-x} \]

      2. *-commutative N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\left(\frac{-1}{96} \cdot {x}^{2} - \frac{1}{4}\right) \cdot {x}^{2} + 1\right)\right) \cdot e^{-x} \]

      3. lower-fma.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot {x}^{2} - \frac{1}{4}, \color{blue}{{x}^{2}}, 1\right)\right)\right) \cdot e^{-x} \]

      4. metadata-eval N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot {x}^{2} - \frac{1}{2} \cdot \frac{1}{2}, {x}^{2}, 1\right)\right)\right) \cdot e^{-x} \]

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

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \frac{1}{2}, {\color{blue}{x}}^{2}, 1\right)\right)\right) \cdot e^{-x} \]

      6. metadata-eval N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot {x}^{2} + \frac{-1}{2} \cdot \frac{1}{2}, {x}^{2}, 1\right)\right)\right) \cdot e^{-x} \]

      7. metadata-eval N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\frac{-1}{96} \cdot {x}^{2} + \frac{-1}{4}, {x}^{2}, 1\right)\right)\right) \cdot e^{-x} \]

      8. lower-fma.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{96}, {x}^{2}, \frac{-1}{4}\right), {\color{blue}{x}}^{2}, 1\right)\right)\right) \cdot e^{-x} \]

      9. unpow2 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{96}, x \cdot x, \frac{-1}{4}\right), {x}^{2}, 1\right)\right)\right) \cdot e^{-x} \]

      10. lower-*.f64 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{96}, x \cdot x, \frac{-1}{4}\right), {x}^{2}, 1\right)\right)\right) \cdot e^{-x} \]

      11. unpow2 N/A

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{96}, x \cdot x, \frac{-1}{4}\right), x \cdot \color{blue}{x}, 1\right)\right)\right) \cdot e^{-x} \]

      12. lower-*.f64 12.4

          \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.010416666666666666, x \cdot x, -0.25\right), x \cdot \color{blue}{x}, 1\right)\right)\right) \cdot e^{-x} \]

   4. Applied rewrites 12.4%

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(-0.010416666666666666, x \cdot x, -0.25\right), x \cdot x, 1\right)\right)}\right) \cdot e^{-x} \]

   5. Taylor expanded in x around 0

      \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{96}, x \cdot x, \frac{-1}{4}\right), x \cdot x, 1\right)\right)\right) \cdot \color{blue}{\left(1 + -1 \cdot x\right)} \]
   6. Step-by-step derivation

      1. metadata-eval N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{96}, x \cdot x, \frac{-1}{4}\right), x \cdot x, 1\right)\right)\right) \cdot \left(1 + \left(\mathsf{neg}\left(1\right)\right) \cdot x\right) \]

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

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{96}, x \cdot x, \frac{-1}{4}\right), x \cdot x, 1\right)\right)\right) \cdot \left(1 - \color{blue}{1 \cdot x}\right) \]

      3. *-lft-identity N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{-1}{96}, x \cdot x, \frac{-1}{4}\right), x \cdot x, 1\right)\right)\right) \cdot \left(1 - x\right) \]

      4. lower--.f64 10.8

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.010416666666666666, x \cdot x, -0.25\right), x \cdot x, 1\right)\right)\right) \cdot \left(1 - \color{blue}{x}\right) \]

   7. Applied rewrites 10.8%

      \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(\mathsf{fma}\left(-0.010416666666666666, x \cdot x, -0.25\right), x \cdot x, 1\right)\right)\right) \cdot \color{blue}{\left(1 - x\right)} \]

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 10.8%

       \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(-0.25, \color{blue}{x} \cdot x, 1\right)\right)\right) \cdot \left(1 - x\right) \]

   if 0.5 < x

   1. Initial program 0.5%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

   2. Taylor expanded in x around 0

      \[\leadsto \left(\color{blue}{1} \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   3. Step-by-step derivation

   4. Applied rewrites 99.5%

      \[\leadsto \left(\color{blue}{1} \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

   5. Taylor expanded in x around 0

      \[\leadsto \left(1 \bmod \left(\sqrt{\color{blue}{1}}\right)\right) \cdot e^{-x} \]
   6. Step-by-step derivation

   7. Applied rewrites 99.3%

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

   8. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

         \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot \left(1 + \left(\mathsf{neg}\left(x\right)\right)\right) \]

      2. lift-neg.f64 N/A

         \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot \left(1 + \left(-x\right)\right) \]

      3. +-commutative N/A

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

      4. metadata-eval N/A

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

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

         \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot \left(\left(-x\right) - \color{blue}{\left(\mathsf{neg}\left(1\right)\right) \cdot 1}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot \left(\left(-x\right) - -1 \cdot 1\right) \]

      7. metadata-eval N/A

         \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot \left(\left(-x\right) - -1\right) \]

      8. lower--.f64 99.3

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

   10. Applied rewrites 99.3%

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

   11. Taylor expanded in x around 0

       \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot 1 \]
   12. Step-by-step derivation

   13. Applied rewrites 99.3%

       \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot 1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 38.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 20:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod 1\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(0.5, x, -1\right), x, 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod \left(\sqrt{1}\right)\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 20.0)
   (* (fmod (exp x) 1.0) (fma (fma 0.5 x -1.0) x 1.0))
   (* (fmod 1.0 (sqrt 1.0)) 1.0)))

C

double code(double x) {
	double tmp;
	if (x <= 20.0) {
		tmp = fmod(exp(x), 1.0) * fma(fma(0.5, x, -1.0), x, 1.0);
	} else {
		tmp = fmod(1.0, sqrt(1.0)) * 1.0;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 20.0)
		tmp = Float64(rem(exp(x), 1.0) * fma(fma(0.5, x, -1.0), x, 1.0));
	else
		tmp = Float64(rem(1.0, sqrt(1.0)) * 1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 20.0], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[(N[(0.5 * x + -1.0), $MachinePrecision] * x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = 1.0, TMP2 = N[Sqrt[1.0], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 20

   1. Initial program 12.7%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

   2. Taylor expanded in x around 0

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{1}\right) \cdot e^{-x} \]
   3. Step-by-step derivation

   4. Applied rewrites 11.8%

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

   5. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. metadata-eval N/A

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

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

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

      6. metadata-eval N/A

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

      7. metadata-eval N/A

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

      8. lower-fma.f64 10.9

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

   7. Applied rewrites 10.9%

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

   if 20 < x

   1. Initial program 0.0%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

   2. Taylor expanded in x around 0

      \[\leadsto \left(\color{blue}{1} \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   3. Step-by-step derivation

   4. Applied rewrites 99.9%

      \[\leadsto \left(\color{blue}{1} \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

   5. Taylor expanded in x around 0

      \[\leadsto \left(1 \bmod \left(\sqrt{\color{blue}{1}}\right)\right) \cdot e^{-x} \]
   6. Step-by-step derivation

   7. Applied rewrites 99.9%

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

   8. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

         \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot \left(1 + \left(\mathsf{neg}\left(x\right)\right)\right) \]

      2. lift-neg.f64 N/A

         \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot \left(1 + \left(-x\right)\right) \]

      3. +-commutative N/A

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

      4. metadata-eval N/A

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

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

         \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot \left(\left(-x\right) - \color{blue}{\left(\mathsf{neg}\left(1\right)\right) \cdot 1}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot \left(\left(-x\right) - -1 \cdot 1\right) \]

      7. metadata-eval N/A

         \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot \left(\left(-x\right) - -1\right) \]

      8. lower--.f64 99.9

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

   10. Applied rewrites 99.9%

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

   11. Taylor expanded in x around 0

       \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot 1 \]
   12. Step-by-step derivation

   13. Applied rewrites 99.9%

       \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot 1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 38.6% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.5)
   (* (fmod (fma (fma 0.5 x 1.0) x 1.0) (sqrt 1.0)) (- (- x) -1.0))
   (* (fmod 1.0 (sqrt 1.0)) 1.0)))

C

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

Julia

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

Wolfram

code[x_] := If[LessEqual[x, 0.5], N[(N[With[{TMP1 = N[(N[(0.5 * x + 1.0), $MachinePrecision] * x + 1.0), $MachinePrecision], TMP2 = N[Sqrt[1.0], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[((-x) - -1.0), $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = 1.0, TMP2 = N[Sqrt[1.0], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.5

   1. Initial program 12.5%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

   2. Taylor expanded in x around 0

      \[\leadsto \left(\color{blue}{1} \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   3. Step-by-step derivation

   4. Applied rewrites 5.7%

      \[\leadsto \left(\color{blue}{1} \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

   5. Taylor expanded in x around 0

      \[\leadsto \left(1 \bmod \left(\sqrt{\color{blue}{1}}\right)\right) \cdot e^{-x} \]
   6. Step-by-step derivation

   7. Applied rewrites 5.2%

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

   8. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

         \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot \left(1 + \left(\mathsf{neg}\left(x\right)\right)\right) \]

      2. lift-neg.f64 N/A

         \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot \left(1 + \left(-x\right)\right) \]

      3. +-commutative N/A

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

      4. metadata-eval N/A

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

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

         \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot \left(\left(-x\right) - \color{blue}{\left(\mathsf{neg}\left(1\right)\right) \cdot 1}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot \left(\left(-x\right) - -1 \cdot 1\right) \]

      7. metadata-eval N/A

         \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot \left(\left(-x\right) - -1\right) \]

      8. lower--.f64 5.3

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

   10. Applied rewrites 5.3%

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

   11. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. lower-fma.f64 N/A

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

       4. +-commutative N/A

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

       5. lower-fma.f64 10.7

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

   13. Applied rewrites 10.7%

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

   if 0.5 < x

   1. Initial program 0.5%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

   2. Taylor expanded in x around 0

      \[\leadsto \left(\color{blue}{1} \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   3. Step-by-step derivation

   4. Applied rewrites 99.5%

      \[\leadsto \left(\color{blue}{1} \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

   5. Taylor expanded in x around 0

      \[\leadsto \left(1 \bmod \left(\sqrt{\color{blue}{1}}\right)\right) \cdot e^{-x} \]
   6. Step-by-step derivation

   7. Applied rewrites 99.3%

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

   8. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

         \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot \left(1 + \left(\mathsf{neg}\left(x\right)\right)\right) \]

      2. lift-neg.f64 N/A

         \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot \left(1 + \left(-x\right)\right) \]

      3. +-commutative N/A

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

      4. metadata-eval N/A

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

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

         \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot \left(\left(-x\right) - \color{blue}{\left(\mathsf{neg}\left(1\right)\right) \cdot 1}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot \left(\left(-x\right) - -1 \cdot 1\right) \]

      7. metadata-eval N/A

         \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot \left(\left(-x\right) - -1\right) \]

      8. lower--.f64 99.3

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

   10. Applied rewrites 99.3%

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

   11. Taylor expanded in x around 0

       \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot 1 \]
   12. Step-by-step derivation

   13. Applied rewrites 99.3%

       \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot 1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 38.5% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.5:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod 1\right) \cdot \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod \left(\sqrt{1}\right)\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.5)
   (* (fmod (exp x) 1.0) (- 1.0 x))
   (* (fmod 1.0 (sqrt 1.0)) 1.0)))

C

double code(double x) {
	double tmp;
	if (x <= 0.5) {
		tmp = fmod(exp(x), 1.0) * (1.0 - x);
	} else {
		tmp = fmod(1.0, sqrt(1.0)) * 1.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.5d0) then
        tmp = mod(exp(x), 1.0d0) * (1.0d0 - x)
    else
        tmp = mod(1.0d0, sqrt(1.0d0)) * 1.0d0
    end if
    code = tmp
end function

Python

def code(x):
	tmp = 0
	if x <= 0.5:
		tmp = math.fmod(math.exp(x), 1.0) * (1.0 - x)
	else:
		tmp = math.fmod(1.0, math.sqrt(1.0)) * 1.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.5)
		tmp = Float64(rem(exp(x), 1.0) * Float64(1.0 - x));
	else
		tmp = Float64(rem(1.0, sqrt(1.0)) * 1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 0.5], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[(1.0 - x), $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = 1.0, TMP2 = N[Sqrt[1.0], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.5

   1. Initial program 12.5%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

   2. Taylor expanded in x around 0

      \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{1}\right) \cdot e^{-x} \]
   3. Step-by-step derivation

   4. Applied rewrites 11.8%

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

   5. Taylor expanded in x around 0

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

      1. metadata-eval N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod 1\right) \cdot \left(1 + \left(\mathsf{neg}\left(1\right)\right) \cdot x\right) \]

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

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

      3. *-lft-identity N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod 1\right) \cdot \left(1 - x\right) \]

      4. lower--.f64 10.6

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

   7. Applied rewrites 10.6%

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

   if 0.5 < x

   1. Initial program 0.5%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

   2. Taylor expanded in x around 0

      \[\leadsto \left(\color{blue}{1} \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   3. Step-by-step derivation

   4. Applied rewrites 99.5%

      \[\leadsto \left(\color{blue}{1} \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

   5. Taylor expanded in x around 0

      \[\leadsto \left(1 \bmod \left(\sqrt{\color{blue}{1}}\right)\right) \cdot e^{-x} \]
   6. Step-by-step derivation

   7. Applied rewrites 99.3%

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

   8. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

         \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot \left(1 + \left(\mathsf{neg}\left(x\right)\right)\right) \]

      2. lift-neg.f64 N/A

         \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot \left(1 + \left(-x\right)\right) \]

      3. +-commutative N/A

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

      4. metadata-eval N/A

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

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

         \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot \left(\left(-x\right) - \color{blue}{\left(\mathsf{neg}\left(1\right)\right) \cdot 1}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot \left(\left(-x\right) - -1 \cdot 1\right) \]

      7. metadata-eval N/A

         \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot \left(\left(-x\right) - -1\right) \]

      8. lower--.f64 99.3

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

   10. Applied rewrites 99.3%

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

   11. Taylor expanded in x around 0

       \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot 1 \]
   12. Step-by-step derivation

   13. Applied rewrites 99.3%

       \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot 1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 38.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \leq 2:\\ \;\;\;\;\left(\left(1 + x\right) \bmod \left(\sqrt{1}\right)\right) \cdot \left(\left(-x\right) - -1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod \left(\sqrt{1}\right)\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* (fmod (exp x) (sqrt (cos x))) (exp (- x))) 2.0)
   (* (fmod (+ 1.0 x) (sqrt 1.0)) (- (- x) -1.0))
   (* (fmod 1.0 (sqrt 1.0)) 1.0)))

C

double code(double x) {
	double tmp;
	if ((fmod(exp(x), sqrt(cos(x))) * exp(-x)) <= 2.0) {
		tmp = fmod((1.0 + x), sqrt(1.0)) * (-x - -1.0);
	} else {
		tmp = fmod(1.0, sqrt(1.0)) * 1.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((mod(exp(x), sqrt(cos(x))) * exp(-x)) <= 2.0d0) then
        tmp = mod((1.0d0 + x), sqrt(1.0d0)) * (-x - (-1.0d0))
    else
        tmp = mod(1.0d0, sqrt(1.0d0)) * 1.0d0
    end if
    code = tmp
end function

Python

def code(x):
	tmp = 0
	if (math.fmod(math.exp(x), math.sqrt(math.cos(x))) * math.exp(-x)) <= 2.0:
		tmp = math.fmod((1.0 + x), math.sqrt(1.0)) * (-x - -1.0)
	else:
		tmp = math.fmod(1.0, math.sqrt(1.0)) * 1.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(rem(exp(x), sqrt(cos(x))) * exp(Float64(-x))) <= 2.0)
		tmp = Float64(rem(Float64(1.0 + x), sqrt(1.0)) * Float64(Float64(-x) - -1.0));
	else
		tmp = Float64(rem(1.0, sqrt(1.0)) * 1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision], 2.0], N[(N[With[{TMP1 = N[(1.0 + x), $MachinePrecision], TMP2 = N[Sqrt[1.0], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[((-x) - -1.0), $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = 1.0, TMP2 = N[Sqrt[1.0], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x))) < 2

   1. Initial program 12.9%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

   2. Taylor expanded in x around 0

      \[\leadsto \left(\color{blue}{1} \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   3. Step-by-step derivation

   4. Applied rewrites 5.7%

      \[\leadsto \left(\color{blue}{1} \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

   5. Taylor expanded in x around 0

      \[\leadsto \left(1 \bmod \left(\sqrt{\color{blue}{1}}\right)\right) \cdot e^{-x} \]
   6. Step-by-step derivation

   7. Applied rewrites 5.3%

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

   8. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

         \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot \left(1 + \left(\mathsf{neg}\left(x\right)\right)\right) \]

      2. lift-neg.f64 N/A

         \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot \left(1 + \left(-x\right)\right) \]

      3. +-commutative N/A

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

      4. metadata-eval N/A

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

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

         \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot \left(\left(-x\right) - \color{blue}{\left(\mathsf{neg}\left(1\right)\right) \cdot 1}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot \left(\left(-x\right) - -1 \cdot 1\right) \]

      7. metadata-eval N/A

         \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot \left(\left(-x\right) - -1\right) \]

      8. lower--.f64 5.3

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

   10. Applied rewrites 5.3%

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

   11. Taylor expanded in x around 0

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

       1. lower-+.f64 10.5

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

   13. Applied rewrites 10.5%

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

   if 2 < (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x)))

   1. Initial program 0.0%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

   2. Taylor expanded in x around 0

      \[\leadsto \left(\color{blue}{1} \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   3. Step-by-step derivation

   4. Applied rewrites 97.9%

      \[\leadsto \left(\color{blue}{1} \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

   5. Taylor expanded in x around 0

      \[\leadsto \left(1 \bmod \left(\sqrt{\color{blue}{1}}\right)\right) \cdot e^{-x} \]
   6. Step-by-step derivation

   7. Applied rewrites 97.8%

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

   8. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

         \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot \left(1 + \left(\mathsf{neg}\left(x\right)\right)\right) \]

      2. lift-neg.f64 N/A

         \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot \left(1 + \left(-x\right)\right) \]

      3. +-commutative N/A

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

      4. metadata-eval N/A

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

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

         \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot \left(\left(-x\right) - \color{blue}{\left(\mathsf{neg}\left(1\right)\right) \cdot 1}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot \left(\left(-x\right) - -1 \cdot 1\right) \]

      7. metadata-eval N/A

         \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot \left(\left(-x\right) - -1\right) \]

      8. lower--.f64 97.9

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

   10. Applied rewrites 97.9%

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

   11. Taylor expanded in x around 0

       \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot 1 \]
   12. Step-by-step derivation

   13. Applied rewrites 97.9%

       \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot 1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 37.8% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 20:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod 1\right)\\ \mathbf{else}:\\ \;\;\;\;\left(1 \bmod \left(\sqrt{1}\right)\right) \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 20.0) (fmod (exp x) 1.0) (* (fmod 1.0 (sqrt 1.0)) 1.0)))

C

double code(double x) {
	double tmp;
	if (x <= 20.0) {
		tmp = fmod(exp(x), 1.0);
	} else {
		tmp = fmod(1.0, sqrt(1.0)) * 1.0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 20.0d0) then
        tmp = mod(exp(x), 1.0d0)
    else
        tmp = mod(1.0d0, sqrt(1.0d0)) * 1.0d0
    end if
    code = tmp
end function

Python

def code(x):
	tmp = 0
	if x <= 20.0:
		tmp = math.fmod(math.exp(x), 1.0)
	else:
		tmp = math.fmod(1.0, math.sqrt(1.0)) * 1.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 20.0)
		tmp = rem(exp(x), 1.0);
	else
		tmp = Float64(rem(1.0, sqrt(1.0)) * 1.0);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 20.0], N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision], N[(N[With[{TMP1 = 1.0, TMP2 = N[Sqrt[1.0], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * 1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 20

   1. Initial program 12.7%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)} \]
   3. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \]

      2. lift-cos.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \]

      3. lift-fmod.f64 N/A

         \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{\left(\sqrt{\cos x}\right)}\right) \]

      4. lift-exp.f64 9.5

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\color{blue}{\cos x}}\right)\right) \]

   4. Applied rewrites 9.5%

      \[\leadsto \color{blue}{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)} \]

   5. Taylor expanded in x around 0

      \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{1}\right)\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 9.5%

      \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{1}\right)\right) \]

   8. Taylor expanded in x around 0

      \[\leadsto \left(\left(e^{x}\right) \bmod 1\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 9.5%

       \[\leadsto \left(\left(e^{x}\right) \bmod 1\right) \]

   if 20 < x

   1. Initial program 0.0%

      \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

   2. Taylor expanded in x around 0

      \[\leadsto \left(\color{blue}{1} \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
   3. Step-by-step derivation

   4. Applied rewrites 99.9%

      \[\leadsto \left(\color{blue}{1} \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

   5. Taylor expanded in x around 0

      \[\leadsto \left(1 \bmod \left(\sqrt{\color{blue}{1}}\right)\right) \cdot e^{-x} \]
   6. Step-by-step derivation

   7. Applied rewrites 99.9%

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

   8. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

         \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot \left(1 + \left(\mathsf{neg}\left(x\right)\right)\right) \]

      2. lift-neg.f64 N/A

         \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot \left(1 + \left(-x\right)\right) \]

      3. +-commutative N/A

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

      4. metadata-eval N/A

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

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

         \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot \left(\left(-x\right) - \color{blue}{\left(\mathsf{neg}\left(1\right)\right) \cdot 1}\right) \]

      6. metadata-eval N/A

         \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot \left(\left(-x\right) - -1 \cdot 1\right) \]

      7. metadata-eval N/A

         \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot \left(\left(-x\right) - -1\right) \]

      8. lower--.f64 99.9

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

   10. Applied rewrites 99.9%

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

   11. Taylor expanded in x around 0

       \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot 1 \]
   12. Step-by-step derivation

   13. Applied rewrites 99.9%

       \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot 1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 34.9% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot 1 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (fmod 1.0 (sqrt 1.0)) 1.0))

C

double code(double x) {
	return fmod(1.0, sqrt(1.0)) * 1.0;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Python

def code(x):
	return math.fmod(1.0, math.sqrt(1.0)) * 1.0

Julia

function code(x)
	return Float64(rem(1.0, sqrt(1.0)) * 1.0)
end

Wolfram

code[x_] := N[(N[With[{TMP1 = 1.0, TMP2 = N[Sqrt[1.0], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * 1.0), $MachinePrecision]

Derivation

1. Initial program 8.8%

   \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

2. Taylor expanded in x around 0

   \[\leadsto \left(\color{blue}{1} \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
3. Step-by-step derivation

4. Applied rewrites 35.2%

   \[\leadsto \left(\color{blue}{1} \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]

5. Taylor expanded in x around 0

   \[\leadsto \left(1 \bmod \left(\sqrt{\color{blue}{1}}\right)\right) \cdot e^{-x} \]
6. Step-by-step derivation

7. Applied rewrites 34.8%

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

8. Taylor expanded in x around 0

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

   1. mul-1-neg N/A

      \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot \left(1 + \left(\mathsf{neg}\left(x\right)\right)\right) \]

   2. lift-neg.f64 N/A

      \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot \left(1 + \left(-x\right)\right) \]

   3. +-commutative N/A

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

   4. metadata-eval N/A

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

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

      \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot \left(\left(-x\right) - \color{blue}{\left(\mathsf{neg}\left(1\right)\right) \cdot 1}\right) \]

   6. metadata-eval N/A

      \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot \left(\left(-x\right) - -1 \cdot 1\right) \]

   7. metadata-eval N/A

      \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot \left(\left(-x\right) - -1\right) \]

   8. lower--.f64 34.9

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

10. Applied rewrites 34.9%

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

11. Taylor expanded in x around 0

    \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot 1 \]
12. Step-by-step derivation

13. Applied rewrites 34.9%

    \[\leadsto \left(1 \bmod \left(\sqrt{1}\right)\right) \cdot 1 \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2025115 
(FPCore (x)
  :name "expfmod (used to be hard to sample)"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))