expfmod (used to be hard to sample)

Percentage Accurate: 6.6% → 61.7%

Time: 12.9s

Alternatives: 6

Speedup: 4.1×

• could not determine a ground truth

  1. x = -1.0769463556697653e+85

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

real(8) function code(x)
    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: 6.6% 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

real(8) function code(x)
    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: 61.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \leq 0.002:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(x, x \cdot 0.5, x\right)\right) \bmod \left(\mathsf{fma}\left(x, x \cdot -0.25, 1\right)\right)\right) \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, -1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(x + 1\right) \bmod 1\right)}{e^{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* (fmod (exp x) (sqrt (cos x))) (exp (- x))) 0.002)
   (*
    (fmod (fma x (* x 0.5) x) (fma x (* x -0.25) 1.0))
    (fma x (fma x 0.5 -1.0) 1.0))
   (/ (fmod (+ x 1.0) 1.0) (exp x))))

C

double code(double x) {
	double tmp;
	if ((fmod(exp(x), sqrt(cos(x))) * exp(-x)) <= 0.002) {
		tmp = fmod(fma(x, (x * 0.5), x), fma(x, (x * -0.25), 1.0)) * fma(x, fma(x, 0.5, -1.0), 1.0);
	} else {
		tmp = fmod((x + 1.0), 1.0) / exp(x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(rem(exp(x), sqrt(cos(x))) * exp(Float64(-x))) <= 0.002)
		tmp = Float64(rem(fma(x, Float64(x * 0.5), x), fma(x, Float64(x * -0.25), 1.0)) * fma(x, fma(x, 0.5, -1.0), 1.0));
	else
		tmp = Float64(rem(Float64(x + 1.0), 1.0) / exp(x));
	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], 0.002], N[(N[With[{TMP1 = N[(x * N[(x * 0.5), $MachinePrecision] + x), $MachinePrecision], TMP2 = N[(x * N[(x * -0.25), $MachinePrecision] + 1.0), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[(x * N[(x * 0.5 + -1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = N[(x + 1.0), $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[Exp[x], $MachinePrecision]), $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))) < 2e-3

   1. Initial program 5.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 5.5

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

   5. Applied rewrites 5.5%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 5.5

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x, x \cdot -0.25, 1\right)\right)\right) \cdot \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.5, -1\right)}, 1\right) \]

   8. Applied rewrites 5.5%

      \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x, x \cdot -0.25, 1\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, -1\right), 1\right)} \]

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 5.4

          \[\leadsto \left(\left(\mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.5, 1\right)}, 1\right)\right) \bmod \left(\mathsf{fma}\left(x, x \cdot -0.25, 1\right)\right)\right) \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, -1\right), 1\right) \]

   11. Applied rewrites 5.4%

       \[\leadsto \left(\color{blue}{\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), 1\right)\right)} \bmod \left(\mathsf{fma}\left(x, x \cdot -0.25, 1\right)\right)\right) \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, -1\right), 1\right) \]

   12. Taylor expanded in x around inf

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

       1. distribute-lft-in N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. rgt-mult-inverse N/A

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

       5. *-rgt-identity N/A

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

       6. unpow2 N/A

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

       7. associate-*r* N/A

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

       8. *-commutative N/A

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

       9. lower-fma.f64 N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 52.6

           \[\leadsto \left(\left(\mathsf{fma}\left(x, \color{blue}{x \cdot 0.5}, x\right)\right) \bmod \left(\mathsf{fma}\left(x, x \cdot -0.25, 1\right)\right)\right) \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, -1\right), 1\right) \]

   14. Applied rewrites 52.6%

       \[\leadsto \left(\color{blue}{\left(\mathsf{fma}\left(x, x \cdot 0.5, x\right)\right)} \bmod \left(\mathsf{fma}\left(x, x \cdot -0.25, 1\right)\right)\right) \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, -1\right), 1\right) \]

   if 2e-3 < (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x)))

   1. Initial program 4.9%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 4.9%

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

      1. lift-exp.f64 N/A

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

      2. lift-fmod.f64 N/A

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

      3. exp-neg N/A

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

      4. lift-exp.f64 N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 4.9

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

   7. Applied rewrites 4.9%

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

   8. Taylor expanded in x around 0

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

      1. lower-+.f64 98.0

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

   10. Applied rewrites 98.0%

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

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \leq 0.002:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(x, x \cdot 0.5, x\right)\right) \bmod \left(\mathsf{fma}\left(x, x \cdot -0.25, 1\right)\right)\right) \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, -1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(x + 1\right) \bmod 1\right)}{e^{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 61.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \leq 0.002:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(x, x \cdot 0.5, x\right)\right) \bmod \left(\mathsf{fma}\left(x, x \cdot -0.25, 1\right)\right)\right) \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, -1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(x + 1\right) \bmod 1\right)}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* (fmod (exp x) (sqrt (cos x))) (exp (- x))) 0.002)
   (*
    (fmod (fma x (* x 0.5) x) (fma x (* x -0.25) 1.0))
    (fma x (fma x 0.5 -1.0) 1.0))
   (/ (fmod (+ x 1.0) 1.0) (+ x 1.0))))

C

double code(double x) {
	double tmp;
	if ((fmod(exp(x), sqrt(cos(x))) * exp(-x)) <= 0.002) {
		tmp = fmod(fma(x, (x * 0.5), x), fma(x, (x * -0.25), 1.0)) * fma(x, fma(x, 0.5, -1.0), 1.0);
	} else {
		tmp = fmod((x + 1.0), 1.0) / (x + 1.0);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(rem(exp(x), sqrt(cos(x))) * exp(Float64(-x))) <= 0.002)
		tmp = Float64(rem(fma(x, Float64(x * 0.5), x), fma(x, Float64(x * -0.25), 1.0)) * fma(x, fma(x, 0.5, -1.0), 1.0));
	else
		tmp = Float64(rem(Float64(x + 1.0), 1.0) / Float64(x + 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], 0.002], N[(N[With[{TMP1 = N[(x * N[(x * 0.5), $MachinePrecision] + x), $MachinePrecision], TMP2 = N[(x * N[(x * -0.25), $MachinePrecision] + 1.0), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[(x * N[(x * 0.5 + -1.0), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = N[(x + 1.0), $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $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))) < 2e-3

   1. Initial program 5.6%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. unpow2 N/A

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

      4. associate-*l* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 5.5

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

   5. Applied rewrites 5.5%

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

   6. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. sub-neg N/A

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

      4. *-commutative N/A

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

      5. metadata-eval N/A

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

      6. lower-fma.f64 5.5

         \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x, x \cdot -0.25, 1\right)\right)\right) \cdot \mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.5, -1\right)}, 1\right) \]

   8. Applied rewrites 5.5%

      \[\leadsto \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x, x \cdot -0.25, 1\right)\right)\right) \cdot \color{blue}{\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, -1\right), 1\right)} \]

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. lower-fma.f64 N/A

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

       3. +-commutative N/A

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

       4. *-commutative N/A

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

       5. lower-fma.f64 5.4

          \[\leadsto \left(\left(\mathsf{fma}\left(x, \color{blue}{\mathsf{fma}\left(x, 0.5, 1\right)}, 1\right)\right) \bmod \left(\mathsf{fma}\left(x, x \cdot -0.25, 1\right)\right)\right) \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, -1\right), 1\right) \]

   11. Applied rewrites 5.4%

       \[\leadsto \left(\color{blue}{\left(\mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, 1\right), 1\right)\right)} \bmod \left(\mathsf{fma}\left(x, x \cdot -0.25, 1\right)\right)\right) \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, -1\right), 1\right) \]

   12. Taylor expanded in x around inf

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

       1. distribute-lft-in N/A

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

       2. unpow2 N/A

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

       3. associate-*l* N/A

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

       4. rgt-mult-inverse N/A

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

       5. *-rgt-identity N/A

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

       6. unpow2 N/A

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

       7. associate-*r* N/A

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

       8. *-commutative N/A

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

       9. lower-fma.f64 N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 52.6

           \[\leadsto \left(\left(\mathsf{fma}\left(x, \color{blue}{x \cdot 0.5}, x\right)\right) \bmod \left(\mathsf{fma}\left(x, x \cdot -0.25, 1\right)\right)\right) \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, -1\right), 1\right) \]

   14. Applied rewrites 52.6%

       \[\leadsto \left(\color{blue}{\left(\mathsf{fma}\left(x, x \cdot 0.5, x\right)\right)} \bmod \left(\mathsf{fma}\left(x, x \cdot -0.25, 1\right)\right)\right) \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, -1\right), 1\right) \]

   if 2e-3 < (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x)))

   1. Initial program 4.9%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 4.9%

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

      1. lift-exp.f64 N/A

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

      2. lift-fmod.f64 N/A

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

      3. exp-neg N/A

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

      4. lift-exp.f64 N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 4.9

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

   7. Applied rewrites 4.9%

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

   8. Taylor expanded in x around 0

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

      1. lower-+.f64 98.0

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

   10. Applied rewrites 98.0%

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

   11. Taylor expanded in x around 0

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

       1. lower-+.f64 95.4

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

   13. Applied rewrites 95.4%

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

4. Final simplification 62.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \leq 0.002:\\ \;\;\;\;\left(\left(\mathsf{fma}\left(x, x \cdot 0.5, x\right)\right) \bmod \left(\mathsf{fma}\left(x, x \cdot -0.25, 1\right)\right)\right) \cdot \mathsf{fma}\left(x, \mathsf{fma}\left(x, 0.5, -1\right), 1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(x + 1\right) \bmod 1\right)}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 25.2% accurate, 3.5× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (/ (fmod (+ x 1.0) 1.0) (+ x 1.0)))

C

double code(double x) {
	return fmod((x + 1.0), 1.0) / (x + 1.0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = mod((x + 1.0d0), 1.0d0) / (x + 1.0d0)
end function

Python

def code(x):
	return math.fmod((x + 1.0), 1.0) / (x + 1.0)

Julia

function code(x)
	return Float64(rem(Float64(x + 1.0), 1.0) / Float64(x + 1.0))
end

Wolfram

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

Derivation

1. Initial program 5.4%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 5.2%

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

   1. lift-exp.f64 N/A

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

   2. lift-fmod.f64 N/A

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

   3. exp-neg N/A

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

   4. lift-exp.f64 N/A

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

   5. un-div-inv N/A

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

   6. lower-/.f64 5.2

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

7. Applied rewrites 5.2%

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

8. Taylor expanded in x around 0

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

   1. lower-+.f64 27.4

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

10. Applied rewrites 27.4%

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

11. Taylor expanded in x around 0

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

    1. lower-+.f64 26.8

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

13. Applied rewrites 26.8%

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

14. Final simplification 26.8%

    \[\leadsto \frac{\left(\left(x + 1\right) \bmod 1\right)}{x + 1} \]
15. Add Preprocessing

Alternative 4: 24.7% accurate, 3.7× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = mod((x + 1.0d0), 1.0d0) * (1.0d0 - x)
end function

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 5.4%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 5.2%

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

   1. lift-exp.f64 N/A

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

   2. lift-fmod.f64 N/A

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

   3. exp-neg N/A

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

   4. lift-exp.f64 N/A

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

   5. un-div-inv N/A

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

   6. lower-/.f64 5.2

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

7. Applied rewrites 5.2%

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

8. Taylor expanded in x around 0

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

   1. lower-+.f64 27.4

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

10. Applied rewrites 27.4%

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

11. Taylor expanded in x around 0

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

    1. associate-*r* N/A

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

    2. distribute-lft1-in N/A

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

    3. +-commutative N/A

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

    4. *-commutative N/A

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

    5. lower-*.f64 N/A

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

    6. lower-fmod.f64 N/A

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

    7. lower-+.f64 N/A

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

    8. mul-1-neg N/A

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

    9. unsub-neg N/A

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

    10. lower--.f64 26.6

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

13. Applied rewrites 26.6%

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

14. Final simplification 26.6%

    \[\leadsto \left(\left(x + 1\right) \bmod 1\right) \cdot \left(1 - x\right) \]
15. Add Preprocessing

Alternative 5: 24.4% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = mod((x + 1.0d0), 1.0d0)
end function

Python

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

Julia

function code(x)
	return rem(Float64(x + 1.0), 1.0)
end

Wolfram

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

Derivation

1. Initial program 5.4%

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

3. Taylor expanded in x around 0

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

   1. lower-fmod.f64 N/A

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

   2. lower-exp.f64 N/A

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

   3. lower-sqrt.f64 N/A

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

   4. lower-cos.f64 4.8

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

5. Applied rewrites 4.8%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 4.8%

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

9. Taylor expanded in x around 0

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

    1. lower-+.f64 26.3

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

11. Applied rewrites 26.3%

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

12. Final simplification 26.3%

    \[\leadsto \left(\left(x + 1\right) \bmod 1\right) \]
13. Add Preprocessing

Alternative 6: 23.4% accurate, 4.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

function code(x)
	return rem(1.0, 1.0)
end

Wolfram

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

Derivation

1. Initial program 5.4%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 25.8%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 25.7%

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

9. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\left(1 \bmod 1\right)} \]
10. Step-by-step derivation

    1. lower-fmod.f64 25.7

       \[\leadsto \color{blue}{\left(1 \bmod 1\right)} \]

11. Applied rewrites 25.7%

    \[\leadsto \color{blue}{\left(1 \bmod 1\right)} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024214 
(FPCore (x)
  :name "expfmod (used to be hard to sample)"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))