expfmod (used to be hard to sample)

Average Accuracy: 7.0% → 62.0%

Time: 19.4s

Precision: binary64

Cost: 6660

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq 1.625 \cdot 10^{-219}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;e^{-x}\\ \end{array} \]

FPCore

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

↓

(FPCore (x) :precision binary64 (if (<= x 1.625e-219) 1.0 (exp (- x))))

C

double code(double x) {
	return fmod(exp(x), sqrt(cos(x))) * exp(-x);
}

↓

double code(double x) {
	double tmp;
	if (x <= 1.625e-219) {
		tmp = 1.0;
	} else {
		tmp = exp(-x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = mod(exp(x), sqrt(cos(x))) * exp(-x)
end function

↓

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.625d-219) then
        tmp = 1.0d0
    else
        tmp = exp(-x)
    end if
    code = tmp
end function

Python

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

↓

def code(x):
	tmp = 0
	if x <= 1.625e-219:
		tmp = 1.0
	else:
		tmp = math.exp(-x)
	return tmp

Julia

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

↓

function code(x)
	tmp = 0.0
	if (x <= 1.625e-219)
		tmp = 1.0;
	else
		tmp = exp(Float64(-x));
	end
	return tmp
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]

↓

code[x_] := If[LessEqual[x, 1.625e-219], 1.0, N[Exp[(-x)], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.62499999999999989e-219

   1. Initial program 8.8%

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

   2. Simplified 8.8%

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

      [Start] 8.8	\[ \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
      exp-neg [=>] 8.8	\[ \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot \color{blue}{\frac{1}{e^{x}}} \]
      associate-*r/ [=>] 8.8	\[ \color{blue}{\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot 1}{e^{x}}} \]
      *-rgt-identity [=>] 8.8	\[ \frac{\color{blue}{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}}{e^{x}} \]

   3. Applied egg-rr 8.8%

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

      [Start] 8.8	\[ \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}} \]
      add-exp-log [=>] 8.8	\[ \frac{\color{blue}{e^{\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}}}{e^{x}} \]
      div-exp [=>] 8.8	\[ \color{blue}{e^{\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x}} \]

   4. Applied egg-rr 8.8%

      \[\leadsto e^{\color{blue}{{\left(\sqrt[3]{\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x}\right)}^{3}}} \]
      Proof

      [Start] 8.8	\[ e^{\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x} \]
      add-cube-cbrt [=>] 8.8	\[ e^{\color{blue}{\left(\sqrt[3]{\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x} \cdot \sqrt[3]{\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x}\right) \cdot \sqrt[3]{\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x}}} \]
      pow3 [=>] 8.8	\[ e^{\color{blue}{{\left(\sqrt[3]{\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x}\right)}^{3}}} \]

   5. Taylor expanded in x around inf 77.8%

      \[\leadsto \color{blue}{1} \]

   if 1.62499999999999989e-219 < x

   1. Initial program 5.1%

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

   2. Simplified 5.1%

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

      [Start] 5.1	\[ \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
      exp-neg [=>] 5.1	\[ \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot \color{blue}{\frac{1}{e^{x}}} \]
      associate-*r/ [=>] 5.1	\[ \color{blue}{\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot 1}{e^{x}}} \]
      *-rgt-identity [=>] 5.1	\[ \frac{\color{blue}{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}}{e^{x}} \]

   3. Applied egg-rr 5.1%

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

      [Start] 5.1	\[ \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}} \]
      add-exp-log [=>] 5.1	\[ \frac{\color{blue}{e^{\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}}}{e^{x}} \]
      div-exp [=>] 5.1	\[ \color{blue}{e^{\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x}} \]

   4. Taylor expanded in x around inf 45.2%

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

   5. Simplified 45.2%

      \[\leadsto e^{\color{blue}{-x}} \]
      Proof

      [Start] 45.2	\[ e^{-1 \cdot x} \]
      mul-1-neg [=>] 45.2	\[ e^{\color{blue}{-x}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 62.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.625 \cdot 10^{-219}:\\ \;\;\;\;1\\ \mathbf{else}:\\ \;\;\;\;e^{-x}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	42.4%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023136 
(FPCore (x)
  :name "expfmod (used to be hard to sample)"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))