expfmod (used to be hard to sample)

Average Accuracy: 7.1% → 63.0%

Time: 20.8s

Precision: binary64

Cost: 84420

Math

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

↓

\[\begin{array}{l} t_0 := \log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\\ \mathbf{if}\;x \leq 0.002:\\ \;\;\;\;{\left(e^{{t_0}^{2} - x \cdot x}\right)}^{\left(\frac{1}{x + 0.3333333333333333 \cdot \left(t_0 \cdot 3\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{-x}\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (let* ((t_0 (log (fmod (exp x) (sqrt (cos x))))))
   (if (<= x 0.002)
     (pow
      (exp (- (pow t_0 2.0) (* x x)))
      (/ 1.0 (+ x (* 0.3333333333333333 (* t_0 3.0)))))
     (exp (- x)))))

C

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

↓

double code(double x) {
	double t_0 = log(fmod(exp(x), sqrt(cos(x))));
	double tmp;
	if (x <= 0.002) {
		tmp = pow(exp((pow(t_0, 2.0) - (x * x))), (1.0 / (x + (0.3333333333333333 * (t_0 * 3.0)))));
	} 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) :: t_0
    real(8) :: tmp
    t_0 = log(mod(exp(x), sqrt(cos(x))))
    if (x <= 0.002d0) then
        tmp = exp(((t_0 ** 2.0d0) - (x * x))) ** (1.0d0 / (x + (0.3333333333333333d0 * (t_0 * 3.0d0))))
    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):
	t_0 = math.log(math.fmod(math.exp(x), math.sqrt(math.cos(x))))
	tmp = 0
	if x <= 0.002:
		tmp = math.pow(math.exp((math.pow(t_0, 2.0) - (x * x))), (1.0 / (x + (0.3333333333333333 * (t_0 * 3.0)))))
	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)
	t_0 = log(rem(exp(x), sqrt(cos(x))))
	tmp = 0.0
	if (x <= 0.002)
		tmp = exp(Float64((t_0 ^ 2.0) - Float64(x * x))) ^ Float64(1.0 / Float64(x + Float64(0.3333333333333333 * Float64(t_0 * 3.0))));
	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_] := Block[{t$95$0 = N[Log[N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]], $MachinePrecision]}, If[LessEqual[x, 0.002], N[Power[N[Exp[N[(N[Power[t$95$0, 2.0], $MachinePrecision] - N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[(1.0 / N[(x + N[(0.3333333333333333 * N[(t$95$0 * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], N[Exp[(-x)], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 2e-3

   1. Initial program 5.8%

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

   2. Simplified 5.8%

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

      [Start] 5.8	\[ \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
      exp-neg [=>] 5.8	\[ \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot \color{blue}{\frac{1}{e^{x}}} \]
      associate-*r/ [=>] 5.8	\[ \color{blue}{\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot 1}{e^{x}}} \]
      *-rgt-identity [=>] 5.8	\[ \frac{\color{blue}{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}}{e^{x}} \]

   3. Applied egg-rr 5.9%

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

      [Start] 5.8	\[ \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}} \]
      add-exp-log [=>] 5.8	\[ \frac{\color{blue}{e^{\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}}}{e^{x}} \]
      div-exp [=>] 5.9	\[ \color{blue}{e^{\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x}} \]

   4. Applied egg-rr 52.4%

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

      [Start] 5.9	\[ e^{\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x} \]
      flip-- [=>] 1.7	\[ e^{\color{blue}{\frac{\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot \log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x \cdot x}{\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) + x}}} \]
      div-inv [=>] 1.7	\[ e^{\color{blue}{\left(\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot \log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x \cdot x\right) \cdot \frac{1}{\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) + x}}} \]
      add-log-exp [=>] 1.6	\[ e^{\color{blue}{\log \left(e^{\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot \log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x \cdot x}\right)} \cdot \frac{1}{\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) + x}} \]
      exp-to-pow [=>] 52.4	\[ \color{blue}{{\left(e^{\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot \log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x \cdot x}\right)}^{\left(\frac{1}{\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) + x}\right)}} \]
      pow2 [=>] 52.4	\[ {\left(e^{\color{blue}{{\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}^{2}} - x \cdot x}\right)}^{\left(\frac{1}{\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) + x}\right)} \]
      +-commutative [=>] 52.4	\[ {\left(e^{{\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}^{2} - x \cdot x}\right)}^{\left(\frac{1}{\color{blue}{x + \log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}}\right)} \]

   5. Applied egg-rr 52.4%

      \[\leadsto {\left(e^{{\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}^{2} - x \cdot x}\right)}^{\left(\frac{1}{x + \color{blue}{0.3333333333333333 \cdot \left(3 \cdot \log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right)}}\right)} \]
      Step-by-step derivation

      [Start] 52.4	\[ {\left(e^{{\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}^{2} - x \cdot x}\right)}^{\left(\frac{1}{x + \log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}\right)} \]
      add-cbrt-cube [=>] 52.4	\[ {\left(e^{{\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}^{2} - x \cdot x}\right)}^{\left(\frac{1}{x + \log \color{blue}{\left(\sqrt[3]{\left(\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right) \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}\right)}}\right)} \]
      pow1/3 [=>] 52.4	\[ {\left(e^{{\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}^{2} - x \cdot x}\right)}^{\left(\frac{1}{x + \log \color{blue}{\left({\left(\left(\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right) \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right)}^{0.3333333333333333}\right)}}\right)} \]
      log-pow [=>] 52.4	\[ {\left(e^{{\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}^{2} - x \cdot x}\right)}^{\left(\frac{1}{x + \color{blue}{0.3333333333333333 \cdot \log \left(\left(\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right) \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right)}}\right)} \]
      pow3 [=>] 52.4	\[ {\left(e^{{\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}^{2} - x \cdot x}\right)}^{\left(\frac{1}{x + 0.3333333333333333 \cdot \log \color{blue}{\left({\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}^{3}\right)}}\right)} \]
      log-pow [=>] 52.4	\[ {\left(e^{{\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}^{2} - x \cdot x}\right)}^{\left(\frac{1}{x + 0.3333333333333333 \cdot \color{blue}{\left(3 \cdot \log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right)}}\right)} \]

   if 2e-3 < x

   1. Initial program 0.0%

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

   2. Simplified 0.0%

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

      [Start] 0.0	\[ \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
      exp-neg [=>] 0.0	\[ \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot \color{blue}{\frac{1}{e^{x}}} \]
      associate-*r/ [=>] 0.0	\[ \color{blue}{\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot 1}{e^{x}}} \]
      *-rgt-identity [=>] 0.0	\[ \frac{\color{blue}{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}}{e^{x}} \]

   3. Applied egg-rr 0.0%

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

      [Start] 0.0	\[ \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}} \]
      add-exp-log [=>] 0.0	\[ \frac{\color{blue}{e^{\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}}}{e^{x}} \]
      div-exp [=>] 0.0	\[ \color{blue}{e^{\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x}} \]

   4. Taylor expanded in x around inf 100.0%

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

   5. Simplified 100.0%

      \[\leadsto e^{\color{blue}{-x}} \]
      Step-by-step derivation

      [Start] 100.0	\[ e^{-1 \cdot x} \]
      neg-mul-1 [<=] 100.0	\[ e^{\color{blue}{-x}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.002:\\ \;\;\;\;{\left(e^{{\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}^{2} - x \cdot x}\right)}^{\left(\frac{1}{x + 0.3333333333333333 \cdot \left(\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot 3\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{-x}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	63.0%
Cost	84164

\[\begin{array}{l} t_0 := \log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\\ \mathbf{if}\;x \leq 0.002:\\ \;\;\;\;{\left(e^{{t_0}^{2} - x \cdot x}\right)}^{\left(\frac{1}{x + t_0}\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{-x}\\ \end{array} \]

Alternative 2
Accuracy	61.0%
Cost	6528

\[e^{-x} \]

Alternative 3
Accuracy	42.8%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023157 
(FPCore (x)
  :name "expfmod (used to be hard to sample)"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))