expfmod (used to be hard to sample)

Average Accuracy: 7.1% → 63.0%

Time: 16.3s

Precision: binary64

Cost: 97028

Math

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

↓

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

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 (cbrt (- (pow t_0 2.0) (* x x))) 3.0)) (/ 1.0 (+ x t_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(cbrt((pow(t_0, 2.0) - (x * x))), 3.0)), (1.0 / (x + t_0)));
	} else {
		tmp = 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((cbrt(Float64((t_0 ^ 2.0) - Float64(x * x))) ^ 3.0)) ^ Float64(1.0 / Float64(x + t_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[Power[N[Power[N[(N[Power[t$95$0, 2.0], $MachinePrecision] - N[(x * x), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision]], $MachinePrecision], N[(1.0 / N[(x + t$95$0), $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%

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

      [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%

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

      [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%

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

      [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%

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

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

   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%

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

      [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%

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

      [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%

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

      [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%

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

Alternatives

Alternative 1
Accuracy	63.0%
Cost	96964

\[\begin{array}{l} t_0 := \log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\\ \mathbf{if}\;x \leq 0.002:\\ \;\;\;\;{\left(e^{\mathsf{expm1}\left(\mathsf{log1p}\left({t_0}^{2} - x \cdot x\right)\right)}\right)}^{\left(\frac{1}{x + t_0}\right)}\\ \mathbf{else}:\\ \;\;\;\;e^{-x}\\ \end{array} \]

Alternative 2
Accuracy	63.0%
Cost	84420

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

Alternative 3
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 4
Accuracy	61.0%
Cost	6528

\[e^{-x} \]

Alternative 5
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))))