expfmod (used to be hard to sample)

Average Accuracy: 6.8% → 63.3%

Time: 17.4s

Precision: binary64

Cost: 122697

Math

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

↓

\[\begin{array}{l} t_0 := \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\\ t_1 := e^{-x}\\ t_2 := t_0 \cdot t_1\\ \mathbf{if}\;t_2 \leq 0 \lor \neg \left(t_2 \leq 2\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt[3]{\frac{\log \left(e^{t_0}\right)}{e^{x}}}\right)}^{3}\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fmod (exp x) (sqrt (cos x))))
        (t_1 (exp (- x)))
        (t_2 (* t_0 t_1)))
   (if (or (<= t_2 0.0) (not (<= t_2 2.0)))
     t_1
     (pow (cbrt (/ (log (exp t_0)) (exp x))) 3.0))))

C

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

↓

double code(double x) {
	double t_0 = fmod(exp(x), sqrt(cos(x)));
	double t_1 = exp(-x);
	double t_2 = t_0 * t_1;
	double tmp;
	if ((t_2 <= 0.0) || !(t_2 <= 2.0)) {
		tmp = t_1;
	} else {
		tmp = pow(cbrt((log(exp(t_0)) / exp(x))), 3.0);
	}
	return tmp;
}

Julia

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

↓

function code(x)
	t_0 = rem(exp(x), sqrt(cos(x)))
	t_1 = exp(Float64(-x))
	t_2 = Float64(t_0 * t_1)
	tmp = 0.0
	if ((t_2 <= 0.0) || !(t_2 <= 2.0))
		tmp = t_1;
	else
		tmp = cbrt(Float64(log(exp(t_0)) / exp(x))) ^ 3.0;
	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[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]}, Block[{t$95$1 = N[Exp[(-x)], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 * t$95$1), $MachinePrecision]}, If[Or[LessEqual[t$95$2, 0.0], N[Not[LessEqual[t$95$2, 2.0]], $MachinePrecision]], t$95$1, N[Power[N[Power[N[(N[Log[N[Exp[t$95$0], $MachinePrecision]], $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 3.0], $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))) < 0.0 or 2 < (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x)))

   1. Initial program 3.4%

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

   2. Simplified 3.4%

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

      [Start] 3.4	\[ \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
      exp-neg [=>] 3.4	\[ \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot \color{blue}{\frac{1}{e^{x}}} \]
      associate-*r/ [=>] 3.4	\[ \color{blue}{\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot 1}{e^{x}}} \]
      *-rgt-identity [=>] 3.4	\[ \frac{\color{blue}{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}}{e^{x}} \]

   3. Applied egg-rr 3.4%

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

      [Start] 3.4	\[ \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}} \]
      add-exp-log [=>] 3.4	\[ \frac{\color{blue}{e^{\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}}}{e^{x}} \]
      div-exp [=>] 3.4	\[ \color{blue}{e^{\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x}} \]

   4. Taylor expanded in x around inf 62.7%

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

   5. Simplified 62.7%

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

      [Start] 62.7	\[ e^{-1 \cdot x} \]
      neg-mul-1 [<=] 62.7	\[ e^{\color{blue}{-x}} \]

   if 0.0 < (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x))) < 2

   1. Initial program 79.7%

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

   2. Simplified 79.9%

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

      [Start] 79.7	\[ \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
      exp-neg [=>] 79.8	\[ \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot \color{blue}{\frac{1}{e^{x}}} \]
      associate-*r/ [=>] 79.9	\[ \color{blue}{\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot 1}{e^{x}}} \]
      *-rgt-identity [=>] 79.9	\[ \frac{\color{blue}{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}}{e^{x}} \]

   3. Applied egg-rr 79.9%

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

      [Start] 79.9	\[ \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}} \]
      add-cube-cbrt [=>] 79.9	\[ \color{blue}{\left(\sqrt[3]{\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}} \cdot \sqrt[3]{\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}}\right) \cdot \sqrt[3]{\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}}} \]
      pow3 [=>] 79.9	\[ \color{blue}{{\left(\sqrt[3]{\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}}\right)}^{3}} \]

   4. Applied egg-rr 77.4%

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

      [Start] 79.9	\[ {\left(\sqrt[3]{\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}}\right)}^{3} \]
      add-log-exp [=>] 77.4	\[ {\left(\sqrt[3]{\frac{\color{blue}{\log \left(e^{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}\right)}}{e^{x}}}\right)}^{3} \]

3. Recombined 2 regimes into one program.

4. Final simplification 63.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \leq 0 \lor \neg \left(\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \leq 2\right):\\ \;\;\;\;e^{-x}\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt[3]{\frac{\log \left(e^{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}\right)}{e^{x}}}\right)}^{3}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	63.4%
Cost	110153

\[\begin{array}{l} t_0 := \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\\ t_1 := e^{-x}\\ t_2 := t_0 \cdot t_1\\ \mathbf{if}\;t_2 \leq 0 \lor \neg \left(t_2 \leq 2\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;{\left(\left(\sqrt[3]{\frac{t_0}{e^{x}}} + 1\right) + -1\right)}^{3}\\ \end{array} \]

Alternative 2
Accuracy	63.4%
Cost	109897

\[\begin{array}{l} t_0 := \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\\ t_1 := e^{-x}\\ t_2 := t_0 \cdot t_1\\ \mathbf{if}\;t_2 \leq 0 \lor \neg \left(t_2 \leq 2\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;{\left(\sqrt[3]{\frac{t_0}{e^{x}}}\right)}^{3}\\ \end{array} \]

Alternative 3
Accuracy	63.3%
Cost	109833

\[\begin{array}{l} t_0 := \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\\ t_1 := e^{-x}\\ t_2 := t_0 \cdot t_1\\ \mathbf{if}\;t_2 \leq 0 \lor \neg \left(t_2 \leq 2\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\log \left(e^{t_0}\right)}{e^{x}}\\ \end{array} \]

Alternative 4
Accuracy	63.4%
Cost	97289

\[\begin{array}{l} t_0 := \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\\ t_1 := e^{-x}\\ t_2 := t_0 \cdot t_1\\ \mathbf{if}\;t_2 \leq 0 \lor \neg \left(t_2 \leq 2\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;-1 + \left(\frac{t_0}{e^{x}} + 1\right)\\ \end{array} \]

Alternative 5
Accuracy	63.4%
Cost	97033

\[\begin{array}{l} t_0 := \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\\ t_1 := e^{-x}\\ t_2 := t_0 \cdot t_1\\ \mathbf{if}\;t_2 \leq 0 \lor \neg \left(t_2 \leq 2\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{e^{x}}\\ \end{array} \]

Alternative 6
Accuracy	61.4%
Cost	6528

\[e^{-x} \]

Reproduce

herbie shell --seed 2023151 
(FPCore (x)
  :name "expfmod (used to be hard to sample)"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))