expfmod (used to be hard to sample)

Average Error: 59.5 → 23.6

Time: 15.3s

Precision: binary64

Cost: 103689

Math

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

↓

\[\begin{array}{l} t_0 := e^{-x}\\ t_1 := \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot t_0\\ \mathbf{if}\;t_1 \leq 0 \lor \neg \left(t_1 \leq 2\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\frac{\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x, x \cdot -0.25, 1\right)\right)\right)}{e^{x}}}\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (- x))) (t_1 (* (fmod (exp x) (sqrt (cos x))) t_0)))
   (if (or (<= t_1 0.0) (not (<= t_1 2.0)))
     t_0
     (log (exp (/ (fmod (exp x) (fma x (* x -0.25) 1.0)) (exp x)))))))

C

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

↓

double code(double x) {
	double t_0 = exp(-x);
	double t_1 = fmod(exp(x), sqrt(cos(x))) * t_0;
	double tmp;
	if ((t_1 <= 0.0) || !(t_1 <= 2.0)) {
		tmp = t_0;
	} else {
		tmp = log(exp((fmod(exp(x), fma(x, (x * -0.25), 1.0)) / 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 = exp(Float64(-x))
	t_1 = Float64(rem(exp(x), sqrt(cos(x))) * t_0)
	tmp = 0.0
	if ((t_1 <= 0.0) || !(t_1 <= 2.0))
		tmp = t_0;
	else
		tmp = log(exp(Float64(rem(exp(x), fma(x, Float64(x * -0.25), 1.0)) / exp(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[Exp[(-x)], $MachinePrecision]}, Block[{t$95$1 = N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * t$95$0), $MachinePrecision]}, If[Or[LessEqual[t$95$1, 0.0], N[Not[LessEqual[t$95$1, 2.0]], $MachinePrecision]], t$95$0, N[Log[N[Exp[N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(x * N[(x * -0.25), $MachinePrecision] + 1.0), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision]], $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))) < 0.0 or 2 < (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x)))

   1. Initial program 61.8

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

   2. Simplified 61.8

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

      [Start] 61.8	\[ \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
      exp-neg [=>] 61.8	\[ \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot \color{blue}{\frac{1}{e^{x}}} \]
      associate-*r/ [=>] 61.8	\[ \color{blue}{\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot 1}{e^{x}}} \]
      *-rgt-identity [=>] 61.8	\[ \frac{\color{blue}{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}}{e^{x}} \]

   3. Taylor expanded in x around 0 61.8

      \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \color{blue}{\left(1 + -0.25 \cdot {x}^{2}\right)}\right)}{e^{x}} \]

   4. Simplified 61.8

      \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \color{blue}{\left(1 + \left(x \cdot x\right) \cdot -0.25\right)}\right)}{e^{x}} \]
      Proof

      [Start] 61.8	\[ \frac{\left(\left(e^{x}\right) \bmod \left(1 + -0.25 \cdot {x}^{2}\right)\right)}{e^{x}} \]
      *-commutative [=>] 61.8	\[ \frac{\left(\left(e^{x}\right) \bmod \left(1 + \color{blue}{{x}^{2} \cdot -0.25}\right)\right)}{e^{x}} \]
      unpow2 [=>] 61.8	\[ \frac{\left(\left(e^{x}\right) \bmod \left(1 + \color{blue}{\left(x \cdot x\right)} \cdot -0.25\right)\right)}{e^{x}} \]

   5. Taylor expanded in x around inf 62.6

      \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \color{blue}{\left(-0.25 \cdot {x}^{2}\right)}\right)}{e^{x}} \]

   6. Simplified 62.6

      \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \color{blue}{\left(x \cdot \left(x \cdot -0.25\right)\right)}\right)}{e^{x}} \]
      Proof

      [Start] 62.6	\[ \frac{\left(\left(e^{x}\right) \bmod \left(-0.25 \cdot {x}^{2}\right)\right)}{e^{x}} \]
      *-commutative [=>] 62.6	\[ \frac{\left(\left(e^{x}\right) \bmod \color{blue}{\left({x}^{2} \cdot -0.25\right)}\right)}{e^{x}} \]
      unpow2 [=>] 62.6	\[ \frac{\left(\left(e^{x}\right) \bmod \left(\color{blue}{\left(x \cdot x\right)} \cdot -0.25\right)\right)}{e^{x}} \]
      associate-*r* [<=] 62.6	\[ \frac{\left(\left(e^{x}\right) \bmod \color{blue}{\left(x \cdot \left(x \cdot -0.25\right)\right)}\right)}{e^{x}} \]

   7. Applied egg-rr 62.6

      \[\leadsto \color{blue}{e^{\log \left(\left(e^{x}\right) \bmod \left(x \cdot \left(x \cdot -0.25\right)\right)\right) - x}} \]

   8. Taylor expanded in x around inf 23.9

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

   9. Simplified 23.9

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

      [Start] 23.9	\[ e^{-1 \cdot x} \]
      mul-1-neg [=>] 23.9	\[ e^{\color{blue}{-x}} \]

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

   1. Initial program 13.3

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

   2. Simplified 13.1

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

      [Start] 13.3	\[ \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
      exp-neg [=>] 13.2	\[ \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot \color{blue}{\frac{1}{e^{x}}} \]
      associate-*r/ [=>] 13.1	\[ \color{blue}{\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot 1}{e^{x}}} \]
      *-rgt-identity [=>] 13.1	\[ \frac{\color{blue}{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}}{e^{x}} \]

   3. Taylor expanded in x around 0 15.7

      \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \color{blue}{\left(1 + -0.25 \cdot {x}^{2}\right)}\right)}{e^{x}} \]

   4. Simplified 15.7

      \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \color{blue}{\left(1 + \left(x \cdot x\right) \cdot -0.25\right)}\right)}{e^{x}} \]
      Proof

      [Start] 15.7	\[ \frac{\left(\left(e^{x}\right) \bmod \left(1 + -0.25 \cdot {x}^{2}\right)\right)}{e^{x}} \]
      *-commutative [=>] 15.7	\[ \frac{\left(\left(e^{x}\right) \bmod \left(1 + \color{blue}{{x}^{2} \cdot -0.25}\right)\right)}{e^{x}} \]
      unpow2 [=>] 15.7	\[ \frac{\left(\left(e^{x}\right) \bmod \left(1 + \color{blue}{\left(x \cdot x\right)} \cdot -0.25\right)\right)}{e^{x}} \]

   5. Applied egg-rr 16.0

      \[\leadsto \color{blue}{\log \left(e^{\frac{\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x, x \cdot -0.25, 1\right)\right)\right)}{e^{x}}}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 23.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \leq 0 \lor \neg \left(\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \leq 2\right):\\ \;\;\;\;e^{-x}\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\frac{\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x, x \cdot -0.25, 1\right)\right)\right)}{e^{x}}}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	23.6
Cost	84617

\[\begin{array}{l} t_0 := e^{-x}\\ t_1 := \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot t_0\\ \mathbf{if}\;t_1 \leq 0 \lor \neg \left(t_1 \leq 2\right):\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(e^{x}\right) \bmod \left(1 + -0.25 \cdot \left(x \cdot x\right)\right)\right)}{e^{x}}\\ \end{array} \]

Alternative 2
Error	24.1
Cost	19588

\[\begin{array}{l} \mathbf{if}\;x \leq -2 \cdot 10^{-16}:\\ \;\;\;\;\frac{\left(\left(e^{x}\right) \bmod 1\right)}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;e^{-x}\\ \end{array} \]

Alternative 3
Error	24.8
Cost	6528

\[e^{-x} \]

Reproduce

herbie shell --seed 2023018 
(FPCore (x)
  :name "expfmod (used to be hard to sample)"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))