expfmod (used to be hard to sample)

Average Error: 59.6 → 56.4

Time: 15.0s

Precision: binary64

Cost: 45188

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{1}{2 - \frac{\left(\left(e^{x}\right) \bmod 1\right)}{e^{x}}} \cdot \left(1 - {\left(\mathsf{expm1}\left(-x\right)\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}}\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x)
 :precision binary64
 (if (<= x -5e-310)
   (*
    (/ 1.0 (- 2.0 (/ (fmod (exp x) 1.0) (exp x))))
    (- 1.0 (pow (expm1 (- x)) 2.0)))
   (log (exp (/ (fmod (exp x) (sqrt (cos x))) (exp x))))))

C

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

↓

double code(double x) {
	double tmp;
	if (x <= -5e-310) {
		tmp = (1.0 / (2.0 - (fmod(exp(x), 1.0) / exp(x)))) * (1.0 - pow(expm1(-x), 2.0));
	} else {
		tmp = log(exp((fmod(exp(x), sqrt(cos(x))) / exp(x))));
	}
	return tmp;
}

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 <= -5e-310:
		tmp = (1.0 / (2.0 - (math.fmod(math.exp(x), 1.0) / math.exp(x)))) * (1.0 - math.pow(math.expm1(-x), 2.0))
	else:
		tmp = math.log(math.exp((math.fmod(math.exp(x), math.sqrt(math.cos(x))) / 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 <= -5e-310)
		tmp = Float64(Float64(1.0 / Float64(2.0 - Float64(rem(exp(x), 1.0) / exp(x)))) * Float64(1.0 - (expm1(Float64(-x)) ^ 2.0)));
	else
		tmp = log(exp(Float64(rem(exp(x), sqrt(cos(x))) / 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_] := If[LessEqual[x, -5e-310], N[(N[(1.0 / N[(2.0 - N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 - N[Power[N[(Exp[(-x)] - 1), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Log[N[Exp[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]], $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.999999999999985e-310

   1. Initial program 58.6

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

   2. Simplified 58.6

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

      [Start] 58.6	\[ \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
      exp-neg [=>] 58.6	\[ \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot \color{blue}{\frac{1}{e^{x}}} \]
      associate-*r/ [=>] 58.6	\[ \color{blue}{\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot 1}{e^{x}}} \]
      *-rgt-identity [=>] 58.6	\[ \frac{\color{blue}{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}}{e^{x}} \]

   3. Taylor expanded in x around 0 58.6

      \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \color{blue}{1}\right)}{e^{x}} \]

   4. Applied egg-rr 58.6

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

   5. Simplified 58.6

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

      [Start] 58.6	\[ \left(1 + \frac{\left(\left(e^{x}\right) \bmod 1\right)}{e^{x}}\right) - 1 \]
      associate--l+ [=>] 58.6	\[ \color{blue}{1 + \left(\frac{\left(\left(e^{x}\right) \bmod 1\right)}{e^{x}} - 1\right)} \]

   6. Applied egg-rr 58.6

      \[\leadsto \color{blue}{\frac{1}{2 - \frac{\left(\left(e^{x}\right) \bmod 1\right)}{e^{x}}} \cdot \left(1 - {\left(\mathsf{expm1}\left(\log \left(\left(e^{x}\right) \bmod 1\right) - x\right)\right)}^{2}\right)} \]

   7. Taylor expanded in x around inf 50.3

      \[\leadsto \frac{1}{2 - \frac{\left(\left(e^{x}\right) \bmod 1\right)}{e^{x}}} \cdot \left(1 - {\left(\mathsf{expm1}\left(\color{blue}{-1 \cdot x}\right)\right)}^{2}\right) \]

   8. Simplified 50.3

      \[\leadsto \frac{1}{2 - \frac{\left(\left(e^{x}\right) \bmod 1\right)}{e^{x}}} \cdot \left(1 - {\left(\mathsf{expm1}\left(\color{blue}{-x}\right)\right)}^{2}\right) \]
      Proof

      [Start] 50.3	\[ \frac{1}{2 - \frac{\left(\left(e^{x}\right) \bmod 1\right)}{e^{x}}} \cdot \left(1 - {\left(\mathsf{expm1}\left(-1 \cdot x\right)\right)}^{2}\right) \]
      mul-1-neg [=>] 50.3	\[ \frac{1}{2 - \frac{\left(\left(e^{x}\right) \bmod 1\right)}{e^{x}}} \cdot \left(1 - {\left(\mathsf{expm1}\left(\color{blue}{-x}\right)\right)}^{2}\right) \]

   if -4.999999999999985e-310 < x

   1. Initial program 60.2

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

   2. Simplified 60.2

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

      [Start] 60.2	\[ \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
      exp-neg [=>] 60.2	\[ \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot \color{blue}{\frac{1}{e^{x}}} \]
      associate-*r/ [=>] 60.2	\[ \color{blue}{\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot 1}{e^{x}}} \]
      *-rgt-identity [=>] 60.2	\[ \frac{\color{blue}{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}}{e^{x}} \]

   3. Applied egg-rr 60.3

      \[\leadsto \color{blue}{\log \left(e^{\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{1}{2 - \frac{\left(\left(e^{x}\right) \bmod 1\right)}{e^{x}}} \cdot \left(1 - {\left(\mathsf{expm1}\left(-x\right)\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\log \left(e^{\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}}\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	56.4
Cost	33028

\[\begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{1}{2 - \frac{\left(\left(e^{x}\right) \bmod 1\right)}{e^{x}}} \cdot \left(1 - {\left(\mathsf{expm1}\left(-x\right)\right)}^{2}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}\\ \end{array} \]

Alternative 2
Error	59.6
Cost	32256

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

Alternative 3
Error	59.8
Cost	26368

\[\frac{\left(\left(e^{x}\right) \bmod \left(\frac{1}{\frac{1}{\mathsf{fma}\left(x, x \cdot -0.25, 1\right)}}\right)\right)}{e^{x}} \]

Alternative 4
Error	59.7
Cost	19840

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

Alternative 5
Error	59.9
Cost	19712

\[1 + \left(\frac{\left(\left(e^{x}\right) \bmod 1\right)}{e^{x}} + -1\right) \]

Alternative 6
Error	59.9
Cost	19712

\[\left(1 + \frac{\left(\left(e^{x}\right) \bmod 1\right)}{e^{x}}\right) + -1 \]

Alternative 7
Error	59.9
Cost	19456

\[\frac{\left(\left(e^{x}\right) \bmod 1\right)}{e^{x}} \]

Alternative 8
Error	60.4
Cost	13440

\[1 + \left(\left(\left(e^{x}\right) \bmod 1\right) \cdot \left(1 - x\right) + -1\right) \]

Alternative 9
Error	60.4
Cost	13184

\[\left(\left(e^{x}\right) \bmod 1\right) \cdot \left(1 - x\right) \]

Alternative 10
Error	60.6
Cost	12928

\[\left(\left(e^{x}\right) \bmod 1\right) \]

Reproduce

herbie shell --seed 2023057 
(FPCore (x)
  :name "expfmod (used to be hard to sample)"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))