expfmod (used to be hard to sample)

Average Accuracy: 6.7% → 12.5%

Time: 19.5s

Precision: binary64

Cost: 155204

Math

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

↓

\[\begin{array}{l} t_0 := \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\\ t_1 := \frac{t_0}{e^{x}}\\ t_2 := 1 - \mathsf{expm1}\left(\log \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x, x \cdot -0.25, 1\right)\right)\right) - x\right)\\ \mathbf{if}\;t_0 \cdot e^{-x} \leq 0:\\ \;\;\;\;\frac{1}{t_2} - \frac{{\left(\mathsf{expm1}\left(-x\right)\right)}^{2}}{t_2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1 + {t_1}^{3}, \frac{1}{\mathsf{fma}\left(t_1, \mathsf{expm1}\left(\log t_0 - x\right), 1\right)}, -1\right)\\ \end{array} \]

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 (/ t_0 (exp x)))
        (t_2
         (- 1.0 (expm1 (- (log (fmod (exp x) (fma x (* x -0.25) 1.0))) x)))))
   (if (<= (* t_0 (exp (- x))) 0.0)
     (- (/ 1.0 t_2) (/ (pow (expm1 (- x)) 2.0) t_2))
     (fma
      (+ 1.0 (pow t_1 3.0))
      (/ 1.0 (fma t_1 (expm1 (- (log t_0) x)) 1.0))
      -1.0))))

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 = t_0 / exp(x);
	double t_2 = 1.0 - expm1((log(fmod(exp(x), fma(x, (x * -0.25), 1.0))) - x));
	double tmp;
	if ((t_0 * exp(-x)) <= 0.0) {
		tmp = (1.0 / t_2) - (pow(expm1(-x), 2.0) / t_2);
	} else {
		tmp = fma((1.0 + pow(t_1, 3.0)), (1.0 / fma(t_1, expm1((log(t_0) - x)), 1.0)), -1.0);
	}
	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 = Float64(t_0 / exp(x))
	t_2 = Float64(1.0 - expm1(Float64(log(rem(exp(x), fma(x, Float64(x * -0.25), 1.0))) - x)))
	tmp = 0.0
	if (Float64(t_0 * exp(Float64(-x))) <= 0.0)
		tmp = Float64(Float64(1.0 / t_2) - Float64((expm1(Float64(-x)) ^ 2.0) / t_2));
	else
		tmp = fma(Float64(1.0 + (t_1 ^ 3.0)), Float64(1.0 / fma(t_1, expm1(Float64(log(t_0) - x)), 1.0)), -1.0);
	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[(t$95$0 / N[Exp[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 - N[(Exp[N[(N[Log[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]], $MachinePrecision] - x), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision], 0.0], N[(N[(1.0 / t$95$2), $MachinePrecision] - N[(N[Power[N[(Exp[(-x)] - 1), $MachinePrecision], 2.0], $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 + N[Power[t$95$1, 3.0], $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(t$95$1 * N[(Exp[N[(N[Log[t$95$0], $MachinePrecision] - x), $MachinePrecision]] - 1), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision] + -1.0), $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

   1. Initial program 4.3%

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

   2. Simplified 4.3%

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

      [Start] 4.3	\[ \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
      exp-neg [=>] 4.3	\[ \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot \color{blue}{\frac{1}{e^{x}}} \]
      associate-*r/ [=>] 4.3	\[ \color{blue}{\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot 1}{e^{x}}} \]
      *-rgt-identity [=>] 4.3	\[ \frac{\color{blue}{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}}{e^{x}} \]

   3. Applied egg-rr 4.3%

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

      [Start] 4.3	\[ \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}} \]
      expm1-log1p-u [=>] 4.3	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}\right)\right)} \]
      expm1-udef [=>] 4.3	\[ \color{blue}{e^{\mathsf{log1p}\left(\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}\right)} - 1} \]
      log1p-udef [=>] 4.3	\[ e^{\color{blue}{\log \left(1 + \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}\right)}} - 1 \]
      add-exp-log [<=] 4.3	\[ \color{blue}{\left(1 + \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}\right)} - 1 \]

   4. Simplified 4.3%

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

      [Start] 4.3	\[ \left(1 + \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}\right) - 1 \]
      associate--l+ [=>] 4.3	\[ \color{blue}{1 + \left(\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}} - 1\right)} \]

   5. Taylor expanded in x around 0 4.3%

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

   6. Simplified 4.3%

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

      [Start] 4.3	\[ 1 + \left(\frac{\left(\left(e^{x}\right) \bmod \left(1 + -0.25 \cdot {x}^{2}\right)\right)}{e^{x}} - 1\right) \]
      *-commutative [=>] 4.3	\[ 1 + \left(\frac{\left(\left(e^{x}\right) \bmod \left(1 + \color{blue}{{x}^{2} \cdot -0.25}\right)\right)}{e^{x}} - 1\right) \]
      unpow2 [=>] 4.3	\[ 1 + \left(\frac{\left(\left(e^{x}\right) \bmod \left(1 + \color{blue}{\left(x \cdot x\right)} \cdot -0.25\right)\right)}{e^{x}} - 1\right) \]

   7. Applied egg-rr 4.3%

      \[\leadsto \color{blue}{\frac{1}{1 - \mathsf{expm1}\left(\log \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x, x \cdot -0.25, 1\right)\right)\right) - x\right)} - \frac{{\left(\mathsf{expm1}\left(\log \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x, x \cdot -0.25, 1\right)\right)\right) - x\right)\right)}^{2}}{1 - \mathsf{expm1}\left(\log \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x, x \cdot -0.25, 1\right)\right)\right) - x\right)}} \]
      Proof

      [Start] 4.3	\[ 1 + \left(\frac{\left(\left(e^{x}\right) \bmod \left(1 + \left(x \cdot x\right) \cdot -0.25\right)\right)}{e^{x}} - 1\right) \]
      flip-+ [=>] 4.3	\[ \color{blue}{\frac{1 \cdot 1 - \left(\frac{\left(\left(e^{x}\right) \bmod \left(1 + \left(x \cdot x\right) \cdot -0.25\right)\right)}{e^{x}} - 1\right) \cdot \left(\frac{\left(\left(e^{x}\right) \bmod \left(1 + \left(x \cdot x\right) \cdot -0.25\right)\right)}{e^{x}} - 1\right)}{1 - \left(\frac{\left(\left(e^{x}\right) \bmod \left(1 + \left(x \cdot x\right) \cdot -0.25\right)\right)}{e^{x}} - 1\right)}} \]
      metadata-eval [=>] 4.3	\[ \frac{\color{blue}{1} - \left(\frac{\left(\left(e^{x}\right) \bmod \left(1 + \left(x \cdot x\right) \cdot -0.25\right)\right)}{e^{x}} - 1\right) \cdot \left(\frac{\left(\left(e^{x}\right) \bmod \left(1 + \left(x \cdot x\right) \cdot -0.25\right)\right)}{e^{x}} - 1\right)}{1 - \left(\frac{\left(\left(e^{x}\right) \bmod \left(1 + \left(x \cdot x\right) \cdot -0.25\right)\right)}{e^{x}} - 1\right)} \]
      div-sub [=>] 4.3	\[ \color{blue}{\frac{1}{1 - \left(\frac{\left(\left(e^{x}\right) \bmod \left(1 + \left(x \cdot x\right) \cdot -0.25\right)\right)}{e^{x}} - 1\right)} - \frac{\left(\frac{\left(\left(e^{x}\right) \bmod \left(1 + \left(x \cdot x\right) \cdot -0.25\right)\right)}{e^{x}} - 1\right) \cdot \left(\frac{\left(\left(e^{x}\right) \bmod \left(1 + \left(x \cdot x\right) \cdot -0.25\right)\right)}{e^{x}} - 1\right)}{1 - \left(\frac{\left(\left(e^{x}\right) \bmod \left(1 + \left(x \cdot x\right) \cdot -0.25\right)\right)}{e^{x}} - 1\right)}} \]

   8. Taylor expanded in x around inf 12.0%

      \[\leadsto \frac{1}{1 - \mathsf{expm1}\left(\log \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x, x \cdot -0.25, 1\right)\right)\right) - x\right)} - \frac{{\left(\mathsf{expm1}\left(\color{blue}{-1 \cdot x}\right)\right)}^{2}}{1 - \mathsf{expm1}\left(\log \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x, x \cdot -0.25, 1\right)\right)\right) - x\right)} \]

   9. Simplified 12.0%

      \[\leadsto \frac{1}{1 - \mathsf{expm1}\left(\log \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x, x \cdot -0.25, 1\right)\right)\right) - x\right)} - \frac{{\left(\mathsf{expm1}\left(\color{blue}{-x}\right)\right)}^{2}}{1 - \mathsf{expm1}\left(\log \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x, x \cdot -0.25, 1\right)\right)\right) - x\right)} \]
      Proof

      [Start] 12.0	\[ \frac{1}{1 - \mathsf{expm1}\left(\log \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x, x \cdot -0.25, 1\right)\right)\right) - x\right)} - \frac{{\left(\mathsf{expm1}\left(-1 \cdot x\right)\right)}^{2}}{1 - \mathsf{expm1}\left(\log \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x, x \cdot -0.25, 1\right)\right)\right) - x\right)} \]
      neg-mul-1 [<=] 12.0	\[ \frac{1}{1 - \mathsf{expm1}\left(\log \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x, x \cdot -0.25, 1\right)\right)\right) - x\right)} - \frac{{\left(\mathsf{expm1}\left(\color{blue}{-x}\right)\right)}^{2}}{1 - \mathsf{expm1}\left(\log \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x, x \cdot -0.25, 1\right)\right)\right) - x\right)} \]

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

   1. Initial program 14.0%

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

   2. Simplified 14.1%

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

      [Start] 14.0	\[ \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
      exp-neg [=>] 14.1	\[ \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot \color{blue}{\frac{1}{e^{x}}} \]
      associate-*r/ [=>] 14.1	\[ \color{blue}{\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot 1}{e^{x}}} \]
      *-rgt-identity [=>] 14.1	\[ \frac{\color{blue}{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}}{e^{x}} \]

   3. Applied egg-rr 14.0%

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

      [Start] 14.1	\[ \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}} \]
      expm1-log1p-u [=>] 14.1	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}\right)\right)} \]
      expm1-udef [=>] 14.0	\[ \color{blue}{e^{\mathsf{log1p}\left(\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}\right)} - 1} \]
      log1p-udef [=>] 14.0	\[ e^{\color{blue}{\log \left(1 + \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}\right)}} - 1 \]
      add-exp-log [<=] 14.0	\[ \color{blue}{\left(1 + \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}\right)} - 1 \]

   4. Applied egg-rr 14.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(1 + {\left(\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}\right)}^{3}, \frac{1}{\mathsf{fma}\left(\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}, \mathsf{expm1}\left(\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x\right), 1\right)}, -1\right)} \]
      Proof

      [Start] 14.0	\[ \left(1 + \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}\right) - 1 \]
      flip3-+ [=>] 14.0	\[ \color{blue}{\frac{{1}^{3} + {\left(\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}\right)}^{3}}{1 \cdot 1 + \left(\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}} \cdot \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}} - 1 \cdot \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}\right)}} - 1 \]
      div-inv [=>] 14.0	\[ \color{blue}{\left({1}^{3} + {\left(\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}\right)}^{3}\right) \cdot \frac{1}{1 \cdot 1 + \left(\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}} \cdot \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}} - 1 \cdot \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}\right)}} - 1 \]
      fma-neg [=>] 14.0	\[ \color{blue}{\mathsf{fma}\left({1}^{3} + {\left(\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}\right)}^{3}, \frac{1}{1 \cdot 1 + \left(\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}} \cdot \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}} - 1 \cdot \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}\right)}, -1\right)} \]

3. Recombined 2 regimes into one program.

4. Final simplification 12.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \leq 0:\\ \;\;\;\;\frac{1}{1 - \mathsf{expm1}\left(\log \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x, x \cdot -0.25, 1\right)\right)\right) - x\right)} - \frac{{\left(\mathsf{expm1}\left(-x\right)\right)}^{2}}{1 - \mathsf{expm1}\left(\log \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x, x \cdot -0.25, 1\right)\right)\right) - x\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(1 + {\left(\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}\right)}^{3}, \frac{1}{\mathsf{fma}\left(\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}, \mathsf{expm1}\left(\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x\right), 1\right)}, -1\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	12.5%
Cost	110916

\[\begin{array}{l} t_0 := \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\\ t_1 := 1 - \mathsf{expm1}\left(\log \left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x, x \cdot -0.25, 1\right)\right)\right) - x\right)\\ \mathbf{if}\;t_0 \cdot e^{-x} \leq 0:\\ \;\;\;\;\frac{1}{t_1} - \frac{{\left(\mathsf{expm1}\left(-x\right)\right)}^{2}}{t_1}\\ \mathbf{else}:\\ \;\;\;\;\left(1 + \left(1 + \left(\frac{t_0}{e^{x}} + -1\right)\right)\right) + -1\\ \end{array} \]

Alternative 2
Accuracy	6.7%
Cost	32768

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

Alternative 3
Accuracy	6.5%
Cost	20096

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

Alternative 4
Accuracy	6.5%
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
Accuracy	6.3%
Cost	19456

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

Alternative 6
Accuracy	5.9%
Cost	13568

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

Alternative 7
Accuracy	5.9%
Cost	13440

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

Alternative 8
Accuracy	5.9%
Cost	13184

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

Alternative 9
Accuracy	5.4%
Cost	12928

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

Reproduce

herbie shell --seed 2023146 
(FPCore (x)
  :name "expfmod (used to be hard to sample)"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))