Average Error: 59.8 → 23.8
Time: 16.4s
Precision: binary64
Cost: 129097
\[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
↓
\[\begin{array}{l} t_0 := \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\\ t_1 := e^{-x}\\ t_2 := t_0 \cdot t_1\\ \mathbf{if}\;t_2 \leq 0 \lor \neg \left(t_2 \leq 2\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;e^{{\left(\sqrt[3]{\log \log \left(e^{t_0}\right)}\right)}^{3} - x}\\ \end{array} \]
(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
     (exp (- (pow (cbrt (log (log (exp t_0)))) 3.0) x)))))
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 = exp((pow(cbrt(log(log(exp(t_0)))), 3.0) - x));
	}
	return tmp;
}

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 = exp(Float64((cbrt(log(log(exp(t_0)))) ^ 3.0) - x));
	end
	return tmp
end

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[Exp[N[(N[Power[N[Power[N[Log[N[Log[N[Exp[t$95$0], $MachinePrecision]], $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision], 3.0], $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision]]]]]

\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}:\\
\;\;\;\;e^{{\left(\sqrt[3]{\log \log \left(e^{t_0}\right)}\right)}^{3} - x}\\


\end{array}

Error

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. Simplified61.8

      \[\leadsto \color{blue}{\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}} \]
      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. Applied egg-rr61.8

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

    4. Taylor expanded in x around inf 24.2

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

    5. Simplified24.2

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

      [Start]24.2

      \[ e^{-1 \cdot x} \]

      mul-1-neg [=>]24.2

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

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

    1. Initial program 13.5

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

    2. Simplified13.4

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

      [Start]13.5

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

      exp-neg [=>]13.5

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

      associate-*r/ [=>]13.4

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

      *-rgt-identity [=>]13.4

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

    3. Applied egg-rr13.4

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

    4. Applied egg-rr13.7

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

    5. Applied egg-rr15.6

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

  4. Final simplification23.8

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

Alternatives

Alternative 1
Error23.8
Cost116489
\[\begin{array}{l} t_0 := \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\\ t_1 := e^{-x}\\ t_2 := t_0 \cdot t_1\\ \mathbf{if}\;t_2 \leq 0 \lor \neg \left(t_2 \leq 2\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{1}{\frac{1}{\log \log \left(e^{t_0}\right) - x}}}\\ \end{array} \]
Alternative 2
Error23.8
Cost116233
\[\begin{array}{l} t_0 := \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\\ t_1 := e^{-x}\\ t_2 := t_0 \cdot t_1\\ \mathbf{if}\;t_2 \leq 0 \lor \neg \left(t_2 \leq 2\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;e^{\log \log \left(e^{t_0}\right) - x}\\ \end{array} \]
Alternative 3
Error23.7
Cost110089
\[\begin{array}{l} t_0 := \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\\ t_1 := e^{-x}\\ t_2 := t_0 \cdot t_1\\ \mathbf{if}\;t_2 \leq 0 \lor \neg \left(t_2 \leq 2\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;e^{\frac{1}{\frac{1}{\log \left(\frac{t_0}{e^{x}}\right)}}}\\ \end{array} \]
Alternative 4
Error23.7
Cost97161
\[\begin{array}{l} t_0 := \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\\ t_1 := e^{-x}\\ t_2 := t_0 \cdot t_1\\ \mathbf{if}\;t_2 \leq 0 \lor \neg \left(t_2 \leq 2\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{e^{x}}{t_0}}\\ \end{array} \]
Alternative 5
Error23.7
Cost97033
\[\begin{array}{l} t_0 := \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\\ t_1 := e^{-x}\\ t_2 := t_0 \cdot t_1\\ \mathbf{if}\;t_2 \leq 0 \lor \neg \left(t_2 \leq 2\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{t_0}{e^{x}}\\ \end{array} \]
Alternative 6
Error24.3
Cost26505
\[\begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-310} \lor \neg \left(x \leq 11.008332471475773\right):\\ \;\;\;\;e^{-x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x, x \cdot -0.25, 1\right)\right)\right) \cdot \frac{1}{e^{x}}\\ \end{array} \]
Alternative 7
Error24.4
Cost20105
\[\begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-310} \lor \neg \left(x \leq 0.52\right):\\ \;\;\;\;e^{-x}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\mathsf{fma}\left(x, x \cdot -0.25, 1\right)\right)\right) \cdot \left(1 - x\right)\\ \end{array} \]
Alternative 8
Error24.3
Cost20105
\[\begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-310} \lor \neg \left(x \leq 11.008332471475773\right):\\ \;\;\;\;e^{-x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(e^{x}\right) \bmod \left(1 + -0.25 \cdot \left(x \cdot x\right)\right)\right)}{e^{x}}\\ \end{array} \]
Alternative 9
Error25.0
Cost6528
\[e^{-x} \]

Error

Reproduce

herbie shell --seed 2022356 
(FPCore (x)
  :name "expfmod (used to be hard to sample)"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))