?

Average Error: 93.25% → 58.09%
Time: 14.1s
Precision: binary64
Cost: 51584

?

\[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
↓
\[\frac{\left(\left(e^{x}\right) \bmod \left(3 \cdot \log \left(\sqrt[3]{e^{\sqrt{\cos x}}}\right)\right)\right)}{e^{x}} \]
(FPCore (x) :precision binary64 (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))
↓
(FPCore (x)
 :precision binary64
 (/ (fmod (exp x) (* 3.0 (log (cbrt (exp (sqrt (cos x))))))) (exp x)))
double code(double x) {
	return fmod(exp(x), sqrt(cos(x))) * exp(-x);
}
↓
double code(double x) {
	return fmod(exp(x), (3.0 * log(cbrt(exp(sqrt(cos(x))))))) / exp(x);
}

function code(x)
	return Float64(rem(exp(x), sqrt(cos(x))) * exp(Float64(-x)))
end
↓
function code(x)
	return Float64(rem(exp(x), Float64(3.0 * log(cbrt(exp(sqrt(cos(x))))))) / exp(x))
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_] := N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(3.0 * N[Log[N[Power[N[Exp[N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision]

\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x}
↓
\frac{\left(\left(e^{x}\right) \bmod \left(3 \cdot \log \left(\sqrt[3]{e^{\sqrt{\cos x}}}\right)\right)\right)}{e^{x}}

Error?

Derivation?

  1. Initial program 93.25

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

  2. Simplified93.24

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

    [Start]93.25

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

    exp-neg [=>]93.24

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

    associate-*r/ [=>]93.24

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

    *-rgt-identity [=>]93.24

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

  3. Applied egg-rr93.24

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

  4. Applied egg-rr58.09

    \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \color{blue}{\left(\log \left({\left(\sqrt[3]{e^{\sqrt{\cos x}}}\right)}^{2}\right) + \log \left(\sqrt[3]{e^{\sqrt{\cos x}}}\right)\right)}\right)}{e^{x}} \]

  5. Simplified58.09

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

    [Start]58.09

    \[ \frac{\left(\left(e^{x}\right) \bmod \left(\log \left({\left(\sqrt[3]{e^{\sqrt{\cos x}}}\right)}^{2}\right) + \log \left(\sqrt[3]{e^{\sqrt{\cos x}}}\right)\right)\right)}{e^{x}} \]

    log-pow [=>]58.09

    \[ \frac{\left(\left(e^{x}\right) \bmod \left(\color{blue}{2 \cdot \log \left(\sqrt[3]{e^{\sqrt{\cos x}}}\right)} + \log \left(\sqrt[3]{e^{\sqrt{\cos x}}}\right)\right)\right)}{e^{x}} \]

    distribute-lft1-in [=>]58.09

    \[ \frac{\left(\left(e^{x}\right) \bmod \color{blue}{\left(\left(2 + 1\right) \cdot \log \left(\sqrt[3]{e^{\sqrt{\cos x}}}\right)\right)}\right)}{e^{x}} \]

    metadata-eval [=>]58.09

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

  6. Final simplification58.09

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

Alternatives

Alternative 1
Error58.45%
Cost38720
\[\frac{\left(\left(e^{x}\right) \bmod \left(3 \cdot \log \left(\sqrt[3]{e}\right)\right)\right)}{e^{x}} \]
Alternative 2
Error91.33%
Cost32448
\[\frac{\left(\left(e^{x}\right) \bmod \left(3 \cdot {\left(\sqrt[3]{0.3333333333333333}\right)}^{3}\right)\right)}{e^{x}} \]
Alternative 3
Error93.46%
Cost19968
\[\frac{\left(\left(e^{x}\right) \bmod \left(3 \cdot \left(0.3333333333333333 + -0.08333333333333333 \cdot \left(x \cdot x\right)\right)\right)\right)}{e^{x}} \]
Alternative 4
Error93.46%
Cost19840
\[\frac{\left(\left(e^{x}\right) \bmod \left(1 + \left(x \cdot x\right) \cdot -0.25\right)\right)}{e^{x}} \]
Alternative 5
Error93.66%
Cost19712
\[\left(1 + \frac{\left(\left(e^{x}\right) \bmod 1\right)}{e^{x}}\right) + -1 \]
Alternative 6
Error93.66%
Cost19456
\[\frac{\left(\left(e^{x}\right) \bmod 1\right)}{e^{x}} \]
Alternative 7
Error94.24%
Cost13440
\[\left(1 + \left(\left(e^{x}\right) \bmod 1\right) \cdot \left(1 - x\right)\right) + -1 \]
Alternative 8
Error94.24%
Cost13184
\[\left(\left(e^{x}\right) \bmod 1\right) \cdot \left(1 - x\right) \]
Alternative 9
Error94.67%
Cost12928
\[\left(\left(e^{x}\right) \bmod 1\right) \]

Error

Reproduce?

herbie shell --seed 2023102 
(FPCore (x)
  :name "expfmod (used to be hard to sample)"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))