?

Average Error: 59.7 → 36.7
Time: 16.5s
Precision: binary64
Cost: 51584

?

\[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
↓
\[\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{3 \cdot \log \left(\sqrt[3]{e^{\cos x}}\right)}\right)\right)}{e^{x}} \]
(FPCore (x) :precision binary64 (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))
↓
(FPCore (x)
 :precision binary64
 (/ (fmod (exp x) (sqrt (* 3.0 (log (cbrt (exp (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), sqrt((3.0 * log(cbrt(exp(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), sqrt(Float64(3.0 * log(cbrt(exp(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[Sqrt[N[(3.0 * N[Log[N[Power[N[Exp[N[Cos[x], $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision]], $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(\sqrt{3 \cdot \log \left(\sqrt[3]{e^{\cos x}}\right)}\right)\right)}{e^{x}}

Error?

Derivation?

  1. Initial program 59.7

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

  2. Simplified59.7

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

    [Start]59.7

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

    exp-neg [=>]59.7

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

    associate-*r/ [=>]59.7

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

    *-rgt-identity [=>]59.7

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

  3. Applied egg-rr59.7

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

  4. Applied egg-rr36.7

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

  5. Simplified36.7

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

    [Start]36.7

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

    log-pow [=>]36.7

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

    distribute-lft1-in [=>]36.7

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

    metadata-eval [=>]36.7

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

  6. Final simplification36.7

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

Alternatives

Alternative 1
Error36.9
Cost45120
\[\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{3 \cdot \log \left(\sqrt[3]{e}\right)}\right)\right)}{e^{x}} \]
Alternative 2
Error58.2
Cost38912
\[\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{3} \cdot \sqrt{\cos x \cdot 0.3333333333333333}\right)\right)}{e^{x}} \]
Alternative 3
Error59.7
Cost32512
\[\left(\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}} + 1\right) + -1 \]
Alternative 4
Error59.7
Cost32384
\[\frac{1}{\frac{e^{x}}{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}} \]
Alternative 5
Error59.7
Cost32256
\[\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}} \]
Alternative 6
Error59.8
Cost20096
\[\frac{\left(\left(e^{x}\right) \bmod \left(1 + \left(1 + \left(-1 + x \cdot \left(x \cdot -0.25\right)\right)\right)\right)\right)}{e^{x}} \]
Alternative 7
Error59.8
Cost19840
\[\frac{\left(\left(e^{x}\right) \bmod \left(1 + -0.25 \cdot \left(x \cdot x\right)\right)\right)}{e^{x}} \]
Alternative 8
Error59.9
Cost19712
\[-1 + \left(1 + \frac{\left(\left(e^{x}\right) \bmod 1\right)}{e^{x}}\right) \]
Alternative 9
Error59.9
Cost19456
\[\frac{\left(\left(e^{x}\right) \bmod 1\right)}{e^{x}} \]
Alternative 10
Error60.3
Cost13440
\[\left(-1 + \left(1 + \left(\left(e^{x}\right) \bmod 1\right)\right)\right) \cdot \left(1 - x\right) \]
Alternative 11
Error60.3
Cost13184
\[\left(\left(e^{x}\right) \bmod 1\right) \cdot \left(1 - x\right) \]
Alternative 12
Error60.6
Cost12928
\[\left(\left(e^{x}\right) \bmod 1\right) \]

Error

Reproduce?

herbie shell --seed 2023073 
(FPCore (x)
  :name "expfmod (used to be hard to sample)"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))