?

Average Error: 59.5 → 24.9
Time: 14.1s
Precision: binary64
Cost: 6528

?

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

real(8) function code(x)
    real(8), intent (in) :: x
    code = mod(exp(x), sqrt(cos(x))) * exp(-x)
end function
↓
real(8) function code(x)
    real(8), intent (in) :: x
    code = exp(-x)
end function

def code(x):
	return math.fmod(math.exp(x), math.sqrt(math.cos(x))) * math.exp(-x)
↓
def code(x):
	return math.exp(-x)

function code(x)
	return Float64(rem(exp(x), sqrt(cos(x))) * exp(Float64(-x)))
end
↓
function code(x)
	return exp(Float64(-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[Exp[(-x)], $MachinePrecision]

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

Error?

Derivation?

  1. Initial program 59.5

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

  2. Simplified59.5

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

    [Start]59.5

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

    exp-neg [=>]59.5

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

    associate-*r/ [=>]59.5

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

    *-rgt-identity [=>]59.5

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

  3. Applied egg-rr59.5

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

  4. Taylor expanded in x around inf 24.9

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

  5. Simplified24.9

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

    [Start]24.9

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

    mul-1-neg [=>]24.9

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

  6. Final simplification24.9

    \[\leadsto e^{-x} \]

Reproduce?

herbie shell --seed 2023041 
(FPCore (x)
  :name "expfmod (used to be hard to sample)"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))