expfmod (used to be hard to sample)

Percentage Accurate: 7.0% → 60.6%

Time: 21.5s

Alternatives: 1

Speedup: 4.9×

• could not determine a ground truth

  1. x = -9.79104082664208e+190

Specification

Math

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

FPCore

(FPCore (x) :precision binary64 (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))

C

double code(double x) {
	return fmod(exp(x), sqrt(cos(x))) * exp(-x);
}

Fortran

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

Python

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

Julia

function code(x)
	return Float64(rem(exp(x), sqrt(cos(x))) * exp(Float64(-x)))
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]

Initial Program: 7.0% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))

C

double code(double x) {
	return fmod(exp(x), sqrt(cos(x))) * exp(-x);
}

Fortran

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

Python

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

Julia

function code(x)
	return Float64(rem(exp(x), sqrt(cos(x))) * exp(Float64(-x)))
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]

Alternative 1: 60.6% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ {e}^{\left(-x\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (pow E (- x)))

C

double code(double x) {
	return pow(((double) M_E), -x);
}

Java

public static double code(double x) {
	return Math.pow(Math.E, -x);
}

Python

def code(x):
	return math.pow(math.e, -x)

Julia

function code(x)
	return exp(1) ^ Float64(-x)
end

MATLAB

function tmp = code(x)
	tmp = 2.71828182845904523536 ^ -x;
end

Wolfram

code[x_] := N[Power[E, (-x)], $MachinePrecision]

Derivation

1. Initial program 5.7%

   \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
2. Step-by-step derivation

   1. /-rgt-identity 5.7%

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

   2. associate-/r/ 5.7%

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

   3. exp-neg 5.8%

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

   4. remove-double-neg 5.8%

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

3. Simplified 5.8%

   \[\leadsto \color{blue}{\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}} \]
4. Add Preprocessing
5. Step-by-step derivation

   1. add-exp-log 5.8%

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

   2. *-un-lft-identity 5.8%

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

   3. exp-prod 5.8%

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

   4. exp-1-e 5.8%

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

   5. log-div 5.8%

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

   6. add-log-exp 5.8%

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

6. Applied egg-rr 5.8%

   \[\leadsto \color{blue}{{e}^{\left(\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) - x\right)}} \]
7. Step-by-step derivation

   1. add-cbrt-cube 5.7%

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

   2. pow1/3 5.8%

      \[\leadsto {e}^{\left(\log \color{blue}{\left({\left(\left(\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right) \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right)}^{0.3333333333333333}\right)} - x\right)} \]

   3. pow-to-exp 5.8%

      \[\leadsto {e}^{\left(\log \color{blue}{\left(e^{\log \left(\left(\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right) \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right) \cdot 0.3333333333333333}\right)} - x\right)} \]

   4. pow3 5.8%

      \[\leadsto {e}^{\left(\log \left(e^{\log \color{blue}{\left({\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}^{3}\right)} \cdot 0.3333333333333333}\right) - x\right)} \]

   5. log-pow 5.8%

      \[\leadsto {e}^{\left(\log \left(e^{\color{blue}{\left(3 \cdot \log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right)} \cdot 0.3333333333333333}\right) - x\right)} \]

8. Applied egg-rr 5.8%

   \[\leadsto {e}^{\left(\log \color{blue}{\left(e^{\left(3 \cdot \log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right) \cdot 0.3333333333333333}\right)} - x\right)} \]
9. Step-by-step derivation

   1. rem-log-exp 5.8%

      \[\leadsto {e}^{\left(\color{blue}{\left(3 \cdot \log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right) \cdot 0.3333333333333333} - x\right)} \]

   2. associate-*l* 5.8%

      \[\leadsto {e}^{\left(\color{blue}{3 \cdot \left(\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot 0.3333333333333333\right)} - x\right)} \]

   3. *-commutative 5.8%

      \[\leadsto {e}^{\left(\color{blue}{\left(\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot 0.3333333333333333\right) \cdot 3} - x\right)} \]

10. Applied egg-rr 5.8%

    \[\leadsto {e}^{\left(\color{blue}{\left(\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot 0.3333333333333333\right) \cdot 3} - x\right)} \]

11. Taylor expanded in x around inf 63.0%

    \[\leadsto {e}^{\color{blue}{\left(-1 \cdot x\right)}} \]
12. Step-by-step derivation

    1. neg-mul-1 63.0%

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

13. Simplified 63.0%

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

14. Final simplification 63.0%

    \[\leadsto {e}^{\left(-x\right)} \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024026 
(FPCore (x)
  :name "expfmod (used to be hard to sample)"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))