expfmod (used to be hard to sample)

Percentage Accurate: 6.6% → 43.2%

Time: 22.5s

Alternatives: 9

Speedup: 1.0×

• could not determine a ground truth

  1. x = -1.7911779839168845e+151

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: 6.6% 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: 43.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt{\cos x}\\ \frac{\left(\left(e^{x}\right) \bmod \left(t\_0 \cdot 0.6666666666666666 + \log \left(\sqrt[3]{e^{t\_0}}\right)\right)\right)}{e^{x}} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (sqrt (cos x))))
   (/
    (fmod (exp x) (+ (* t_0 0.6666666666666666) (log (cbrt (exp t_0)))))
    (exp x))))

C

double code(double x) {
	double t_0 = sqrt(cos(x));
	return fmod(exp(x), ((t_0 * 0.6666666666666666) + log(cbrt(exp(t_0))))) / exp(x);
}

Julia

function code(x)
	t_0 = sqrt(cos(x))
	return Float64(rem(exp(x), Float64(Float64(t_0 * 0.6666666666666666) + log(cbrt(exp(t_0))))) / exp(x))
end

Wolfram

code[x_] := Block[{t$95$0 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(N[(t$95$0 * 0.6666666666666666), $MachinePrecision] + N[Log[N[Power[N[Exp[t$95$0], $MachinePrecision], 1/3], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 7.7%

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

   1. /-rgt-identity 7.7%

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

   2. associate-/r/ 7.7%

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

   3. exp-neg 7.7%

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

   4. remove-double-neg 7.7%

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

3. Simplified 7.7%

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

   1. add-log-exp 7.7%

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

   2. add-cube-cbrt 44.6%

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

   3. log-prod 44.6%

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

   4. pow2 44.6%

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

6. Applied egg-rr 44.6%

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

7. Taylor expanded in x around inf 44.7%

   \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\color{blue}{\log \left(\sqrt[3]{{\left(e^{\sqrt{\cos x}}\right)}^{2}}\right)} + \log \left(\sqrt[3]{e^{\sqrt{\cos x}}}\right)\right)\right)}{e^{x}} \]
8. Step-by-step derivation

   1. unpow1/3 44.7%

      \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\log \color{blue}{\left({\left({\left(e^{\sqrt{\cos x}}\right)}^{2}\right)}^{0.3333333333333333}\right)} + \log \left(\sqrt[3]{e^{\sqrt{\cos x}}}\right)\right)\right)}{e^{x}} \]

   2. unpow2 44.7%

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

   3. prod-exp 44.7%

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

   4. exp-prod 44.7%

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

   5. count-2 44.7%

      \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\log \left(e^{\color{blue}{\left(2 \cdot \sqrt{\cos x}\right)} \cdot 0.3333333333333333}\right) + \log \left(\sqrt[3]{e^{\sqrt{\cos x}}}\right)\right)\right)}{e^{x}} \]

   6. associate-*r* 44.7%

      \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\log \left(e^{\color{blue}{2 \cdot \left(\sqrt{\cos x} \cdot 0.3333333333333333\right)}}\right) + \log \left(\sqrt[3]{e^{\sqrt{\cos x}}}\right)\right)\right)}{e^{x}} \]

   7. *-commutative 44.7%

      \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\log \left(e^{2 \cdot \color{blue}{\left(0.3333333333333333 \cdot \sqrt{\cos x}\right)}}\right) + \log \left(\sqrt[3]{e^{\sqrt{\cos x}}}\right)\right)\right)}{e^{x}} \]

   8. log1p-expm1 44.7%

      \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\log \left(e^{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(2 \cdot \left(0.3333333333333333 \cdot \sqrt{\cos x}\right)\right)\right)}}\right) + \log \left(\sqrt[3]{e^{\sqrt{\cos x}}}\right)\right)\right)}{e^{x}} \]

   9. log1p-define 44.7%

      \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\log \left(e^{\color{blue}{\log \left(1 + \mathsf{expm1}\left(2 \cdot \left(0.3333333333333333 \cdot \sqrt{\cos x}\right)\right)\right)}}\right) + \log \left(\sqrt[3]{e^{\sqrt{\cos x}}}\right)\right)\right)}{e^{x}} \]

   10. rem-exp-log 44.7%

       \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\log \color{blue}{\left(1 + \mathsf{expm1}\left(2 \cdot \left(0.3333333333333333 \cdot \sqrt{\cos x}\right)\right)\right)} + \log \left(\sqrt[3]{e^{\sqrt{\cos x}}}\right)\right)\right)}{e^{x}} \]

   11. log1p-define 44.7%

       \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(2 \cdot \left(0.3333333333333333 \cdot \sqrt{\cos x}\right)\right)\right)} + \log \left(\sqrt[3]{e^{\sqrt{\cos x}}}\right)\right)\right)}{e^{x}} \]

   12. log1p-expm1 44.7%

       \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\color{blue}{2 \cdot \left(0.3333333333333333 \cdot \sqrt{\cos x}\right)} + \log \left(\sqrt[3]{e^{\sqrt{\cos x}}}\right)\right)\right)}{e^{x}} \]

   13. count-2 44.7%

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

   14. distribute-rgt-out 44.7%

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

   15. metadata-eval 44.7%

       \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x} \cdot \color{blue}{0.6666666666666666} + \log \left(\sqrt[3]{e^{\sqrt{\cos x}}}\right)\right)\right)}{e^{x}} \]

9. Simplified 44.7%

   \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\color{blue}{\sqrt{\cos x} \cdot 0.6666666666666666} + \log \left(\sqrt[3]{e^{\sqrt{\cos x}}}\right)\right)\right)}{e^{x}} \]
10. Add Preprocessing

Alternative 2: 42.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x} \cdot 0.6666666666666666 + \log \left(\sqrt[3]{e}\right)\right)\right)}{e^{x}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  (fmod (exp x) (+ (* (sqrt (cos x)) 0.6666666666666666) (log (cbrt E))))
  (exp x)))

C

double code(double x) {
	return fmod(exp(x), ((sqrt(cos(x)) * 0.6666666666666666) + log(cbrt(((double) M_E))))) / exp(x);
}

Julia

function code(x)
	return Float64(rem(exp(x), Float64(Float64(sqrt(cos(x)) * 0.6666666666666666) + log(cbrt(exp(1))))) / exp(x))
end

Wolfram

code[x_] := N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(N[(N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision] * 0.6666666666666666), $MachinePrecision] + N[Log[N[Power[E, 1/3], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.7%

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

   1. /-rgt-identity 7.7%

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

   2. associate-/r/ 7.7%

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

   3. exp-neg 7.7%

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

   4. remove-double-neg 7.7%

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

3. Simplified 7.7%

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

   1. add-log-exp 7.7%

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

   2. add-cube-cbrt 44.6%

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

   3. log-prod 44.6%

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

   4. pow2 44.6%

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

6. Applied egg-rr 44.6%

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

7. Taylor expanded in x around inf 44.7%

   \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\color{blue}{\log \left(\sqrt[3]{{\left(e^{\sqrt{\cos x}}\right)}^{2}}\right)} + \log \left(\sqrt[3]{e^{\sqrt{\cos x}}}\right)\right)\right)}{e^{x}} \]
8. Step-by-step derivation

   1. unpow1/3 44.7%

      \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\log \color{blue}{\left({\left({\left(e^{\sqrt{\cos x}}\right)}^{2}\right)}^{0.3333333333333333}\right)} + \log \left(\sqrt[3]{e^{\sqrt{\cos x}}}\right)\right)\right)}{e^{x}} \]

   2. unpow2 44.7%

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

   3. prod-exp 44.7%

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

   4. exp-prod 44.7%

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

   5. count-2 44.7%

      \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\log \left(e^{\color{blue}{\left(2 \cdot \sqrt{\cos x}\right)} \cdot 0.3333333333333333}\right) + \log \left(\sqrt[3]{e^{\sqrt{\cos x}}}\right)\right)\right)}{e^{x}} \]

   6. associate-*r* 44.7%

      \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\log \left(e^{\color{blue}{2 \cdot \left(\sqrt{\cos x} \cdot 0.3333333333333333\right)}}\right) + \log \left(\sqrt[3]{e^{\sqrt{\cos x}}}\right)\right)\right)}{e^{x}} \]

   7. *-commutative 44.7%

      \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\log \left(e^{2 \cdot \color{blue}{\left(0.3333333333333333 \cdot \sqrt{\cos x}\right)}}\right) + \log \left(\sqrt[3]{e^{\sqrt{\cos x}}}\right)\right)\right)}{e^{x}} \]

   8. log1p-expm1 44.7%

      \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\log \left(e^{\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(2 \cdot \left(0.3333333333333333 \cdot \sqrt{\cos x}\right)\right)\right)}}\right) + \log \left(\sqrt[3]{e^{\sqrt{\cos x}}}\right)\right)\right)}{e^{x}} \]

   9. log1p-define 44.7%

      \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\log \left(e^{\color{blue}{\log \left(1 + \mathsf{expm1}\left(2 \cdot \left(0.3333333333333333 \cdot \sqrt{\cos x}\right)\right)\right)}}\right) + \log \left(\sqrt[3]{e^{\sqrt{\cos x}}}\right)\right)\right)}{e^{x}} \]

   10. rem-exp-log 44.7%

       \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\log \color{blue}{\left(1 + \mathsf{expm1}\left(2 \cdot \left(0.3333333333333333 \cdot \sqrt{\cos x}\right)\right)\right)} + \log \left(\sqrt[3]{e^{\sqrt{\cos x}}}\right)\right)\right)}{e^{x}} \]

   11. log1p-define 44.7%

       \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(2 \cdot \left(0.3333333333333333 \cdot \sqrt{\cos x}\right)\right)\right)} + \log \left(\sqrt[3]{e^{\sqrt{\cos x}}}\right)\right)\right)}{e^{x}} \]

   12. log1p-expm1 44.7%

       \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\color{blue}{2 \cdot \left(0.3333333333333333 \cdot \sqrt{\cos x}\right)} + \log \left(\sqrt[3]{e^{\sqrt{\cos x}}}\right)\right)\right)}{e^{x}} \]

   13. count-2 44.7%

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

   14. distribute-rgt-out 44.7%

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

   15. metadata-eval 44.7%

       \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x} \cdot \color{blue}{0.6666666666666666} + \log \left(\sqrt[3]{e^{\sqrt{\cos x}}}\right)\right)\right)}{e^{x}} \]

9. Simplified 44.7%

   \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\color{blue}{\sqrt{\cos x} \cdot 0.6666666666666666} + \log \left(\sqrt[3]{e^{\sqrt{\cos x}}}\right)\right)\right)}{e^{x}} \]

10. Taylor expanded in x around 0 44.0%

    \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x} \cdot 0.6666666666666666 + \color{blue}{\log \left(\sqrt[3]{e^{1}}\right)}\right)\right)}{e^{x}} \]
11. Step-by-step derivation

    1. exp-1-e 44.0%

       \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x} \cdot 0.6666666666666666 + \log \left(\sqrt[3]{\color{blue}{e}}\right)\right)\right)}{e^{x}} \]

12. Simplified 44.0%

    \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x} \cdot 0.6666666666666666 + \color{blue}{\log \left(\sqrt[3]{e}\right)}\right)\right)}{e^{x}} \]
13. Add Preprocessing

Alternative 3: 6.6% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

function code(x)
	return exp(Float64(log(rem(exp(x), sqrt(cos(x)))) - x))
end

Wolfram

code[x_] := N[Exp[N[(N[Log[N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]], $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 7.7%

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

   1. /-rgt-identity 7.7%

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

   2. associate-/r/ 7.7%

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

   3. exp-neg 7.7%

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

   4. remove-double-neg 7.7%

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

3. Simplified 7.7%

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

   1. add-exp-log 7.7%

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

   2. div-exp 7.8%

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

6. Applied egg-rr 7.8%

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

Alternative 4: 6.6% accurate, 1.0× speedup

Math

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

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(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]

Derivation

1. Initial program 7.7%

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

   1. /-rgt-identity 7.7%

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

   2. associate-/r/ 7.7%

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

   3. exp-neg 7.7%

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

   4. remove-double-neg 7.7%

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

3. Simplified 7.7%

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

Alternative 5: 6.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \frac{\left(\left(e^{x}\right) \bmod \left(1 + -0.25 \cdot {x}^{2}\right)\right)}{e^{x}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (fmod (exp x) (+ 1.0 (* -0.25 (pow x 2.0)))) (exp x)))

C

double code(double x) {
	return fmod(exp(x), (1.0 + (-0.25 * pow(x, 2.0)))) / exp(x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = mod(exp(x), (1.0d0 + ((-0.25d0) * (x ** 2.0d0)))) / exp(x)
end function

Python

def code(x):
	return math.fmod(math.exp(x), (1.0 + (-0.25 * math.pow(x, 2.0)))) / math.exp(x)

Julia

function code(x)
	return Float64(rem(exp(x), Float64(1.0 + Float64(-0.25 * (x ^ 2.0)))) / exp(x))
end

Wolfram

code[x_] := N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(1.0 + N[(-0.25 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.7%

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

   1. /-rgt-identity 7.7%

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

   2. associate-/r/ 7.7%

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

   3. exp-neg 7.7%

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

   4. remove-double-neg 7.7%

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

3. Simplified 7.7%

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

5. Taylor expanded in x around 0 7.3%

   \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \color{blue}{\left(1 + -0.25 \cdot {x}^{2}\right)}\right)}{e^{x}} \]
6. Add Preprocessing

Alternative 6: 6.3% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ e^{\log \left(\left(e^{x}\right) \bmod 1\right) - x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (exp (- (log (fmod (exp x) 1.0)) x)))

C

double code(double x) {
	return exp((log(fmod(exp(x), 1.0)) - x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = exp((log(mod(exp(x), 1.0d0)) - x))
end function

Python

def code(x):
	return math.exp((math.log(math.fmod(math.exp(x), 1.0)) - x))

Julia

function code(x)
	return exp(Float64(log(rem(exp(x), 1.0)) - x))
end

Wolfram

code[x_] := N[Exp[N[(N[Log[N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]], $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 7.7%

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

   1. /-rgt-identity 7.7%

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

   2. associate-/r/ 7.7%

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

   3. exp-neg 7.7%

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

   4. remove-double-neg 7.7%

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

3. Simplified 7.7%

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

5. Taylor expanded in x around 0 7.0%

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

   1. add-exp-log 7.0%

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

   2. div-exp 7.0%

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

7. Applied egg-rr 7.0%

   \[\leadsto \color{blue}{e^{\log \left(\left(e^{x}\right) \bmod 1\right) - x}} \]
8. Add Preprocessing

Alternative 7: 6.2% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \frac{\left(\left(e^{x}\right) \bmod 1\right)}{e^{x}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (fmod (exp x) 1.0) (exp x)))

C

double code(double x) {
	return fmod(exp(x), 1.0) / exp(x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = mod(exp(x), 1.0d0) / exp(x)
end function

Python

def code(x):
	return math.fmod(math.exp(x), 1.0) / math.exp(x)

Julia

function code(x)
	return Float64(rem(exp(x), 1.0) / exp(x))
end

Wolfram

code[x_] := N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.7%

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

   1. /-rgt-identity 7.7%

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

   2. associate-/r/ 7.7%

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

   3. exp-neg 7.7%

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

   4. remove-double-neg 7.7%

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

3. Simplified 7.7%

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

5. Taylor expanded in x around 0 7.0%

   \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \color{blue}{1}\right)}{e^{x}} \]
6. Add Preprocessing

Alternative 8: 5.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \left(\left(e^{x}\right) \bmod 1\right) \cdot \left(1 - x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (fmod (exp x) 1.0) (- 1.0 x)))

C

double code(double x) {
	return fmod(exp(x), 1.0) * (1.0 - x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = mod(exp(x), 1.0d0) * (1.0d0 - x)
end function

Python

def code(x):
	return math.fmod(math.exp(x), 1.0) * (1.0 - x)

Julia

function code(x)
	return Float64(rem(exp(x), 1.0) * Float64(1.0 - x))
end

Wolfram

code[x_] := N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[(1.0 - x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.7%

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

   1. /-rgt-identity 7.7%

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

   2. associate-/r/ 7.7%

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

   3. exp-neg 7.7%

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

   4. remove-double-neg 7.7%

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

3. Simplified 7.7%

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

5. Taylor expanded in x around 0 7.0%

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

6. Taylor expanded in x around 0 6.5%

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

   1. +-commutative 6.5%

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

   2. *-lft-identity 6.5%

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

   3. associate-*r* 6.5%

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

   4. neg-mul-1 6.5%

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

   5. distribute-rgt-out 6.5%

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

   6. sub-neg 6.5%

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

8. Simplified 6.5%

   \[\leadsto \color{blue}{\left(\left(e^{x}\right) \bmod 1\right) \cdot \left(1 - x\right)} \]
9. Add Preprocessing

Alternative 9: 5.2% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \left(\left(e^{x}\right) \bmod 1\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (fmod (exp x) 1.0))

C

double code(double x) {
	return fmod(exp(x), 1.0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = mod(exp(x), 1.0d0)
end function

Python

def code(x):
	return math.fmod(math.exp(x), 1.0)

Julia

function code(x)
	return rem(exp(x), 1.0)
end

Wolfram

code[x_] := N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]

Derivation

1. Initial program 7.7%

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

   1. /-rgt-identity 7.7%

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

   2. associate-/r/ 7.7%

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

   3. exp-neg 7.7%

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

   4. remove-double-neg 7.7%

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

3. Simplified 7.7%

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

5. Taylor expanded in x around 0 7.0%

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

6. Taylor expanded in x around 0 6.0%

   \[\leadsto \color{blue}{\left(\left(e^{x}\right) \bmod 1\right)} \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024096 
(FPCore (x)
  :name "expfmod (used to be hard to sample)"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))