expfmod (used to be hard to sample)

Percentage Accurate: 6.9% → 9.4%

Time: 24.8s

Alternatives: 7

Speedup: 1.0×

• could not determine a ground truth

  1. x = -9.555308485206832e+192

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: 6.9% 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: 9.4% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.5%

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

   1. /-rgt-identity 6.5%

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

   2. associate-/r/ 6.5%

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

   3. exp-neg 6.5%

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

   4. remove-double-neg 6.5%

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

3. Simplified 6.5%

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

   1. add-log-exp 6.4%

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

6. Applied egg-rr 6.4%

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

   1. add-log-exp 6.4%

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

   2. add-cube-cbrt 8.4%

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

   3. log-prod 8.3%

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

8. Applied egg-rr 8.3%

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

   1. log-prod 8.4%

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

   2. *-un-lft-identity 8.4%

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

   3. *-un-lft-identity 8.4%

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

   4. distribute-rgt-out 8.4%

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

   5. metadata-eval 8.4%

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

   6. pow1/3 8.5%

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

   7. log-pow 8.4%

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

   8. add-log-exp 8.4%

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

10. Applied egg-rr 8.4%

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

11. Taylor expanded in x around 0 8.4%

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

    1. *-commutative 8.4%

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

13. Simplified 8.4%

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

Alternative 2: 6.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ {\left(e^{\mathsf{log1p}\left(\sqrt[3]{\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}}\right)} + -1\right)}^{3} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (pow
  (+ (exp (log1p (cbrt (/ (fmod (exp x) (sqrt (cos x))) (exp x))))) -1.0)
  3.0))

C

double code(double x) {
	return pow((exp(log1p(cbrt((fmod(exp(x), sqrt(cos(x))) / exp(x))))) + -1.0), 3.0);
}

Julia

function code(x)
	return Float64(exp(log1p(cbrt(Float64(rem(exp(x), sqrt(cos(x))) / exp(x))))) + -1.0) ^ 3.0
end

Wolfram

code[x_] := N[Power[N[(N[Exp[N[Log[1 + N[Power[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], 1/3], $MachinePrecision]], $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision], 3.0], $MachinePrecision]

Derivation

1. Initial program 6.5%

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

   1. /-rgt-identity 6.5%

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

   2. associate-/r/ 6.5%

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

   3. exp-neg 6.5%

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

   4. remove-double-neg 6.5%

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

3. Simplified 6.5%

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

   1. add-cube-cbrt 6.5%

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

   2. pow3 6.5%

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

6. Applied egg-rr 6.5%

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

   1. expm1-log1p-u 6.5%

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

   2. expm1-undefine 6.5%

      \[\leadsto {\color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt[3]{\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}}\right)} - 1\right)}}^{3} \]

8. Applied egg-rr 6.5%

   \[\leadsto {\color{blue}{\left(e^{\mathsf{log1p}\left(\sqrt[3]{\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}}\right)} - 1\right)}}^{3} \]

9. Final simplification 6.5%

   \[\leadsto {\left(e^{\mathsf{log1p}\left(\sqrt[3]{\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}}\right)} + -1\right)}^{3} \]
10. Add Preprocessing

Alternative 3: 7.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ {\left(\frac{1}{\sqrt[3]{\frac{e^{x}}{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}}}\right)}^{3} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (pow (/ 1.0 (cbrt (/ (exp x) (fmod (exp x) (sqrt (cos x)))))) 3.0))

C

double code(double x) {
	return pow((1.0 / cbrt((exp(x) / fmod(exp(x), sqrt(cos(x)))))), 3.0);
}

Julia

function code(x)
	return Float64(1.0 / cbrt(Float64(exp(x) / rem(exp(x), sqrt(cos(x)))))) ^ 3.0
end

Wolfram

code[x_] := N[Power[N[(1.0 / N[Power[N[(N[Exp[x], $MachinePrecision] / N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]

Derivation

1. Initial program 6.5%

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

   1. /-rgt-identity 6.5%

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

   2. associate-/r/ 6.5%

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

   3. exp-neg 6.5%

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

   4. remove-double-neg 6.5%

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

3. Simplified 6.5%

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

   1. add-cube-cbrt 6.5%

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

   2. pow3 6.5%

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

6. Applied egg-rr 6.5%

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

   1. clear-num 6.5%

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

   2. cbrt-div 6.5%

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

   3. metadata-eval 6.5%

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

8. Applied egg-rr 6.5%

   \[\leadsto {\color{blue}{\left(\frac{1}{\sqrt[3]{\frac{e^{x}}{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}}}\right)}}^{3} \]
9. Add Preprocessing

Alternative 4: 7.0% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (pow (cbrt (/ (fmod (exp x) (sqrt (cos x))) (exp x))) 3.0))

C

double code(double x) {
	return pow(cbrt((fmod(exp(x), sqrt(cos(x))) / exp(x))), 3.0);
}

Julia

function code(x)
	return cbrt(Float64(rem(exp(x), sqrt(cos(x))) / exp(x))) ^ 3.0
end

Wolfram

code[x_] := N[Power[N[Power[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], 1/3], $MachinePrecision], 3.0], $MachinePrecision]

Derivation

1. Initial program 6.5%

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

   1. /-rgt-identity 6.5%

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

   2. associate-/r/ 6.5%

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

   3. exp-neg 6.5%

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

   4. remove-double-neg 6.5%

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

3. Simplified 6.5%

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

   1. add-cube-cbrt 6.5%

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

   2. pow3 6.5%

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

6. Applied egg-rr 6.5%

   \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}}\right)}^{3}} \]
7. Add Preprocessing

Alternative 5: 7.0% 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 6.5%

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

   1. /-rgt-identity 6.5%

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

   2. associate-/r/ 6.5%

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

   3. exp-neg 6.5%

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

   4. remove-double-neg 6.5%

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

3. Simplified 6.5%

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

Alternative 6: 6.1% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

function code(x)
	return Float64(rem(exp(x), sqrt(cos(x))) * Float64(1.0 - 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[(1.0 - x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.5%

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

   1. /-rgt-identity 6.5%

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

   2. associate-/r/ 6.5%

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

   3. exp-neg 6.5%

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

   4. remove-double-neg 6.5%

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

3. Simplified 6.5%

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

5. Taylor expanded in x around 0 6.1%

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

   1. +-commutative 6.1%

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

   2. mul-1-neg 6.1%

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

   3. unsub-neg 6.1%

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

   4. *-lft-identity 6.1%

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

   5. distribute-rgt-out-- 6.1%

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

7. Simplified 6.1%

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

Alternative 7: 5.4% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.5%

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

   1. /-rgt-identity 6.5%

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

   2. associate-/r/ 6.5%

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

   3. exp-neg 6.5%

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

   4. remove-double-neg 6.5%

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

3. Simplified 6.5%

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

5. Taylor expanded in x around 0 5.4%

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

Reproduce

herbie shell --seed 2024108 
(FPCore (x)
  :name "expfmod (used to be hard to sample)"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))