expfmod (used to be hard to sample)

Percentage Accurate: 7.2% → 9.5%

Time: 17.3s

Alternatives: 11

Speedup: 1.0×

• could not determine a ground truth

  1. x = -6.103375971316035e+105

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: 7.2% 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: 9.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{e^{e^{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}}}\\ \frac{\log \left(2 \cdot \log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(t_0\right)\right)\right) + \log t_0\right)}{e^{x}} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (cbrt (exp (exp (fmod (exp x) (sqrt (cos x))))))))
   (/ (log (+ (* 2.0 (log (expm1 (log1p t_0)))) (log t_0))) (exp x))))

C

double code(double x) {
	double t_0 = cbrt(exp(exp(fmod(exp(x), sqrt(cos(x))))));
	return log(((2.0 * log(expm1(log1p(t_0)))) + log(t_0))) / exp(x);
}

Julia

function code(x)
	t_0 = cbrt(exp(exp(rem(exp(x), sqrt(cos(x))))))
	return Float64(log(Float64(Float64(2.0 * log(expm1(log1p(t_0)))) + log(t_0))) / exp(x))
end

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[Exp[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]], $MachinePrecision], 1/3], $MachinePrecision]}, N[(N[Log[N[(N[(2.0 * N[Log[N[(Exp[N[Log[1 + t$95$0], $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] + N[Log[t$95$0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 6.3%

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

   1. exp-neg 6.4%

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

   2. associate-*r/ 6.4%

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

   3. *-rgt-identity 6.4%

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

3. Simplified 6.4%

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

   1. add-log-exp 6.3%

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

5. Applied egg-rr 6.3%

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

   1. add-log-exp 6.3%

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

   2. add-cube-cbrt 9.2%

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

   3. log-prod 9.2%

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

7. Applied egg-rr 9.2%

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

   1. log-prod 9.2%

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

   2. count-2 9.2%

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

9. Simplified 9.2%

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

    1. expm1-log1p-u 9.3%

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

11. Applied egg-rr 9.3%

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

12. Final simplification 9.3%

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

Alternative 2: 9.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}\\ \frac{\log \left(\log \left(\sqrt[3]{e^{t_0}}\right) + 2 \cdot \log \left(e^{t_0 \cdot 0.3333333333333333}\right)\right)}{e^{x}} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (fmod (exp x) (sqrt (cos x))))))
   (/
    (log
     (+ (log (cbrt (exp t_0))) (* 2.0 (log (exp (* t_0 0.3333333333333333))))))
    (exp x))))

C

double code(double x) {
	double t_0 = exp(fmod(exp(x), sqrt(cos(x))));
	return log((log(cbrt(exp(t_0))) + (2.0 * log(exp((t_0 * 0.3333333333333333)))))) / 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(2.0 * log(exp(Float64(t_0 * 0.3333333333333333)))))) / 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[(2.0 * N[Log[N[Exp[N[(t$95$0 * 0.3333333333333333), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 6.3%

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

   1. exp-neg 6.4%

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

   2. associate-*r/ 6.4%

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

   3. *-rgt-identity 6.4%

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

3. Simplified 6.4%

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

   1. add-log-exp 6.3%

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

5. Applied egg-rr 6.3%

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

   1. add-log-exp 6.3%

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

   2. add-cube-cbrt 9.2%

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

   3. log-prod 9.2%

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

7. Applied egg-rr 9.2%

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

   1. log-prod 9.2%

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

   2. count-2 9.2%

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

9. Simplified 9.2%

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

    1. pow1/3 9.3%

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

    2. pow-exp 9.2%

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

11. Applied egg-rr 9.2%

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

12. Final simplification 9.2%

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

Alternative 3: 9.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \frac{\log \left(\log \left(\mathsf{expm1}\left(\mathsf{log1p}\left(\sqrt[3]{e^{e^{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}}}\right)\right)\right) \cdot 3\right)}{e^{x}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/
  (log
   (*
    (log (expm1 (log1p (cbrt (exp (exp (fmod (exp x) (sqrt (cos x)))))))))
    3.0))
  (exp x)))

C

double code(double x) {
	return log((log(expm1(log1p(cbrt(exp(exp(fmod(exp(x), sqrt(cos(x))))))))) * 3.0)) / exp(x);
}

Julia

function code(x)
	return Float64(log(Float64(log(expm1(log1p(cbrt(exp(exp(rem(exp(x), sqrt(cos(x))))))))) * 3.0)) / exp(x))
end

Wolfram

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

Derivation

1. Initial program 6.3%

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

   1. exp-neg 6.4%

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

   2. associate-*r/ 6.4%

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

   3. *-rgt-identity 6.4%

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

3. Simplified 6.4%

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

   1. add-log-exp 6.3%

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

5. Applied egg-rr 6.3%

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

   1. add-log-exp 6.3%

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

   2. add-cube-cbrt 9.2%

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

   3. log-prod 9.2%

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

7. Applied egg-rr 9.2%

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

   1. log-prod 9.2%

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

   2. count-2 9.2%

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

9. Simplified 9.2%

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

10. Taylor expanded in x around 0 6.2%

    \[\leadsto \frac{\color{blue}{\log \left(\log \left({\left(e^{e^{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}}\right)}^{0.3333333333333333}\right) + 2 \cdot \log \left({\left(e^{e^{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}}\right)}^{0.3333333333333333}\right)\right)}}{e^{x}} \]
11. Step-by-step derivation

    1. distribute-rgt1-in 6.2%

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

    2. metadata-eval 6.2%

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

    3. unpow1/3 9.2%

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

12. Simplified 9.2%

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

    1. expm1-log1p-u 9.3%

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

14. Applied egg-rr 9.2%

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

15. Final simplification 9.2%

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

Alternative 4: 9.5% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(N[Log[N[(N[Log[N[Power[N[Exp[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]], $MachinePrecision], 1/3], $MachinePrecision]], $MachinePrecision] * 3.0), $MachinePrecision]], $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.3%

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

   1. exp-neg 6.4%

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

   2. associate-*r/ 6.4%

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

   3. *-rgt-identity 6.4%

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

3. Simplified 6.4%

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

   1. add-log-exp 6.3%

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

5. Applied egg-rr 6.3%

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

   1. add-log-exp 6.3%

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

   2. add-cube-cbrt 9.2%

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

   3. log-prod 9.2%

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

7. Applied egg-rr 9.2%

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

   1. log-prod 9.2%

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

   2. count-2 9.2%

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

9. Simplified 9.2%

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

10. Taylor expanded in x around 0 6.2%

    \[\leadsto \frac{\color{blue}{\log \left(\log \left({\left(e^{e^{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}}\right)}^{0.3333333333333333}\right) + 2 \cdot \log \left({\left(e^{e^{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}}\right)}^{0.3333333333333333}\right)\right)}}{e^{x}} \]
11. Step-by-step derivation

    1. distribute-rgt1-in 6.2%

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

    2. metadata-eval 6.2%

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

    3. unpow1/3 9.2%

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

12. Simplified 9.2%

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

13. Final simplification 9.2%

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

Alternative 5: 7.2% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

code[x_] := N[Log[N[Exp[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]], $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 6.3%

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

   1. exp-neg 6.4%

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

   2. associate-*r/ 6.4%

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

   3. *-rgt-identity 6.4%

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

3. Simplified 6.4%

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

   1. add-log-exp 6.4%

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

5. Applied egg-rr 6.4%

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

6. Final simplification 6.4%

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

Alternative 6: 7.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

code[x_] := N[(N[(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] + 1.0), $MachinePrecision] + -1.0), $MachinePrecision]

Derivation

1. Initial program 6.3%

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

   1. exp-neg 6.4%

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

   2. associate-*r/ 6.4%

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

   3. *-rgt-identity 6.4%

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

3. Simplified 6.4%

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

   1. expm1-log1p-u 6.4%

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

   2. expm1-udef 6.4%

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

   3. log1p-udef 6.4%

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

   4. add-exp-log 6.4%

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

5. Applied egg-rr 6.4%

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

6. Final simplification 6.4%

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

Alternative 7: 7.2% 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 6.3%

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

   1. exp-neg 6.4%

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

   2. associate-*r/ 6.4%

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

   3. *-rgt-identity 6.4%

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

3. Simplified 6.4%

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

4. Final simplification 6.4%

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

Alternative 8: 6.1% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 6.3%

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

   1. exp-neg 6.4%

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

   2. associate-*r/ 6.4%

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

   3. *-rgt-identity 6.4%

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

3. Simplified 6.4%

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

4. Taylor expanded in x around 0 5.3%

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

   1. mul-1-neg 5.3%

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

   2. unsub-neg 5.3%

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

   3. *-rgt-identity 5.3%

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

   4. distribute-lft-out-- 5.3%

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

6. Simplified 5.3%

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

   1. expm1-log1p-u 5.3%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right)\right)} \cdot \left(1 - x\right) \]

   2. expm1-udef 5.3%

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

   3. log1p-udef 5.3%

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

   4. add-exp-log 5.3%

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

8. Applied egg-rr 5.3%

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

9. Final simplification 5.3%

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

Alternative 9: 6.1% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

code[x_] := N[(N[(1.0 + 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]), $MachinePrecision] + -1.0), $MachinePrecision]

Derivation

1. Initial program 6.3%

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

   1. exp-neg 6.4%

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

   2. associate-*r/ 6.4%

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

   3. *-rgt-identity 6.4%

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

3. Simplified 6.4%

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

   1. expm1-log1p-u 6.4%

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

   2. expm1-udef 6.4%

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

   3. log1p-udef 6.4%

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

   4. add-exp-log 6.4%

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

5. Applied egg-rr 6.4%

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

6. Taylor expanded in x around 0 5.3%

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

   1. mul-1-neg 5.3%

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

   2. unsub-neg 5.3%

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

   3. *-rgt-identity 5.3%

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

   4. distribute-lft-out-- 5.3%

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

8. Simplified 5.3%

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

9. Final simplification 5.3%

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

Alternative 10: 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 6.3%

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

   1. exp-neg 6.4%

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

   2. associate-*r/ 6.4%

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

   3. *-rgt-identity 6.4%

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

3. Simplified 6.4%

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

4. Taylor expanded in x around 0 5.3%

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

   1. mul-1-neg 5.3%

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

   2. unsub-neg 5.3%

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

   3. *-rgt-identity 5.3%

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

   4. distribute-lft-out-- 5.3%

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

6. Simplified 5.3%

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

7. Final simplification 5.3%

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

Alternative 11: 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 6.3%

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

   1. exp-neg 6.4%

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

   2. associate-*r/ 6.4%

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

   3. *-rgt-identity 6.4%

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

3. Simplified 6.4%

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

4. Taylor expanded in x around 0 4.7%

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

5. Final simplification 4.7%

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

Reproduce

herbie shell --seed 2023189 
(FPCore (x)
  :name "expfmod (used to be hard to sample)"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))