expfmod (used to be hard to sample)

Percentage Accurate: 7.1% → 7.0%

Time: 29.5s

Alternatives: 4

Speedup: 1.0×

• could not determine a ground truth

  1. x = -3.5573763822042146e+117

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: 7.1% 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: 7.0% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\\ t_1 := \sqrt[3]{\log t_0}\\ \frac{{\left(e^{{\left(\sqrt[3]{\sqrt[3]{{t_1}^{4}} \cdot \left(t_1 \cdot {\left(\sqrt[3]{\sqrt[3]{\log \left(\left(t_0 + 1\right) + -1\right)}}\right)}^{2}\right)}\right)}^{2}}\right)}^{\left(\sqrt[3]{\log \log \left(e^{t_0}\right)}\right)}}{e^{x}} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fmod (exp x) (sqrt (cos x)))) (t_1 (cbrt (log t_0))))
   (/
    (pow
     (exp
      (pow
       (cbrt
        (*
         (cbrt (pow t_1 4.0))
         (* t_1 (pow (cbrt (cbrt (log (+ (+ t_0 1.0) -1.0)))) 2.0))))
       2.0))
     (cbrt (log (log (exp t_0)))))
    (exp x))))

C

double code(double x) {
	double t_0 = fmod(exp(x), sqrt(cos(x)));
	double t_1 = cbrt(log(t_0));
	return pow(exp(pow(cbrt((cbrt(pow(t_1, 4.0)) * (t_1 * pow(cbrt(cbrt(log(((t_0 + 1.0) + -1.0)))), 2.0)))), 2.0)), cbrt(log(log(exp(t_0))))) / exp(x);
}

Julia

function code(x)
	t_0 = rem(exp(x), sqrt(cos(x)))
	t_1 = cbrt(log(t_0))
	return Float64((exp((cbrt(Float64(cbrt((t_1 ^ 4.0)) * Float64(t_1 * (cbrt(cbrt(log(Float64(Float64(t_0 + 1.0) + -1.0)))) ^ 2.0)))) ^ 2.0)) ^ cbrt(log(log(exp(t_0))))) / exp(x))
end

Wolfram

code[x_] := Block[{t$95$0 = N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]}, Block[{t$95$1 = N[Power[N[Log[t$95$0], $MachinePrecision], 1/3], $MachinePrecision]}, N[(N[Power[N[Exp[N[Power[N[Power[N[(N[Power[N[Power[t$95$1, 4.0], $MachinePrecision], 1/3], $MachinePrecision] * N[(t$95$1 * N[Power[N[Power[N[Power[N[Log[N[(N[(t$95$0 + 1.0), $MachinePrecision] + -1.0), $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision], 1/3], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision], N[Power[N[Log[N[Log[N[Exp[t$95$0], $MachinePrecision]], $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision]], $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Initial program 6.8%

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

   1. /-rgt-identity 6.8%

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

   2. associate-/r/ 6.8%

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

   3. exp-neg 6.8%

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

   4. remove-double-neg 6.8%

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

3. Simplified 6.8%

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

   1. add-exp-log 6.8%

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

   2. add-cube-cbrt 6.8%

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

   3. exp-prod 6.8%

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

   4. pow2 6.8%

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

6. Applied egg-rr 6.8%

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

   1. add-log-exp 6.8%

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

8. Applied egg-rr 6.8%

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

   1. add-cube-cbrt 6.8%

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

   2. unpow2 6.8%

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

   3. add-cube-cbrt 6.8%

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

   4. associate-*l* 6.8%

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

10. Applied egg-rr 6.8%

    \[\leadsto \frac{{\left(e^{{\left(\sqrt[3]{\color{blue}{\sqrt[3]{{\left(\sqrt[3]{\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}\right)}^{4}} \cdot \left({\left(\sqrt[3]{\sqrt[3]{\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}}\right)}^{2} \cdot \sqrt[3]{\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}\right)}}\right)}^{2}}\right)}^{\left(\sqrt[3]{\log \log \left(e^{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}\right)}\right)}}{e^{x}} \]
11. Step-by-step derivation

    1. expm1-log1p-u 6.8%

       \[\leadsto \frac{{\left(e^{{\left(\sqrt[3]{\sqrt[3]{{\left(\sqrt[3]{\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}\right)}^{4}} \cdot \left({\left(\sqrt[3]{\sqrt[3]{\log \color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right)\right)\right)}}}\right)}^{2} \cdot \sqrt[3]{\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}\right)}\right)}^{2}}\right)}^{\left(\sqrt[3]{\log \log \left(e^{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}\right)}\right)}}{e^{x}} \]

    2. expm1-udef 6.8%

       \[\leadsto \frac{{\left(e^{{\left(\sqrt[3]{\sqrt[3]{{\left(\sqrt[3]{\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}\right)}^{4}} \cdot \left({\left(\sqrt[3]{\sqrt[3]{\log \color{blue}{\left(e^{\mathsf{log1p}\left(\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right)} - 1\right)}}}\right)}^{2} \cdot \sqrt[3]{\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}\right)}\right)}^{2}}\right)}^{\left(\sqrt[3]{\log \log \left(e^{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}\right)}\right)}}{e^{x}} \]

    3. log1p-udef 6.8%

       \[\leadsto \frac{{\left(e^{{\left(\sqrt[3]{\sqrt[3]{{\left(\sqrt[3]{\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}\right)}^{4}} \cdot \left({\left(\sqrt[3]{\sqrt[3]{\log \left(e^{\color{blue}{\log \left(1 + \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right)}} - 1\right)}}\right)}^{2} \cdot \sqrt[3]{\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}\right)}\right)}^{2}}\right)}^{\left(\sqrt[3]{\log \log \left(e^{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}\right)}\right)}}{e^{x}} \]

    4. rem-exp-log 6.8%

       \[\leadsto \frac{{\left(e^{{\left(\sqrt[3]{\sqrt[3]{{\left(\sqrt[3]{\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}\right)}^{4}} \cdot \left({\left(\sqrt[3]{\sqrt[3]{\log \left(\color{blue}{\left(1 + \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right)} - 1\right)}}\right)}^{2} \cdot \sqrt[3]{\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}\right)}\right)}^{2}}\right)}^{\left(\sqrt[3]{\log \log \left(e^{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}\right)}\right)}}{e^{x}} \]

12. Applied egg-rr 6.8%

    \[\leadsto \frac{{\left(e^{{\left(\sqrt[3]{\sqrt[3]{{\left(\sqrt[3]{\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}\right)}^{4}} \cdot \left({\left(\sqrt[3]{\sqrt[3]{\log \color{blue}{\left(\left(1 + \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right) - 1\right)}}}\right)}^{2} \cdot \sqrt[3]{\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}\right)}\right)}^{2}}\right)}^{\left(\sqrt[3]{\log \log \left(e^{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}\right)}\right)}}{e^{x}} \]

13. Final simplification 6.8%

    \[\leadsto \frac{{\left(e^{{\left(\sqrt[3]{\sqrt[3]{{\left(\sqrt[3]{\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}\right)}^{4}} \cdot \left(\sqrt[3]{\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)} \cdot {\left(\sqrt[3]{\sqrt[3]{\log \left(\left(\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) + 1\right) + -1\right)}}\right)}^{2}\right)}\right)}^{2}}\right)}^{\left(\sqrt[3]{\log \log \left(e^{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}\right)}\right)}}{e^{x}} \]
14. Add Preprocessing

Alternative 2: 7.1% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	t_0 = cbrt(log(rem(exp(x), sqrt(cos(x)))))
	return Float64((exp((t_0 ^ 2.0)) ^ t_0) / exp(x))
end

Wolfram

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

Derivation

1. Initial program 6.8%

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

   1. /-rgt-identity 6.8%

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

   2. associate-/r/ 6.8%

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

   3. exp-neg 6.8%

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

   4. remove-double-neg 6.8%

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

3. Simplified 6.8%

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

   1. add-exp-log 6.8%

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

   2. add-cube-cbrt 6.8%

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

   3. exp-prod 6.8%

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

   4. pow2 6.8%

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

6. Applied egg-rr 6.8%

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

7. Final simplification 6.8%

   \[\leadsto \frac{{\left(e^{{\left(\sqrt[3]{\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}\right)}^{2}}\right)}^{\left(\sqrt[3]{\log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}\right)}}{e^{x}} \]
8. Add Preprocessing

Alternative 3: 7.1% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

code[x_] := N[(N[Exp[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]], $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 6.8%

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

   1. /-rgt-identity 6.8%

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

   2. associate-/r/ 6.8%

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

   3. exp-neg 6.8%

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

   4. remove-double-neg 6.8%

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

3. Simplified 6.8%

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

   1. add-exp-log 6.8%

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

6. Applied egg-rr 6.8%

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

7. Final simplification 6.8%

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

Alternative 4: 7.1% 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 6.8%

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

   1. /-rgt-identity 6.8%

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

   2. associate-/r/ 6.8%

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

   3. exp-neg 6.8%

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

   4. remove-double-neg 6.8%

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

3. Simplified 6.8%

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

5. Final simplification 6.8%

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

Reproduce

herbie shell --seed 2024021 
(FPCore (x)
  :name "expfmod (used to be hard to sample)"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))