expfmod (used to be hard to sample)

Percentage Accurate: 6.8% → 43.0%

Time: 20.1s

Alternatives: 13

Speedup: 1.0×

• could not determine a ground truth

  1. x = -2.5282304878579024e+102

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: 6.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

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

Alternative 1: 43.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \leq 0:\\ \;\;\;\;\left(1 - x\right) \cdot \left(\left(e^{x}\right) \bmod \left(\log \left(\sqrt[3]{e^{2}}\right) + \log \left(\sqrt[3]{e}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(\left(e^{x}\right) \bmod \left({\left(\sqrt[3]{\cos x}\right)}^{1.5}\right)\right) - x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (* (fmod (exp x) (sqrt (cos x))) (exp (- x))) 0.0)
   (* (- 1.0 x) (fmod (exp x) (+ (log (cbrt (exp 2.0))) (log (cbrt E)))))
   (exp (- (log (fmod (exp x) (pow (cbrt (cos x)) 1.5))) x))))

C

double code(double x) {
	double tmp;
	if ((fmod(exp(x), sqrt(cos(x))) * exp(-x)) <= 0.0) {
		tmp = (1.0 - x) * fmod(exp(x), (log(cbrt(exp(2.0))) + log(cbrt(((double) M_E)))));
	} else {
		tmp = exp((log(fmod(exp(x), pow(cbrt(cos(x)), 1.5))) - x));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(rem(exp(x), sqrt(cos(x))) * exp(Float64(-x))) <= 0.0)
		tmp = Float64(Float64(1.0 - x) * rem(exp(x), Float64(log(cbrt(exp(2.0))) + log(cbrt(exp(1))))));
	else
		tmp = exp(Float64(log(rem(exp(x), (cbrt(cos(x)) ^ 1.5))) - x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[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], 0.0], N[(N[(1.0 - x), $MachinePrecision] * N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(N[Log[N[Power[N[Exp[2.0], $MachinePrecision], 1/3], $MachinePrecision]], $MachinePrecision] + N[Log[N[Power[E, 1/3], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]), $MachinePrecision], N[Exp[N[(N[Log[N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Power[N[Power[N[Cos[x], $MachinePrecision], 1/3], $MachinePrecision], 1.5], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]], $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x))) < 0.0

   1. Initial program 4.6%

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

      1. /-rgt-identity 4.6%

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

      2. associate-/r/ 4.6%

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

      3. exp-neg 4.6%

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

      4. remove-double-neg 4.6%

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

   3. Simplified 4.6%

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

      1. add-log-exp 4.6%

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

      2. add-cube-cbrt 47.5%

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

      3. log-prod 47.5%

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

      4. pow2 47.5%

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

   6. Applied egg-rr 47.5%

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

   7. Taylor expanded in x around 0 47.5%

      \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\log \color{blue}{\left(\sqrt[3]{{\left(e^{1}\right)}^{2}}\right)} + \log \left(\sqrt[3]{e^{\sqrt{\cos x}}}\right)\right)\right)}{e^{x}} \]
   8. Step-by-step derivation

      1. unpow2 47.5%

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

      2. prod-exp 47.5%

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

      3. metadata-eval 47.5%

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

   9. Simplified 47.5%

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

   10. Taylor expanded in x around 0 47.5%

       \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\log \left(\sqrt[3]{e^{2}}\right) + \log \color{blue}{\left(\sqrt[3]{e^{1}}\right)}\right)\right)}{e^{x}} \]
   11. Step-by-step derivation

       1. exp-1-e 47.5%

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

   12. Simplified 47.5%

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

   13. Taylor expanded in x around 0 47.5%

       \[\leadsto \color{blue}{-1 \cdot \left(x \cdot \left(\left(e^{x}\right) \bmod \left(\log \left(\sqrt[3]{e}\right) + \log \left(\sqrt[3]{e^{2}}\right)\right)\right)\right) + \left(\left(e^{x}\right) \bmod \left(\log \left(\sqrt[3]{e}\right) + \log \left(\sqrt[3]{e^{2}}\right)\right)\right)} \]
   14. Step-by-step derivation

       1. associate-*r* 47.5%

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

       2. neg-mul-1 47.5%

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

       3. distribute-lft1-in 47.5%

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

       4. +-commutative 47.5%

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

       5. sub-neg 47.5%

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

   15. Simplified 47.5%

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

   if 0.0 < (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x)))

   1. Initial program 17.3%

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

      1. /-rgt-identity 17.3%

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

      2. associate-/r/ 17.3%

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

      3. exp-neg 17.3%

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

      4. remove-double-neg 17.3%

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

   3. Simplified 17.3%

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

      1. add-exp-log 17.3%

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

      2. div-exp 17.6%

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

   6. Applied egg-rr 17.6%

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

      1. pow1/2 17.6%

         \[\leadsto e^{\log \left(\left(e^{x}\right) \bmod \color{blue}{\left({\cos x}^{0.5}\right)}\right) - x} \]

      2. add-cube-cbrt 17.7%

         \[\leadsto e^{\log \left(\left(e^{x}\right) \bmod \left({\color{blue}{\left(\left(\sqrt[3]{\cos x} \cdot \sqrt[3]{\cos x}\right) \cdot \sqrt[3]{\cos x}\right)}}^{0.5}\right)\right) - x} \]

      3. pow3 17.7%

         \[\leadsto e^{\log \left(\left(e^{x}\right) \bmod \left({\color{blue}{\left({\left(\sqrt[3]{\cos x}\right)}^{3}\right)}}^{0.5}\right)\right) - x} \]

      4. pow-pow 17.6%

         \[\leadsto e^{\log \left(\left(e^{x}\right) \bmod \color{blue}{\left({\left(\sqrt[3]{\cos x}\right)}^{\left(3 \cdot 0.5\right)}\right)}\right) - x} \]

      5. metadata-eval 17.6%

         \[\leadsto e^{\log \left(\left(e^{x}\right) \bmod \left({\left(\sqrt[3]{\cos x}\right)}^{\color{blue}{1.5}}\right)\right) - x} \]

   8. Applied egg-rr 17.6%

      \[\leadsto e^{\log \left(\left(e^{x}\right) \bmod \color{blue}{\left({\left(\sqrt[3]{\cos x}\right)}^{1.5}\right)}\right) - x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 39.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \leq 0:\\ \;\;\;\;\left(1 - x\right) \cdot \left(\left(e^{x}\right) \bmod \left(\log \left(\sqrt[3]{e^{2}}\right) + \log \left(\sqrt[3]{e}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(\left(e^{x}\right) \bmod \left({\left(\sqrt[3]{\cos x}\right)}^{1.5}\right)\right) - x}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 42.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \sqrt[3]{e^{\sqrt{\cos x}}}\\ \frac{\left(\left(e^{x}\right) \bmod \left(\log \left({t\_0}^{2}\right) + \log t\_0\right)\right)}{e^{x}} \end{array} \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.9%

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

   1. /-rgt-identity 7.9%

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

   2. associate-/r/ 7.9%

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

   3. exp-neg 7.9%

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

   4. remove-double-neg 7.9%

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

3. Simplified 7.9%

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

   1. add-log-exp 7.9%

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

   2. add-cube-cbrt 39.6%

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

   3. log-prod 39.6%

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

   4. pow2 39.6%

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

6. Applied egg-rr 39.6%

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

7. Final simplification 39.6%

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

Alternative 3: 42.5% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.9%

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

   1. /-rgt-identity 7.9%

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

   2. associate-/r/ 7.9%

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

   3. exp-neg 7.9%

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

   4. remove-double-neg 7.9%

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

3. Simplified 7.9%

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

   1. add-log-exp 7.9%

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

   2. add-cube-cbrt 39.6%

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

   3. log-prod 39.6%

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

   4. pow2 39.6%

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

6. Applied egg-rr 39.6%

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

7. Taylor expanded in x around 0 38.8%

   \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\log \color{blue}{\left(\sqrt[3]{{\left(e^{1}\right)}^{2}}\right)} + \log \left(\sqrt[3]{e^{\sqrt{\cos x}}}\right)\right)\right)}{e^{x}} \]
8. Step-by-step derivation

   1. unpow2 38.8%

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

   2. prod-exp 38.8%

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

   3. metadata-eval 38.8%

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

9. Simplified 38.8%

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

10. Final simplification 38.8%

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

Alternative 4: 42.5% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (/ (fmod (exp x) (+ (log (cbrt (exp 2.0))) (log (cbrt E)))) (exp x)))

C

double code(double x) {
	return fmod(exp(x), (log(cbrt(exp(2.0))) + log(cbrt(((double) M_E))))) / exp(x);
}

Julia

function code(x)
	return Float64(rem(exp(x), Float64(log(cbrt(exp(2.0))) + log(cbrt(exp(1))))) / exp(x))
end

Wolfram

code[x_] := N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(N[Log[N[Power[N[Exp[2.0], $MachinePrecision], 1/3], $MachinePrecision]], $MachinePrecision] + N[Log[N[Power[E, 1/3], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 7.9%

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

   1. /-rgt-identity 7.9%

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

   2. associate-/r/ 7.9%

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

   3. exp-neg 7.9%

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

   4. remove-double-neg 7.9%

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

3. Simplified 7.9%

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

   1. add-log-exp 7.9%

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

   2. add-cube-cbrt 39.6%

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

   3. log-prod 39.6%

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

   4. pow2 39.6%

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

6. Applied egg-rr 39.6%

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

7. Taylor expanded in x around 0 38.8%

   \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\log \color{blue}{\left(\sqrt[3]{{\left(e^{1}\right)}^{2}}\right)} + \log \left(\sqrt[3]{e^{\sqrt{\cos x}}}\right)\right)\right)}{e^{x}} \]
8. Step-by-step derivation

   1. unpow2 38.8%

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

   2. prod-exp 38.8%

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

   3. metadata-eval 38.8%

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

9. Simplified 38.8%

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

10. Taylor expanded in x around 0 38.7%

    \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\log \left(\sqrt[3]{e^{2}}\right) + \log \color{blue}{\left(\sqrt[3]{e^{1}}\right)}\right)\right)}{e^{x}} \]
11. Step-by-step derivation

    1. exp-1-e 38.7%

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

12. Simplified 38.7%

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

13. Final simplification 38.7%

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

Alternative 5: 6.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ e^{\log \left(\left(e^{x}\right) \bmod \left({\left(\sqrt[3]{\cos x}\right)}^{1.5}\right)\right) - x} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (exp (- (log (fmod (exp x) (pow (cbrt (cos x)) 1.5))) x)))

C

double code(double x) {
	return exp((log(fmod(exp(x), pow(cbrt(cos(x)), 1.5))) - x));
}

Julia

function code(x)
	return exp(Float64(log(rem(exp(x), (cbrt(cos(x)) ^ 1.5))) - x))
end

Wolfram

code[x_] := N[Exp[N[(N[Log[N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Power[N[Power[N[Cos[x], $MachinePrecision], 1/3], $MachinePrecision], 1.5], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]], $MachinePrecision] - x), $MachinePrecision]], $MachinePrecision]

Derivation

1. Initial program 7.9%

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

   1. /-rgt-identity 7.9%

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

   2. associate-/r/ 7.9%

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

   3. exp-neg 7.9%

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

   4. remove-double-neg 7.9%

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

3. Simplified 7.9%

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

   1. add-exp-log 7.9%

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

   2. div-exp 7.9%

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

6. Applied egg-rr 7.9%

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

   1. pow1/2 7.9%

      \[\leadsto e^{\log \left(\left(e^{x}\right) \bmod \color{blue}{\left({\cos x}^{0.5}\right)}\right) - x} \]

   2. add-cube-cbrt 8.0%

      \[\leadsto e^{\log \left(\left(e^{x}\right) \bmod \left({\color{blue}{\left(\left(\sqrt[3]{\cos x} \cdot \sqrt[3]{\cos x}\right) \cdot \sqrt[3]{\cos x}\right)}}^{0.5}\right)\right) - x} \]

   3. pow3 8.0%

      \[\leadsto e^{\log \left(\left(e^{x}\right) \bmod \left({\color{blue}{\left({\left(\sqrt[3]{\cos x}\right)}^{3}\right)}}^{0.5}\right)\right) - x} \]

   4. pow-pow 8.0%

      \[\leadsto e^{\log \left(\left(e^{x}\right) \bmod \color{blue}{\left({\left(\sqrt[3]{\cos x}\right)}^{\left(3 \cdot 0.5\right)}\right)}\right) - x} \]

   5. metadata-eval 8.0%

      \[\leadsto e^{\log \left(\left(e^{x}\right) \bmod \left({\left(\sqrt[3]{\cos x}\right)}^{\color{blue}{1.5}}\right)\right) - x} \]

8. Applied egg-rr 8.0%

   \[\leadsto e^{\log \left(\left(e^{x}\right) \bmod \color{blue}{\left({\left(\sqrt[3]{\cos x}\right)}^{1.5}\right)}\right) - x} \]

9. Final simplification 8.0%

   \[\leadsto e^{\log \left(\left(e^{x}\right) \bmod \left({\left(\sqrt[3]{\cos x}\right)}^{1.5}\right)\right) - x} \]
10. Add Preprocessing

Alternative 6: 6.8% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.9%

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

   1. /-rgt-identity 7.9%

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

   2. associate-/r/ 7.9%

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

   3. exp-neg 7.9%

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

   4. remove-double-neg 7.9%

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

3. Simplified 7.9%

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

   1. add-exp-log 7.9%

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

   2. div-exp 7.9%

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

6. Applied egg-rr 7.9%

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

7. Final simplification 7.9%

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

Alternative 7: 6.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.9%

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

   1. /-rgt-identity 7.9%

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

   2. associate-/r/ 7.9%

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

   3. exp-neg 7.9%

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

   4. remove-double-neg 7.9%

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

3. Simplified 7.9%

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

   1. clear-num 7.9%

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

   2. inv-pow 7.9%

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

6. Applied egg-rr 7.9%

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

   1. unpow-1 7.9%

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

8. Applied egg-rr 7.9%

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

9. Final simplification 7.9%

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

Alternative 8: 6.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.9%

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

   1. /-rgt-identity 7.9%

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

   2. associate-/r/ 7.9%

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

   3. exp-neg 7.9%

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

   4. remove-double-neg 7.9%

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

3. Simplified 7.9%

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

5. Final simplification 7.9%

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

Alternative 9: 6.6% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.9%

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

   1. /-rgt-identity 7.9%

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

   2. associate-/r/ 7.9%

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

   3. exp-neg 7.9%

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

   4. remove-double-neg 7.9%

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

3. Simplified 7.9%

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

5. Taylor expanded in x around 0 7.4%

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

6. Final simplification 7.4%

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

Alternative 10: 6.4% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.9%

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

   1. /-rgt-identity 7.9%

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

   2. associate-/r/ 7.9%

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

   3. exp-neg 7.9%

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

   4. remove-double-neg 7.9%

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

3. Simplified 7.9%

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

5. Taylor expanded in x around 0 6.9%

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

   1. add-exp-log 6.9%

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

   2. div-exp 7.0%

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

7. Applied egg-rr 7.0%

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

8. Final simplification 7.0%

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

Alternative 11: 6.3% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.9%

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

   1. /-rgt-identity 7.9%

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

   2. associate-/r/ 7.9%

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

   3. exp-neg 7.9%

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

   4. remove-double-neg 7.9%

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

3. Simplified 7.9%

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

5. Taylor expanded in x around 0 6.9%

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

6. Final simplification 6.9%

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

Alternative 12: 5.9% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.9%

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

   1. /-rgt-identity 7.9%

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

   2. associate-/r/ 7.9%

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

   3. exp-neg 7.9%

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

   4. remove-double-neg 7.9%

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

3. Simplified 7.9%

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

5. Taylor expanded in x around 0 6.6%

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

   1. +-commutative 6.6%

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

   2. mul-1-neg 6.6%

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

   3. unsub-neg 6.6%

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

   4. *-lft-identity 6.6%

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

   5. distribute-rgt-out-- 6.6%

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

7. Simplified 6.6%

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

8. Taylor expanded in x around 0 6.4%

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

9. Final simplification 6.4%

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

Alternative 13: 5.4% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 7.9%

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

   1. /-rgt-identity 7.9%

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

   2. associate-/r/ 7.9%

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

   3. exp-neg 7.9%

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

   4. remove-double-neg 7.9%

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

3. Simplified 7.9%

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

5. Taylor expanded in x around 0 6.9%

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

6. Taylor expanded in x around 0 5.9%

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

7. Final simplification 5.9%

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

Reproduce

herbie shell --seed 2024077 
(FPCore (x)
  :name "expfmod (used to be hard to sample)"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))