expfmod (used to be hard to sample)

Percentage Accurate: 13.6% → 19.5%

Time: 1.5min

Alternatives: 12

Speedup: 1.0×

• could not determine a ground truth

  1. x = 1.9813490492686707e-261

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: 13.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

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

Alternative 1: 19.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\\ \mathbf{if}\;t\_0 \cdot e^{-x} \leq 0:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\log \left(\sqrt[3]{e^{2}}\right) + \log \left(\sqrt[3]{e}\right)\right)\right) \cdot \left(1 - x\right)\\ \mathbf{else}:\\ \;\;\;\;e^{\log t\_0 - x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fmod (exp x) (sqrt (cos x)))))
   (if (<= (* t_0 (exp (- x))) 0.0)
     (* (fmod (exp x) (+ (log (cbrt (exp 2.0))) (log (cbrt E)))) (- 1.0 x))
     (exp (- (log t_0) x)))))

C

double code(double x) {
	double t_0 = fmod(exp(x), sqrt(cos(x)));
	double tmp;
	if ((t_0 * exp(-x)) <= 0.0) {
		tmp = fmod(exp(x), (log(cbrt(exp(2.0))) + log(cbrt(((double) M_E))))) * (1.0 - x);
	} else {
		tmp = exp((log(t_0) - x));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = rem(exp(x), sqrt(cos(x)))
	tmp = 0.0
	if (Float64(t_0 * exp(Float64(-x))) <= 0.0)
		tmp = Float64(rem(exp(x), Float64(log(cbrt(exp(2.0))) + log(cbrt(exp(1))))) * Float64(1.0 - x));
	else
		tmp = exp(Float64(log(t_0) - x));
	end
	return tmp
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]}, If[LessEqual[N[(t$95$0 * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision], 0.0], 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[(1.0 - x), $MachinePrecision]), $MachinePrecision], N[Exp[N[(N[Log[t$95$0], $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 3.1%

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

      1. /-rgt-identity 3.1%

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

      2. associate-/r/ 3.1%

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

      3. exp-neg 3.1%

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

      4. remove-double-neg 3.1%

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

   3. Simplified 3.1%

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

      1. add-log-exp 3.1%

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

      2. add-cube-cbrt 100.0%

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

      3. log-prod 100.0%

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

      4. pow2 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around 0 100.0%

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

      1. unpow2 100.0%

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

      2. prod-exp 100.0%

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

      3. metadata-eval 100.0%

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

   9. Simplified 100.0%

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

   10. Taylor expanded in x around 0 100.0%

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

       1. exp-1-e 100.0%

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

   12. Simplified 100.0%

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

   13. Taylor expanded in x around 0 100.0%

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

       1. associate-*r* 100.0%

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

       2. neg-mul-1 100.0%

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

       3. distribute-lft1-in 100.0%

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

       4. +-commutative 100.0%

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

       5. sub-neg 100.0%

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

   15. Simplified 100.0%

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

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

   1. Initial program 15.8%

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

      1. exp-neg 15.8%

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

      2. div-inv 15.9%

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

      3. clear-num 15.9%

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

      4. inv-pow 15.9%

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

      5. pow-to-exp 15.9%

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

      6. diff-log 15.9%

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

      7. add-log-exp 16.0%

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

   4. Applied egg-rr 16.0%

      \[\leadsto \color{blue}{e^{\left(x - \log \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right) \cdot -1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 21.2%

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

Alternative 2: 19.2% 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 15.0%

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

   1. /-rgt-identity 15.0%

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

   2. associate-/r/ 15.0%

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

   3. exp-neg 15.1%

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

   4. remove-double-neg 15.1%

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

3. Simplified 15.1%

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

   1. add-log-exp 15.1%

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

   2. add-cube-cbrt 20.9%

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

   3. log-prod 20.9%

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

   4. pow2 20.9%

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

6. Applied egg-rr 20.9%

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

Alternative 3: 17.6% 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 15.0%

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

   1. /-rgt-identity 15.0%

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

   2. associate-/r/ 15.0%

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

   3. exp-neg 15.1%

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

   4. remove-double-neg 15.1%

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

3. Simplified 15.1%

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

   1. add-log-exp 15.1%

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

   2. add-cube-cbrt 20.9%

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

   3. log-prod 20.9%

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

   4. pow2 20.9%

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

6. Applied egg-rr 20.9%

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

7. Taylor expanded in x around 0 19.7%

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

   1. unpow2 19.7%

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

   2. prod-exp 19.7%

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

   3. metadata-eval 19.7%

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

9. Simplified 19.7%

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

10. Final simplification 19.7%

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

Alternative 4: 17.6% 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 15.0%

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

   1. /-rgt-identity 15.0%

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

   2. associate-/r/ 15.0%

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

   3. exp-neg 15.1%

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

   4. remove-double-neg 15.1%

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

3. Simplified 15.1%

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

   1. add-log-exp 15.1%

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

   2. add-cube-cbrt 20.9%

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

   3. log-prod 20.9%

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

   4. pow2 20.9%

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

6. Applied egg-rr 20.9%

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

7. Taylor expanded in x around 0 19.7%

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

   1. unpow2 19.7%

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

   2. prod-exp 19.7%

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

   3. metadata-eval 19.7%

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

9. Simplified 19.7%

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

10. Taylor expanded in x around 0 19.7%

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

    1. exp-1-e 19.7%

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

12. Simplified 19.7%

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

Alternative 5: 13.7% 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 15.0%

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

   1. exp-neg 15.1%

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

   2. div-inv 15.1%

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

   3. clear-num 15.1%

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

   4. inv-pow 15.1%

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

   5. pow-to-exp 15.1%

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

   6. diff-log 15.1%

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

   7. add-log-exp 15.2%

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

4. Applied egg-rr 15.2%

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

5. Final simplification 15.2%

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

Alternative 6: 13.7% 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 15.0%

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

   1. /-rgt-identity 15.0%

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

   2. associate-/r/ 15.0%

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

   3. exp-neg 15.1%

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

   4. remove-double-neg 15.1%

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

3. Simplified 15.1%

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

Alternative 7: 12.7% 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 15.0%

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

   1. /-rgt-identity 15.0%

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

   2. associate-/r/ 15.0%

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

   3. exp-neg 15.1%

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

   4. remove-double-neg 15.1%

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

3. Simplified 15.1%

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

5. Taylor expanded in x around 0 14.4%

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

Alternative 8: 11.9% 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 15.0%

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

   1. /-rgt-identity 15.0%

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

   2. associate-/r/ 15.0%

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

   3. exp-neg 15.1%

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

   4. remove-double-neg 15.1%

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

3. Simplified 15.1%

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

   1. add-log-exp 15.1%

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

   2. add-cube-cbrt 20.9%

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

   3. log-prod 20.9%

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

   4. pow2 20.9%

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

6. Applied egg-rr 20.9%

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

7. Taylor expanded in x around 0 19.7%

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

   1. unpow2 19.7%

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

   2. prod-exp 19.7%

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

   3. metadata-eval 19.7%

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

9. Simplified 19.7%

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

10. Taylor expanded in x around 0 19.7%

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

    1. exp-1-e 19.7%

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

12. Simplified 19.7%

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

    1. metadata-eval 19.7%

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

    2. prod-exp 19.7%

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

    3. e-exp-1 19.7%

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

    4. e-exp-1 19.7%

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

    5. cbrt-unprod 19.6%

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

    6. log-prod 19.6%

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

    7. add-cube-cbrt 13.7%

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

    8. log-E 13.7%

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

    9. add-exp-log 13.7%

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

    10. log-div 13.7%

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

    11. add-log-exp 13.8%

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

14. Applied egg-rr 13.8%

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

Alternative 9: 11.9% 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 15.0%

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

   1. /-rgt-identity 15.0%

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

   2. associate-/r/ 15.0%

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

   3. exp-neg 15.1%

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

   4. remove-double-neg 15.1%

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

3. Simplified 15.1%

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

5. Taylor expanded in x around 0 13.7%

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

Alternative 10: 10.4% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \left(\left(e^{x}\right) \bmod 1\right) \cdot \left(\left(1 - x\right) + x \cdot \left(x \cdot 0.5\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (fmod (exp x) 1.0) (+ (- 1.0 x) (* x (* x 0.5)))))

C

double code(double x) {
	return fmod(exp(x), 1.0) * ((1.0 - x) + (x * (x * 0.5)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = mod(exp(x), 1.0d0) * ((1.0d0 - x) + (x * (x * 0.5d0)))
end function

Python

def code(x):
	return math.fmod(math.exp(x), 1.0) * ((1.0 - x) + (x * (x * 0.5)))

Julia

function code(x)
	return Float64(rem(exp(x), 1.0) * Float64(Float64(1.0 - x) + Float64(x * Float64(x * 0.5))))
end

Wolfram

code[x_] := N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] * N[(N[(1.0 - x), $MachinePrecision] + N[(x * N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 15.0%

   \[\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0 12.4%

   \[\leadsto \color{blue}{x \cdot \left(-1 \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) + 0.5 \cdot \left(x \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right)\right) + \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)} \]
4. Step-by-step derivation

   1. +-commutative 12.4%

      \[\leadsto \color{blue}{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) + x \cdot \left(-1 \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) + 0.5 \cdot \left(x \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right)\right)} \]

   2. neg-mul-1 12.4%

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

   3. distribute-lft-in 12.4%

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

   4. distribute-rgt-neg-in 12.4%

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

   5. mul-1-neg 12.4%

      \[\leadsto \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) + \left(\color{blue}{-1 \cdot \left(x \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right)} + x \cdot \left(0.5 \cdot \left(x \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right)\right)\right) \]

   6. associate-+r+ 12.4%

      \[\leadsto \color{blue}{\left(\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) + -1 \cdot \left(x \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right)\right) + x \cdot \left(0.5 \cdot \left(x \cdot \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\right)\right)} \]

5. Simplified 12.4%

   \[\leadsto \color{blue}{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot \left(\left(\left(-x\right) + 1\right) + x \cdot \left(x \cdot 0.5\right)\right)} \]

6. Taylor expanded in x around 0 11.8%

   \[\leadsto \left(\left(e^{x}\right) \bmod \color{blue}{1}\right) \cdot \left(\left(\left(-x\right) + 1\right) + x \cdot \left(x \cdot 0.5\right)\right) \]

7. Final simplification 11.8%

   \[\leadsto \left(\left(e^{x}\right) \bmod 1\right) \cdot \left(\left(1 - x\right) + x \cdot \left(x \cdot 0.5\right)\right) \]
8. Add Preprocessing

Alternative 11: 9.7% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 15.0%

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

   1. /-rgt-identity 15.0%

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

   2. associate-/r/ 15.0%

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

   3. exp-neg 15.1%

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

   4. remove-double-neg 15.1%

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

3. Simplified 15.1%

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

5. Taylor expanded in x around 0 13.7%

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

6. Taylor expanded in x around 0 10.9%

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

   1. +-commutative 10.9%

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

   2. *-lft-identity 10.9%

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

   3. associate-*r* 10.9%

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

   4. distribute-rgt-out 10.9%

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

   5. mul-1-neg 10.9%

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

   6. unsub-neg 10.9%

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

8. Simplified 10.9%

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

Alternative 12: 7.9% 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 15.0%

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

   1. /-rgt-identity 15.0%

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

   2. associate-/r/ 15.0%

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

   3. exp-neg 15.1%

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

   4. remove-double-neg 15.1%

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

3. Simplified 15.1%

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

5. Taylor expanded in x around 0 13.7%

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

6. Taylor expanded in x around 0 8.9%

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

Reproduce

herbie shell --seed 2024089 
(FPCore (x)
  :name "expfmod (used to be hard to sample)"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))