expfmod (used to be hard to sample)

Percentage Accurate: 7.2% → 63.0%

Time: 22.1s

Alternatives: 11

Speedup: 4.8×

• could not determine a ground truth

  1. x = -4.6308622968988173e+89

Specification

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

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

Initial Program: 7.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

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

Alternative 1: 63.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\\ t_1 := t\_0 \cdot e^{-x}\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\log \left(\sqrt[3]{e}\right) + \log \left({\left(\sqrt[3]{e}\right)}^{2}\right)}\right)\right)\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;\frac{{\left({t\_0}^{3}\right)}^{0.3333333333333333}}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 \bmod 1\right)}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fmod (exp x) (sqrt (cos x)))) (t_1 (* t_0 (exp (- x)))))
   (if (<= t_1 0.0)
     (fmod (exp x) (sqrt (+ (log (cbrt E)) (log (pow (cbrt E) 2.0)))))
     (if (<= t_1 2.0)
       (/ (pow (pow t_0 3.0) 0.3333333333333333) (exp x))
       (/ (fmod 1.0 1.0) (+ x 1.0))))))

C

double code(double x) {
	double t_0 = fmod(exp(x), sqrt(cos(x)));
	double t_1 = t_0 * exp(-x);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = fmod(exp(x), sqrt((log(cbrt(((double) M_E))) + log(pow(cbrt(((double) M_E)), 2.0)))));
	} else if (t_1 <= 2.0) {
		tmp = pow(pow(t_0, 3.0), 0.3333333333333333) / exp(x);
	} else {
		tmp = fmod(1.0, 1.0) / (x + 1.0);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = rem(exp(x), sqrt(cos(x)))
	t_1 = Float64(t_0 * exp(Float64(-x)))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = rem(exp(x), sqrt(Float64(log(cbrt(exp(1))) + log((cbrt(exp(1)) ^ 2.0)))));
	elseif (t_1 <= 2.0)
		tmp = Float64(((t_0 ^ 3.0) ^ 0.3333333333333333) / exp(x));
	else
		tmp = Float64(rem(1.0, 1.0) / Float64(x + 1.0));
	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]}, Block[{t$95$1 = N[(t$95$0 * N[Exp[(-x)], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Sqrt[N[(N[Log[N[Power[E, 1/3], $MachinePrecision]], $MachinePrecision] + N[Log[N[Power[N[Power[E, 1/3], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[(N[Power[N[Power[t$95$0, 3.0], $MachinePrecision], 0.3333333333333333], $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = 1.0, TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 4.1%

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

      1. /-rgt-identity 4.1%

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

      2. associate-/r/ 4.1%

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

      3. exp-neg 4.1%

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

      4. remove-double-neg 4.1%

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

   3. Simplified 4.1%

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

      1. add-log-exp 4.1%

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

      2. add-cube-cbrt 55.7%

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

      3. log-prod 55.7%

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

      4. pow2 55.7%

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

   6. Applied egg-rr 55.7%

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

   7. Taylor expanded in x around 0 55.7%

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

      1. exp-1-e 55.7%

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

   9. Simplified 55.7%

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

   10. Taylor expanded in x around 0 55.7%

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

       1. exp-1-e 55.7%

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

   12. Simplified 55.7%

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

   13. Taylor expanded in x around 0 55.7%

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

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

   1. Initial program 86.1%

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

      1. /-rgt-identity 86.1%

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

      2. associate-/r/ 85.6%

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

      3. exp-neg 86.2%

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

      4. remove-double-neg 86.2%

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

   3. Simplified 86.2%

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

      1. add-cbrt-cube 85.3%

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

      2. pow1/3 86.2%

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

      3. pow3 86.2%

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

   6. Applied egg-rr 86.2%

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

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

   1. Initial program 0.0%

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

      1. /-rgt-identity 0.0%

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

      2. associate-/r/ 0.0%

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

      3. exp-neg 0.0%

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

      4. remove-double-neg 0.0%

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

   3. Simplified 0.0%

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

      1. add-cbrt-cube 0.0%

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

      2. pow3 0.0%

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

      3. pow1/2 0.0%

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

      4. pow-pow 0.0%

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

      5. metadata-eval 0.0%

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

   6. Applied egg-rr 0.0%

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

   7. Taylor expanded in x around 0 0.0%

      \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt[3]{{\cos x}^{1.5}}\right)\right)}{\color{blue}{1 + x}} \]
   8. Step-by-step derivation

      1. +-commutative 0.0%

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt[3]{{\cos x}^{1.5}}\right)\right)}{\color{blue}{x + 1}} \]

   9. Simplified 0.0%

      \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt[3]{{\cos x}^{1.5}}\right)\right)}{\color{blue}{x + 1}} \]

   10. Taylor expanded in x around 0 0.0%

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

   11. Taylor expanded in x around 0 100.0%

       \[\leadsto \frac{\left(\color{blue}{1} \bmod 1\right)}{x + 1} \]
3. Recombined 3 regimes into one program.

4. Final simplification 65.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \leq 0:\\ \;\;\;\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\log \left(\sqrt[3]{e}\right) + \log \left({\left(\sqrt[3]{e}\right)}^{2}\right)}\right)\right)\\ \mathbf{elif}\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \leq 2:\\ \;\;\;\;\frac{{\left({\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}^{3}\right)}^{0.3333333333333333}}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 \bmod 1\right)}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 63.0% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	t_0 = cbrt(exp(cos(x)))
	tmp = 0.0
	if (Float64(rem(exp(x), sqrt(cos(x))) * exp(Float64(-x))) <= 2.0)
		tmp = Float64(rem(exp(x), sqrt(Float64(log((t_0 ^ 2.0)) + log(t_0)))) / exp(x));
	else
		tmp = Float64(rem(1.0, 1.0) / Float64(x + 1.0));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Power[N[Exp[N[Cos[x], $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision]}, 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], 2.0], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Sqrt[N[(N[Log[N[Power[t$95$0, 2.0], $MachinePrecision]], $MachinePrecision] + N[Log[t$95$0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = 1.0, TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[(x + 1.0), $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))) < 2

   1. Initial program 9.9%

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

      1. /-rgt-identity 9.9%

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

      2. associate-/r/ 9.9%

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

      3. exp-neg 9.9%

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

      4. remove-double-neg 9.9%

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

   3. Simplified 9.9%

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

      1. add-log-exp 9.9%

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

      2. add-cube-cbrt 57.7%

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

      3. log-prod 57.7%

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

      4. pow2 57.7%

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

   6. Applied egg-rr 57.7%

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

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

   1. Initial program 0.0%

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

      1. /-rgt-identity 0.0%

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

      2. associate-/r/ 0.0%

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

      3. exp-neg 0.0%

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

      4. remove-double-neg 0.0%

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

   3. Simplified 0.0%

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

      1. add-cbrt-cube 0.0%

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

      2. pow3 0.0%

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

      3. pow1/2 0.0%

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

      4. pow-pow 0.0%

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

      5. metadata-eval 0.0%

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

   6. Applied egg-rr 0.0%

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

   7. Taylor expanded in x around 0 0.0%

      \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt[3]{{\cos x}^{1.5}}\right)\right)}{\color{blue}{1 + x}} \]
   8. Step-by-step derivation

      1. +-commutative 0.0%

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt[3]{{\cos x}^{1.5}}\right)\right)}{\color{blue}{x + 1}} \]

   9. Simplified 0.0%

      \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt[3]{{\cos x}^{1.5}}\right)\right)}{\color{blue}{x + 1}} \]

   10. Taylor expanded in x around 0 0.0%

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

   11. Taylor expanded in x around 0 100.0%

       \[\leadsto \frac{\left(\color{blue}{1} \bmod 1\right)}{x + 1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 32.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\\ t_1 := e^{-x}\\ t_2 := t\_0 \cdot t\_1\\ \mathbf{if}\;t\_2 \leq 4 \cdot 10^{-14}:\\ \;\;\;\;t\_1 \cdot \log \left(e^{\left(\left(x \cdot \left(1 + \frac{1}{x}\right)\right) \bmod \left(1 + -0.25 \cdot {x}^{2}\right)\right)}\right)\\ \mathbf{elif}\;t\_2 \leq 2:\\ \;\;\;\;\frac{{\left({t\_0}^{3}\right)}^{0.3333333333333333}}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 \bmod 1\right)}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fmod (exp x) (sqrt (cos x))))
        (t_1 (exp (- x)))
        (t_2 (* t_0 t_1)))
   (if (<= t_2 4e-14)
     (*
      t_1
      (log (exp (fmod (* x (+ 1.0 (/ 1.0 x))) (+ 1.0 (* -0.25 (pow x 2.0)))))))
     (if (<= t_2 2.0)
       (/ (pow (pow t_0 3.0) 0.3333333333333333) (exp x))
       (/ (fmod 1.0 1.0) (+ x 1.0))))))

C

double code(double x) {
	double t_0 = fmod(exp(x), sqrt(cos(x)));
	double t_1 = exp(-x);
	double t_2 = t_0 * t_1;
	double tmp;
	if (t_2 <= 4e-14) {
		tmp = t_1 * log(exp(fmod((x * (1.0 + (1.0 / x))), (1.0 + (-0.25 * pow(x, 2.0))))));
	} else if (t_2 <= 2.0) {
		tmp = pow(pow(t_0, 3.0), 0.3333333333333333) / exp(x);
	} else {
		tmp = fmod(1.0, 1.0) / (x + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = mod(exp(x), sqrt(cos(x)))
    t_1 = exp(-x)
    t_2 = t_0 * t_1
    if (t_2 <= 4d-14) then
        tmp = t_1 * log(exp(mod((x * (1.0d0 + (1.0d0 / x))), (1.0d0 + ((-0.25d0) * (x ** 2.0d0))))))
    else if (t_2 <= 2.0d0) then
        tmp = ((t_0 ** 3.0d0) ** 0.3333333333333333d0) / exp(x)
    else
        tmp = mod(1.0d0, 1.0d0) / (x + 1.0d0)
    end if
    code = tmp
end function

Python

def code(x):
	t_0 = math.fmod(math.exp(x), math.sqrt(math.cos(x)))
	t_1 = math.exp(-x)
	t_2 = t_0 * t_1
	tmp = 0
	if t_2 <= 4e-14:
		tmp = t_1 * math.log(math.exp(math.fmod((x * (1.0 + (1.0 / x))), (1.0 + (-0.25 * math.pow(x, 2.0))))))
	elif t_2 <= 2.0:
		tmp = math.pow(math.pow(t_0, 3.0), 0.3333333333333333) / math.exp(x)
	else:
		tmp = math.fmod(1.0, 1.0) / (x + 1.0)
	return tmp

Julia

function code(x)
	t_0 = rem(exp(x), sqrt(cos(x)))
	t_1 = exp(Float64(-x))
	t_2 = Float64(t_0 * t_1)
	tmp = 0.0
	if (t_2 <= 4e-14)
		tmp = Float64(t_1 * log(exp(rem(Float64(x * Float64(1.0 + Float64(1.0 / x))), Float64(1.0 + Float64(-0.25 * (x ^ 2.0)))))));
	elseif (t_2 <= 2.0)
		tmp = Float64(((t_0 ^ 3.0) ^ 0.3333333333333333) / exp(x));
	else
		tmp = Float64(rem(1.0, 1.0) / Float64(x + 1.0));
	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]}, Block[{t$95$1 = N[Exp[(-x)], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 * t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, 4e-14], N[(t$95$1 * N[Log[N[Exp[N[With[{TMP1 = N[(x * N[(1.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], TMP2 = N[(1.0 + N[(-0.25 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2.0], N[(N[Power[N[Power[t$95$0, 3.0], $MachinePrecision], 0.3333333333333333], $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = 1.0, TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x))) < 4e-14

   1. Initial program 4.2%

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

      1. add-log-exp 4.2%

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

   4. Applied egg-rr 4.2%

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

   5. Taylor expanded in x around 0 4.2%

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

   6. Taylor expanded in x around 0 4.2%

      \[\leadsto \log \left(e^{\left(\color{blue}{\left(1 + x\right)} \bmod \left(1 + -0.25 \cdot {x}^{2}\right)\right)}\right) \cdot e^{-x} \]
   7. Step-by-step derivation

      1. +-commutative 4.2%

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt[3]{{\cos x}^{1.5}}\right)\right)}{\color{blue}{x + 1}} \]

   8. Simplified 4.2%

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

   9. Taylor expanded in x around inf 13.1%

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

   if 4e-14 < (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x))) < 2

   1. Initial program 90.0%

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

      1. /-rgt-identity 90.0%

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

      2. associate-/r/ 89.6%

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

      3. exp-neg 90.1%

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

      4. remove-double-neg 90.1%

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

   3. Simplified 90.1%

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

      1. add-cbrt-cube 89.3%

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

      2. pow1/3 90.1%

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

      3. pow3 90.1%

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

   6. Applied egg-rr 90.1%

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

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

   1. Initial program 0.0%

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

      1. /-rgt-identity 0.0%

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

      2. associate-/r/ 0.0%

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

      3. exp-neg 0.0%

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

      4. remove-double-neg 0.0%

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

   3. Simplified 0.0%

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

      1. add-cbrt-cube 0.0%

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

      2. pow3 0.0%

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

      3. pow1/2 0.0%

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

      4. pow-pow 0.0%

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

      5. metadata-eval 0.0%

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

   6. Applied egg-rr 0.0%

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

   7. Taylor expanded in x around 0 0.0%

      \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt[3]{{\cos x}^{1.5}}\right)\right)}{\color{blue}{1 + x}} \]
   8. Step-by-step derivation

      1. +-commutative 0.0%

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt[3]{{\cos x}^{1.5}}\right)\right)}{\color{blue}{x + 1}} \]

   9. Simplified 0.0%

      \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt[3]{{\cos x}^{1.5}}\right)\right)}{\color{blue}{x + 1}} \]

   10. Taylor expanded in x around 0 0.0%

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

   11. Taylor expanded in x around 0 100.0%

       \[\leadsto \frac{\left(\color{blue}{1} \bmod 1\right)}{x + 1} \]
3. Recombined 3 regimes into one program.

4. Final simplification 32.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \leq 4 \cdot 10^{-14}:\\ \;\;\;\;e^{-x} \cdot \log \left(e^{\left(\left(x \cdot \left(1 + \frac{1}{x}\right)\right) \bmod \left(1 + -0.25 \cdot {x}^{2}\right)\right)}\right)\\ \mathbf{elif}\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \leq 2:\\ \;\;\;\;\frac{{\left({\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}^{3}\right)}^{0.3333333333333333}}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 \bmod 1\right)}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 62.6% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	tmp = 0.0
	if (Float64(rem(exp(x), sqrt(cos(x))) * exp(Float64(-x))) <= 2.0)
		tmp = Float64(rem(exp(x), sqrt(Float64(log((cbrt(exp(cos(x))) ^ 2.0)) + log(cbrt(exp(1)))))) / exp(x));
	else
		tmp = Float64(rem(1.0, 1.0) / Float64(x + 1.0));
	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], 2.0], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Sqrt[N[(N[Log[N[Power[N[Power[N[Exp[N[Cos[x], $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision] + N[Log[N[Power[E, 1/3], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = 1.0, TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[(x + 1.0), $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))) < 2

   1. Initial program 9.9%

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

      1. /-rgt-identity 9.9%

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

      2. associate-/r/ 9.9%

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

      3. exp-neg 9.9%

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

      4. remove-double-neg 9.9%

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

   3. Simplified 9.9%

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

      1. add-log-exp 9.9%

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

      2. add-cube-cbrt 57.7%

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

      3. log-prod 57.7%

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

      4. pow2 57.7%

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

   6. Applied egg-rr 57.7%

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

   7. Taylor expanded in x around 0 56.4%

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

      1. exp-1-e 56.4%

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

   9. Simplified 56.4%

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

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

   1. Initial program 0.0%

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

      1. /-rgt-identity 0.0%

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

      2. associate-/r/ 0.0%

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

      3. exp-neg 0.0%

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

      4. remove-double-neg 0.0%

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

   3. Simplified 0.0%

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

      1. add-cbrt-cube 0.0%

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

      2. pow3 0.0%

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

      3. pow1/2 0.0%

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

      4. pow-pow 0.0%

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

      5. metadata-eval 0.0%

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

   6. Applied egg-rr 0.0%

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

   7. Taylor expanded in x around 0 0.0%

      \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt[3]{{\cos x}^{1.5}}\right)\right)}{\color{blue}{1 + x}} \]
   8. Step-by-step derivation

      1. +-commutative 0.0%

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt[3]{{\cos x}^{1.5}}\right)\right)}{\color{blue}{x + 1}} \]

   9. Simplified 0.0%

      \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt[3]{{\cos x}^{1.5}}\right)\right)}{\color{blue}{x + 1}} \]

   10. Taylor expanded in x around 0 0.0%

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

   11. Taylor expanded in x around 0 100.0%

       \[\leadsto \frac{\left(\color{blue}{1} \bmod 1\right)}{x + 1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 32.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\\ t_1 := e^{-x}\\ t_2 := t\_0 \cdot t\_1\\ \mathbf{if}\;t\_2 \leq 4 \cdot 10^{-14}:\\ \;\;\;\;t\_1 \cdot \log \left(e^{\left(\left(x \cdot \left(1 + \frac{1}{x}\right)\right) \bmod \left(1 + -0.25 \cdot {x}^{2}\right)\right)}\right)\\ \mathbf{elif}\;t\_2 \leq 2:\\ \;\;\;\;\frac{t\_0}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 \bmod 1\right)}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fmod (exp x) (sqrt (cos x))))
        (t_1 (exp (- x)))
        (t_2 (* t_0 t_1)))
   (if (<= t_2 4e-14)
     (*
      t_1
      (log (exp (fmod (* x (+ 1.0 (/ 1.0 x))) (+ 1.0 (* -0.25 (pow x 2.0)))))))
     (if (<= t_2 2.0) (/ t_0 (exp x)) (/ (fmod 1.0 1.0) (+ x 1.0))))))

C

double code(double x) {
	double t_0 = fmod(exp(x), sqrt(cos(x)));
	double t_1 = exp(-x);
	double t_2 = t_0 * t_1;
	double tmp;
	if (t_2 <= 4e-14) {
		tmp = t_1 * log(exp(fmod((x * (1.0 + (1.0 / x))), (1.0 + (-0.25 * pow(x, 2.0))))));
	} else if (t_2 <= 2.0) {
		tmp = t_0 / exp(x);
	} else {
		tmp = fmod(1.0, 1.0) / (x + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_0 = mod(exp(x), sqrt(cos(x)))
    t_1 = exp(-x)
    t_2 = t_0 * t_1
    if (t_2 <= 4d-14) then
        tmp = t_1 * log(exp(mod((x * (1.0d0 + (1.0d0 / x))), (1.0d0 + ((-0.25d0) * (x ** 2.0d0))))))
    else if (t_2 <= 2.0d0) then
        tmp = t_0 / exp(x)
    else
        tmp = mod(1.0d0, 1.0d0) / (x + 1.0d0)
    end if
    code = tmp
end function

Python

def code(x):
	t_0 = math.fmod(math.exp(x), math.sqrt(math.cos(x)))
	t_1 = math.exp(-x)
	t_2 = t_0 * t_1
	tmp = 0
	if t_2 <= 4e-14:
		tmp = t_1 * math.log(math.exp(math.fmod((x * (1.0 + (1.0 / x))), (1.0 + (-0.25 * math.pow(x, 2.0))))))
	elif t_2 <= 2.0:
		tmp = t_0 / math.exp(x)
	else:
		tmp = math.fmod(1.0, 1.0) / (x + 1.0)
	return tmp

Julia

function code(x)
	t_0 = rem(exp(x), sqrt(cos(x)))
	t_1 = exp(Float64(-x))
	t_2 = Float64(t_0 * t_1)
	tmp = 0.0
	if (t_2 <= 4e-14)
		tmp = Float64(t_1 * log(exp(rem(Float64(x * Float64(1.0 + Float64(1.0 / x))), Float64(1.0 + Float64(-0.25 * (x ^ 2.0)))))));
	elseif (t_2 <= 2.0)
		tmp = Float64(t_0 / exp(x));
	else
		tmp = Float64(rem(1.0, 1.0) / Float64(x + 1.0));
	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]}, Block[{t$95$1 = N[Exp[(-x)], $MachinePrecision]}, Block[{t$95$2 = N[(t$95$0 * t$95$1), $MachinePrecision]}, If[LessEqual[t$95$2, 4e-14], N[(t$95$1 * N[Log[N[Exp[N[With[{TMP1 = N[(x * N[(1.0 + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], TMP2 = N[(1.0 + N[(-0.25 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2.0], N[(t$95$0 / N[Exp[x], $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = 1.0, TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x))) < 4e-14

   1. Initial program 4.2%

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

      1. add-log-exp 4.2%

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

   4. Applied egg-rr 4.2%

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

   5. Taylor expanded in x around 0 4.2%

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

   6. Taylor expanded in x around 0 4.2%

      \[\leadsto \log \left(e^{\left(\color{blue}{\left(1 + x\right)} \bmod \left(1 + -0.25 \cdot {x}^{2}\right)\right)}\right) \cdot e^{-x} \]
   7. Step-by-step derivation

      1. +-commutative 4.2%

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt[3]{{\cos x}^{1.5}}\right)\right)}{\color{blue}{x + 1}} \]

   8. Simplified 4.2%

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

   9. Taylor expanded in x around inf 13.1%

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

   if 4e-14 < (*.f64 (fmod.f64 (exp.f64 x) (sqrt.f64 (cos.f64 x))) (exp.f64 (neg.f64 x))) < 2

   1. Initial program 90.0%

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

      1. /-rgt-identity 90.0%

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

      2. associate-/r/ 89.6%

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

      3. exp-neg 90.1%

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

      4. remove-double-neg 90.1%

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

   3. Simplified 90.1%

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

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

   1. Initial program 0.0%

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

      1. /-rgt-identity 0.0%

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

      2. associate-/r/ 0.0%

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

      3. exp-neg 0.0%

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

      4. remove-double-neg 0.0%

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

   3. Simplified 0.0%

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

      1. add-cbrt-cube 0.0%

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

      2. pow3 0.0%

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

      3. pow1/2 0.0%

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

      4. pow-pow 0.0%

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

      5. metadata-eval 0.0%

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

   6. Applied egg-rr 0.0%

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

   7. Taylor expanded in x around 0 0.0%

      \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt[3]{{\cos x}^{1.5}}\right)\right)}{\color{blue}{1 + x}} \]
   8. Step-by-step derivation

      1. +-commutative 0.0%

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt[3]{{\cos x}^{1.5}}\right)\right)}{\color{blue}{x + 1}} \]

   9. Simplified 0.0%

      \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt[3]{{\cos x}^{1.5}}\right)\right)}{\color{blue}{x + 1}} \]

   10. Taylor expanded in x around 0 0.0%

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

   11. Taylor expanded in x around 0 100.0%

       \[\leadsto \frac{\left(\color{blue}{1} \bmod 1\right)}{x + 1} \]
3. Recombined 3 regimes into one program.

4. Final simplification 32.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \leq 4 \cdot 10^{-14}:\\ \;\;\;\;e^{-x} \cdot \log \left(e^{\left(\left(x \cdot \left(1 + \frac{1}{x}\right)\right) \bmod \left(1 + -0.25 \cdot {x}^{2}\right)\right)}\right)\\ \mathbf{elif}\;\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right) \cdot e^{-x} \leq 2:\\ \;\;\;\;\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 \bmod 1\right)}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 27.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = mod(exp(x), sqrt(cos(x)))
    if ((t_0 * exp(-x)) <= 2.0d0) then
        tmp = t_0 / exp(x)
    else
        tmp = mod(1.0d0, 1.0d0) / (x + 1.0d0)
    end if
    code = tmp
end function

Python

def code(x):
	t_0 = math.fmod(math.exp(x), math.sqrt(math.cos(x)))
	tmp = 0
	if (t_0 * math.exp(-x)) <= 2.0:
		tmp = t_0 / math.exp(x)
	else:
		tmp = math.fmod(1.0, 1.0) / (x + 1.0)
	return tmp

Julia

function code(x)
	t_0 = rem(exp(x), sqrt(cos(x)))
	tmp = 0.0
	if (Float64(t_0 * exp(Float64(-x))) <= 2.0)
		tmp = Float64(t_0 / exp(x));
	else
		tmp = Float64(rem(1.0, 1.0) / Float64(x + 1.0));
	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], 2.0], N[(t$95$0 / N[Exp[x], $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = 1.0, TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[(x + 1.0), $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))) < 2

   1. Initial program 9.9%

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

      1. /-rgt-identity 9.9%

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

      2. associate-/r/ 9.9%

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

      3. exp-neg 9.9%

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

      4. remove-double-neg 9.9%

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

   3. Simplified 9.9%

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

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

   1. Initial program 0.0%

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

      1. /-rgt-identity 0.0%

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

      2. associate-/r/ 0.0%

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

      3. exp-neg 0.0%

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

      4. remove-double-neg 0.0%

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

   3. Simplified 0.0%

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

      1. add-cbrt-cube 0.0%

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

      2. pow3 0.0%

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

      3. pow1/2 0.0%

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

      4. pow-pow 0.0%

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

      5. metadata-eval 0.0%

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

   6. Applied egg-rr 0.0%

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

   7. Taylor expanded in x around 0 0.0%

      \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt[3]{{\cos x}^{1.5}}\right)\right)}{\color{blue}{1 + x}} \]
   8. Step-by-step derivation

      1. +-commutative 0.0%

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt[3]{{\cos x}^{1.5}}\right)\right)}{\color{blue}{x + 1}} \]

   9. Simplified 0.0%

      \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt[3]{{\cos x}^{1.5}}\right)\right)}{\color{blue}{x + 1}} \]

   10. Taylor expanded in x around 0 0.0%

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

   11. Taylor expanded in x around 0 100.0%

       \[\leadsto \frac{\left(\color{blue}{1} \bmod 1\right)}{x + 1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 62.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\log \left(\sqrt[3]{e}\right) + \log \left({\left(\sqrt[3]{e}\right)}^{2}\right)}\right)\right)}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;e^{-x} \cdot \log \left(e^{\left(\left(x + 1\right) \bmod \left(\sqrt{\cos x}\right)\right)}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x -5e-310)
   (/
    (fmod (exp x) (sqrt (+ (log (cbrt E)) (log (pow (cbrt E) 2.0)))))
    (exp x))
   (* (exp (- x)) (log (exp (fmod (+ x 1.0) (sqrt (cos x))))))))

C

double code(double x) {
	double tmp;
	if (x <= -5e-310) {
		tmp = fmod(exp(x), sqrt((log(cbrt(((double) M_E))) + log(pow(cbrt(((double) M_E)), 2.0))))) / exp(x);
	} else {
		tmp = exp(-x) * log(exp(fmod((x + 1.0), sqrt(cos(x)))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= -5e-310)
		tmp = Float64(rem(exp(x), sqrt(Float64(log(cbrt(exp(1))) + log((cbrt(exp(1)) ^ 2.0))))) / exp(x));
	else
		tmp = Float64(exp(Float64(-x)) * log(exp(rem(Float64(x + 1.0), sqrt(cos(x))))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, -5e-310], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[Sqrt[N[(N[Log[N[Power[E, 1/3], $MachinePrecision]], $MachinePrecision] + N[Log[N[Power[N[Power[E, 1/3], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision], N[(N[Exp[(-x)], $MachinePrecision] * N[Log[N[Exp[N[With[{TMP1 = N[(x + 1.0), $MachinePrecision], TMP2 = N[Sqrt[N[Cos[x], $MachinePrecision]], $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -4.999999999999985e-310

   1. Initial program 8.4%

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

      1. /-rgt-identity 8.4%

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

      2. associate-/r/ 8.4%

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

      3. exp-neg 8.4%

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

      4. remove-double-neg 8.4%

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

   3. Simplified 8.4%

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

      1. add-log-exp 8.4%

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

      2. add-cube-cbrt 100.0%

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

      3. log-prod 100.0%

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

      4. pow2 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around 0 100.0%

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

      1. exp-1-e 100.0%

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

   9. Simplified 100.0%

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

   10. Taylor expanded in x around 0 100.0%

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

       1. exp-1-e 100.0%

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

   12. Simplified 100.0%

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

   if -4.999999999999985e-310 < x

   1. Initial program 8.0%

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

      1. add-log-exp 7.9%

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

   4. Applied egg-rr 7.9%

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

   5. Taylor expanded in x around 0 37.1%

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

      1. +-commutative 6.2%

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt[3]{{\cos x}^{1.5}}\right)\right)}{\color{blue}{x + 1}} \]

   7. Simplified 37.1%

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

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5 \cdot 10^{-310}:\\ \;\;\;\;\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\log \left(\sqrt[3]{e}\right) + \log \left({\left(\sqrt[3]{e}\right)}^{2}\right)}\right)\right)}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;e^{-x} \cdot \log \left(e^{\left(\left(x + 1\right) \bmod \left(\sqrt{\cos x}\right)\right)}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 26.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.5:\\ \;\;\;\;\frac{\left(\left(e^{x}\right) \bmod \left(1 + -0.25 \cdot \left(x \cdot x\right)\right)\right)}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 \bmod 1\right)}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.5)
   (/ (fmod (exp x) (+ 1.0 (* -0.25 (* x x)))) (exp x))
   (/ (fmod 1.0 1.0) (+ x 1.0))))

C

double code(double x) {
	double tmp;
	if (x <= 0.5) {
		tmp = fmod(exp(x), (1.0 + (-0.25 * (x * x)))) / exp(x);
	} else {
		tmp = fmod(1.0, 1.0) / (x + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.5d0) then
        tmp = mod(exp(x), (1.0d0 + ((-0.25d0) * (x * x)))) / exp(x)
    else
        tmp = mod(1.0d0, 1.0d0) / (x + 1.0d0)
    end if
    code = tmp
end function

Python

def code(x):
	tmp = 0
	if x <= 0.5:
		tmp = math.fmod(math.exp(x), (1.0 + (-0.25 * (x * x)))) / math.exp(x)
	else:
		tmp = math.fmod(1.0, 1.0) / (x + 1.0)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.5)
		tmp = Float64(rem(exp(x), Float64(1.0 + Float64(-0.25 * Float64(x * x)))) / exp(x));
	else
		tmp = Float64(rem(1.0, 1.0) / Float64(x + 1.0));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 0.5], N[(N[With[{TMP1 = N[Exp[x], $MachinePrecision], TMP2 = N[(1.0 + N[(-0.25 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[Exp[x], $MachinePrecision]), $MachinePrecision], N[(N[With[{TMP1 = 1.0, TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.5

   1. Initial program 9.9%

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

      1. /-rgt-identity 9.9%

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

      2. associate-/r/ 9.9%

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

      3. exp-neg 9.9%

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

      4. remove-double-neg 9.9%

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

   3. Simplified 9.9%

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

   5. Taylor expanded in x around 0 9.0%

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

      1. unpow2 9.0%

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

   7. Applied egg-rr 9.0%

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

   if 0.5 < x

   1. Initial program 0.0%

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

      1. /-rgt-identity 0.0%

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

      2. associate-/r/ 0.0%

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

      3. exp-neg 0.0%

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

      4. remove-double-neg 0.0%

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

   3. Simplified 0.0%

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

      1. add-cbrt-cube 0.0%

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

      2. pow3 0.0%

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

      3. pow1/2 0.0%

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

      4. pow-pow 0.0%

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

      5. metadata-eval 0.0%

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

   6. Applied egg-rr 0.0%

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

   7. Taylor expanded in x around 0 0.0%

      \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt[3]{{\cos x}^{1.5}}\right)\right)}{\color{blue}{1 + x}} \]
   8. Step-by-step derivation

      1. +-commutative 0.0%

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt[3]{{\cos x}^{1.5}}\right)\right)}{\color{blue}{x + 1}} \]

   9. Simplified 0.0%

      \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt[3]{{\cos x}^{1.5}}\right)\right)}{\color{blue}{x + 1}} \]

   10. Taylor expanded in x around 0 0.0%

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

   11. Taylor expanded in x around 0 100.0%

       \[\leadsto \frac{\left(\color{blue}{1} \bmod 1\right)}{x + 1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 26.6% accurate, 3.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + x \cdot \left(1 + x \cdot 0.5\right)\\ \mathbf{if}\;x \leq 0.5:\\ \;\;\;\;\frac{\left(t\_0 \bmod \left(1 + -0.25 \cdot \left(x \cdot x\right)\right)\right)}{t\_0}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 \bmod 1\right)}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* x (+ 1.0 (* x 0.5))))))
   (if (<= x 0.5)
     (/ (fmod t_0 (+ 1.0 (* -0.25 (* x x)))) t_0)
     (/ (fmod 1.0 1.0) (+ x 1.0)))))

C

double code(double x) {
	double t_0 = 1.0 + (x * (1.0 + (x * 0.5)));
	double tmp;
	if (x <= 0.5) {
		tmp = fmod(t_0, (1.0 + (-0.25 * (x * x)))) / t_0;
	} else {
		tmp = fmod(1.0, 1.0) / (x + 1.0);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 1.0d0 + (x * (1.0d0 + (x * 0.5d0)))
    if (x <= 0.5d0) then
        tmp = mod(t_0, (1.0d0 + ((-0.25d0) * (x * x)))) / t_0
    else
        tmp = mod(1.0d0, 1.0d0) / (x + 1.0d0)
    end if
    code = tmp
end function

Python

def code(x):
	t_0 = 1.0 + (x * (1.0 + (x * 0.5)))
	tmp = 0
	if x <= 0.5:
		tmp = math.fmod(t_0, (1.0 + (-0.25 * (x * x)))) / t_0
	else:
		tmp = math.fmod(1.0, 1.0) / (x + 1.0)
	return tmp

Julia

function code(x)
	t_0 = Float64(1.0 + Float64(x * Float64(1.0 + Float64(x * 0.5))))
	tmp = 0.0
	if (x <= 0.5)
		tmp = Float64(rem(t_0, Float64(1.0 + Float64(-0.25 * Float64(x * x)))) / t_0);
	else
		tmp = Float64(rem(1.0, 1.0) / Float64(x + 1.0));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 + N[(x * N[(1.0 + N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 0.5], N[(N[With[{TMP1 = t$95$0, TMP2 = N[(1.0 + N[(-0.25 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / t$95$0), $MachinePrecision], N[(N[With[{TMP1 = 1.0, TMP2 = 1.0}, Mod[Abs[TMP1], Abs[TMP2]] * Sign[TMP1]], $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 0.5

   1. Initial program 9.9%

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

      1. /-rgt-identity 9.9%

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

      2. associate-/r/ 9.9%

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

      3. exp-neg 9.9%

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

      4. remove-double-neg 9.9%

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

   3. Simplified 9.9%

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

   5. Taylor expanded in x around 0 9.0%

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

      1. unpow2 9.0%

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

   7. Applied egg-rr 9.0%

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

   8. Taylor expanded in x around 0 8.1%

      \[\leadsto \frac{\left(\color{blue}{\left(1 + x \cdot \left(1 + 0.5 \cdot x\right)\right)} \bmod \left(1 + -0.25 \cdot \left(x \cdot x\right)\right)\right)}{e^{x}} \]
   9. Step-by-step derivation

      1. *-commutative 8.1%

         \[\leadsto \frac{\left(\left(1 + x \cdot \left(1 + \color{blue}{x \cdot 0.5}\right)\right) \bmod \left(1 + -0.25 \cdot \left(x \cdot x\right)\right)\right)}{e^{x}} \]

   10. Simplified 8.1%

       \[\leadsto \frac{\left(\color{blue}{\left(1 + x \cdot \left(1 + x \cdot 0.5\right)\right)} \bmod \left(1 + -0.25 \cdot \left(x \cdot x\right)\right)\right)}{e^{x}} \]

   11. Taylor expanded in x around 0 8.8%

       \[\leadsto \frac{\left(\left(1 + x \cdot \left(1 + x \cdot 0.5\right)\right) \bmod \left(1 + -0.25 \cdot \left(x \cdot x\right)\right)\right)}{\color{blue}{1 + x \cdot \left(1 + 0.5 \cdot x\right)}} \]
   12. Step-by-step derivation

       1. *-commutative 8.1%

          \[\leadsto \frac{\left(\left(1 + x \cdot \left(1 + \color{blue}{x \cdot 0.5}\right)\right) \bmod \left(1 + -0.25 \cdot \left(x \cdot x\right)\right)\right)}{e^{x}} \]

   13. Simplified 8.8%

       \[\leadsto \frac{\left(\left(1 + x \cdot \left(1 + x \cdot 0.5\right)\right) \bmod \left(1 + -0.25 \cdot \left(x \cdot x\right)\right)\right)}{\color{blue}{1 + x \cdot \left(1 + x \cdot 0.5\right)}} \]

   if 0.5 < x

   1. Initial program 0.0%

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

      1. /-rgt-identity 0.0%

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

      2. associate-/r/ 0.0%

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

      3. exp-neg 0.0%

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

      4. remove-double-neg 0.0%

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

   3. Simplified 0.0%

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

      1. add-cbrt-cube 0.0%

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

      2. pow3 0.0%

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

      3. pow1/2 0.0%

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

      4. pow-pow 0.0%

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

      5. metadata-eval 0.0%

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

   6. Applied egg-rr 0.0%

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

   7. Taylor expanded in x around 0 0.0%

      \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt[3]{{\cos x}^{1.5}}\right)\right)}{\color{blue}{1 + x}} \]
   8. Step-by-step derivation

      1. +-commutative 0.0%

         \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt[3]{{\cos x}^{1.5}}\right)\right)}{\color{blue}{x + 1}} \]

   9. Simplified 0.0%

      \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt[3]{{\cos x}^{1.5}}\right)\right)}{\color{blue}{x + 1}} \]

   10. Taylor expanded in x around 0 0.0%

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

   11. Taylor expanded in x around 0 100.0%

       \[\leadsto \frac{\left(\color{blue}{1} \bmod 1\right)}{x + 1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 25.8% accurate, 4.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 8.2%

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

   1. /-rgt-identity 8.2%

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

   2. associate-/r/ 8.1%

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

   3. exp-neg 8.2%

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

   4. remove-double-neg 8.2%

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

3. Simplified 8.2%

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

   1. add-cbrt-cube 8.1%

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

   2. pow3 8.1%

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

   3. pow1/2 8.1%

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

   4. pow-pow 8.1%

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

   5. metadata-eval 8.1%

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

6. Applied egg-rr 8.1%

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

7. Taylor expanded in x around 0 6.4%

   \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt[3]{{\cos x}^{1.5}}\right)\right)}{\color{blue}{1 + x}} \]
8. Step-by-step derivation

   1. +-commutative 6.4%

      \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt[3]{{\cos x}^{1.5}}\right)\right)}{\color{blue}{x + 1}} \]

9. Simplified 6.4%

   \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt[3]{{\cos x}^{1.5}}\right)\right)}{\color{blue}{x + 1}} \]

10. Taylor expanded in x around 0 6.1%

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

11. Taylor expanded in x around 0 23.2%

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

    1. +-commutative 6.4%

       \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt[3]{{\cos x}^{1.5}}\right)\right)}{\color{blue}{x + 1}} \]

13. Simplified 23.2%

    \[\leadsto \frac{\left(\color{blue}{\left(x + 1\right)} \bmod 1\right)}{x + 1} \]
14. Add Preprocessing

Alternative 11: 23.3% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 8.2%

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

   1. /-rgt-identity 8.2%

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

   2. associate-/r/ 8.1%

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

   3. exp-neg 8.2%

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

   4. remove-double-neg 8.2%

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

3. Simplified 8.2%

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

   1. add-cbrt-cube 8.1%

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

   2. pow3 8.1%

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

   3. pow1/2 8.1%

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

   4. pow-pow 8.1%

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

   5. metadata-eval 8.1%

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

6. Applied egg-rr 8.1%

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

7. Taylor expanded in x around 0 6.4%

   \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt[3]{{\cos x}^{1.5}}\right)\right)}{\color{blue}{1 + x}} \]
8. Step-by-step derivation

   1. +-commutative 6.4%

      \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt[3]{{\cos x}^{1.5}}\right)\right)}{\color{blue}{x + 1}} \]

9. Simplified 6.4%

   \[\leadsto \frac{\left(\left(e^{x}\right) \bmod \left(\sqrt[3]{{\cos x}^{1.5}}\right)\right)}{\color{blue}{x + 1}} \]

10. Taylor expanded in x around 0 6.1%

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

11. Taylor expanded in x around 0 21.3%

    \[\leadsto \frac{\left(\color{blue}{1} \bmod 1\right)}{x + 1} \]
12. Add Preprocessing

Reproduce

herbie shell --seed 2024139 
(FPCore (x)
  :name "expfmod (used to be hard to sample)"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))