expfmod (used to be hard to sample)

Percentage Accurate: 7.2% → 63.2%

Time: 33.7s

Alternatives: 5

Speedup: 5.0×

• could not determine a ground truth

  1. x = -2.2091352157069e+295

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: 63.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\\ t_1 := e^{-x}\\ t_2 := t\_0 \cdot t\_1\\ \mathbf{if}\;t\_2 \leq 0:\\ \;\;\;\;e^{x}\\ \mathbf{elif}\;t\_2 \leq 2:\\ \;\;\;\;\frac{1}{\frac{e^{x}}{t\_0}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 0.0) (exp x) (if (<= t_2 2.0) (/ 1.0 (/ (exp x) t_0)) t_1))))

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 <= 0.0) {
		tmp = exp(x);
	} else if (t_2 <= 2.0) {
		tmp = 1.0 / (exp(x) / t_0);
	} else {
		tmp = t_1;
	}
	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 <= 0.0d0) then
        tmp = exp(x)
    else if (t_2 <= 2.0d0) then
        tmp = 1.0d0 / (exp(x) / t_0)
    else
        tmp = t_1
    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 <= 0.0:
		tmp = math.exp(x)
	elif t_2 <= 2.0:
		tmp = 1.0 / (math.exp(x) / t_0)
	else:
		tmp = t_1
	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 <= 0.0)
		tmp = exp(x);
	elseif (t_2 <= 2.0)
		tmp = Float64(1.0 / Float64(exp(x) / t_0));
	else
		tmp = t_1;
	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, 0.0], N[Exp[x], $MachinePrecision], If[LessEqual[t$95$2, 2.0], N[(1.0 / N[(N[Exp[x], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

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-cbrt-cube 4.1%

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

      2. pow1/3 4.1%

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

      3. pow-to-exp 4.1%

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

      4. pow3 4.1%

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

      5. log-pow 4.1%

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

      6. log-div 4.1%

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

      7. add-log-exp 4.1%

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

   6. Applied egg-rr 4.1%

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

   7. Taylor expanded in x around inf 58.8%

      \[\leadsto e^{\left(3 \cdot \color{blue}{\left(-1 \cdot x\right)}\right) \cdot 0.3333333333333333} \]
   8. Step-by-step derivation

      1. neg-mul-1 58.8%

         \[\leadsto e^{\left(3 \cdot \color{blue}{\left(-x\right)}\right) \cdot 0.3333333333333333} \]

   9. Simplified 58.8%

      \[\leadsto e^{\left(3 \cdot \color{blue}{\left(-x\right)}\right) \cdot 0.3333333333333333} \]
   10. Step-by-step derivation

       1. *-commutative 58.8%

          \[\leadsto e^{\color{blue}{0.3333333333333333 \cdot \left(3 \cdot \left(-x\right)\right)}} \]

       2. associate-*r* 58.8%

          \[\leadsto e^{\color{blue}{\left(0.3333333333333333 \cdot 3\right) \cdot \left(-x\right)}} \]

       3. metadata-eval 58.8%

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

       4. *-un-lft-identity 58.8%

          \[\leadsto e^{\color{blue}{-x}} \]

       5. add-sqr-sqrt 56.5%

          \[\leadsto e^{\color{blue}{\sqrt{-x} \cdot \sqrt{-x}}} \]

       6. sqrt-unprod 58.8%

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

       7. sqr-neg 58.8%

          \[\leadsto e^{\sqrt{\color{blue}{x \cdot x}}} \]

       8. sqrt-unprod 2.2%

          \[\leadsto e^{\color{blue}{\sqrt{x} \cdot \sqrt{x}}} \]

       9. add-sqr-sqrt 58.8%

          \[\leadsto e^{\color{blue}{x}} \]

       10. add-log-exp 58.8%

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

       11. *-un-lft-identity 58.8%

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

       12. log-prod 58.8%

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

       13. metadata-eval 58.8%

           \[\leadsto e^{\color{blue}{0} + \log \left(e^{x}\right)} \]

       14. add-log-exp 58.8%

           \[\leadsto e^{0 + \color{blue}{x}} \]

   11. Applied egg-rr 58.8%

       \[\leadsto e^{\color{blue}{0 + x}} \]
   12. Step-by-step derivation

       1. +-lft-identity 58.8%

          \[\leadsto e^{\color{blue}{x}} \]

   13. Simplified 58.8%

       \[\leadsto e^{\color{blue}{x}} \]

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

   1. Initial program 78.9%

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

      1. /-rgt-identity 78.9%

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

      2. associate-/r/ 78.8%

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

      3. exp-neg 79.1%

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

      4. remove-double-neg 79.1%

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

   3. Simplified 79.1%

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

      1. add-exp-log 79.1%

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

      2. div-exp 79.0%

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

   6. Applied egg-rr 79.0%

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

      1. exp-diff 79.1%

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

      2. add-exp-log 79.1%

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

      3. clear-num 79.1%

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

   8. Applied egg-rr 79.1%

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

   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-exp-log 0.0%

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

      2. div-exp 0.0%

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

   6. Applied egg-rr 0.0%

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

   7. Taylor expanded in x around inf 100.0%

      \[\leadsto e^{\color{blue}{-1 \cdot x}} \]
   8. Step-by-step derivation

      1. neg-mul-1 100.0%

         \[\leadsto e^{\left(3 \cdot \color{blue}{\left(-x\right)}\right) \cdot 0.3333333333333333} \]

   9. Simplified 100.0%

      \[\leadsto e^{\color{blue}{-x}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 68.1%

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

Alternative 2: 63.1% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	t_0 = sqrt(cos(x))
	t_1 = cbrt(exp(t_0))
	t_2 = exp(Float64(-x))
	tmp = 0.0
	if (Float64(rem(exp(x), t_0) * t_2) <= 2.0)
		tmp = Float64(rem(exp(x), Float64(log((t_1 ^ 2.0)) + log(t_1))) / exp(x));
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

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

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 9.2%

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

      1. /-rgt-identity 9.2%

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

      2. associate-/r/ 9.2%

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

      3. exp-neg 9.2%

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

      4. remove-double-neg 9.2%

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

   3. Simplified 9.2%

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

      1. add-log-exp 9.2%

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

      2. add-cube-cbrt 59.9%

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

      3. log-prod 59.9%

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

      4. pow2 59.9%

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

   6. Applied egg-rr 59.9%

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

   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-exp-log 0.0%

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

      2. div-exp 0.0%

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

   6. Applied egg-rr 0.0%

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

   7. Taylor expanded in x around inf 100.0%

      \[\leadsto e^{\color{blue}{-1 \cdot x}} \]
   8. Step-by-step derivation

      1. neg-mul-1 100.0%

         \[\leadsto e^{\left(3 \cdot \color{blue}{\left(-x\right)}\right) \cdot 0.3333333333333333} \]

   9. Simplified 100.0%

      \[\leadsto e^{\color{blue}{-x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.9%

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

Alternative 3: 63.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)\\ t_1 := e^{-x}\\ t_2 := t\_0 \cdot t\_1\\ \mathbf{if}\;t\_2 \leq 0:\\ \;\;\;\;e^{x}\\ \mathbf{elif}\;t\_2 \leq 2:\\ \;\;\;\;\frac{t\_0}{e^{x}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 0.0) (exp x) (if (<= t_2 2.0) (/ t_0 (exp x)) t_1))))

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 <= 0.0) {
		tmp = exp(x);
	} else if (t_2 <= 2.0) {
		tmp = t_0 / exp(x);
	} else {
		tmp = t_1;
	}
	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 <= 0.0d0) then
        tmp = exp(x)
    else if (t_2 <= 2.0d0) then
        tmp = t_0 / exp(x)
    else
        tmp = t_1
    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 <= 0.0:
		tmp = math.exp(x)
	elif t_2 <= 2.0:
		tmp = t_0 / math.exp(x)
	else:
		tmp = t_1
	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 <= 0.0)
		tmp = exp(x);
	elseif (t_2 <= 2.0)
		tmp = Float64(t_0 / exp(x));
	else
		tmp = t_1;
	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, 0.0], N[Exp[x], $MachinePrecision], If[LessEqual[t$95$2, 2.0], N[(t$95$0 / N[Exp[x], $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

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-cbrt-cube 4.1%

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

      2. pow1/3 4.1%

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

      3. pow-to-exp 4.1%

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

      4. pow3 4.1%

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

      5. log-pow 4.1%

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

      6. log-div 4.1%

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

      7. add-log-exp 4.1%

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

   6. Applied egg-rr 4.1%

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

   7. Taylor expanded in x around inf 58.8%

      \[\leadsto e^{\left(3 \cdot \color{blue}{\left(-1 \cdot x\right)}\right) \cdot 0.3333333333333333} \]
   8. Step-by-step derivation

      1. neg-mul-1 58.8%

         \[\leadsto e^{\left(3 \cdot \color{blue}{\left(-x\right)}\right) \cdot 0.3333333333333333} \]

   9. Simplified 58.8%

      \[\leadsto e^{\left(3 \cdot \color{blue}{\left(-x\right)}\right) \cdot 0.3333333333333333} \]
   10. Step-by-step derivation

       1. *-commutative 58.8%

          \[\leadsto e^{\color{blue}{0.3333333333333333 \cdot \left(3 \cdot \left(-x\right)\right)}} \]

       2. associate-*r* 58.8%

          \[\leadsto e^{\color{blue}{\left(0.3333333333333333 \cdot 3\right) \cdot \left(-x\right)}} \]

       3. metadata-eval 58.8%

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

       4. *-un-lft-identity 58.8%

          \[\leadsto e^{\color{blue}{-x}} \]

       5. add-sqr-sqrt 56.5%

          \[\leadsto e^{\color{blue}{\sqrt{-x} \cdot \sqrt{-x}}} \]

       6. sqrt-unprod 58.8%

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

       7. sqr-neg 58.8%

          \[\leadsto e^{\sqrt{\color{blue}{x \cdot x}}} \]

       8. sqrt-unprod 2.2%

          \[\leadsto e^{\color{blue}{\sqrt{x} \cdot \sqrt{x}}} \]

       9. add-sqr-sqrt 58.8%

          \[\leadsto e^{\color{blue}{x}} \]

       10. add-log-exp 58.8%

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

       11. *-un-lft-identity 58.8%

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

       12. log-prod 58.8%

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

       13. metadata-eval 58.8%

           \[\leadsto e^{\color{blue}{0} + \log \left(e^{x}\right)} \]

       14. add-log-exp 58.8%

           \[\leadsto e^{0 + \color{blue}{x}} \]

   11. Applied egg-rr 58.8%

       \[\leadsto e^{\color{blue}{0 + x}} \]
   12. Step-by-step derivation

       1. +-lft-identity 58.8%

          \[\leadsto e^{\color{blue}{x}} \]

   13. Simplified 58.8%

       \[\leadsto e^{\color{blue}{x}} \]

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

   1. Initial program 78.9%

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

      1. /-rgt-identity 78.9%

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

      2. associate-/r/ 78.8%

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

      3. exp-neg 79.1%

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

      4. remove-double-neg 79.1%

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

   3. Simplified 79.1%

      \[\leadsto \color{blue}{\frac{\left(\left(e^{x}\right) \bmod \left(\sqrt{\cos x}\right)\right)}{e^{x}}} \]
   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-exp-log 0.0%

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

      2. div-exp 0.0%

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

   6. Applied egg-rr 0.0%

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

   7. Taylor expanded in x around inf 100.0%

      \[\leadsto e^{\color{blue}{-1 \cdot x}} \]
   8. Step-by-step derivation

      1. neg-mul-1 100.0%

         \[\leadsto e^{\left(3 \cdot \color{blue}{\left(-x\right)}\right) \cdot 0.3333333333333333} \]

   9. Simplified 100.0%

      \[\leadsto e^{\color{blue}{-x}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 68.1%

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

Alternative 4: 60.9% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ e^{-x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (exp (- x)))

C

double code(double x) {
	return exp(-x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = exp(-x)
end function

Java

public static double code(double x) {
	return Math.exp(-x);
}

Python

def code(x):
	return math.exp(-x)

Julia

function code(x)
	return exp(Float64(-x))
end

MATLAB

function tmp = code(x)
	tmp = exp(-x);
end

Wolfram

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

Derivation

1. Initial program 7.4%

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

   1. /-rgt-identity 7.4%

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

   2. associate-/r/ 7.3%

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

   3. exp-neg 7.4%

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

   4. remove-double-neg 7.4%

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

3. Simplified 7.4%

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

   1. add-exp-log 7.4%

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

   2. div-exp 7.4%

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

6. Applied egg-rr 7.4%

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

7. Taylor expanded in x around inf 65.2%

   \[\leadsto e^{\color{blue}{-1 \cdot x}} \]
8. Step-by-step derivation

   1. neg-mul-1 65.2%

      \[\leadsto e^{\left(3 \cdot \color{blue}{\left(-x\right)}\right) \cdot 0.3333333333333333} \]

9. Simplified 65.2%

   \[\leadsto e^{\color{blue}{-x}} \]

10. Final simplification 65.2%

    \[\leadsto e^{-x} \]
11. Add Preprocessing

Alternative 5: 41.0% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ e^{x} \end{array} \]

FPCore

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

C

double code(double x) {
	return exp(x);
}

Fortran

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

Java

public static double code(double x) {
	return Math.exp(x);
}

Python

def code(x):
	return math.exp(x)

Julia

function code(x)
	return exp(x)
end

MATLAB

function tmp = code(x)
	tmp = exp(x);
end

Wolfram

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

Derivation

1. Initial program 7.4%

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

   1. /-rgt-identity 7.4%

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

   2. associate-/r/ 7.3%

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

   3. exp-neg 7.4%

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

   4. remove-double-neg 7.4%

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

3. Simplified 7.4%

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

   1. add-cbrt-cube 7.4%

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

   2. pow1/3 7.4%

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

   3. pow-to-exp 7.4%

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

   4. pow3 7.4%

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

   5. log-pow 7.4%

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

   6. log-div 7.4%

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

   7. add-log-exp 7.4%

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

6. Applied egg-rr 7.4%

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

7. Taylor expanded in x around inf 65.2%

   \[\leadsto e^{\left(3 \cdot \color{blue}{\left(-1 \cdot x\right)}\right) \cdot 0.3333333333333333} \]
8. Step-by-step derivation

   1. neg-mul-1 65.2%

      \[\leadsto e^{\left(3 \cdot \color{blue}{\left(-x\right)}\right) \cdot 0.3333333333333333} \]

9. Simplified 65.2%

   \[\leadsto e^{\left(3 \cdot \color{blue}{\left(-x\right)}\right) \cdot 0.3333333333333333} \]
10. Step-by-step derivation

    1. *-commutative 65.2%

       \[\leadsto e^{\color{blue}{0.3333333333333333 \cdot \left(3 \cdot \left(-x\right)\right)}} \]

    2. associate-*r* 65.2%

       \[\leadsto e^{\color{blue}{\left(0.3333333333333333 \cdot 3\right) \cdot \left(-x\right)}} \]

    3. metadata-eval 65.2%

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

    4. *-un-lft-identity 65.2%

       \[\leadsto e^{\color{blue}{-x}} \]

    5. add-sqr-sqrt 43.2%

       \[\leadsto e^{\color{blue}{\sqrt{-x} \cdot \sqrt{-x}}} \]

    6. sqrt-unprod 45.5%

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

    7. sqr-neg 45.5%

       \[\leadsto e^{\sqrt{\color{blue}{x \cdot x}}} \]

    8. sqrt-unprod 2.4%

       \[\leadsto e^{\color{blue}{\sqrt{x} \cdot \sqrt{x}}} \]

    9. add-sqr-sqrt 45.5%

       \[\leadsto e^{\color{blue}{x}} \]

    10. add-log-exp 45.5%

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

    11. *-un-lft-identity 45.5%

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

    12. log-prod 45.5%

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

    13. metadata-eval 45.5%

        \[\leadsto e^{\color{blue}{0} + \log \left(e^{x}\right)} \]

    14. add-log-exp 45.5%

        \[\leadsto e^{0 + \color{blue}{x}} \]

11. Applied egg-rr 45.5%

    \[\leadsto e^{\color{blue}{0 + x}} \]
12. Step-by-step derivation

    1. +-lft-identity 45.5%

       \[\leadsto e^{\color{blue}{x}} \]

13. Simplified 45.5%

    \[\leadsto e^{\color{blue}{x}} \]

14. Final simplification 45.5%

    \[\leadsto e^{x} \]
15. Add Preprocessing

Reproduce

herbie shell --seed 2024055 
(FPCore (x)
  :name "expfmod (used to be hard to sample)"
  :precision binary64
  (* (fmod (exp x) (sqrt (cos x))) (exp (- x))))