ENA, Section 1.4, Exercise 1

Percentage Accurate: 94.5% → 99.4%

Time: 8.3s

Alternatives: 10

Speedup: 1.0×

Specification

\[1.99 \leq x \land x \leq 2.01\]

Math

\[\begin{array}{l} \\ \cos x \cdot e^{10 \cdot \left(x \cdot x\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (cos x) (exp (* 10.0 (* x x)))))

C

double code(double x) {
	return cos(x) * exp((10.0 * (x * x)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = cos(x) * exp((10.0d0 * (x * x)))
end function

Java

public static double code(double x) {
	return Math.cos(x) * Math.exp((10.0 * (x * x)));
}

Python

def code(x):
	return math.cos(x) * math.exp((10.0 * (x * x)))

Julia

function code(x)
	return Float64(cos(x) * exp(Float64(10.0 * Float64(x * x))))
end

MATLAB

function tmp = code(x)
	tmp = cos(x) * exp((10.0 * (x * x)));
end

Wolfram

code[x_] := N[(N[Cos[x], $MachinePrecision] * N[Exp[N[(10.0 * N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Initial Program: 94.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \cos x \cdot e^{10 \cdot \left(x \cdot x\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (cos x) (exp (* 10.0 (* x x)))))

C

double code(double x) {
	return cos(x) * exp((10.0 * (x * x)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = cos(x) * exp((10.0d0 * (x * x)))
end function

Java

public static double code(double x) {
	return Math.cos(x) * Math.exp((10.0 * (x * x)));
}

Python

def code(x):
	return math.cos(x) * math.exp((10.0 * (x * x)))

Julia

function code(x)
	return Float64(cos(x) * exp(Float64(10.0 * Float64(x * x))))
end

MATLAB

function tmp = code(x)
	tmp = cos(x) * exp((10.0 * (x * x)));
end

Wolfram

code[x_] := N[(N[Cos[x], $MachinePrecision] * N[Exp[N[(10.0 * N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \cos x \cdot \sqrt{{\left({\left(e^{20}\right)}^{x}\right)}^{x}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (cos x) (sqrt (pow (pow (exp 20.0) x) x))))

C

double code(double x) {
	return cos(x) * sqrt(pow(pow(exp(20.0), x), x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = cos(x) * sqrt(((exp(20.0d0) ** x) ** x))
end function

Java

public static double code(double x) {
	return Math.cos(x) * Math.sqrt(Math.pow(Math.pow(Math.exp(20.0), x), x));
}

Python

def code(x):
	return math.cos(x) * math.sqrt(math.pow(math.pow(math.exp(20.0), x), x))

Julia

function code(x)
	return Float64(cos(x) * sqrt(((exp(20.0) ^ x) ^ x)))
end

MATLAB

function tmp = code(x)
	tmp = cos(x) * sqrt(((exp(20.0) ^ x) ^ x));
end

Wolfram

code[x_] := N[(N[Cos[x], $MachinePrecision] * N[Sqrt[N[Power[N[Power[N[Exp[20.0], $MachinePrecision], x], $MachinePrecision], x], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.4%

   \[\cos x \cdot e^{10 \cdot \left(x \cdot x\right)} \]
2. Step-by-step derivation

   1. associate-*r* 94.2%

      \[\leadsto \cos x \cdot e^{\color{blue}{\left(10 \cdot x\right) \cdot x}} \]

   2. exp-prod 94.8%

      \[\leadsto \cos x \cdot \color{blue}{{\left(e^{10 \cdot x}\right)}^{x}} \]

   3. sqr-pow 94.7%

      \[\leadsto \cos x \cdot \color{blue}{\left({\left(e^{10 \cdot x}\right)}^{\left(\frac{x}{2}\right)} \cdot {\left(e^{10 \cdot x}\right)}^{\left(\frac{x}{2}\right)}\right)} \]

   4. sqr-pow 94.8%

      \[\leadsto \cos x \cdot \color{blue}{{\left(e^{10 \cdot x}\right)}^{x}} \]

   5. exp-prod 97.9%

      \[\leadsto \cos x \cdot {\color{blue}{\left({\left(e^{10}\right)}^{x}\right)}}^{x} \]

3. Simplified 97.9%

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

   1. add-cube-cbrt 95.4%

      \[\leadsto \cos x \cdot {\left({\color{blue}{\left(\left(\sqrt[3]{e^{10}} \cdot \sqrt[3]{e^{10}}\right) \cdot \sqrt[3]{e^{10}}\right)}}^{x}\right)}^{x} \]

   2. unpow-prod-down 95.7%

      \[\leadsto \cos x \cdot {\color{blue}{\left({\left(\sqrt[3]{e^{10}} \cdot \sqrt[3]{e^{10}}\right)}^{x} \cdot {\left(\sqrt[3]{e^{10}}\right)}^{x}\right)}}^{x} \]

   3. pow-to-exp 93.9%

      \[\leadsto \cos x \cdot {\left(\color{blue}{e^{\log \left(\sqrt[3]{e^{10}} \cdot \sqrt[3]{e^{10}}\right) \cdot x}} \cdot {\left(\sqrt[3]{e^{10}}\right)}^{x}\right)}^{x} \]

   4. pow2 93.9%

      \[\leadsto \cos x \cdot {\left(e^{\log \color{blue}{\left({\left(\sqrt[3]{e^{10}}\right)}^{2}\right)} \cdot x} \cdot {\left(\sqrt[3]{e^{10}}\right)}^{x}\right)}^{x} \]

   5. log-pow 96.0%

      \[\leadsto \cos x \cdot {\left(e^{\color{blue}{\left(2 \cdot \log \left(\sqrt[3]{e^{10}}\right)\right)} \cdot x} \cdot {\left(\sqrt[3]{e^{10}}\right)}^{x}\right)}^{x} \]

   6. pow1/3 93.9%

      \[\leadsto \cos x \cdot {\left(e^{\left(2 \cdot \log \color{blue}{\left({\left(e^{10}\right)}^{0.3333333333333333}\right)}\right) \cdot x} \cdot {\left(\sqrt[3]{e^{10}}\right)}^{x}\right)}^{x} \]

   7. log-pow 93.9%

      \[\leadsto \cos x \cdot {\left(e^{\left(2 \cdot \color{blue}{\left(0.3333333333333333 \cdot \log \left(e^{10}\right)\right)}\right) \cdot x} \cdot {\left(\sqrt[3]{e^{10}}\right)}^{x}\right)}^{x} \]

   8. add-log-exp 93.9%

      \[\leadsto \cos x \cdot {\left(e^{\left(2 \cdot \left(0.3333333333333333 \cdot \color{blue}{10}\right)\right) \cdot x} \cdot {\left(\sqrt[3]{e^{10}}\right)}^{x}\right)}^{x} \]

   9. metadata-eval 96.0%

      \[\leadsto \cos x \cdot {\left(e^{\left(2 \cdot \color{blue}{3.3333333333333335}\right) \cdot x} \cdot {\left(\sqrt[3]{e^{10}}\right)}^{x}\right)}^{x} \]

   10. metadata-eval 96.0%

       \[\leadsto \cos x \cdot {\left(e^{\color{blue}{6.666666666666667} \cdot x} \cdot {\left(\sqrt[3]{e^{10}}\right)}^{x}\right)}^{x} \]

   11. pow1/3 96.0%

       \[\leadsto \cos x \cdot {\left(e^{6.666666666666667 \cdot x} \cdot {\color{blue}{\left({\left(e^{10}\right)}^{0.3333333333333333}\right)}}^{x}\right)}^{x} \]

   12. pow-exp 96.1%

       \[\leadsto \cos x \cdot {\left(e^{6.666666666666667 \cdot x} \cdot {\color{blue}{\left(e^{10 \cdot 0.3333333333333333}\right)}}^{x}\right)}^{x} \]

   13. metadata-eval 95.0%

       \[\leadsto \cos x \cdot {\left(e^{6.666666666666667 \cdot x} \cdot {\left(e^{\color{blue}{3.3333333333333335}}\right)}^{x}\right)}^{x} \]

5. Applied egg-rr 95.0%

   \[\leadsto \cos x \cdot {\color{blue}{\left(e^{6.666666666666667 \cdot x} \cdot {\left(e^{3.3333333333333335}\right)}^{x}\right)}}^{x} \]
6. Step-by-step derivation

   1. add-sqr-sqrt 94.9%

      \[\leadsto \cos x \cdot {\color{blue}{\left(\sqrt{e^{6.666666666666667 \cdot x} \cdot {\left(e^{3.3333333333333335}\right)}^{x}} \cdot \sqrt{e^{6.666666666666667 \cdot x} \cdot {\left(e^{3.3333333333333335}\right)}^{x}}\right)}}^{x} \]

   2. sqrt-unprod 95.0%

      \[\leadsto \cos x \cdot {\color{blue}{\left(\sqrt{\left(e^{6.666666666666667 \cdot x} \cdot {\left(e^{3.3333333333333335}\right)}^{x}\right) \cdot \left(e^{6.666666666666667 \cdot x} \cdot {\left(e^{3.3333333333333335}\right)}^{x}\right)}\right)}}^{x} \]

   3. associate-*l* 95.0%

      \[\leadsto \cos x \cdot {\left(\sqrt{\color{blue}{e^{6.666666666666667 \cdot x} \cdot \left({\left(e^{3.3333333333333335}\right)}^{x} \cdot \left(e^{6.666666666666667 \cdot x} \cdot {\left(e^{3.3333333333333335}\right)}^{x}\right)\right)}}\right)}^{x} \]

   4. sqr-pow 94.9%

      \[\leadsto \cos x \cdot {\left(\sqrt{e^{6.666666666666667 \cdot x} \cdot \left(\color{blue}{\left({\left(e^{3.3333333333333335}\right)}^{\left(\frac{x}{2}\right)} \cdot {\left(e^{3.3333333333333335}\right)}^{\left(\frac{x}{2}\right)}\right)} \cdot \left(e^{6.666666666666667 \cdot x} \cdot {\left(e^{3.3333333333333335}\right)}^{x}\right)\right)}\right)}^{x} \]

   5. pow-prod-down 94.9%

      \[\leadsto \cos x \cdot {\left(\sqrt{e^{6.666666666666667 \cdot x} \cdot \left(\color{blue}{{\left(e^{3.3333333333333335} \cdot e^{3.3333333333333335}\right)}^{\left(\frac{x}{2}\right)}} \cdot \left(e^{6.666666666666667 \cdot x} \cdot {\left(e^{3.3333333333333335}\right)}^{x}\right)\right)}\right)}^{x} \]

   6. prod-exp 95.1%

      \[\leadsto \cos x \cdot {\left(\sqrt{e^{6.666666666666667 \cdot x} \cdot \left({\color{blue}{\left(e^{3.3333333333333335 + 3.3333333333333335}\right)}}^{\left(\frac{x}{2}\right)} \cdot \left(e^{6.666666666666667 \cdot x} \cdot {\left(e^{3.3333333333333335}\right)}^{x}\right)\right)}\right)}^{x} \]

   7. metadata-eval 95.1%

      \[\leadsto \cos x \cdot {\left(\sqrt{e^{6.666666666666667 \cdot x} \cdot \left({\left(e^{\color{blue}{6.666666666666667}}\right)}^{\left(\frac{x}{2}\right)} \cdot \left(e^{6.666666666666667 \cdot x} \cdot {\left(e^{3.3333333333333335}\right)}^{x}\right)\right)}\right)}^{x} \]

   8. sqrt-pow1 95.1%

      \[\leadsto \cos x \cdot {\left(\sqrt{e^{6.666666666666667 \cdot x} \cdot \left(\color{blue}{\sqrt{{\left(e^{6.666666666666667}\right)}^{x}}} \cdot \left(e^{6.666666666666667 \cdot x} \cdot {\left(e^{3.3333333333333335}\right)}^{x}\right)\right)}\right)}^{x} \]

   9. exp-prod 95.1%

      \[\leadsto \cos x \cdot {\left(\sqrt{e^{6.666666666666667 \cdot x} \cdot \left(\sqrt{\color{blue}{e^{6.666666666666667 \cdot x}}} \cdot \left(e^{6.666666666666667 \cdot x} \cdot {\left(e^{3.3333333333333335}\right)}^{x}\right)\right)}\right)}^{x} \]

   10. *-commutative 95.1%

       \[\leadsto \cos x \cdot {\left(\sqrt{e^{6.666666666666667 \cdot x} \cdot \left(\sqrt{e^{6.666666666666667 \cdot x}} \cdot \color{blue}{\left({\left(e^{3.3333333333333335}\right)}^{x} \cdot e^{6.666666666666667 \cdot x}\right)}\right)}\right)}^{x} \]

   11. sqr-pow 95.1%

       \[\leadsto \cos x \cdot {\left(\sqrt{e^{6.666666666666667 \cdot x} \cdot \left(\sqrt{e^{6.666666666666667 \cdot x}} \cdot \left(\color{blue}{\left({\left(e^{3.3333333333333335}\right)}^{\left(\frac{x}{2}\right)} \cdot {\left(e^{3.3333333333333335}\right)}^{\left(\frac{x}{2}\right)}\right)} \cdot e^{6.666666666666667 \cdot x}\right)\right)}\right)}^{x} \]

   12. pow-prod-down 95.1%

       \[\leadsto \cos x \cdot {\left(\sqrt{e^{6.666666666666667 \cdot x} \cdot \left(\sqrt{e^{6.666666666666667 \cdot x}} \cdot \left(\color{blue}{{\left(e^{3.3333333333333335} \cdot e^{3.3333333333333335}\right)}^{\left(\frac{x}{2}\right)}} \cdot e^{6.666666666666667 \cdot x}\right)\right)}\right)}^{x} \]

   13. prod-exp 95.3%

       \[\leadsto \cos x \cdot {\left(\sqrt{e^{6.666666666666667 \cdot x} \cdot \left(\sqrt{e^{6.666666666666667 \cdot x}} \cdot \left({\color{blue}{\left(e^{3.3333333333333335 + 3.3333333333333335}\right)}}^{\left(\frac{x}{2}\right)} \cdot e^{6.666666666666667 \cdot x}\right)\right)}\right)}^{x} \]

   14. metadata-eval 95.3%

       \[\leadsto \cos x \cdot {\left(\sqrt{e^{6.666666666666667 \cdot x} \cdot \left(\sqrt{e^{6.666666666666667 \cdot x}} \cdot \left({\left(e^{\color{blue}{6.666666666666667}}\right)}^{\left(\frac{x}{2}\right)} \cdot e^{6.666666666666667 \cdot x}\right)\right)}\right)}^{x} \]

   15. sqrt-pow1 95.3%

       \[\leadsto \cos x \cdot {\left(\sqrt{e^{6.666666666666667 \cdot x} \cdot \left(\sqrt{e^{6.666666666666667 \cdot x}} \cdot \left(\color{blue}{\sqrt{{\left(e^{6.666666666666667}\right)}^{x}}} \cdot e^{6.666666666666667 \cdot x}\right)\right)}\right)}^{x} \]

   16. exp-prod 95.1%

       \[\leadsto \cos x \cdot {\left(\sqrt{e^{6.666666666666667 \cdot x} \cdot \left(\sqrt{e^{6.666666666666667 \cdot x}} \cdot \left(\sqrt{\color{blue}{e^{6.666666666666667 \cdot x}}} \cdot e^{6.666666666666667 \cdot x}\right)\right)}\right)}^{x} \]

7. Applied egg-rr 99.1%

   \[\leadsto \cos x \cdot {\color{blue}{\left(\sqrt{{\left(e^{20}\right)}^{x}}\right)}}^{x} \]
8. Step-by-step derivation

   1. pow1/2 99.1%

      \[\leadsto \cos x \cdot {\color{blue}{\left({\left({\left(e^{20}\right)}^{x}\right)}^{0.5}\right)}}^{x} \]

   2. pow-pow 99.2%

      \[\leadsto \cos x \cdot {\color{blue}{\left({\left(e^{20}\right)}^{\left(x \cdot 0.5\right)}\right)}}^{x} \]

9. Applied egg-rr 99.2%

   \[\leadsto \cos x \cdot {\color{blue}{\left({\left(e^{20}\right)}^{\left(x \cdot 0.5\right)}\right)}}^{x} \]
10. Step-by-step derivation

    1. add-sqr-sqrt 98.8%

       \[\leadsto \cos x \cdot \color{blue}{\left(\sqrt{{\left({\left(e^{20}\right)}^{\left(x \cdot 0.5\right)}\right)}^{x}} \cdot \sqrt{{\left({\left(e^{20}\right)}^{\left(x \cdot 0.5\right)}\right)}^{x}}\right)} \]

    2. sqrt-unprod 99.2%

       \[\leadsto \cos x \cdot \color{blue}{\sqrt{{\left({\left(e^{20}\right)}^{\left(x \cdot 0.5\right)}\right)}^{x} \cdot {\left({\left(e^{20}\right)}^{\left(x \cdot 0.5\right)}\right)}^{x}}} \]

    3. pow-prod-down 99.1%

       \[\leadsto \cos x \cdot \sqrt{\color{blue}{{\left({\left(e^{20}\right)}^{\left(x \cdot 0.5\right)} \cdot {\left(e^{20}\right)}^{\left(x \cdot 0.5\right)}\right)}^{x}}} \]

    4. pow-unpow 99.2%

       \[\leadsto \cos x \cdot \sqrt{{\left(\color{blue}{{\left({\left(e^{20}\right)}^{x}\right)}^{0.5}} \cdot {\left(e^{20}\right)}^{\left(x \cdot 0.5\right)}\right)}^{x}} \]

    5. pow-unpow 99.0%

       \[\leadsto \cos x \cdot \sqrt{{\left({\left({\left(e^{20}\right)}^{x}\right)}^{0.5} \cdot \color{blue}{{\left({\left(e^{20}\right)}^{x}\right)}^{0.5}}\right)}^{x}} \]

    6. pow-prod-up 99.3%

       \[\leadsto \cos x \cdot \sqrt{{\color{blue}{\left({\left({\left(e^{20}\right)}^{x}\right)}^{\left(0.5 + 0.5\right)}\right)}}^{x}} \]

    7. metadata-eval 99.3%

       \[\leadsto \cos x \cdot \sqrt{{\left({\left({\left(e^{20}\right)}^{x}\right)}^{\color{blue}{1}}\right)}^{x}} \]

    8. pow1 99.3%

       \[\leadsto \cos x \cdot \sqrt{{\color{blue}{\left({\left(e^{20}\right)}^{x}\right)}}^{x}} \]

11. Applied egg-rr 99.3%

    \[\leadsto \cos x \cdot \color{blue}{\sqrt{{\left({\left(e^{20}\right)}^{x}\right)}^{x}}} \]

12. Final simplification 99.3%

    \[\leadsto \cos x \cdot \sqrt{{\left({\left(e^{20}\right)}^{x}\right)}^{x}} \]

Alternative 2: 98.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \cos x \cdot {\left({\left(e^{5}\right)}^{x}\right)}^{\left(x + x\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (cos x) (pow (pow (exp 5.0) x) (+ x x))))

C

double code(double x) {
	return cos(x) * pow(pow(exp(5.0), x), (x + x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = cos(x) * ((exp(5.0d0) ** x) ** (x + x))
end function

Java

public static double code(double x) {
	return Math.cos(x) * Math.pow(Math.pow(Math.exp(5.0), x), (x + x));
}

Python

def code(x):
	return math.cos(x) * math.pow(math.pow(math.exp(5.0), x), (x + x))

Julia

function code(x)
	return Float64(cos(x) * ((exp(5.0) ^ x) ^ Float64(x + x)))
end

MATLAB

function tmp = code(x)
	tmp = cos(x) * ((exp(5.0) ^ x) ^ (x + x));
end

Wolfram

code[x_] := N[(N[Cos[x], $MachinePrecision] * N[Power[N[Power[N[Exp[5.0], $MachinePrecision], x], $MachinePrecision], N[(x + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.4%

   \[\cos x \cdot e^{10 \cdot \left(x \cdot x\right)} \]
2. Step-by-step derivation

   1. associate-*r* 94.2%

      \[\leadsto \cos x \cdot e^{\color{blue}{\left(10 \cdot x\right) \cdot x}} \]

   2. add-log-exp 94.2%

      \[\leadsto \cos x \cdot e^{\left(10 \cdot x\right) \cdot \color{blue}{\log \left(e^{x}\right)}} \]

   3. log-pow 94.2%

      \[\leadsto \cos x \cdot e^{\color{blue}{\log \left({\left(e^{x}\right)}^{\left(10 \cdot x\right)}\right)}} \]

   4. pow-pow 94.8%

      \[\leadsto \cos x \cdot e^{\log \color{blue}{\left({\left({\left(e^{x}\right)}^{10}\right)}^{x}\right)}} \]

   5. add-exp-log 96.8%

      \[\leadsto \cos x \cdot \color{blue}{{\left({\left(e^{x}\right)}^{10}\right)}^{x}} \]

   6. add-sqr-sqrt 96.7%

      \[\leadsto \cos x \cdot {\color{blue}{\left(\sqrt{{\left(e^{x}\right)}^{10}} \cdot \sqrt{{\left(e^{x}\right)}^{10}}\right)}}^{x} \]

   7. unpow-prod-down 96.8%

      \[\leadsto \cos x \cdot \color{blue}{\left({\left(\sqrt{{\left(e^{x}\right)}^{10}}\right)}^{x} \cdot {\left(\sqrt{{\left(e^{x}\right)}^{10}}\right)}^{x}\right)} \]

   8. pow-prod-up 96.8%

      \[\leadsto \cos x \cdot \color{blue}{{\left(\sqrt{{\left(e^{x}\right)}^{10}}\right)}^{\left(x + x\right)}} \]

   9. sqrt-pow1 96.7%

      \[\leadsto \cos x \cdot {\color{blue}{\left({\left(e^{x}\right)}^{\left(\frac{10}{2}\right)}\right)}}^{\left(x + x\right)} \]

   10. add-exp-log 94.7%

       \[\leadsto \cos x \cdot {\color{blue}{\left(e^{\log \left({\left(e^{x}\right)}^{\left(\frac{10}{2}\right)}\right)}\right)}}^{\left(x + x\right)} \]

   11. log-pow 94.7%

       \[\leadsto \cos x \cdot {\left(e^{\color{blue}{\frac{10}{2} \cdot \log \left(e^{x}\right)}}\right)}^{\left(x + x\right)} \]

   12. metadata-eval 94.7%

       \[\leadsto \cos x \cdot {\left(e^{\color{blue}{5} \cdot \log \left(e^{x}\right)}\right)}^{\left(x + x\right)} \]

   13. add-log-exp 94.7%

       \[\leadsto \cos x \cdot {\left(e^{5 \cdot \color{blue}{x}}\right)}^{\left(x + x\right)} \]

3. Applied egg-rr 94.7%

   \[\leadsto \cos x \cdot \color{blue}{{\left(e^{5 \cdot x}\right)}^{\left(x + x\right)}} \]
4. Step-by-step derivation

   1. exp-prod 98.2%

      \[\leadsto \cos x \cdot {\color{blue}{\left({\left(e^{5}\right)}^{x}\right)}}^{\left(x + x\right)} \]

5. Applied egg-rr 98.2%

   \[\leadsto \cos x \cdot {\color{blue}{\left({\left(e^{5}\right)}^{x}\right)}}^{\left(x + x\right)} \]

6. Final simplification 98.2%

   \[\leadsto \cos x \cdot {\left({\left(e^{5}\right)}^{x}\right)}^{\left(x + x\right)} \]

Alternative 3: 99.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \cos x \cdot {\left({\left(e^{20}\right)}^{\left(x \cdot 0.5\right)}\right)}^{x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (cos x) (pow (pow (exp 20.0) (* x 0.5)) x)))

C

double code(double x) {
	return cos(x) * pow(pow(exp(20.0), (x * 0.5)), x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = cos(x) * ((exp(20.0d0) ** (x * 0.5d0)) ** x)
end function

Java

public static double code(double x) {
	return Math.cos(x) * Math.pow(Math.pow(Math.exp(20.0), (x * 0.5)), x);
}

Python

def code(x):
	return math.cos(x) * math.pow(math.pow(math.exp(20.0), (x * 0.5)), x)

Julia

function code(x)
	return Float64(cos(x) * ((exp(20.0) ^ Float64(x * 0.5)) ^ x))
end

MATLAB

function tmp = code(x)
	tmp = cos(x) * ((exp(20.0) ^ (x * 0.5)) ^ x);
end

Wolfram

code[x_] := N[(N[Cos[x], $MachinePrecision] * N[Power[N[Power[N[Exp[20.0], $MachinePrecision], N[(x * 0.5), $MachinePrecision]], $MachinePrecision], x], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.4%

   \[\cos x \cdot e^{10 \cdot \left(x \cdot x\right)} \]
2. Step-by-step derivation

   1. associate-*r* 94.2%

      \[\leadsto \cos x \cdot e^{\color{blue}{\left(10 \cdot x\right) \cdot x}} \]

   2. exp-prod 94.8%

      \[\leadsto \cos x \cdot \color{blue}{{\left(e^{10 \cdot x}\right)}^{x}} \]

   3. sqr-pow 94.7%

      \[\leadsto \cos x \cdot \color{blue}{\left({\left(e^{10 \cdot x}\right)}^{\left(\frac{x}{2}\right)} \cdot {\left(e^{10 \cdot x}\right)}^{\left(\frac{x}{2}\right)}\right)} \]

   4. sqr-pow 94.8%

      \[\leadsto \cos x \cdot \color{blue}{{\left(e^{10 \cdot x}\right)}^{x}} \]

   5. exp-prod 97.9%

      \[\leadsto \cos x \cdot {\color{blue}{\left({\left(e^{10}\right)}^{x}\right)}}^{x} \]

3. Simplified 97.9%

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

   1. add-cube-cbrt 95.4%

      \[\leadsto \cos x \cdot {\left({\color{blue}{\left(\left(\sqrt[3]{e^{10}} \cdot \sqrt[3]{e^{10}}\right) \cdot \sqrt[3]{e^{10}}\right)}}^{x}\right)}^{x} \]

   2. unpow-prod-down 95.7%

      \[\leadsto \cos x \cdot {\color{blue}{\left({\left(\sqrt[3]{e^{10}} \cdot \sqrt[3]{e^{10}}\right)}^{x} \cdot {\left(\sqrt[3]{e^{10}}\right)}^{x}\right)}}^{x} \]

   3. pow-to-exp 93.9%

      \[\leadsto \cos x \cdot {\left(\color{blue}{e^{\log \left(\sqrt[3]{e^{10}} \cdot \sqrt[3]{e^{10}}\right) \cdot x}} \cdot {\left(\sqrt[3]{e^{10}}\right)}^{x}\right)}^{x} \]

   4. pow2 93.9%

      \[\leadsto \cos x \cdot {\left(e^{\log \color{blue}{\left({\left(\sqrt[3]{e^{10}}\right)}^{2}\right)} \cdot x} \cdot {\left(\sqrt[3]{e^{10}}\right)}^{x}\right)}^{x} \]

   5. log-pow 96.0%

      \[\leadsto \cos x \cdot {\left(e^{\color{blue}{\left(2 \cdot \log \left(\sqrt[3]{e^{10}}\right)\right)} \cdot x} \cdot {\left(\sqrt[3]{e^{10}}\right)}^{x}\right)}^{x} \]

   6. pow1/3 93.9%

      \[\leadsto \cos x \cdot {\left(e^{\left(2 \cdot \log \color{blue}{\left({\left(e^{10}\right)}^{0.3333333333333333}\right)}\right) \cdot x} \cdot {\left(\sqrt[3]{e^{10}}\right)}^{x}\right)}^{x} \]

   7. log-pow 93.9%

      \[\leadsto \cos x \cdot {\left(e^{\left(2 \cdot \color{blue}{\left(0.3333333333333333 \cdot \log \left(e^{10}\right)\right)}\right) \cdot x} \cdot {\left(\sqrt[3]{e^{10}}\right)}^{x}\right)}^{x} \]

   8. add-log-exp 93.9%

      \[\leadsto \cos x \cdot {\left(e^{\left(2 \cdot \left(0.3333333333333333 \cdot \color{blue}{10}\right)\right) \cdot x} \cdot {\left(\sqrt[3]{e^{10}}\right)}^{x}\right)}^{x} \]

   9. metadata-eval 96.0%

      \[\leadsto \cos x \cdot {\left(e^{\left(2 \cdot \color{blue}{3.3333333333333335}\right) \cdot x} \cdot {\left(\sqrt[3]{e^{10}}\right)}^{x}\right)}^{x} \]

   10. metadata-eval 96.0%

       \[\leadsto \cos x \cdot {\left(e^{\color{blue}{6.666666666666667} \cdot x} \cdot {\left(\sqrt[3]{e^{10}}\right)}^{x}\right)}^{x} \]

   11. pow1/3 96.0%

       \[\leadsto \cos x \cdot {\left(e^{6.666666666666667 \cdot x} \cdot {\color{blue}{\left({\left(e^{10}\right)}^{0.3333333333333333}\right)}}^{x}\right)}^{x} \]

   12. pow-exp 96.1%

       \[\leadsto \cos x \cdot {\left(e^{6.666666666666667 \cdot x} \cdot {\color{blue}{\left(e^{10 \cdot 0.3333333333333333}\right)}}^{x}\right)}^{x} \]

   13. metadata-eval 95.0%

       \[\leadsto \cos x \cdot {\left(e^{6.666666666666667 \cdot x} \cdot {\left(e^{\color{blue}{3.3333333333333335}}\right)}^{x}\right)}^{x} \]

5. Applied egg-rr 95.0%

   \[\leadsto \cos x \cdot {\color{blue}{\left(e^{6.666666666666667 \cdot x} \cdot {\left(e^{3.3333333333333335}\right)}^{x}\right)}}^{x} \]
6. Step-by-step derivation

   1. add-sqr-sqrt 94.9%

      \[\leadsto \cos x \cdot {\color{blue}{\left(\sqrt{e^{6.666666666666667 \cdot x} \cdot {\left(e^{3.3333333333333335}\right)}^{x}} \cdot \sqrt{e^{6.666666666666667 \cdot x} \cdot {\left(e^{3.3333333333333335}\right)}^{x}}\right)}}^{x} \]

   2. sqrt-unprod 95.0%

      \[\leadsto \cos x \cdot {\color{blue}{\left(\sqrt{\left(e^{6.666666666666667 \cdot x} \cdot {\left(e^{3.3333333333333335}\right)}^{x}\right) \cdot \left(e^{6.666666666666667 \cdot x} \cdot {\left(e^{3.3333333333333335}\right)}^{x}\right)}\right)}}^{x} \]

   3. associate-*l* 95.0%

      \[\leadsto \cos x \cdot {\left(\sqrt{\color{blue}{e^{6.666666666666667 \cdot x} \cdot \left({\left(e^{3.3333333333333335}\right)}^{x} \cdot \left(e^{6.666666666666667 \cdot x} \cdot {\left(e^{3.3333333333333335}\right)}^{x}\right)\right)}}\right)}^{x} \]

   4. sqr-pow 94.9%

      \[\leadsto \cos x \cdot {\left(\sqrt{e^{6.666666666666667 \cdot x} \cdot \left(\color{blue}{\left({\left(e^{3.3333333333333335}\right)}^{\left(\frac{x}{2}\right)} \cdot {\left(e^{3.3333333333333335}\right)}^{\left(\frac{x}{2}\right)}\right)} \cdot \left(e^{6.666666666666667 \cdot x} \cdot {\left(e^{3.3333333333333335}\right)}^{x}\right)\right)}\right)}^{x} \]

   5. pow-prod-down 94.9%

      \[\leadsto \cos x \cdot {\left(\sqrt{e^{6.666666666666667 \cdot x} \cdot \left(\color{blue}{{\left(e^{3.3333333333333335} \cdot e^{3.3333333333333335}\right)}^{\left(\frac{x}{2}\right)}} \cdot \left(e^{6.666666666666667 \cdot x} \cdot {\left(e^{3.3333333333333335}\right)}^{x}\right)\right)}\right)}^{x} \]

   6. prod-exp 95.1%

      \[\leadsto \cos x \cdot {\left(\sqrt{e^{6.666666666666667 \cdot x} \cdot \left({\color{blue}{\left(e^{3.3333333333333335 + 3.3333333333333335}\right)}}^{\left(\frac{x}{2}\right)} \cdot \left(e^{6.666666666666667 \cdot x} \cdot {\left(e^{3.3333333333333335}\right)}^{x}\right)\right)}\right)}^{x} \]

   7. metadata-eval 95.1%

      \[\leadsto \cos x \cdot {\left(\sqrt{e^{6.666666666666667 \cdot x} \cdot \left({\left(e^{\color{blue}{6.666666666666667}}\right)}^{\left(\frac{x}{2}\right)} \cdot \left(e^{6.666666666666667 \cdot x} \cdot {\left(e^{3.3333333333333335}\right)}^{x}\right)\right)}\right)}^{x} \]

   8. sqrt-pow1 95.1%

      \[\leadsto \cos x \cdot {\left(\sqrt{e^{6.666666666666667 \cdot x} \cdot \left(\color{blue}{\sqrt{{\left(e^{6.666666666666667}\right)}^{x}}} \cdot \left(e^{6.666666666666667 \cdot x} \cdot {\left(e^{3.3333333333333335}\right)}^{x}\right)\right)}\right)}^{x} \]

   9. exp-prod 95.1%

      \[\leadsto \cos x \cdot {\left(\sqrt{e^{6.666666666666667 \cdot x} \cdot \left(\sqrt{\color{blue}{e^{6.666666666666667 \cdot x}}} \cdot \left(e^{6.666666666666667 \cdot x} \cdot {\left(e^{3.3333333333333335}\right)}^{x}\right)\right)}\right)}^{x} \]

   10. *-commutative 95.1%

       \[\leadsto \cos x \cdot {\left(\sqrt{e^{6.666666666666667 \cdot x} \cdot \left(\sqrt{e^{6.666666666666667 \cdot x}} \cdot \color{blue}{\left({\left(e^{3.3333333333333335}\right)}^{x} \cdot e^{6.666666666666667 \cdot x}\right)}\right)}\right)}^{x} \]

   11. sqr-pow 95.1%

       \[\leadsto \cos x \cdot {\left(\sqrt{e^{6.666666666666667 \cdot x} \cdot \left(\sqrt{e^{6.666666666666667 \cdot x}} \cdot \left(\color{blue}{\left({\left(e^{3.3333333333333335}\right)}^{\left(\frac{x}{2}\right)} \cdot {\left(e^{3.3333333333333335}\right)}^{\left(\frac{x}{2}\right)}\right)} \cdot e^{6.666666666666667 \cdot x}\right)\right)}\right)}^{x} \]

   12. pow-prod-down 95.1%

       \[\leadsto \cos x \cdot {\left(\sqrt{e^{6.666666666666667 \cdot x} \cdot \left(\sqrt{e^{6.666666666666667 \cdot x}} \cdot \left(\color{blue}{{\left(e^{3.3333333333333335} \cdot e^{3.3333333333333335}\right)}^{\left(\frac{x}{2}\right)}} \cdot e^{6.666666666666667 \cdot x}\right)\right)}\right)}^{x} \]

   13. prod-exp 95.3%

       \[\leadsto \cos x \cdot {\left(\sqrt{e^{6.666666666666667 \cdot x} \cdot \left(\sqrt{e^{6.666666666666667 \cdot x}} \cdot \left({\color{blue}{\left(e^{3.3333333333333335 + 3.3333333333333335}\right)}}^{\left(\frac{x}{2}\right)} \cdot e^{6.666666666666667 \cdot x}\right)\right)}\right)}^{x} \]

   14. metadata-eval 95.3%

       \[\leadsto \cos x \cdot {\left(\sqrt{e^{6.666666666666667 \cdot x} \cdot \left(\sqrt{e^{6.666666666666667 \cdot x}} \cdot \left({\left(e^{\color{blue}{6.666666666666667}}\right)}^{\left(\frac{x}{2}\right)} \cdot e^{6.666666666666667 \cdot x}\right)\right)}\right)}^{x} \]

   15. sqrt-pow1 95.3%

       \[\leadsto \cos x \cdot {\left(\sqrt{e^{6.666666666666667 \cdot x} \cdot \left(\sqrt{e^{6.666666666666667 \cdot x}} \cdot \left(\color{blue}{\sqrt{{\left(e^{6.666666666666667}\right)}^{x}}} \cdot e^{6.666666666666667 \cdot x}\right)\right)}\right)}^{x} \]

   16. exp-prod 95.1%

       \[\leadsto \cos x \cdot {\left(\sqrt{e^{6.666666666666667 \cdot x} \cdot \left(\sqrt{e^{6.666666666666667 \cdot x}} \cdot \left(\sqrt{\color{blue}{e^{6.666666666666667 \cdot x}}} \cdot e^{6.666666666666667 \cdot x}\right)\right)}\right)}^{x} \]

7. Applied egg-rr 99.1%

   \[\leadsto \cos x \cdot {\color{blue}{\left(\sqrt{{\left(e^{20}\right)}^{x}}\right)}}^{x} \]
8. Step-by-step derivation

   1. pow1/2 99.1%

      \[\leadsto \cos x \cdot {\color{blue}{\left({\left({\left(e^{20}\right)}^{x}\right)}^{0.5}\right)}}^{x} \]

   2. pow-pow 99.2%

      \[\leadsto \cos x \cdot {\color{blue}{\left({\left(e^{20}\right)}^{\left(x \cdot 0.5\right)}\right)}}^{x} \]

9. Applied egg-rr 99.2%

   \[\leadsto \cos x \cdot {\color{blue}{\left({\left(e^{20}\right)}^{\left(x \cdot 0.5\right)}\right)}}^{x} \]

10. Final simplification 99.2%

    \[\leadsto \cos x \cdot {\left({\left(e^{20}\right)}^{\left(x \cdot 0.5\right)}\right)}^{x} \]

Alternative 4: 98.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \cos x \cdot {\left({\left(e^{10}\right)}^{x}\right)}^{x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (cos x) (pow (pow (exp 10.0) x) x)))

C

double code(double x) {
	return cos(x) * pow(pow(exp(10.0), x), x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = cos(x) * ((exp(10.0d0) ** x) ** x)
end function

Java

public static double code(double x) {
	return Math.cos(x) * Math.pow(Math.pow(Math.exp(10.0), x), x);
}

Python

def code(x):
	return math.cos(x) * math.pow(math.pow(math.exp(10.0), x), x)

Julia

function code(x)
	return Float64(cos(x) * ((exp(10.0) ^ x) ^ x))
end

MATLAB

function tmp = code(x)
	tmp = cos(x) * ((exp(10.0) ^ x) ^ x);
end

Wolfram

code[x_] := N[(N[Cos[x], $MachinePrecision] * N[Power[N[Power[N[Exp[10.0], $MachinePrecision], x], $MachinePrecision], x], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.4%

   \[\cos x \cdot e^{10 \cdot \left(x \cdot x\right)} \]
2. Step-by-step derivation

   1. associate-*r* 94.2%

      \[\leadsto \cos x \cdot e^{\color{blue}{\left(10 \cdot x\right) \cdot x}} \]

   2. exp-prod 94.8%

      \[\leadsto \cos x \cdot \color{blue}{{\left(e^{10 \cdot x}\right)}^{x}} \]

   3. sqr-pow 94.7%

      \[\leadsto \cos x \cdot \color{blue}{\left({\left(e^{10 \cdot x}\right)}^{\left(\frac{x}{2}\right)} \cdot {\left(e^{10 \cdot x}\right)}^{\left(\frac{x}{2}\right)}\right)} \]

   4. sqr-pow 94.8%

      \[\leadsto \cos x \cdot \color{blue}{{\left(e^{10 \cdot x}\right)}^{x}} \]

   5. exp-prod 97.9%

      \[\leadsto \cos x \cdot {\color{blue}{\left({\left(e^{10}\right)}^{x}\right)}}^{x} \]

3. Simplified 97.9%

   \[\leadsto \color{blue}{\cos x \cdot {\left({\left(e^{10}\right)}^{x}\right)}^{x}} \]

4. Final simplification 97.9%

   \[\leadsto \cos x \cdot {\left({\left(e^{10}\right)}^{x}\right)}^{x} \]

Alternative 5: 95.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \cos x \cdot \sqrt[3]{e^{x \cdot \left(x \cdot 30\right)}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (cos x) (cbrt (exp (* x (* x 30.0))))))

C

double code(double x) {
	return cos(x) * cbrt(exp((x * (x * 30.0))));
}

Java

public static double code(double x) {
	return Math.cos(x) * Math.cbrt(Math.exp((x * (x * 30.0))));
}

Julia

function code(x)
	return Float64(cos(x) * cbrt(exp(Float64(x * Float64(x * 30.0)))))
end

Wolfram

code[x_] := N[(N[Cos[x], $MachinePrecision] * N[Power[N[Exp[N[(x * N[(x * 30.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.4%

   \[\cos x \cdot e^{10 \cdot \left(x \cdot x\right)} \]

3. Applied egg-rr 98.8%

   \[\leadsto \cos x \cdot \color{blue}{\sqrt[3]{{\left({\left(e^{30}\right)}^{x}\right)}^{x}}} \]
4. Step-by-step derivation

   1. add-exp-log 95.6%

      \[\leadsto \cos x \cdot \sqrt[3]{\color{blue}{e^{\log \left({\left({\left(e^{30}\right)}^{x}\right)}^{x}\right)}}} \]

   2. log-pow 95.3%

      \[\leadsto \cos x \cdot \sqrt[3]{e^{\color{blue}{x \cdot \log \left({\left(e^{30}\right)}^{x}\right)}}} \]

   3. pow-exp 95.3%

      \[\leadsto \cos x \cdot \sqrt[3]{e^{x \cdot \log \color{blue}{\left(e^{30 \cdot x}\right)}}} \]

   4. add-log-exp 95.3%

      \[\leadsto \cos x \cdot \sqrt[3]{e^{x \cdot \color{blue}{\left(30 \cdot x\right)}}} \]

5. Applied egg-rr 95.3%

   \[\leadsto \cos x \cdot \sqrt[3]{\color{blue}{e^{x \cdot \left(30 \cdot x\right)}}} \]

6. Final simplification 95.3%

   \[\leadsto \cos x \cdot \sqrt[3]{e^{x \cdot \left(x \cdot 30\right)}} \]

Alternative 6: 95.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \cos x \cdot {\left(e^{10}\right)}^{\left(x \cdot x\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (cos x) (pow (exp 10.0) (* x x))))

C

double code(double x) {
	return cos(x) * pow(exp(10.0), (x * x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = cos(x) * (exp(10.0d0) ** (x * x))
end function

Java

public static double code(double x) {
	return Math.cos(x) * Math.pow(Math.exp(10.0), (x * x));
}

Python

def code(x):
	return math.cos(x) * math.pow(math.exp(10.0), (x * x))

Julia

function code(x)
	return Float64(cos(x) * (exp(10.0) ^ Float64(x * x)))
end

MATLAB

function tmp = code(x)
	tmp = cos(x) * (exp(10.0) ^ (x * x));
end

Wolfram

code[x_] := N[(N[Cos[x], $MachinePrecision] * N[Power[N[Exp[10.0], $MachinePrecision], N[(x * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.4%

   \[\cos x \cdot e^{10 \cdot \left(x \cdot x\right)} \]
2. Step-by-step derivation

   1. exp-prod 95.2%

      \[\leadsto \cos x \cdot \color{blue}{{\left(e^{10}\right)}^{\left(x \cdot x\right)}} \]

3. Simplified 95.2%

   \[\leadsto \color{blue}{\cos x \cdot {\left(e^{10}\right)}^{\left(x \cdot x\right)}} \]

4. Final simplification 95.2%

   \[\leadsto \cos x \cdot {\left(e^{10}\right)}^{\left(x \cdot x\right)} \]

Alternative 7: 94.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \cos x \cdot e^{10 \cdot \left(x \cdot x\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (cos x) (exp (* 10.0 (* x x)))))

C

double code(double x) {
	return cos(x) * exp((10.0 * (x * x)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = cos(x) * exp((10.0d0 * (x * x)))
end function

Java

public static double code(double x) {
	return Math.cos(x) * Math.exp((10.0 * (x * x)));
}

Python

def code(x):
	return math.cos(x) * math.exp((10.0 * (x * x)))

Julia

function code(x)
	return Float64(cos(x) * exp(Float64(10.0 * Float64(x * x))))
end

MATLAB

function tmp = code(x)
	tmp = cos(x) * exp((10.0 * (x * x)));
end

Wolfram

code[x_] := N[(N[Cos[x], $MachinePrecision] * N[Exp[N[(10.0 * N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.4%

   \[\cos x \cdot e^{10 \cdot \left(x \cdot x\right)} \]

2. Final simplification 94.4%

   \[\leadsto \cos x \cdot e^{10 \cdot \left(x \cdot x\right)} \]

Alternative 8: 10.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \cos x \cdot e^{x \cdot 5} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (cos x) (exp (* x 5.0))))

C

double code(double x) {
	return cos(x) * exp((x * 5.0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = cos(x) * exp((x * 5.0d0))
end function

Java

public static double code(double x) {
	return Math.cos(x) * Math.exp((x * 5.0));
}

Python

def code(x):
	return math.cos(x) * math.exp((x * 5.0))

Julia

function code(x)
	return Float64(cos(x) * exp(Float64(x * 5.0)))
end

MATLAB

function tmp = code(x)
	tmp = cos(x) * exp((x * 5.0));
end

Wolfram

code[x_] := N[(N[Cos[x], $MachinePrecision] * N[Exp[N[(x * 5.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.4%

   \[\cos x \cdot e^{10 \cdot \left(x \cdot x\right)} \]
2. Step-by-step derivation

   1. associate-*r* 94.2%

      \[\leadsto \cos x \cdot e^{\color{blue}{\left(10 \cdot x\right) \cdot x}} \]

   2. add-log-exp 94.2%

      \[\leadsto \cos x \cdot e^{\left(10 \cdot x\right) \cdot \color{blue}{\log \left(e^{x}\right)}} \]

   3. log-pow 94.2%

      \[\leadsto \cos x \cdot e^{\color{blue}{\log \left({\left(e^{x}\right)}^{\left(10 \cdot x\right)}\right)}} \]

   4. pow-pow 94.8%

      \[\leadsto \cos x \cdot e^{\log \color{blue}{\left({\left({\left(e^{x}\right)}^{10}\right)}^{x}\right)}} \]

   5. add-exp-log 96.8%

      \[\leadsto \cos x \cdot \color{blue}{{\left({\left(e^{x}\right)}^{10}\right)}^{x}} \]

   6. add-sqr-sqrt 96.7%

      \[\leadsto \cos x \cdot {\color{blue}{\left(\sqrt{{\left(e^{x}\right)}^{10}} \cdot \sqrt{{\left(e^{x}\right)}^{10}}\right)}}^{x} \]

   7. unpow-prod-down 96.8%

      \[\leadsto \cos x \cdot \color{blue}{\left({\left(\sqrt{{\left(e^{x}\right)}^{10}}\right)}^{x} \cdot {\left(\sqrt{{\left(e^{x}\right)}^{10}}\right)}^{x}\right)} \]

   8. pow-prod-up 96.8%

      \[\leadsto \cos x \cdot \color{blue}{{\left(\sqrt{{\left(e^{x}\right)}^{10}}\right)}^{\left(x + x\right)}} \]

   9. sqrt-pow1 96.7%

      \[\leadsto \cos x \cdot {\color{blue}{\left({\left(e^{x}\right)}^{\left(\frac{10}{2}\right)}\right)}}^{\left(x + x\right)} \]

   10. add-exp-log 94.7%

       \[\leadsto \cos x \cdot {\color{blue}{\left(e^{\log \left({\left(e^{x}\right)}^{\left(\frac{10}{2}\right)}\right)}\right)}}^{\left(x + x\right)} \]

   11. log-pow 94.7%

       \[\leadsto \cos x \cdot {\left(e^{\color{blue}{\frac{10}{2} \cdot \log \left(e^{x}\right)}}\right)}^{\left(x + x\right)} \]

   12. metadata-eval 94.7%

       \[\leadsto \cos x \cdot {\left(e^{\color{blue}{5} \cdot \log \left(e^{x}\right)}\right)}^{\left(x + x\right)} \]

   13. add-log-exp 94.7%

       \[\leadsto \cos x \cdot {\left(e^{5 \cdot \color{blue}{x}}\right)}^{\left(x + x\right)} \]

3. Applied egg-rr 94.7%

   \[\leadsto \cos x \cdot \color{blue}{{\left(e^{5 \cdot x}\right)}^{\left(x + x\right)}} \]

5. Applied egg-rr 10.3%

   \[\leadsto \cos x \cdot \color{blue}{e^{x \cdot 5}} \]

6. Final simplification 10.3%

   \[\leadsto \cos x \cdot e^{x \cdot 5} \]

Alternative 9: 9.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \cos x \end{array} \]

FPCore

(FPCore (x) :precision binary64 (cos x))

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x)
	return cos(x)
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.4%

   \[\cos x \cdot e^{10 \cdot \left(x \cdot x\right)} \]

2. Taylor expanded in x around 0 9.6%

   \[\leadsto \cos x \cdot \color{blue}{1} \]

3. Final simplification 9.6%

   \[\leadsto \cos x \]

Alternative 10: 1.5% accurate, 29.6× speedup

Math

\[\begin{array}{l} \\ 1 + \left(x \cdot x\right) \cdot 9.5 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (+ 1.0 (* (* x x) 9.5)))

C

double code(double x) {
	return 1.0 + ((x * x) * 9.5);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 + ((x * x) * 9.5d0)
end function

Java

public static double code(double x) {
	return 1.0 + ((x * x) * 9.5);
}

Python

def code(x):
	return 1.0 + ((x * x) * 9.5)

Julia

function code(x)
	return Float64(1.0 + Float64(Float64(x * x) * 9.5))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 + ((x * x) * 9.5);
end

Wolfram

code[x_] := N[(1.0 + N[(N[(x * x), $MachinePrecision] * 9.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.4%

   \[\cos x \cdot e^{10 \cdot \left(x \cdot x\right)} \]

2. Taylor expanded in x around 0 1.5%

   \[\leadsto \color{blue}{1 + 9.5 \cdot {x}^{2}} \]
3. Step-by-step derivation

   1. unpow2 1.5%

      \[\leadsto 1 + 9.5 \cdot \color{blue}{\left(x \cdot x\right)} \]

4. Simplified 1.5%

   \[\leadsto \color{blue}{1 + 9.5 \cdot \left(x \cdot x\right)} \]

5. Final simplification 1.5%

   \[\leadsto 1 + \left(x \cdot x\right) \cdot 9.5 \]

Reproduce

herbie shell --seed 2023224 
(FPCore (x)
  :name "ENA, Section 1.4, Exercise 1"
  :precision binary64
  :pre (and (<= 1.99 x) (<= x 2.01))
  (* (cos x) (exp (* 10.0 (* x x)))))