ENA, Section 1.4, Exercise 1

Percentage Accurate: 94.5% → 99.4%

Time: 8.6s

Alternatives: 9

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, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.5%

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

   1. pow-exp 95.2%

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

   2. add-cbrt-cube 95.2%

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

   3. pow3 95.2%

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

   4. pow2 95.2%

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

3. Applied egg-rr 95.2%

   \[\leadsto \cos x \cdot \color{blue}{\sqrt[3]{{\left({\left(e^{10}\right)}^{\left({x}^{2}\right)}\right)}^{3}}} \]
4. Step-by-step derivation

   1. rem-cbrt-cube 95.2%

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

   2. unpow2 95.2%

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

   3. pow-pow 97.9%

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

   4. sqr-pow 98.0%

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

   5. pow-prod-down 97.9%

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

   6. pow-prod-down 98.0%

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

   7. prod-exp 99.4%

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

   8. metadata-eval 99.4%

      \[\leadsto \cos x \cdot {\left({\left(e^{\color{blue}{20}}\right)}^{x}\right)}^{\left(\frac{x}{2}\right)} \]

   9. div-inv 99.4%

      \[\leadsto \cos x \cdot {\left({\left(e^{20}\right)}^{x}\right)}^{\color{blue}{\left(x \cdot \frac{1}{2}\right)}} \]

   10. metadata-eval 99.4%

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

5. Applied egg-rr 99.4%

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

6. Final simplification 99.4%

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

Alternative 2: 95.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \cos x \cdot {\left(e^{{x}^{2}}\right)}^{10} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (cos x) (pow (exp (pow x 2.0)) 10.0)))

C

double code(double x) {
	return cos(x) * pow(exp(pow(x, 2.0)), 10.0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = cos(x) * (exp((x ** 2.0d0)) ** 10.0d0)
end function

Java

public static double code(double x) {
	return Math.cos(x) * Math.pow(Math.exp(Math.pow(x, 2.0)), 10.0);
}

Python

def code(x):
	return math.cos(x) * math.pow(math.exp(math.pow(x, 2.0)), 10.0)

Julia

function code(x)
	return Float64(cos(x) * (exp((x ^ 2.0)) ^ 10.0))
end

MATLAB

function tmp = code(x)
	tmp = cos(x) * (exp((x ^ 2.0)) ^ 10.0);
end

Wolfram

code[x_] := N[(N[Cos[x], $MachinePrecision] * N[Power[N[Exp[N[Power[x, 2.0], $MachinePrecision]], $MachinePrecision], 10.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.5%

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

   1. cos-neg 94.5%

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

   2. sqr-neg 94.5%

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

   3. *-commutative 94.5%

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

   4. distribute-lft-neg-out 94.5%

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

   5. distribute-lft-neg-out 94.5%

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

   6. exp-neg 94.5%

      \[\leadsto \cos \left(-x\right) \cdot \color{blue}{\frac{1}{e^{\left(x \cdot \left(-x\right)\right) \cdot 10}}} \]

   7. metadata-eval 94.5%

      \[\leadsto \cos \left(-x\right) \cdot \frac{\color{blue}{-1 \cdot -1}}{e^{\left(x \cdot \left(-x\right)\right) \cdot 10}} \]

   8. associate-*r/ 94.5%

      \[\leadsto \color{blue}{\frac{\cos \left(-x\right) \cdot \left(-1 \cdot -1\right)}{e^{\left(x \cdot \left(-x\right)\right) \cdot 10}}} \]

   9. metadata-eval 94.5%

      \[\leadsto \frac{\cos \left(-x\right) \cdot \color{blue}{1}}{e^{\left(x \cdot \left(-x\right)\right) \cdot 10}} \]

   10. *-rgt-identity 94.5%

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

   11. cos-neg 94.5%

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

   12. metadata-eval 94.5%

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

   13. metadata-eval 94.5%

       \[\leadsto \frac{\cos x}{e^{\left(x \cdot \left(-x\right)\right) \cdot \left(-\color{blue}{10 \cdot -1}\right)}} \]

   14. distribute-rgt-neg-in 94.5%

       \[\leadsto \frac{\cos x}{e^{\color{blue}{-\left(x \cdot \left(-x\right)\right) \cdot \left(10 \cdot -1\right)}}} \]

   15. distribute-lft-neg-in 94.5%

       \[\leadsto \frac{\cos x}{e^{\color{blue}{\left(-x \cdot \left(-x\right)\right) \cdot \left(10 \cdot -1\right)}}} \]

   16. distribute-lft-neg-out 94.5%

       \[\leadsto \frac{\cos x}{e^{\color{blue}{\left(\left(-x\right) \cdot \left(-x\right)\right)} \cdot \left(10 \cdot -1\right)}} \]

   17. exp-prod 95.2%

       \[\leadsto \frac{\cos x}{\color{blue}{{\left(e^{\left(-x\right) \cdot \left(-x\right)}\right)}^{\left(10 \cdot -1\right)}}} \]

   18. sqr-neg 95.2%

       \[\leadsto \frac{\cos x}{{\left(e^{\color{blue}{x \cdot x}}\right)}^{\left(10 \cdot -1\right)}} \]

   19. exp-prod 96.7%

       \[\leadsto \frac{\cos x}{{\color{blue}{\left({\left(e^{x}\right)}^{x}\right)}}^{\left(10 \cdot -1\right)}} \]

3. Simplified 96.7%

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

4. Taylor expanded in x around inf 95.3%

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

5. Final simplification 95.3%

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

Alternative 3: 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 94.5%

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

   1. pow-exp 95.2%

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

   2. pow-unpow 97.9%

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

3. Applied egg-rr 97.9%

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

4. Final simplification 97.9%

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

Alternative 4: 98.1% accurate, 0.5× speedup

Math

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

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 94.5%

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

   1. cos-neg 94.5%

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

   2. sqr-neg 94.5%

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

   3. *-commutative 94.5%

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

   4. distribute-lft-neg-out 94.5%

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

   5. distribute-lft-neg-out 94.5%

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

   6. exp-neg 94.5%

      \[\leadsto \cos \left(-x\right) \cdot \color{blue}{\frac{1}{e^{\left(x \cdot \left(-x\right)\right) \cdot 10}}} \]

   7. metadata-eval 94.5%

      \[\leadsto \cos \left(-x\right) \cdot \frac{\color{blue}{-1 \cdot -1}}{e^{\left(x \cdot \left(-x\right)\right) \cdot 10}} \]

   8. associate-*r/ 94.5%

      \[\leadsto \color{blue}{\frac{\cos \left(-x\right) \cdot \left(-1 \cdot -1\right)}{e^{\left(x \cdot \left(-x\right)\right) \cdot 10}}} \]

   9. metadata-eval 94.5%

      \[\leadsto \frac{\cos \left(-x\right) \cdot \color{blue}{1}}{e^{\left(x \cdot \left(-x\right)\right) \cdot 10}} \]

   10. *-rgt-identity 94.5%

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

   11. cos-neg 94.5%

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

   12. associate-*l* 94.4%

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

   13. *-commutative 94.4%

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

   14. exp-prod 94.8%

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

   15. distribute-rgt-neg-out 94.8%

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

   16. *-commutative 94.8%

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

   17. distribute-rgt-neg-in 94.8%

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

   18. metadata-eval 94.8%

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

3. Simplified 94.8%

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

   1. add-sqr-sqrt 94.3%

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

   2. unpow-prod-down 94.2%

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

5. Applied egg-rr 94.2%

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

   1. pow-sqr 94.2%

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

   2. *-commutative 94.2%

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

   3. associate-*r* 94.2%

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

   4. metadata-eval 94.2%

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

7. Simplified 94.2%

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

   1. sqrt-pow2 94.8%

      \[\leadsto \frac{\cos x}{\color{blue}{{\left(e^{x}\right)}^{\left(\frac{-20 \cdot x}{2}\right)}}} \]

   2. pow-exp 94.4%

      \[\leadsto \frac{\cos x}{\color{blue}{e^{x \cdot \frac{-20 \cdot x}{2}}}} \]

   3. *-commutative 94.4%

      \[\leadsto \frac{\cos x}{e^{x \cdot \frac{\color{blue}{x \cdot -20}}{2}}} \]

   4. associate-/l* 93.8%

      \[\leadsto \frac{\cos x}{e^{x \cdot \color{blue}{\frac{x}{\frac{2}{-20}}}}} \]

   5. metadata-eval 93.8%

      \[\leadsto \frac{\cos x}{e^{x \cdot \frac{x}{\color{blue}{-0.1}}}} \]

9. Applied egg-rr 93.8%

   \[\leadsto \frac{\cos x}{\color{blue}{e^{x \cdot \frac{x}{-0.1}}}} \]
10. Step-by-step derivation

    1. associate-*r/ 94.2%

       \[\leadsto \frac{\cos x}{e^{\color{blue}{\frac{x \cdot x}{-0.1}}}} \]

    2. unpow2 94.2%

       \[\leadsto \frac{\cos x}{e^{\frac{\color{blue}{{x}^{2}}}{-0.1}}} \]

    3. div-inv 94.5%

       \[\leadsto \frac{\cos x}{e^{\color{blue}{{x}^{2} \cdot \frac{1}{-0.1}}}} \]

    4. metadata-eval 94.5%

       \[\leadsto \frac{\cos x}{e^{{x}^{2} \cdot \color{blue}{-10}}} \]

    5. rem-log-exp 94.5%

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

    6. log-pow 94.5%

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

    7. add-exp-log 95.2%

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

    8. unpow2 95.2%

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

    9. pow-unpow 98.1%

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

11. Applied egg-rr 98.1%

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

12. Final simplification 98.1%

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

Alternative 5: 95.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.5%

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

   1. pow-exp 95.2%

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

   2. pow-unpow 97.9%

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

3. Applied egg-rr 97.9%

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

   1. add-exp-log 95.0%

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

   2. log-pow 95.0%

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

   3. rem-log-exp 95.0%

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

5. Applied egg-rr 95.0%

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

6. Final simplification 95.0%

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

Alternative 6: 95.3% 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 94.5%

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

   1. cos-neg 94.5%

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

   2. *-commutative 94.5%

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

   3. exp-prod 95.2%

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

   4. cos-neg 95.2%

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

3. Simplified 95.2%

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

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 94.5%

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

2. Final simplification 94.5%

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

Alternative 8: 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 94.5%

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

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 9: 1.5% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x) :precision binary64 1.0)

C

double code(double x) {
	return 1.0;
}

Fortran

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

Java

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

Python

def code(x):
	return 1.0

Julia

function code(x)
	return 1.0
end

MATLAB

function tmp = code(x)
	tmp = 1.0;
end

Wolfram

code[x_] := 1.0

Derivation

1. Initial program 94.5%

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

   1. pow-exp 95.2%

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

   2. add-cbrt-cube 95.2%

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

   3. pow3 95.2%

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

   4. pow2 95.2%

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

3. Applied egg-rr 95.2%

   \[\leadsto \cos x \cdot \color{blue}{\sqrt[3]{{\left({\left(e^{10}\right)}^{\left({x}^{2}\right)}\right)}^{3}}} \]
4. Step-by-step derivation

   1. rem-cbrt-cube 95.2%

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

   2. unpow2 95.2%

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

   3. pow-pow 97.9%

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

   4. sqr-pow 98.0%

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

   5. pow-prod-down 97.9%

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

   6. pow-prod-down 98.0%

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

   7. prod-exp 99.4%

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

   8. metadata-eval 99.4%

      \[\leadsto \cos x \cdot {\left({\left(e^{\color{blue}{20}}\right)}^{x}\right)}^{\left(\frac{x}{2}\right)} \]

   9. div-inv 99.4%

      \[\leadsto \cos x \cdot {\left({\left(e^{20}\right)}^{x}\right)}^{\color{blue}{\left(x \cdot \frac{1}{2}\right)}} \]

   10. metadata-eval 99.4%

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

5. Applied egg-rr 99.4%

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

6. Taylor expanded in x around 0 1.5%

   \[\leadsto \color{blue}{1} \]

7. Final simplification 1.5%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023336 
(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)))))