ENA, Section 1.4, Exercise 1

Percentage Accurate: 94.5% → 99.4%

Time: 9.7s

Alternatives: 8

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 94.4%

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

   1. pow-exp 95.1%

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

   2. add-sqr-sqrt 95.1%

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

   3. sqrt-unprod 95.1%

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

   4. add-exp-log 94.4%

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

   5. pow-prod-down 94.4%

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

   6. log-pow 94.4%

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

   7. pow-exp 95.2%

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

   8. pow2 95.2%

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

   9. log-prod 95.2%

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

   10. rem-log-exp 95.2%

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

   11. rem-log-exp 95.2%

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

   12. metadata-eval 95.2%

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

4. Applied egg-rr 95.2%

   \[\leadsto \cos x \cdot \color{blue}{\sqrt{{\left(e^{{x}^{2}}\right)}^{20}}} \]
5. Step-by-step derivation

   1. add-exp-log 94.4%

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

   2. sqrt-pow1 94.4%

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

   3. metadata-eval 94.4%

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

   4. log-pow 94.4%

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

   5. add-log-exp 94.4%

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

   6. pow-exp 95.1%

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

   7. unpow2 95.1%

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

   8. pow-pow 98.0%

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

   9. sqr-pow 98.0%

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

   10. pow-prod-down 98.0%

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

   11. pow-prod-down 98.0%

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

   12. prod-exp 99.4%

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

   13. metadata-eval 99.4%

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

   14. div-inv 99.4%

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

   15. metadata-eval 99.4%

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

6. Applied egg-rr 99.4%

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

7. Final simplification 99.4%

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

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 94.4%

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

   1. *-commutative 94.4%

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

   2. exp-prod 95.2%

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

   3. pow2 95.2%

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

4. Applied egg-rr 95.2%

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

5. Final simplification 95.2%

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

Alternative 3: 97.9% 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. Add Preprocessing
3. Step-by-step derivation

   1. pow-exp 95.1%

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

   2. pow-unpow 98.0%

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

4. Applied egg-rr 98.0%

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

5. Final simplification 98.0%

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

Alternative 4: 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 94.4%

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

   1. exp-prod 95.1%

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

3. Simplified 95.1%

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

5. Final simplification 95.1%

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

Alternative 5: 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. Add Preprocessing

3. Final simplification 94.4%

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

Alternative 6: 9.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ {x}^{2} \cdot -0.5 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (pow x 2.0) -0.5))

C

double code(double x) {
	return pow(x, 2.0) * -0.5;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x ** 2.0d0) * (-0.5d0)
end function

Java

public static double code(double x) {
	return Math.pow(x, 2.0) * -0.5;
}

Python

def code(x):
	return math.pow(x, 2.0) * -0.5

Julia

function code(x)
	return Float64((x ^ 2.0) * -0.5)
end

MATLAB

function tmp = code(x)
	tmp = (x ^ 2.0) * -0.5;
end

Wolfram

code[x_] := N[(N[Power[x, 2.0], $MachinePrecision] * -0.5), $MachinePrecision]

Derivation

1. Initial program 94.4%

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

3. Taylor expanded in x around 0 9.6%

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

4. Taylor expanded in x around 0 9.7%

   \[\leadsto \color{blue}{\left(1 + -0.5 \cdot {x}^{2}\right)} \cdot 1 \]
5. Step-by-step derivation

   1. *-commutative 9.7%

      \[\leadsto \left(1 + \color{blue}{{x}^{2} \cdot -0.5}\right) \cdot 1 \]

6. Simplified 9.7%

   \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot -0.5\right)} \cdot 1 \]

7. Taylor expanded in x around inf 9.7%

   \[\leadsto \color{blue}{\left(-0.5 \cdot {x}^{2}\right)} \cdot 1 \]
8. Step-by-step derivation

   1. *-commutative 9.7%

      \[\leadsto \color{blue}{\left({x}^{2} \cdot -0.5\right)} \cdot 1 \]

9. Simplified 9.7%

   \[\leadsto \color{blue}{\left({x}^{2} \cdot -0.5\right)} \cdot 1 \]

10. Final simplification 9.7%

    \[\leadsto {x}^{2} \cdot -0.5 \]
11. Add Preprocessing

Alternative 7: 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. Add Preprocessing

3. Taylor expanded in x around 0 9.6%

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

4. Final simplification 9.6%

   \[\leadsto \cos x \]
5. Add Preprocessing

Alternative 8: 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 94.4%

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

   1. pow-exp 95.1%

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

   2. add-sqr-sqrt 95.1%

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

   3. sqrt-unprod 95.1%

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

   4. add-exp-log 94.4%

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

   5. pow-prod-down 94.4%

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

   6. log-pow 94.4%

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

   7. pow-exp 95.2%

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

   8. pow2 95.2%

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

   9. log-prod 95.2%

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

   10. rem-log-exp 95.2%

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

   11. rem-log-exp 95.2%

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

   12. metadata-eval 95.2%

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

4. Applied egg-rr 95.2%

   \[\leadsto \cos x \cdot \color{blue}{\sqrt{{\left(e^{{x}^{2}}\right)}^{20}}} \]
5. Step-by-step derivation

   1. add-exp-log 94.4%

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

   2. sqrt-pow1 94.4%

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

   3. metadata-eval 94.4%

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

   4. log-pow 94.4%

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

   5. add-log-exp 94.4%

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

   6. pow-exp 95.1%

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

   7. unpow2 95.1%

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

   8. pow-pow 98.0%

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

   9. sqr-pow 98.0%

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

   10. pow-prod-down 98.0%

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

   11. pow-prod-down 98.0%

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

   12. prod-exp 99.4%

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

   13. metadata-eval 99.4%

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

   14. div-inv 99.4%

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

   15. metadata-eval 99.4%

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

6. Applied egg-rr 99.4%

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

7. Taylor expanded in x around 0 1.5%

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

8. Final simplification 1.5%

   \[\leadsto 1 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024035 
(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)))))