ENA, Section 1.4, Exercise 1

Percentage Accurate: 94.5% → 99.5%

Time: 9.8s

Alternatives: 11

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: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \cos x \cdot {\left({\left(e^{20}\right)}^{\left(x \cdot 2\right)}\right)}^{\left(x \cdot 0.25\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (cos x) (pow (pow (exp 20.0) (* x 2.0)) (* x 0.25))))

C

double code(double x) {
	return cos(x) * pow(pow(exp(20.0), (x * 2.0)), (x * 0.25));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = cos(x) * ((exp(20.0d0) ** (x * 2.0d0)) ** (x * 0.25d0))
end function

Java

public static double code(double x) {
	return Math.cos(x) * Math.pow(Math.pow(Math.exp(20.0), (x * 2.0)), (x * 0.25));
}

Python

def code(x):
	return math.cos(x) * math.pow(math.pow(math.exp(20.0), (x * 2.0)), (x * 0.25))

Julia

function code(x)
	return Float64(cos(x) * ((exp(20.0) ^ Float64(x * 2.0)) ^ Float64(x * 0.25)))
end

MATLAB

function tmp = code(x)
	tmp = cos(x) * ((exp(20.0) ^ (x * 2.0)) ^ (x * 0.25));
end

Wolfram

code[x_] := N[(N[Cos[x], $MachinePrecision] * N[Power[N[Power[N[Exp[20.0], $MachinePrecision], N[(x * 2.0), $MachinePrecision]], $MachinePrecision], N[(x * 0.25), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.6%

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

   1. associate-*r* 94.4%

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

   2. *-commutative 94.4%

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

   3. exp-prod 94.9%

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

   4. expm1-log1p-u 92.8%

      \[\leadsto \cos x \cdot {\left(e^{x}\right)}^{\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(10 \cdot x\right)\right)\right)}} \]

   5. expm1-undefine 92.8%

      \[\leadsto \cos x \cdot {\left(e^{x}\right)}^{\color{blue}{\left(e^{\mathsf{log1p}\left(10 \cdot x\right)} - 1\right)}} \]

   6. pow-sub 92.8%

      \[\leadsto \cos x \cdot \color{blue}{\frac{{\left(e^{x}\right)}^{\left(e^{\mathsf{log1p}\left(10 \cdot x\right)}\right)}}{{\left(e^{x}\right)}^{1}}} \]

   7. *-commutative 92.8%

      \[\leadsto \cos x \cdot \frac{{\left(e^{x}\right)}^{\left(e^{\mathsf{log1p}\left(\color{blue}{x \cdot 10}\right)}\right)}}{{\left(e^{x}\right)}^{1}} \]

   8. pow1 92.8%

      \[\leadsto \cos x \cdot \frac{{\left(e^{x}\right)}^{\left(e^{\mathsf{log1p}\left(x \cdot 10\right)}\right)}}{\color{blue}{e^{x}}} \]

4. Applied egg-rr 92.8%

   \[\leadsto \cos x \cdot \color{blue}{\frac{{\left(e^{x}\right)}^{\left(e^{\mathsf{log1p}\left(x \cdot 10\right)}\right)}}{e^{x}}} \]
5. Step-by-step derivation

   1. log1p-undefine 92.9%

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

   2. rem-exp-log 94.9%

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

   3. *-commutative 94.9%

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

6. Simplified 94.9%

   \[\leadsto \cos x \cdot \color{blue}{\frac{{\left(e^{x}\right)}^{\left(1 + 10 \cdot x\right)}}{e^{x}}} \]
7. Step-by-step derivation

   1. pow1 94.9%

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

   2. pow-div 94.9%

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

   3. add-exp-log 92.9%

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

   4. expm1-undefine 92.9%

      \[\leadsto \cos x \cdot {\left(e^{x}\right)}^{\color{blue}{\left(\mathsf{expm1}\left(\log \left(1 + 10 \cdot x\right)\right)\right)}} \]

   5. log1p-define 92.8%

      \[\leadsto \cos x \cdot {\left(e^{x}\right)}^{\left(\mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(10 \cdot x\right)}\right)\right)} \]

   6. expm1-log1p-u 94.9%

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

   7. pow-unpow 96.7%

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

   8. exp-prod 95.1%

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

   9. *-commutative 95.1%

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

   10. pow-exp 97.9%

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

   11. metadata-eval 97.9%

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

   12. pow-exp 93.8%

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

   13. pow-unpow 94.2%

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

   14. sqr-pow 94.2%

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

   15. pow-prod-down 94.3%

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

   16. pow-exp 94.9%

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

   17. metadata-eval 94.9%

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

   18. pow-exp 95.1%

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

   19. metadata-eval 95.1%

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

   20. prod-exp 95.1%

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

   21. metadata-eval 95.1%

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

   22. associate-*r/ 95.1%

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

8. Applied egg-rr 99.2%

   \[\leadsto \cos x \cdot \color{blue}{{\left({\left(e^{40}\right)}^{x}\right)}^{\left(\left(x \cdot 0.5\right) \cdot 0.5\right)}} \]
9. Step-by-step derivation

   1. associate-*l* 99.2%

      \[\leadsto \cos x \cdot {\left({\left(e^{40}\right)}^{x}\right)}^{\color{blue}{\left(x \cdot \left(0.5 \cdot 0.5\right)\right)}} \]

   2. metadata-eval 99.2%

      \[\leadsto \cos x \cdot {\left({\left(e^{40}\right)}^{x}\right)}^{\left(x \cdot \color{blue}{0.25}\right)} \]

10. Simplified 99.2%

    \[\leadsto \cos x \cdot \color{blue}{{\left({\left(e^{40}\right)}^{x}\right)}^{\left(x \cdot 0.25\right)}} \]
11. Step-by-step derivation

    1. metadata-eval 99.2%

       \[\leadsto \cos x \cdot {\left({\left(e^{\color{blue}{20 + 20}}\right)}^{x}\right)}^{\left(x \cdot 0.25\right)} \]

    2. prod-exp 99.2%

       \[\leadsto \cos x \cdot {\left({\color{blue}{\left(e^{20} \cdot e^{20}\right)}}^{x}\right)}^{\left(x \cdot 0.25\right)} \]

    3. pow-prod-down 99.3%

       \[\leadsto \cos x \cdot {\color{blue}{\left({\left(e^{20}\right)}^{x} \cdot {\left(e^{20}\right)}^{x}\right)}}^{\left(x \cdot 0.25\right)} \]

12. Applied egg-rr 99.3%

    \[\leadsto \cos x \cdot {\color{blue}{\left({\left(e^{20}\right)}^{x} \cdot {\left(e^{20}\right)}^{x}\right)}}^{\left(x \cdot 0.25\right)} \]
13. Step-by-step derivation

    1. pow-sqr 99.5%

       \[\leadsto \cos x \cdot {\color{blue}{\left({\left(e^{20}\right)}^{\left(2 \cdot x\right)}\right)}}^{\left(x \cdot 0.25\right)} \]

    2. *-commutative 99.5%

       \[\leadsto \cos x \cdot {\left({\left(e^{20}\right)}^{\color{blue}{\left(x \cdot 2\right)}}\right)}^{\left(x \cdot 0.25\right)} \]

14. Simplified 99.5%

    \[\leadsto \cos x \cdot {\color{blue}{\left({\left(e^{20}\right)}^{\left(x \cdot 2\right)}\right)}}^{\left(x \cdot 0.25\right)} \]
15. Add Preprocessing

Alternative 2: 99.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \cos x \cdot {\left({\left(e^{20}\right)}^{x}\right)}^{\left(\frac{x}{2}\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (cos x) (pow (pow (exp 20.0) x) (/ x 2.0))))

C

double code(double x) {
	return cos(x) * pow(pow(exp(20.0), x), (x / 2.0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = cos(x) * ((exp(20.0d0) ** x) ** (x / 2.0d0))
end function

Java

public static double code(double x) {
	return Math.cos(x) * Math.pow(Math.pow(Math.exp(20.0), x), (x / 2.0));
}

Python

def code(x):
	return math.cos(x) * math.pow(math.pow(math.exp(20.0), x), (x / 2.0))

Julia

function code(x)
	return Float64(cos(x) * ((exp(20.0) ^ x) ^ Float64(x / 2.0)))
end

MATLAB

function tmp = code(x)
	tmp = cos(x) * ((exp(20.0) ^ x) ^ (x / 2.0));
end

Wolfram

code[x_] := N[(N[Cos[x], $MachinePrecision] * N[Power[N[Power[N[Exp[20.0], $MachinePrecision], x], $MachinePrecision], N[(x / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.6%

   \[\cos x \cdot e^{10 \cdot \left(x \cdot x\right)} \]
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. sqr-pow 95.1%

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

   3. pow-prod-down 95.1%

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

   4. associate-/l* 95.1%

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

   5. pow-unpow 97.9%

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

   6. prod-exp 99.4%

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

   7. metadata-eval 99.4%

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

4. Applied egg-rr 99.4%

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

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 94.6%

   \[\cos x \cdot e^{10 \cdot \left(x \cdot x\right)} \]
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 97.9%

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

4. Applied egg-rr 97.9%

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

Alternative 4: 95.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \cos x \cdot {\left(e^{x \cdot 40}\right)}^{\left(x \cdot 0.25\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (cos x) (pow (exp (* x 40.0)) (* x 0.25))))

C

double code(double x) {
	return cos(x) * pow(exp((x * 40.0)), (x * 0.25));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = cos(x) * (exp((x * 40.0d0)) ** (x * 0.25d0))
end function

Java

public static double code(double x) {
	return Math.cos(x) * Math.pow(Math.exp((x * 40.0)), (x * 0.25));
}

Python

def code(x):
	return math.cos(x) * math.pow(math.exp((x * 40.0)), (x * 0.25))

Julia

function code(x)
	return Float64(cos(x) * (exp(Float64(x * 40.0)) ^ Float64(x * 0.25)))
end

MATLAB

function tmp = code(x)
	tmp = cos(x) * (exp((x * 40.0)) ^ (x * 0.25));
end

Wolfram

code[x_] := N[(N[Cos[x], $MachinePrecision] * N[Power[N[Exp[N[(x * 40.0), $MachinePrecision]], $MachinePrecision], N[(x * 0.25), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.6%

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

   1. associate-*r* 94.4%

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

   2. *-commutative 94.4%

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

   3. exp-prod 94.9%

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

   4. expm1-log1p-u 92.8%

      \[\leadsto \cos x \cdot {\left(e^{x}\right)}^{\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(10 \cdot x\right)\right)\right)}} \]

   5. expm1-undefine 92.8%

      \[\leadsto \cos x \cdot {\left(e^{x}\right)}^{\color{blue}{\left(e^{\mathsf{log1p}\left(10 \cdot x\right)} - 1\right)}} \]

   6. pow-sub 92.8%

      \[\leadsto \cos x \cdot \color{blue}{\frac{{\left(e^{x}\right)}^{\left(e^{\mathsf{log1p}\left(10 \cdot x\right)}\right)}}{{\left(e^{x}\right)}^{1}}} \]

   7. *-commutative 92.8%

      \[\leadsto \cos x \cdot \frac{{\left(e^{x}\right)}^{\left(e^{\mathsf{log1p}\left(\color{blue}{x \cdot 10}\right)}\right)}}{{\left(e^{x}\right)}^{1}} \]

   8. pow1 92.8%

      \[\leadsto \cos x \cdot \frac{{\left(e^{x}\right)}^{\left(e^{\mathsf{log1p}\left(x \cdot 10\right)}\right)}}{\color{blue}{e^{x}}} \]

4. Applied egg-rr 92.8%

   \[\leadsto \cos x \cdot \color{blue}{\frac{{\left(e^{x}\right)}^{\left(e^{\mathsf{log1p}\left(x \cdot 10\right)}\right)}}{e^{x}}} \]
5. Step-by-step derivation

   1. log1p-undefine 92.9%

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

   2. rem-exp-log 94.9%

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

   3. *-commutative 94.9%

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

6. Simplified 94.9%

   \[\leadsto \cos x \cdot \color{blue}{\frac{{\left(e^{x}\right)}^{\left(1 + 10 \cdot x\right)}}{e^{x}}} \]
7. Step-by-step derivation

   1. pow1 94.9%

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

   2. pow-div 94.9%

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

   3. add-exp-log 92.9%

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

   4. expm1-undefine 92.9%

      \[\leadsto \cos x \cdot {\left(e^{x}\right)}^{\color{blue}{\left(\mathsf{expm1}\left(\log \left(1 + 10 \cdot x\right)\right)\right)}} \]

   5. log1p-define 92.8%

      \[\leadsto \cos x \cdot {\left(e^{x}\right)}^{\left(\mathsf{expm1}\left(\color{blue}{\mathsf{log1p}\left(10 \cdot x\right)}\right)\right)} \]

   6. expm1-log1p-u 94.9%

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

   7. pow-unpow 96.7%

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

   8. exp-prod 95.1%

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

   9. *-commutative 95.1%

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

   10. pow-exp 97.9%

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

   11. metadata-eval 97.9%

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

   12. pow-exp 93.8%

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

   13. pow-unpow 94.2%

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

   14. sqr-pow 94.2%

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

   15. pow-prod-down 94.3%

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

   16. pow-exp 94.9%

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

   17. metadata-eval 94.9%

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

   18. pow-exp 95.1%

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

   19. metadata-eval 95.1%

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

   20. prod-exp 95.1%

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

   21. metadata-eval 95.1%

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

   22. associate-*r/ 95.1%

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

8. Applied egg-rr 99.2%

   \[\leadsto \cos x \cdot \color{blue}{{\left({\left(e^{40}\right)}^{x}\right)}^{\left(\left(x \cdot 0.5\right) \cdot 0.5\right)}} \]
9. Step-by-step derivation

   1. associate-*l* 99.2%

      \[\leadsto \cos x \cdot {\left({\left(e^{40}\right)}^{x}\right)}^{\color{blue}{\left(x \cdot \left(0.5 \cdot 0.5\right)\right)}} \]

   2. metadata-eval 99.2%

      \[\leadsto \cos x \cdot {\left({\left(e^{40}\right)}^{x}\right)}^{\left(x \cdot \color{blue}{0.25}\right)} \]

10. Simplified 99.2%

    \[\leadsto \cos x \cdot \color{blue}{{\left({\left(e^{40}\right)}^{x}\right)}^{\left(x \cdot 0.25\right)}} \]
11. Step-by-step derivation

    1. add-exp-log 95.2%

       \[\leadsto \cos x \cdot {\color{blue}{\left(e^{\log \left({\left(e^{40}\right)}^{x}\right)}\right)}}^{\left(x \cdot 0.25\right)} \]

    2. log-pow 95.2%

       \[\leadsto \cos x \cdot {\left(e^{\color{blue}{x \cdot \log \left(e^{40}\right)}}\right)}^{\left(x \cdot 0.25\right)} \]

    3. rem-log-exp 95.2%

       \[\leadsto \cos x \cdot {\left(e^{x \cdot \color{blue}{40}}\right)}^{\left(x \cdot 0.25\right)} \]

12. Applied egg-rr 95.2%

    \[\leadsto \cos x \cdot {\color{blue}{\left(e^{x \cdot 40}\right)}}^{\left(x \cdot 0.25\right)} \]
13. Add Preprocessing

Alternative 5: 95.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \cos x \cdot {\left(e^{x \cdot 20}\right)}^{\left(\frac{x}{2}\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (cos x) (pow (exp (* x 20.0)) (/ x 2.0))))

C

double code(double x) {
	return cos(x) * pow(exp((x * 20.0)), (x / 2.0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = cos(x) * (exp((x * 20.0d0)) ** (x / 2.0d0))
end function

Java

public static double code(double x) {
	return Math.cos(x) * Math.pow(Math.exp((x * 20.0)), (x / 2.0));
}

Python

def code(x):
	return math.cos(x) * math.pow(math.exp((x * 20.0)), (x / 2.0))

Julia

function code(x)
	return Float64(cos(x) * (exp(Float64(x * 20.0)) ^ Float64(x / 2.0)))
end

MATLAB

function tmp = code(x)
	tmp = cos(x) * (exp((x * 20.0)) ^ (x / 2.0));
end

Wolfram

code[x_] := N[(N[Cos[x], $MachinePrecision] * N[Power[N[Exp[N[(x * 20.0), $MachinePrecision]], $MachinePrecision], N[(x / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.6%

   \[\cos x \cdot e^{10 \cdot \left(x \cdot x\right)} \]
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. sqr-pow 95.1%

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

   3. pow-prod-down 95.1%

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

   4. associate-/l* 95.1%

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

   5. pow-unpow 97.9%

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

   6. prod-exp 99.4%

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

   7. metadata-eval 99.4%

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

4. Applied egg-rr 99.4%

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

   1. add-exp-log 95.2%

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

   2. log-pow 95.2%

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

   3. rem-log-exp 95.2%

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

6. Applied egg-rr 95.2%

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

Alternative 6: 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 94.6%

   \[\cos x \cdot e^{10 \cdot \left(x \cdot x\right)} \]
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 97.9%

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

4. Applied egg-rr 97.9%

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

   1. add-exp-log 95.1%

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

   2. log-pow 95.1%

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

   3. rem-log-exp 95.1%

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

6. Applied egg-rr 95.1%

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

Alternative 7: 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 94.6%

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

Alternative 8: 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 94.6%

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

Alternative 9: 9.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ 1 + {x}^{2} \cdot -0.5 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (+ 1.0 (* (pow x 2.0) -0.5)))

C

double code(double x) {
	return 1.0 + (pow(x, 2.0) * -0.5);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 + ((x ** 2.0d0) * (-0.5d0))
end function

Java

public static double code(double x) {
	return 1.0 + (Math.pow(x, 2.0) * -0.5);
}

Python

def code(x):
	return 1.0 + (math.pow(x, 2.0) * -0.5)

Julia

function code(x)
	return Float64(1.0 + Float64((x ^ 2.0) * -0.5))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 + ((x ^ 2.0) * -0.5);
end

Wolfram

code[x_] := N[(1.0 + N[(N[Power[x, 2.0], $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.6%

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

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

   1. *-commutative 9.7%

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

6. Simplified 9.7%

   \[\leadsto \color{blue}{1 + {x}^{2} \cdot -0.5} \]
7. Add Preprocessing

Alternative 10: 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 94.6%

   \[\cos x \cdot e^{10 \cdot \left(x \cdot x\right)} \]
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 inf 9.6%

   \[\leadsto \color{blue}{\cos x} \]
5. Add Preprocessing

Alternative 11: 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 94.6%

   \[\cos x \cdot e^{10 \cdot \left(x \cdot x\right)} \]
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. sqr-pow 95.1%

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

   3. pow-prod-down 95.1%

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

   4. associate-/l* 95.1%

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

   5. pow-unpow 97.9%

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

   6. prod-exp 99.4%

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

   7. metadata-eval 99.4%

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

4. Applied egg-rr 99.4%

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

5. Taylor expanded in x around 0 1.5%

   \[\leadsto \color{blue}{1} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024135 
(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)))))