ENA, Section 1.4, Exercise 1

Percentage Accurate: 94.5% → 99.4%

Time: 9.0s

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

\[\begin{array}{l} \\ \cos x \cdot {\left({\left(e^{80}\right)}^{\left(x \cdot 0.5\right)}\right)}^{\left(x \cdot 0.25\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (cos x) (pow (pow (exp 80.0) (* x 0.5)) (* x 0.25))))

C

double code(double x) {
	return cos(x) * pow(pow(exp(80.0), (x * 0.5)), (x * 0.25));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = cos(x) * ((exp(80.0d0) ** (x * 0.5d0)) ** (x * 0.25d0))
end function

Java

public static double code(double x) {
	return Math.cos(x) * Math.pow(Math.pow(Math.exp(80.0), (x * 0.5)), (x * 0.25));
}

Python

def code(x):
	return math.cos(x) * math.pow(math.pow(math.exp(80.0), (x * 0.5)), (x * 0.25))

Julia

function code(x)
	return Float64(cos(x) * ((exp(80.0) ^ Float64(x * 0.5)) ^ Float64(x * 0.25)))
end

MATLAB

function tmp = code(x)
	tmp = cos(x) * ((exp(80.0) ^ (x * 0.5)) ^ (x * 0.25));
end

Wolfram

code[x_] := N[(N[Cos[x], $MachinePrecision] * N[Power[N[Power[N[Exp[80.0], $MachinePrecision], N[(x * 0.5), $MachinePrecision]], $MachinePrecision], N[(x * 0.25), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.3%

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

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

   3. expm1-log1p-u 94.4%

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

   4. expm1-undefine 94.7%

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

   5. pow-sub 94.7%

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

   6. pow1 94.7%

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

4. Applied egg-rr 94.7%

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

   1. log1p-undefine 94.5%

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

   2. rem-exp-log 96.2%

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

6. Simplified 96.2%

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

   1. pow1 96.2%

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

   2. pow-div 96.2%

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

   3. add-exp-log 94.5%

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

   4. expm1-undefine 94.6%

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

   5. log1p-define 94.4%

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

   6. expm1-log1p-u 98.0%

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

   7. pow-pow 95.1%

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

   8. add-sqr-sqrt 95.2%

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

   9. sqrt-unprod 95.1%

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

   10. prod-exp 95.1%

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

   11. metadata-eval 95.1%

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

   12. pow1/2 95.1%

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

   13. unpow2 95.1%

       \[\leadsto \cos x \cdot {\left({\left(e^{20}\right)}^{0.5}\right)}^{\color{blue}{\left({x}^{2}\right)}} \]

   14. pow-unpow 95.1%

       \[\leadsto \cos x \cdot \color{blue}{{\left(e^{20}\right)}^{\left(0.5 \cdot {x}^{2}\right)}} \]

   15. *-commutative 95.1%

       \[\leadsto \cos x \cdot {\left(e^{20}\right)}^{\color{blue}{\left({x}^{2} \cdot 0.5\right)}} \]

   16. unpow2 95.1%

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

   17. associate-*r* 95.1%

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

   18. metadata-eval 95.1%

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

   19. div-inv 95.1%

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

   20. pow-pow 99.4%

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

   21. sqr-pow 99.1%

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

   22. pow-prod-down 99.3%

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

8. Applied egg-rr 99.2%

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

   1. add-sqr-sqrt 99.2%

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

   2. sqrt-unprod 99.2%

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

   3. pow-prod-down 99.2%

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

   4. prod-exp 99.4%

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

   5. metadata-eval 99.4%

      \[\leadsto \cos x \cdot {\left(\sqrt{{\left(e^{\color{blue}{80}}\right)}^{x}}\right)}^{\left(x \cdot 0.25\right)} \]

10. Applied egg-rr 99.4%

    \[\leadsto \cos x \cdot {\color{blue}{\left(\sqrt{{\left(e^{80}\right)}^{x}}\right)}}^{\left(x \cdot 0.25\right)} \]
11. Step-by-step derivation

    1. pow1/2 99.4%

       \[\leadsto \cos x \cdot {\color{blue}{\left({\left({\left(e^{80}\right)}^{x}\right)}^{0.5}\right)}}^{\left(x \cdot 0.25\right)} \]

    2. pow-pow 99.4%

       \[\leadsto \cos x \cdot {\color{blue}{\left({\left(e^{80}\right)}^{\left(x \cdot 0.5\right)}\right)}}^{\left(x \cdot 0.25\right)} \]

12. Applied egg-rr 99.4%

    \[\leadsto \cos x \cdot {\color{blue}{\left({\left(e^{80}\right)}^{\left(x \cdot 0.5\right)}\right)}}^{\left(x \cdot 0.25\right)} \]

13. Final simplification 99.4%

    \[\leadsto \cos x \cdot {\left({\left(e^{80}\right)}^{\left(x \cdot 0.5\right)}\right)}^{\left(x \cdot 0.25\right)} \]
14. 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 94.3%

   \[\cos x \cdot e^{10 \cdot \left(x \cdot x\right)} \]
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. Final simplification 99.4%

   \[\leadsto \cos x \cdot {\left({\left(e^{20}\right)}^{x}\right)}^{\left(\frac{x}{2}\right)} \]
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 94.3%

   \[\cos x \cdot e^{10 \cdot \left(x \cdot x\right)} \]
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: 95.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \cos x \cdot {\left(e^{80}\right)}^{\left(x \cdot \left(x \cdot 0.125\right)\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (cos x) (pow (exp 80.0) (* x (* x 0.125)))))

C

double code(double x) {
	return cos(x) * pow(exp(80.0), (x * (x * 0.125)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = cos(x) * (exp(80.0d0) ** (x * (x * 0.125d0)))
end function

Java

public static double code(double x) {
	return Math.cos(x) * Math.pow(Math.exp(80.0), (x * (x * 0.125)));
}

Python

def code(x):
	return math.cos(x) * math.pow(math.exp(80.0), (x * (x * 0.125)))

Julia

function code(x)
	return Float64(cos(x) * (exp(80.0) ^ Float64(x * Float64(x * 0.125))))
end

MATLAB

function tmp = code(x)
	tmp = cos(x) * (exp(80.0) ^ (x * (x * 0.125)));
end

Wolfram

code[x_] := N[(N[Cos[x], $MachinePrecision] * N[Power[N[Exp[80.0], $MachinePrecision], N[(x * N[(x * 0.125), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.3%

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

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

   3. expm1-log1p-u 94.4%

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

   4. expm1-undefine 94.7%

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

   5. pow-sub 94.7%

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

   6. pow1 94.7%

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

4. Applied egg-rr 94.7%

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

   1. log1p-undefine 94.5%

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

   2. rem-exp-log 96.2%

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

6. Simplified 96.2%

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

   1. pow1 96.2%

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

   2. pow-div 96.2%

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

   3. add-exp-log 94.5%

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

   4. expm1-undefine 94.6%

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

   5. log1p-define 94.4%

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

   6. expm1-log1p-u 98.0%

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

   7. pow-pow 95.1%

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

   8. add-sqr-sqrt 95.2%

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

   9. sqrt-unprod 95.1%

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

   10. prod-exp 95.1%

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

   11. metadata-eval 95.1%

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

   12. pow1/2 95.1%

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

   13. unpow2 95.1%

       \[\leadsto \cos x \cdot {\left({\left(e^{20}\right)}^{0.5}\right)}^{\color{blue}{\left({x}^{2}\right)}} \]

   14. pow-unpow 95.1%

       \[\leadsto \cos x \cdot \color{blue}{{\left(e^{20}\right)}^{\left(0.5 \cdot {x}^{2}\right)}} \]

   15. *-commutative 95.1%

       \[\leadsto \cos x \cdot {\left(e^{20}\right)}^{\color{blue}{\left({x}^{2} \cdot 0.5\right)}} \]

   16. unpow2 95.1%

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

   17. associate-*r* 95.1%

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

   18. metadata-eval 95.1%

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

   19. div-inv 95.1%

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

   20. pow-pow 99.4%

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

   21. sqr-pow 99.1%

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

   22. pow-prod-down 99.3%

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

8. Applied egg-rr 99.2%

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

   1. add-sqr-sqrt 99.2%

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

   2. sqrt-unprod 99.2%

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

   3. pow-prod-down 99.2%

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

   4. prod-exp 99.4%

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

   5. metadata-eval 99.4%

      \[\leadsto \cos x \cdot {\left(\sqrt{{\left(e^{\color{blue}{80}}\right)}^{x}}\right)}^{\left(x \cdot 0.25\right)} \]

10. Applied egg-rr 99.4%

    \[\leadsto \cos x \cdot {\color{blue}{\left(\sqrt{{\left(e^{80}\right)}^{x}}\right)}}^{\left(x \cdot 0.25\right)} \]
11. Step-by-step derivation

    1. *-un-lft-identity 99.4%

       \[\leadsto \cos x \cdot \color{blue}{\left(1 \cdot {\left(\sqrt{{\left(e^{80}\right)}^{x}}\right)}^{\left(x \cdot 0.25\right)}\right)} \]

    2. *-commutative 99.4%

       \[\leadsto \cos x \cdot \color{blue}{\left({\left(\sqrt{{\left(e^{80}\right)}^{x}}\right)}^{\left(x \cdot 0.25\right)} \cdot 1\right)} \]

    3. sqrt-pow2 99.4%

       \[\leadsto \cos x \cdot \left(\color{blue}{{\left({\left(e^{80}\right)}^{x}\right)}^{\left(\frac{x \cdot 0.25}{2}\right)}} \cdot 1\right) \]

    4. pow-pow 95.2%

       \[\leadsto \cos x \cdot \left(\color{blue}{{\left(e^{80}\right)}^{\left(x \cdot \frac{x \cdot 0.25}{2}\right)}} \cdot 1\right) \]

    5. associate-/l* 95.2%

       \[\leadsto \cos x \cdot \left({\left(e^{80}\right)}^{\left(x \cdot \color{blue}{\left(x \cdot \frac{0.25}{2}\right)}\right)} \cdot 1\right) \]

    6. metadata-eval 95.2%

       \[\leadsto \cos x \cdot \left({\left(e^{80}\right)}^{\left(x \cdot \left(x \cdot \color{blue}{0.125}\right)\right)} \cdot 1\right) \]

12. Applied egg-rr 95.2%

    \[\leadsto \cos x \cdot \color{blue}{\left({\left(e^{80}\right)}^{\left(x \cdot \left(x \cdot 0.125\right)\right)} \cdot 1\right)} \]

13. Final simplification 95.2%

    \[\leadsto \cos x \cdot {\left(e^{80}\right)}^{\left(x \cdot \left(x \cdot 0.125\right)\right)} \]
14. Add Preprocessing

Alternative 5: 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 94.3%

   \[\cos x \cdot e^{10 \cdot \left(x \cdot x\right)} \]
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 6: 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 94.3%

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

3. Final simplification 94.3%

   \[\leadsto \cos x \cdot e^{10 \cdot \left(x \cdot x\right)} \]
4. 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 94.3%

   \[\cos x \cdot e^{10 \cdot \left(x \cdot x\right)} \]
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 94.3%

   \[\cos x \cdot e^{10 \cdot \left(x \cdot x\right)} \]
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. Final simplification 1.5%

   \[\leadsto 1 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024067 
(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)))))