ENA, Section 1.4, Exercise 1

Percentage Accurate: 94.5% → 99.4%

Time: 10.1s

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, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.8%

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

   1. pow-exp 95.4%

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

   2. sqr-pow 95.4%

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

   3. pow-prod-down 95.4%

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

   4. associate-/l* 95.4%

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

   5. pow-unpow 98.0%

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

   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 inf 94.6%

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

   1. *-commutative 94.6%

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

   2. exp-prod 94.6%

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

   3. unpow1/2 94.6%

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

   4. *-commutative 94.6%

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

   5. exp-to-pow 99.4%

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

7. Simplified 99.4%

   \[\leadsto \color{blue}{\cos x \cdot \sqrt{{\left({\left(e^{20}\right)}^{x}\right)}^{x}}} \]
8. 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 94.8%

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

   1. pow-exp 95.4%

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

   2. sqr-pow 95.4%

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

   3. pow-prod-down 95.4%

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

   4. associate-/l* 95.4%

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

   5. pow-unpow 98.0%

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

   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: 98.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \cos x \cdot {\left({\left(e^{5}\right)}^{x}\right)}^{\left(x \cdot 2\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (cos x) (pow (pow (exp 5.0) x) (* x 2.0))))

C

double code(double x) {
	return cos(x) * pow(pow(exp(5.0), x), (x * 2.0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = cos(x) * ((exp(5.0d0) ** x) ** (x * 2.0d0))
end function

Java

public static double code(double x) {
	return Math.cos(x) * Math.pow(Math.pow(Math.exp(5.0), x), (x * 2.0));
}

Python

def code(x):
	return math.cos(x) * math.pow(math.pow(math.exp(5.0), x), (x * 2.0))

Julia

function code(x)
	return Float64(cos(x) * ((exp(5.0) ^ x) ^ Float64(x * 2.0)))
end

MATLAB

function tmp = code(x)
	tmp = cos(x) * ((exp(5.0) ^ x) ^ (x * 2.0));
end

Wolfram

code[x_] := N[(N[Cos[x], $MachinePrecision] * N[Power[N[Power[N[Exp[5.0], $MachinePrecision], x], $MachinePrecision], N[(x * 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.8%

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

   1. pow-exp 95.4%

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

   2. sqr-pow 95.4%

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

   3. pow-prod-down 95.4%

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

   4. associate-/l* 95.4%

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

   5. pow-unpow 98.0%

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

   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 inf 94.6%

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

   1. *-commutative 94.6%

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

   2. exp-prod 94.6%

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

   3. unpow1/2 94.6%

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

   4. *-commutative 94.6%

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

   5. exp-to-pow 99.4%

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

7. Simplified 99.4%

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

   1. sqrt-pow1 99.4%

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

   2. add-sqr-sqrt 99.2%

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

   3. unpow-prod-down 98.9%

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

   4. div-inv 98.9%

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

   5. metadata-eval 98.9%

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

   6. div-inv 98.9%

      \[\leadsto \cos x \cdot \left({\left(\sqrt{{\left(e^{20}\right)}^{x}}\right)}^{\left(x \cdot 0.5\right)} \cdot {\left(\sqrt{{\left(e^{20}\right)}^{x}}\right)}^{\color{blue}{\left(x \cdot \frac{1}{2}\right)}}\right) \]

   7. metadata-eval 98.9%

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

9. Applied egg-rr 98.9%

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

    1. pow-sqr 99.3%

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

    2. *-commutative 99.3%

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

    3. associate-*r* 99.3%

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

    4. metadata-eval 99.3%

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

    5. *-lft-identity 99.3%

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

11. Simplified 99.3%

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

    1. sqrt-pow2 99.4%

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

    2. pow-pow 95.4%

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

    3. metadata-eval 95.4%

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

    4. prod-exp 95.4%

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

    5. associate-/l* 95.4%

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

    6. pow-prod-down 95.4%

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

    7. sqr-pow 95.4%

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

    8. pow-unpow 98.0%

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

    9. add-sqr-sqrt 98.1%

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

    10. unpow-prod-down 97.9%

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

    11. sqrt-pow1 98.0%

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

    12. pow-to-exp 96.1%

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

    13. associate-/l* 96.1%

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

    14. *-commutative 96.1%

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

    15. associate-/l* 96.1%

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

    16. rem-log-exp 96.1%

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

    17. metadata-eval 96.1%

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

13. Applied egg-rr 94.9%

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

    1. pow-sqr 94.9%

       \[\leadsto \cos x \cdot \color{blue}{{\left(e^{x \cdot 5}\right)}^{\left(2 \cdot x\right)}} \]

    2. *-commutative 94.9%

       \[\leadsto \cos x \cdot {\left(e^{\color{blue}{5 \cdot x}}\right)}^{\left(2 \cdot x\right)} \]

    3. exp-prod 98.1%

       \[\leadsto \cos x \cdot {\color{blue}{\left({\left(e^{5}\right)}^{x}\right)}}^{\left(2 \cdot x\right)} \]

    4. *-commutative 98.1%

       \[\leadsto \cos x \cdot {\left({\left(e^{5}\right)}^{x}\right)}^{\color{blue}{\left(x \cdot 2\right)}} \]

15. Simplified 98.1%

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

Alternative 4: 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 94.8%

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

   1. pow-exp 95.4%

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

   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. Add Preprocessing

Alternative 5: 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 94.8%

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

   1. exp-prod 95.4%

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

3. Simplified 95.4%

   \[\leadsto \color{blue}{\cos x \cdot {\left(e^{10}\right)}^{\left(x \cdot x\right)}} \]
4. Add Preprocessing
5. 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 94.8%

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

Alternative 7: 9.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \frac{1}{1 + {x}^{2} \cdot -9.5} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 1.0 (+ 1.0 (* (pow x 2.0) -9.5))))

C

double code(double x) {
	return 1.0 / (1.0 + (pow(x, 2.0) * -9.5));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 / (1.0d0 + ((x ** 2.0d0) * (-9.5d0)))
end function

Java

public static double code(double x) {
	return 1.0 / (1.0 + (Math.pow(x, 2.0) * -9.5));
}

Python

def code(x):
	return 1.0 / (1.0 + (math.pow(x, 2.0) * -9.5))

Julia

function code(x)
	return Float64(1.0 / Float64(1.0 + Float64((x ^ 2.0) * -9.5)))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 / (1.0 + ((x ^ 2.0) * -9.5));
end

Wolfram

code[x_] := N[(1.0 / N[(1.0 + N[(N[Power[x, 2.0], $MachinePrecision] * -9.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.8%

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

   1. pow-exp 95.4%

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

   2. sqr-pow 95.4%

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

   3. pow-prod-down 95.4%

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

   4. associate-/l* 95.4%

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

   5. pow-unpow 98.0%

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

   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. pow-pow 95.4%

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

   2. metadata-eval 95.4%

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

   3. prod-exp 95.4%

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

   4. metadata-eval 95.4%

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

   5. pow-exp 95.1%

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

   6. metadata-eval 95.1%

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

   7. pow-exp 94.5%

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

   8. associate-*r/ 94.5%

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

   9. pow-prod-down 94.4%

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

   10. sqr-pow 94.4%

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

   11. pow-unpow 93.9%

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

   12. pow-exp 98.0%

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

   13. metadata-eval 98.0%

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

   14. expm1-log1p-u 94.5%

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

   15. log1p-define 94.6%

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

   16. expm1-undefine 94.7%

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

   17. add-exp-log 96.3%

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

   18. pow-div 96.3%

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

6. Applied egg-rr 96.3%

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

7. Taylor expanded in x around 0 9.5%

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

   1. *-commutative 9.5%

      \[\leadsto \frac{1}{1 + \color{blue}{{x}^{2} \cdot -9.5}} \]

9. Simplified 9.5%

   \[\leadsto \frac{1}{\color{blue}{1 + {x}^{2} \cdot -9.5}} \]
10. 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 94.8%

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

   1. pow-exp 95.4%

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

   2. sqr-pow 95.4%

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

   3. pow-prod-down 95.4%

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

   4. associate-/l* 95.4%

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

   5. pow-unpow 98.0%

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

   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 2024163 
(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)))))