ENA, Section 1.4, Exercise 1

Percentage Accurate: 94.5% → 99.4%

Time: 9.2s

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 94.5%

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

   1. associate-*r* 94.4%

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

   2. add-log-exp 94.4%

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

   3. log-pow 94.3%

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

   4. pow-pow 94.9%

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

   5. add-exp-log 96.8%

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

   6. sqr-pow 96.8%

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

   7. pow-prod-down 96.8%

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

   8. div-inv 96.8%

      \[\leadsto \cos x \cdot {\left({\left(e^{x}\right)}^{10} \cdot {\left(e^{x}\right)}^{10}\right)}^{\color{blue}{\left(x \cdot \frac{1}{2}\right)}} \]

   9. metadata-eval 96.8%

      \[\leadsto \cos x \cdot {\left({\left(e^{x}\right)}^{10} \cdot {\left(e^{x}\right)}^{10}\right)}^{\left(x \cdot \color{blue}{0.5}\right)} \]

   10. pow-prod-up 96.8%

       \[\leadsto \cos x \cdot {\color{blue}{\left({\left(e^{x}\right)}^{\left(10 + 10\right)}\right)}}^{\left(x \cdot 0.5\right)} \]

   11. metadata-eval 96.8%

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

3. Applied egg-rr 96.8%

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

   1. add-sqr-sqrt 94.9%

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

   2. unpow-prod-down 94.8%

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

5. Applied egg-rr 94.8%

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

   1. pow-sqr 94.8%

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

   2. metadata-eval 94.8%

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

7. Simplified 94.8%

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

8. Taylor expanded in x around inf 96.8%

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

   1. exp-prod 95.2%

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

   2. *-commutative 95.2%

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

   3. exp-prod 99.4%

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

10. Simplified 99.4%

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

11. Final simplification 99.4%

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

Alternative 2: 98.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.5%

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

   1. associate-*r* 94.4%

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

   2. exp-prod 95.2%

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

   3. sqr-pow 95.1%

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

   4. sqr-pow 95.2%

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

   5. exp-prod 97.9%

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

3. Simplified 97.9%

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

4. Final simplification 97.9%

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

Alternative 3: 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 94.5%

   \[\cos x \cdot e^{10 \cdot \left(x \cdot x\right)} \]
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. Final simplification 95.4%

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

Alternative 4: 95.2% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.5%

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

   1. associate-*r* 94.4%

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

   2. add-log-exp 94.4%

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

   3. log-pow 94.3%

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

   4. pow-pow 94.9%

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

   5. add-exp-log 96.8%

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

   6. sqr-pow 96.8%

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

   7. pow-prod-down 96.8%

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

   8. div-inv 96.8%

      \[\leadsto \cos x \cdot {\left({\left(e^{x}\right)}^{10} \cdot {\left(e^{x}\right)}^{10}\right)}^{\color{blue}{\left(x \cdot \frac{1}{2}\right)}} \]

   9. metadata-eval 96.8%

      \[\leadsto \cos x \cdot {\left({\left(e^{x}\right)}^{10} \cdot {\left(e^{x}\right)}^{10}\right)}^{\left(x \cdot \color{blue}{0.5}\right)} \]

   10. pow-prod-up 96.8%

       \[\leadsto \cos x \cdot {\color{blue}{\left({\left(e^{x}\right)}^{\left(10 + 10\right)}\right)}}^{\left(x \cdot 0.5\right)} \]

   11. metadata-eval 96.8%

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

3. Applied egg-rr 96.8%

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

   1. add-sqr-sqrt 94.9%

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

   2. unpow-prod-down 94.8%

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

5. Applied egg-rr 94.8%

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

   1. pow-sqr 94.8%

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

   2. metadata-eval 94.8%

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

7. Simplified 94.8%

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

8. Taylor expanded in x around inf 96.8%

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

   1. exp-prod 95.2%

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

   2. *-commutative 95.2%

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

   3. exp-prod 99.4%

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

10. Simplified 99.4%

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

11. Taylor expanded in x around inf 94.4%

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

    1. *-commutative 94.4%

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

    2. rem-exp-log 0.0%

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

    3. prod-exp 0.0%

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

    4. associate-*r* 0.0%

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

    5. exp-prod 0.0%

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

    6. *-commutative 0.0%

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

    7. exp-prod 0.0%

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

    8. log-pow 0.0%

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

    9. rem-log-exp 0.0%

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

    10. associate-*r* 0.0%

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

    11. metadata-eval 0.0%

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

    12. *-commutative 0.0%

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

    13. metadata-eval 0.0%

        \[\leadsto e^{\left(x \cdot \color{blue}{\left(30 \cdot 0.3333333333333333\right)}\right) \cdot x + \log \cos x} \]

    14. associate-*l* 0.0%

        \[\leadsto e^{\color{blue}{\left(\left(x \cdot 30\right) \cdot 0.3333333333333333\right)} \cdot x + \log \cos x} \]

    15. *-commutative 0.0%

        \[\leadsto e^{\color{blue}{\left(0.3333333333333333 \cdot \left(x \cdot 30\right)\right)} \cdot x + \log \cos x} \]

    16. associate-*r* 0.0%

        \[\leadsto e^{\color{blue}{0.3333333333333333 \cdot \left(\left(x \cdot 30\right) \cdot x\right)} + \log \cos x} \]

    17. *-commutative 0.0%

        \[\leadsto e^{0.3333333333333333 \cdot \color{blue}{\left(x \cdot \left(x \cdot 30\right)\right)} + \log \cos x} \]

13. Simplified 95.5%

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

14. Final simplification 95.5%

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

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 94.5%

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

2. Final simplification 94.5%

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

Alternative 6: 18.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ e^{10 \cdot \left(x \cdot x\right)} \cdot \left(1 + \left(x \cdot x\right) \cdot -0.5\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (exp (* 10.0 (* x x))) (+ 1.0 (* (* x x) -0.5))))

C

double code(double x) {
	return exp((10.0 * (x * x))) * (1.0 + ((x * x) * -0.5));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = exp((10.0d0 * (x * x))) * (1.0d0 + ((x * x) * (-0.5d0)))
end function

Java

public static double code(double x) {
	return Math.exp((10.0 * (x * x))) * (1.0 + ((x * x) * -0.5));
}

Python

def code(x):
	return math.exp((10.0 * (x * x))) * (1.0 + ((x * x) * -0.5))

Julia

function code(x)
	return Float64(exp(Float64(10.0 * Float64(x * x))) * Float64(1.0 + Float64(Float64(x * x) * -0.5)))
end

MATLAB

function tmp = code(x)
	tmp = exp((10.0 * (x * x))) * (1.0 + ((x * x) * -0.5));
end

Wolfram

code[x_] := N[(N[Exp[N[(10.0 * N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(N[(x * x), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.5%

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

2. Taylor expanded in x around 0 18.2%

   \[\leadsto \color{blue}{\left(1 + -0.5 \cdot {x}^{2}\right)} \cdot e^{10 \cdot \left(x \cdot x\right)} \]
3. Step-by-step derivation

   1. unpow2 18.2%

      \[\leadsto \left(1 + -0.5 \cdot \color{blue}{\left(x \cdot x\right)}\right) \cdot e^{10 \cdot \left(x \cdot x\right)} \]

4. Simplified 18.2%

   \[\leadsto \color{blue}{\left(1 + -0.5 \cdot \left(x \cdot x\right)\right)} \cdot e^{10 \cdot \left(x \cdot x\right)} \]

5. Final simplification 18.2%

   \[\leadsto e^{10 \cdot \left(x \cdot x\right)} \cdot \left(1 + \left(x \cdot x\right) \cdot -0.5\right) \]

Alternative 7: 9.7% accurate, 41.4× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(x \cdot -0.5\right) \end{array} \]

FPCore

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

C

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

Fortran

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

Java

public static double code(double x) {
	return x * (x * -0.5);
}

Python

def code(x):
	return x * (x * -0.5)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.5%

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

   1. *-commutative 94.5%

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

   2. exp-prod 95.5%

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

   3. exp-prod 96.8%

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

   4. pow-unpow 94.7%

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

   5. add-sqr-sqrt 93.3%

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

   6. associate-*l* 93.4%

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

   7. pow-unpow 93.7%

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

3. Applied egg-rr 93.7%

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

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%

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

   2. unpow2 9.7%

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

6. Simplified 9.7%

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

7. Taylor expanded in x around inf 9.7%

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

   1. *-commutative 9.7%

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

   2. unpow2 9.7%

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

   3. associate-*r* 9.7%

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

9. Simplified 9.7%

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

10. Final simplification 9.7%

    \[\leadsto x \cdot \left(x \cdot -0.5\right) \]

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 94.5%

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

   1. associate-*r* 94.4%

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

   2. add-log-exp 94.4%

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

   3. log-pow 94.3%

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

   4. pow-pow 94.9%

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

   5. add-exp-log 96.8%

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

   6. sqr-pow 96.8%

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

   7. pow-prod-down 96.8%

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

   8. div-inv 96.8%

      \[\leadsto \cos x \cdot {\left({\left(e^{x}\right)}^{10} \cdot {\left(e^{x}\right)}^{10}\right)}^{\color{blue}{\left(x \cdot \frac{1}{2}\right)}} \]

   9. metadata-eval 96.8%

      \[\leadsto \cos x \cdot {\left({\left(e^{x}\right)}^{10} \cdot {\left(e^{x}\right)}^{10}\right)}^{\left(x \cdot \color{blue}{0.5}\right)} \]

   10. pow-prod-up 96.8%

       \[\leadsto \cos x \cdot {\color{blue}{\left({\left(e^{x}\right)}^{\left(10 + 10\right)}\right)}}^{\left(x \cdot 0.5\right)} \]

   11. metadata-eval 96.8%

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

3. Applied egg-rr 96.8%

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

   1. add-sqr-sqrt 94.9%

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

   2. unpow-prod-down 94.8%

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

5. Applied egg-rr 94.8%

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

   1. pow-sqr 94.8%

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

   2. metadata-eval 94.8%

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

7. Simplified 94.8%

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

8. Taylor expanded in x around inf 96.8%

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

   1. exp-prod 95.2%

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

   2. *-commutative 95.2%

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

   3. exp-prod 99.4%

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

10. Simplified 99.4%

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

11. Taylor expanded in x around 0 1.5%

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

12. Final simplification 1.5%

    \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023240 
(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)))))