ENA, Section 1.4, Exercise 1

Percentage Accurate: 94.5% → 99.4%

Time: 16.0s

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 94.4%

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

   1. pow-exp 95.2%

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

   2. sqr-pow 95.3%

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

   3. pow-prod-down 95.3%

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

   4. associate-/l* 95.3%

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

   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-log-exp 98.9%

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

   2. add-sqr-sqrt 98.2%

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

   3. log-prod 98.4%

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

6. Applied egg-rr 98.4%

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

   1. count-2 98.4%

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

8. Simplified 98.4%

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

   1. log1p-expm1-u 98.4%

      \[\leadsto \left(2 \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\log \left(\sqrt{e^{\cos x}}\right)\right)\right)}\right) \cdot {\left({\left(e^{20}\right)}^{x}\right)}^{\left(\frac{x}{2}\right)} \]

   2. pow1/2 98.4%

      \[\leadsto \left(2 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\log \color{blue}{\left({\left(e^{\cos x}\right)}^{0.5}\right)}\right)\right)\right) \cdot {\left({\left(e^{20}\right)}^{x}\right)}^{\left(\frac{x}{2}\right)} \]

   3. log-pow 98.9%

      \[\leadsto \left(2 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(\color{blue}{0.5 \cdot \log \left(e^{\cos x}\right)}\right)\right)\right) \cdot {\left({\left(e^{20}\right)}^{x}\right)}^{\left(\frac{x}{2}\right)} \]

   4. add-log-exp 99.3%

      \[\leadsto \left(2 \cdot \mathsf{log1p}\left(\mathsf{expm1}\left(0.5 \cdot \color{blue}{\cos x}\right)\right)\right) \cdot {\left({\left(e^{20}\right)}^{x}\right)}^{\left(\frac{x}{2}\right)} \]

10. Applied egg-rr 99.3%

    \[\leadsto \left(2 \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(0.5 \cdot \cos x\right)\right)}\right) \cdot {\left({\left(e^{20}\right)}^{x}\right)}^{\left(\frac{x}{2}\right)} \]

11. Taylor expanded in x around inf 94.3%

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

    1. *-commutative 94.3%

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

    2. exp-prod 94.3%

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

    3. *-commutative 94.3%

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

    4. exp-to-pow 99.4%

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

    5. unpow1/2 99.4%

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

13. Simplified 99.4%

    \[\leadsto \color{blue}{\cos x \cdot \sqrt{{\left({\left(e^{20}\right)}^{x}\right)}^{x}}} \]
14. 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. sqr-pow 99.1%

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

    3. pow-prod-down 99.4%

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

    4. pow-prod-down 99.3%

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

    5. prod-exp 99.3%

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

    6. metadata-eval 99.3%

       \[\leadsto \cos x \cdot {\left({\left(e^{\color{blue}{40}}\right)}^{x}\right)}^{\left(\frac{\frac{x}{2}}{2}\right)} \]

    7. div-inv 99.3%

       \[\leadsto \cos x \cdot {\left({\left(e^{40}\right)}^{x}\right)}^{\left(\frac{\color{blue}{x \cdot \frac{1}{2}}}{2}\right)} \]

    8. metadata-eval 99.3%

       \[\leadsto \cos x \cdot {\left({\left(e^{40}\right)}^{x}\right)}^{\left(\frac{x \cdot \color{blue}{0.5}}{2}\right)} \]

    9. associate-/l* 99.3%

       \[\leadsto \cos x \cdot {\left({\left(e^{40}\right)}^{x}\right)}^{\color{blue}{\left(x \cdot \frac{0.5}{2}\right)}} \]

    10. metadata-eval 99.3%

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

15. Applied egg-rr 99.3%

    \[\leadsto \cos x \cdot \color{blue}{{\left({\left(e^{40}\right)}^{x}\right)}^{\left(x \cdot 0.25\right)}} \]
16. 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 99.3%

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

    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%

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

17. Applied egg-rr 99.4%

    \[\leadsto \cos x \cdot {\color{blue}{\left(\sqrt{{\left(e^{80}\right)}^{x}}\right)}}^{\left(x \cdot 0.25\right)} \]
18. 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 99.5%

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

19. Applied egg-rr 99.5%

    \[\leadsto \cos x \cdot {\color{blue}{\left({\left(e^{80}\right)}^{\left(x \cdot 0.5\right)}\right)}}^{\left(x \cdot 0.25\right)} \]
20. 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 94.4%

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

   1. pow-exp 95.2%

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

   2. sqr-pow 95.3%

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

   3. pow-prod-down 95.3%

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

   4. associate-/l* 95.3%

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

   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 94.4%

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

   1. pow-exp 95.2%

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

   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: 95.2% 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 94.4%

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

   1. exp-prod 95.2%

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

3. Simplified 95.2%

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

Alternative 5: 94.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.4%

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

Alternative 6: 9.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.4%

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

3. Taylor expanded in x around 0 9.6%

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

4. Taylor expanded in x around 0 9.7%

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

   1. *-commutative 9.7%

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

6. Simplified 9.7%

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

7. Taylor expanded in x around inf 9.7%

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

   1. *-commutative 9.7%

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

9. Simplified 9.7%

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

10. Final simplification 9.7%

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

Alternative 7: 9.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \cos x \end{array} \]

FPCore

(FPCore (x) :precision binary64 (cos x))

C

double code(double x) {
	return cos(x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = cos(x)
end function

Java

public static double code(double x) {
	return Math.cos(x);
}

Python

def code(x):
	return math.cos(x)

Julia

function code(x)
	return cos(x)
end

MATLAB

function tmp = code(x)
	tmp = cos(x);
end

Wolfram

code[x_] := N[Cos[x], $MachinePrecision]

Derivation

1. Initial program 94.4%

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

3. Taylor expanded in x around 0 9.6%

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

4. Final simplification 9.6%

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

Alternative 8: 1.5% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x) :precision binary64 1.0)

C

double code(double x) {
	return 1.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0
end function

Java

public static double code(double x) {
	return 1.0;
}

Python

def code(x):
	return 1.0

Julia

function code(x)
	return 1.0
end

MATLAB

function tmp = code(x)
	tmp = 1.0;
end

Wolfram

code[x_] := 1.0

Derivation

1. Initial program 94.4%

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

   1. pow-exp 95.2%

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

   2. sqr-pow 95.3%

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

   3. pow-prod-down 95.3%

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

   4. associate-/l* 95.3%

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

   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 2024107 
(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)))))