ENA, Section 1.4, Exercise 1

Percentage Accurate: 94.5% → 99.4%

Time: 8.8s

Alternatives: 7

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

\[\begin{array}{l} \\ \cos x \cdot {\left(\sqrt{{\left(e^{80}\right)}^{x}}\right)}^{\left(\frac{x}{4}\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (cos x) (pow (sqrt (pow (exp 80.0) x)) (/ x 4.0))))

C

double code(double x) {
	return cos(x) * pow(sqrt(pow(exp(80.0), x)), (x / 4.0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = cos(x) * (sqrt((exp(80.0d0) ** x)) ** (x / 4.0d0))
end function

Java

public static double code(double x) {
	return Math.cos(x) * Math.pow(Math.sqrt(Math.pow(Math.exp(80.0), x)), (x / 4.0));
}

Python

def code(x):
	return math.cos(x) * math.pow(math.sqrt(math.pow(math.exp(80.0), x)), (x / 4.0))

Julia

function code(x)
	return Float64(cos(x) * (sqrt((exp(80.0) ^ x)) ^ Float64(x / 4.0)))
end

MATLAB

function tmp = code(x)
	tmp = cos(x) * (sqrt((exp(80.0) ^ x)) ^ (x / 4.0));
end

Wolfram

code[x_] := N[(N[Cos[x], $MachinePrecision] * N[Power[N[Sqrt[N[Power[N[Exp[80.0], $MachinePrecision], x], $MachinePrecision]], $MachinePrecision], N[(x / 4.0), $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 95.3%

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

3. Simplified 95.3%

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

   1. pow-exp 94.4%

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

   2. *-commutative 94.4%

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

   3. exp-prod 95.3%

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

   4. pow-exp 96.7%

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

   5. pow-unpow 94.6%

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

   6. sqr-pow 94.7%

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

   7. pow2 94.7%

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

   8. pow-exp 94.3%

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

   9. pow-exp 94.3%

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

   10. associate-/l* 93.8%

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

   11. metadata-eval 93.8%

       \[\leadsto \cos x \cdot e^{\left(x \cdot \frac{x}{\color{blue}{0.2}}\right) \cdot 2} \]

5. Applied egg-rr 93.8%

   \[\leadsto \cos x \cdot \color{blue}{e^{\left(x \cdot \frac{x}{0.2}\right) \cdot 2}} \]
6. Step-by-step derivation

   1. associate-*l* 93.8%

      \[\leadsto \cos x \cdot e^{\color{blue}{x \cdot \left(\frac{x}{0.2} \cdot 2\right)}} \]

   2. exp-prod 94.1%

      \[\leadsto \cos x \cdot \color{blue}{{\left(e^{x}\right)}^{\left(\frac{x}{0.2} \cdot 2\right)}} \]

   3. metadata-eval 94.1%

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

   4. associate-/l* 94.6%

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

   5. *-commutative 94.6%

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

   6. pow-pow 94.7%

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

   7. sqrt-pow2 94.4%

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

   8. pow2 94.4%

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

   9. pow-prod-up 94.5%

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

   10. add-exp-log 93.9%

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

   11. log-pow 93.9%

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

   12. count-2 93.9%

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

   13. associate-*r* 93.9%

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

   14. metadata-eval 93.9%

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

   15. rem-log-exp 93.9%

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

   16. pow1/2 93.9%

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

   17. pow-exp 93.8%

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

   18. add-log-exp 94.3%

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

   19. metadata-eval 94.3%

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

   20. div-inv 94.3%

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

7. Applied egg-rr 99.2%

   \[\leadsto \cos x \cdot \color{blue}{{\left({\left(e^{40}\right)}^{x}\right)}^{\left(\frac{x}{4}\right)}} \]
8. Step-by-step derivation

   1. add-sqr-sqrt 99.1%

      \[\leadsto \cos x \cdot {\color{blue}{\left(\sqrt{{\left(e^{40}\right)}^{x}} \cdot \sqrt{{\left(e^{40}\right)}^{x}}\right)}}^{\left(\frac{x}{4}\right)} \]

   2. sqrt-unprod 99.2%

      \[\leadsto \cos x \cdot {\color{blue}{\left(\sqrt{{\left(e^{40}\right)}^{x} \cdot {\left(e^{40}\right)}^{x}}\right)}}^{\left(\frac{x}{4}\right)} \]

   3. pow-prod-down 99.2%

      \[\leadsto \cos x \cdot {\left(\sqrt{\color{blue}{{\left(e^{40} \cdot e^{40}\right)}^{x}}}\right)}^{\left(\frac{x}{4}\right)} \]

   4. prod-exp 99.4%

      \[\leadsto \cos x \cdot {\left(\sqrt{{\color{blue}{\left(e^{40 + 40}\right)}}^{x}}\right)}^{\left(\frac{x}{4}\right)} \]

   5. metadata-eval 99.4%

      \[\leadsto \cos x \cdot {\left(\sqrt{{\left(e^{\color{blue}{80}}\right)}^{x}}\right)}^{\left(\frac{x}{4}\right)} \]

9. Applied egg-rr 99.4%

   \[\leadsto \cos x \cdot {\color{blue}{\left(\sqrt{{\left(e^{80}\right)}^{x}}\right)}}^{\left(\frac{x}{4}\right)} \]

10. Final simplification 99.4%

    \[\leadsto \cos x \cdot {\left(\sqrt{{\left(e^{80}\right)}^{x}}\right)}^{\left(\frac{x}{4}\right)} \]

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. Step-by-step derivation

   1. associate-*r* 94.3%

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

   2. exp-prod 94.9%

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

   3. sqr-pow 94.9%

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

   4. pow-prod-down 94.9%

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

   5. *-commutative 94.9%

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

   6. exp-prod 96.0%

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

   7. *-commutative 96.0%

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

   8. exp-prod 96.8%

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

   9. pow-prod-up 96.8%

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

   10. metadata-eval 96.8%

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

3. Applied egg-rr 96.8%

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

   1. expm1-log1p-u 94.9%

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

   2. expm1-udef 94.9%

      \[\leadsto \cos x \cdot {\color{blue}{\left(e^{\mathsf{log1p}\left({\left(e^{x}\right)}^{20}\right)} - 1\right)}}^{\left(\frac{x}{2}\right)} \]

5. Applied egg-rr 94.9%

   \[\leadsto \cos x \cdot {\color{blue}{\left(e^{\mathsf{log1p}\left({\left(e^{x}\right)}^{20}\right)} - 1\right)}}^{\left(\frac{x}{2}\right)} \]
6. Step-by-step derivation

   1. expm1-def 94.9%

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

   2. expm1-log1p 96.8%

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

   3. exp-prod 94.9%

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

   4. *-commutative 94.9%

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

   5. exp-prod 99.4%

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

7. Simplified 99.4%

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

8. Final simplification 99.4%

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

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 94.4%

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

   1. pow-exp 95.3%

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

   2. pow-unpow 97.9%

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

3. Applied egg-rr 97.9%

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

4. Final simplification 97.9%

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

Alternative 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 95.3%

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

3. Simplified 95.3%

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

4. Final simplification 95.3%

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

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. Final simplification 94.4%

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

Alternative 6: 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. Taylor expanded in x around 0 9.6%

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

3. Final simplification 9.6%

   \[\leadsto \cos x \]

Alternative 7: 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. Step-by-step derivation

   1. associate-*r* 94.3%

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

   2. exp-prod 94.9%

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

   3. sqr-pow 94.9%

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

   4. pow-prod-down 94.9%

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

   5. *-commutative 94.9%

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

   6. exp-prod 96.0%

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

   7. *-commutative 96.0%

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

   8. exp-prod 96.8%

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

   9. pow-prod-up 96.8%

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

   10. metadata-eval 96.8%

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

3. Applied egg-rr 96.8%

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

   1. expm1-log1p-u 94.9%

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

   2. expm1-udef 94.9%

      \[\leadsto \cos x \cdot {\color{blue}{\left(e^{\mathsf{log1p}\left({\left(e^{x}\right)}^{20}\right)} - 1\right)}}^{\left(\frac{x}{2}\right)} \]

5. Applied egg-rr 94.9%

   \[\leadsto \cos x \cdot {\color{blue}{\left(e^{\mathsf{log1p}\left({\left(e^{x}\right)}^{20}\right)} - 1\right)}}^{\left(\frac{x}{2}\right)} \]
6. Step-by-step derivation

   1. expm1-def 94.9%

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

   2. expm1-log1p 96.8%

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

   3. exp-prod 94.9%

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

   4. *-commutative 94.9%

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

   5. exp-prod 99.4%

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

7. Simplified 99.4%

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

8. Taylor expanded in x around 0 1.5%

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

9. Final simplification 1.5%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023315 
(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)))))