ENA, Section 1.4, Exercise 1

Percentage Accurate: 94.5% → 99.4%

Time: 8.1s

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

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

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 94.7%

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

   1. pow-exp 94.7%

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

   2. *-commutative 94.7%

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

   3. exp-prod 95.4%

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

   4. pow-exp 96.7%

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

   5. pow-unpow 94.9%

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

   6. sqr-pow 94.8%

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

   7. pow2 94.8%

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

   8. pow-exp 94.5%

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

   9. pow-exp 94.5%

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

   10. associate-/l* 93.9%

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

   11. metadata-eval 93.9%

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

5. Applied egg-rr 93.9%

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

   1. associate-*l* 93.9%

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

   2. *-commutative 93.9%

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

   3. add-log-exp 93.9%

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

   4. div-inv 94.5%

      \[\leadsto \cos x \cdot e^{\log \left(e^{\color{blue}{\left(x \cdot \frac{1}{0.2}\right)} \cdot 2}\right) \cdot x} \]

   5. associate-*l* 94.5%

      \[\leadsto \cos x \cdot e^{\log \left(e^{\color{blue}{x \cdot \left(\frac{1}{0.2} \cdot 2\right)}}\right) \cdot x} \]

   6. metadata-eval 94.5%

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

   7. metadata-eval 94.5%

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

   8. pow-exp 94.4%

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

   9. metadata-eval 94.4%

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

   10. sqrt-pow1 94.4%

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

   11. pow-to-exp 96.8%

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

   12. sqrt-pow2 96.8%

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

   13. pow-to-exp 94.4%

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

   14. log-pow 94.5%

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

   15. rem-log-exp 94.5%

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

   16. add-log-exp 94.5%

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

   17. pow-exp 95.3%

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

   18. pow-to-exp 99.4%

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

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 99.2%

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

   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 99.3%

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

   4. prod-exp 99.5%

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

   5. metadata-eval 99.5%

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

9. Applied egg-rr 99.5%

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

10. Final simplification 99.5%

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

Alternative 2: 98.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.7%

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

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

3. Applied egg-rr 98.0%

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

   1. pow-exp 95.2%

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

   2. *-commutative 95.2%

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

   3. pow-exp 96.8%

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

   4. metadata-eval 96.8%

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

   5. sqrt-pow1 96.8%

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

5. Applied egg-rr 96.8%

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

   1. add-exp-log 95.2%

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

   2. log-pow 95.2%

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

   3. add-log-exp 95.2%

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

   4. pow-exp 99.1%

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

   5. add-sqr-sqrt 98.0%

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

   6. unpow-prod-down 98.0%

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

   7. metadata-eval 98.0%

      \[\leadsto \cos x \cdot {\left(\sqrt{{\left(\sqrt{e^{\color{blue}{10 + 10}}}\right)}^{x} \cdot {\left(\sqrt{e^{20}}\right)}^{x}}\right)}^{x} \]

   8. prod-exp 98.0%

      \[\leadsto \cos x \cdot {\left(\sqrt{{\left(\sqrt{\color{blue}{e^{10} \cdot e^{10}}}\right)}^{x} \cdot {\left(\sqrt{e^{20}}\right)}^{x}}\right)}^{x} \]

   9. sqrt-unprod 98.9%

      \[\leadsto \cos x \cdot {\left(\sqrt{{\color{blue}{\left(\sqrt{e^{10}} \cdot \sqrt{e^{10}}\right)}}^{x} \cdot {\left(\sqrt{e^{20}}\right)}^{x}}\right)}^{x} \]

   10. pow2 98.9%

       \[\leadsto \cos x \cdot {\left(\sqrt{{\color{blue}{\left({\left(\sqrt{e^{10}}\right)}^{2}\right)}}^{x} \cdot {\left(\sqrt{e^{20}}\right)}^{x}}\right)}^{x} \]

   11. pow-unpow 99.1%

       \[\leadsto \cos x \cdot {\left(\sqrt{\color{blue}{{\left(\sqrt{e^{10}}\right)}^{\left(2 \cdot x\right)}} \cdot {\left(\sqrt{e^{20}}\right)}^{x}}\right)}^{x} \]

   12. *-commutative 99.1%

       \[\leadsto \cos x \cdot {\left(\sqrt{{\left(\sqrt{e^{10}}\right)}^{\color{blue}{\left(x \cdot 2\right)}} \cdot {\left(\sqrt{e^{20}}\right)}^{x}}\right)}^{x} \]

   13. metadata-eval 99.1%

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

   14. prod-exp 99.1%

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

   15. sqrt-unprod 97.8%

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

   16. pow2 97.8%

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

   17. pow-unpow 98.4%

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

   18. *-commutative 98.4%

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

7. Applied egg-rr 98.4%

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

8. Final simplification 98.4%

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

Alternative 3: 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 94.7%

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

   1. associate-*r* 94.5%

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

   2. exp-prod 95.2%

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

   3. sqr-pow 95.2%

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

   4. pow-prod-down 95.2%

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

   5. *-commutative 95.2%

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

   6. exp-prod 96.2%

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

   7. *-commutative 96.2%

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

   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 95.3%

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

   2. expm1-udef 95.3%

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

5. Applied egg-rr 95.3%

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

   1. expm1-def 95.3%

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

   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 95.3%

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

   4. *-commutative 95.3%

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

   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 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 94.7%

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

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

3. Applied egg-rr 98.0%

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

4. Final simplification 98.0%

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

Alternative 5: 95.2% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.7%

   \[\cos x \cdot e^{10 \cdot \left(x \cdot x\right)} \]
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 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 94.7%

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

2. Final simplification 94.7%

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

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 94.7%

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

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 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 94.7%

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

   1. associate-*r* 94.5%

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

   2. exp-prod 95.2%

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

   3. sqr-pow 95.2%

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

   4. pow-prod-down 95.2%

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

   5. *-commutative 95.2%

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

   6. exp-prod 96.2%

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

   7. *-commutative 96.2%

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

   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 95.3%

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

   2. expm1-udef 95.3%

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

5. Applied egg-rr 95.3%

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

   1. expm1-def 95.3%

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

   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 95.3%

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

   4. *-commutative 95.3%

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

   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 2023332 
(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)))))