ENA, Section 1.4, Exercise 1

Percentage Accurate: 94.5% → 99.4%

Time: 8.4s

Alternatives: 16

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

\[\begin{array}{l} \\ \cos x \cdot {\left({\left(e^{80}\right)}^{x}\right)}^{\left(\frac{x}{8}\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (cos x) (pow (pow (exp 80.0) x) (/ x 8.0))))

C

double code(double x) {
	return cos(x) * pow(pow(exp(80.0), x), (x / 8.0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = cos(x) * ((exp(80.0d0) ** x) ** (x / 8.0d0))
end function

Java

public static double code(double x) {
	return Math.cos(x) * Math.pow(Math.pow(Math.exp(80.0), x), (x / 8.0));
}

Python

def code(x):
	return math.cos(x) * math.pow(math.pow(math.exp(80.0), x), (x / 8.0))

Julia

function code(x)
	return Float64(cos(x) * ((exp(80.0) ^ x) ^ Float64(x / 8.0)))
end

MATLAB

function tmp = code(x)
	tmp = cos(x) * ((exp(80.0) ^ x) ^ (x / 8.0));
end

Wolfram

code[x_] := N[(N[Cos[x], $MachinePrecision] * N[Power[N[Power[N[Exp[80.0], $MachinePrecision], x], $MachinePrecision], N[(x / 8.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.3%

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

   1. lift-exp.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. exp-prod N/A

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

   4. lift-*.f64 N/A

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

   5. sqr-neg-rev N/A

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

   6. pow-unpow N/A

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

   7. lower-pow.f64 N/A

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

   8. lower-pow.f64 N/A

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

   9. lower-exp.f64 N/A

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

   10. lower-neg.f64 N/A

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

   11. lower-neg.f64 98.1

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

4. Applied rewrites 98.1%

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

   1. lift-pow.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. pow-pow N/A

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

   4. lift-neg.f64 N/A

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

   5. lift-neg.f64 N/A

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

   6. sqr-neg-rev N/A

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

   7. lift-*.f64 N/A

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

   8. sqr-pow N/A

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

   9. pow-prod-down N/A

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

   10. lift-*.f64 N/A

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

   11. associate-*r/ N/A

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

   12. pow-unpow N/A

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

   13. lower-pow.f64 N/A

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

   14. lower-pow.f64 N/A

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

   15. lift-exp.f64 N/A

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

   16. lift-exp.f64 N/A

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

   17. exp-lft-sqr-rev N/A

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

   18. metadata-eval N/A

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

   19. metadata-eval N/A

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

   20. lower-exp.f64 N/A

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

   21. metadata-eval N/A

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

   22. lower-/.f64 99.3

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

6. Applied rewrites 99.3%

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

   1. lift-pow.f64 N/A

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

   2. sqr-pow N/A

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

   3. pow-prod-down N/A

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

   4. lower-pow.f64 N/A

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

   5. lift-pow.f64 N/A

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

   6. lift-pow.f64 N/A

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

   7. pow-prod-down N/A

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

   8. lower-pow.f64 N/A

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

   9. lift-exp.f64 N/A

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

   10. lift-exp.f64 N/A

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

   11. exp-lft-sqr-rev N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval N/A

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

   14. lower-exp.f64 N/A

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

   15. metadata-eval N/A

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

   16. lift-/.f64 N/A

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

   17. associate-/l/ N/A

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

   18. metadata-eval N/A

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

   19. lift-/.f64 99.2

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

8. Applied rewrites 99.2%

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

   1. lift-pow.f64 N/A

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

   2. sqr-pow N/A

      \[\leadsto \cos x \cdot \color{blue}{\left({\left({\left(e^{40}\right)}^{x}\right)}^{\left(\frac{\frac{x}{4}}{2}\right)} \cdot {\left({\left(e^{40}\right)}^{x}\right)}^{\left(\frac{\frac{x}{4}}{2}\right)}\right)} \]

   3. pow-prod-down N/A

      \[\leadsto \cos x \cdot \color{blue}{{\left({\left(e^{40}\right)}^{x} \cdot {\left(e^{40}\right)}^{x}\right)}^{\left(\frac{\frac{x}{4}}{2}\right)}} \]

   4. lower-pow.f64 N/A

      \[\leadsto \cos x \cdot \color{blue}{{\left({\left(e^{40}\right)}^{x} \cdot {\left(e^{40}\right)}^{x}\right)}^{\left(\frac{\frac{x}{4}}{2}\right)}} \]

   5. lift-pow.f64 N/A

      \[\leadsto \cos x \cdot {\left(\color{blue}{{\left(e^{40}\right)}^{x}} \cdot {\left(e^{40}\right)}^{x}\right)}^{\left(\frac{\frac{x}{4}}{2}\right)} \]

   6. lift-pow.f64 N/A

      \[\leadsto \cos x \cdot {\left({\left(e^{40}\right)}^{x} \cdot \color{blue}{{\left(e^{40}\right)}^{x}}\right)}^{\left(\frac{\frac{x}{4}}{2}\right)} \]

   7. pow-prod-down N/A

      \[\leadsto \cos x \cdot {\color{blue}{\left({\left(e^{40} \cdot e^{40}\right)}^{x}\right)}}^{\left(\frac{\frac{x}{4}}{2}\right)} \]

   8. lower-pow.f64 N/A

      \[\leadsto \cos x \cdot {\color{blue}{\left({\left(e^{40} \cdot e^{40}\right)}^{x}\right)}}^{\left(\frac{\frac{x}{4}}{2}\right)} \]

   9. lift-exp.f64 N/A

      \[\leadsto \cos x \cdot {\left({\left(\color{blue}{e^{40}} \cdot e^{40}\right)}^{x}\right)}^{\left(\frac{\frac{x}{4}}{2}\right)} \]

   10. lift-exp.f64 N/A

       \[\leadsto \cos x \cdot {\left({\left(e^{40} \cdot \color{blue}{e^{40}}\right)}^{x}\right)}^{\left(\frac{\frac{x}{4}}{2}\right)} \]

   11. exp-lft-sqr-rev N/A

       \[\leadsto \cos x \cdot {\left({\color{blue}{\left(e^{40 \cdot 2}\right)}}^{x}\right)}^{\left(\frac{\frac{x}{4}}{2}\right)} \]

   12. lower-exp.f64 N/A

       \[\leadsto \cos x \cdot {\left({\color{blue}{\left(e^{40 \cdot 2}\right)}}^{x}\right)}^{\left(\frac{\frac{x}{4}}{2}\right)} \]

   13. metadata-eval N/A

       \[\leadsto \cos x \cdot {\left({\left(e^{\color{blue}{80}}\right)}^{x}\right)}^{\left(\frac{\frac{x}{4}}{2}\right)} \]

   14. lift-/.f64 N/A

       \[\leadsto \cos x \cdot {\left({\left(e^{80}\right)}^{x}\right)}^{\left(\frac{\color{blue}{\frac{x}{4}}}{2}\right)} \]

   15. associate-/l/ N/A

       \[\leadsto \cos x \cdot {\left({\left(e^{80}\right)}^{x}\right)}^{\color{blue}{\left(\frac{x}{4 \cdot 2}\right)}} \]

   16. metadata-eval N/A

       \[\leadsto \cos x \cdot {\left({\left(e^{80}\right)}^{x}\right)}^{\left(\frac{x}{\color{blue}{8}}\right)} \]

   17. metadata-eval N/A

       \[\leadsto \cos x \cdot {\left({\left(e^{80}\right)}^{x}\right)}^{\left(\frac{x}{\color{blue}{{2}^{3}}}\right)} \]

   18. lower-/.f64 N/A

       \[\leadsto \cos x \cdot {\left({\left(e^{80}\right)}^{x}\right)}^{\color{blue}{\left(\frac{x}{{2}^{3}}\right)}} \]

   19. metadata-eval 99.4

       \[\leadsto \cos x \cdot {\left({\left(e^{80}\right)}^{x}\right)}^{\left(\frac{x}{\color{blue}{8}}\right)} \]

10. Applied rewrites 99.4%

    \[\leadsto \cos x \cdot \color{blue}{{\left({\left(e^{80}\right)}^{x}\right)}^{\left(\frac{x}{8}\right)}} \]
11. Add Preprocessing

Alternative 2: 99.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \cos x \cdot {\left({\left(e^{20}\right)}^{\left(-x\right)}\right)}^{\left(-0.5 \cdot x\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (cos x) (pow (pow (exp 20.0) (- x)) (* -0.5 x))))

C

double code(double x) {
	return cos(x) * pow(pow(exp(20.0), -x), (-0.5 * x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = cos(x) * ((exp(20.0d0) ** -x) ** ((-0.5d0) * x))
end function

Java

public static double code(double x) {
	return Math.cos(x) * Math.pow(Math.pow(Math.exp(20.0), -x), (-0.5 * x));
}

Python

def code(x):
	return math.cos(x) * math.pow(math.pow(math.exp(20.0), -x), (-0.5 * x))

Julia

function code(x)
	return Float64(cos(x) * ((exp(20.0) ^ Float64(-x)) ^ Float64(-0.5 * x)))
end

MATLAB

function tmp = code(x)
	tmp = cos(x) * ((exp(20.0) ^ -x) ^ (-0.5 * x));
end

Wolfram

code[x_] := N[(N[Cos[x], $MachinePrecision] * N[Power[N[Power[N[Exp[20.0], $MachinePrecision], (-x)], $MachinePrecision], N[(-0.5 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.3%

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

   1. lift-exp.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. exp-prod N/A

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

   5. lower-pow.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. exp-prod N/A

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

   8. lower-pow.f64 N/A

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

   9. lower-exp.f64 96.7

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

4. Applied rewrites 96.7%

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

   1. lift-pow.f64 N/A

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

   2. lift-exp.f64 N/A

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

   3. pow-exp N/A

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

   4. lift-*.f64 N/A

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

   5. lower-exp.f64 95.2

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

6. Applied rewrites 95.2%

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

   1. lift-pow.f64 N/A

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

   2. lift-exp.f64 N/A

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

   3. pow-exp N/A

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

   4. *-commutative N/A

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

   5. pow-exp N/A

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

   6. lift-*.f64 N/A

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

   7. sqr-neg-rev N/A

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

   8. pow-pow N/A

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

   9. sqr-pow N/A

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

   10. pow-prod-down N/A

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

   11. lower-pow.f64 N/A

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

   12. pow-prod-down N/A

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

   13. lower-pow.f64 N/A

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

   14. exp-lft-sqr-rev N/A

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

   15. metadata-eval N/A

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

   16. lower-exp.f64 N/A

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

   17. lower-neg.f64 N/A

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

   18. mul-1-neg N/A

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

   19. *-commutative N/A

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

   20. associate-/l* N/A

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

   21. metadata-eval N/A

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

   22. *-commutative N/A

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

   23. lower-*.f64 99.3

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

8. Applied rewrites 99.3%

   \[\leadsto \cos x \cdot \color{blue}{{\left({\left(e^{20}\right)}^{\left(-x\right)}\right)}^{\left(-0.5 \cdot x\right)}} \]
9. Add Preprocessing

Alternative 3: 95.2% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (* (cos x) (pow (pow (exp 10.0) -1.0) (* (- x) x))))

C

double code(double x) {
	return cos(x) * pow(pow(exp(10.0), -1.0), (-x * x));
}

Fortran

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

Java

public static double code(double x) {
	return Math.cos(x) * Math.pow(Math.pow(Math.exp(10.0), -1.0), (-x * x));
}

Python

def code(x):
	return math.cos(x) * math.pow(math.pow(math.exp(10.0), -1.0), (-x * x))

Julia

function code(x)
	return Float64(cos(x) * ((exp(10.0) ^ -1.0) ^ Float64(Float64(-x) * x)))
end

MATLAB

function tmp = code(x)
	tmp = cos(x) * ((exp(10.0) ^ -1.0) ^ (-x * x));
end

Wolfram

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

Derivation

1. Initial program 94.3%

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

   1. lift-exp.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. exp-prod N/A

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

   4. lift-*.f64 N/A

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

   5. sqr-neg-rev N/A

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

   6. pow-unpow N/A

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

   7. lower-pow.f64 N/A

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

   8. lower-pow.f64 N/A

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

   9. lower-exp.f64 N/A

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

   10. lower-neg.f64 N/A

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

   11. lower-neg.f64 98.1

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

4. Applied rewrites 98.1%

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

   1. lift-pow.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. lift-exp.f64 N/A

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

   4. pow-exp N/A

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

   5. pow-exp N/A

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

   6. associate-*r* N/A

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

   7. lift-neg.f64 N/A

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

   8. lift-neg.f64 N/A

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

   9. sqr-neg-rev N/A

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

   10. metadata-eval N/A

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

   11. lift-*.f64 N/A

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

   12. distribute-lft-neg-in N/A

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

   13. distribute-rgt-neg-in N/A

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

   14. pow-exp N/A

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

   15. lower-pow.f64 N/A

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

   16. lower-exp.f64 N/A

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

   17. lift-*.f64 N/A

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

   18. distribute-lft-neg-in N/A

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

   19. lift-neg.f64 N/A

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

   20. lower-*.f64 95.2

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

6. Applied rewrites 95.2%

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

   1. lift-exp.f64 N/A

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

   2. sinh-+-cosh-rev N/A

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

   3. flip-+ N/A

      \[\leadsto \cos x \cdot {\color{blue}{\left(\frac{\cosh -10 \cdot \cosh -10 - \sinh -10 \cdot \sinh -10}{\cosh -10 - \sinh -10}\right)}}^{\left(\left(-x\right) \cdot x\right)} \]

   4. sinh-cosh N/A

      \[\leadsto \cos x \cdot {\left(\frac{\color{blue}{1}}{\cosh -10 - \sinh -10}\right)}^{\left(\left(-x\right) \cdot x\right)} \]

   5. sinh---cosh N/A

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

   6. metadata-eval N/A

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

   7. lower-/.f64 N/A

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

   8. lower-exp.f64 95.4

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

8. Applied rewrites 95.4%

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

9. Final simplification 95.4%

   \[\leadsto \cos x \cdot {\left({\left(e^{10}\right)}^{-1}\right)}^{\left(\left(-x\right) \cdot x\right)} \]
10. Add Preprocessing

Alternative 4: 99.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \cos x \cdot {\left({\left(e^{40}\right)}^{x}\right)}^{\left(x \cdot 0.25\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (cos x) (pow (pow (exp 40.0) x) (* x 0.25))))

C

double code(double x) {
	return cos(x) * pow(pow(exp(40.0), x), (x * 0.25));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = cos(x) * ((exp(40.0d0) ** x) ** (x * 0.25d0))
end function

Java

public static double code(double x) {
	return Math.cos(x) * Math.pow(Math.pow(Math.exp(40.0), x), (x * 0.25));
}

Python

def code(x):
	return math.cos(x) * math.pow(math.pow(math.exp(40.0), x), (x * 0.25))

Julia

function code(x)
	return Float64(cos(x) * ((exp(40.0) ^ x) ^ Float64(x * 0.25)))
end

MATLAB

function tmp = code(x)
	tmp = cos(x) * ((exp(40.0) ^ x) ^ (x * 0.25));
end

Wolfram

code[x_] := N[(N[Cos[x], $MachinePrecision] * N[Power[N[Power[N[Exp[40.0], $MachinePrecision], x], $MachinePrecision], N[(x * 0.25), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.3%

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

   1. lift-exp.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. exp-prod N/A

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

   4. lift-*.f64 N/A

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

   5. sqr-neg-rev N/A

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

   6. pow-unpow N/A

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

   7. lower-pow.f64 N/A

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

   8. lower-pow.f64 N/A

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

   9. lower-exp.f64 N/A

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

   10. lower-neg.f64 N/A

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

   11. lower-neg.f64 98.1

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

4. Applied rewrites 98.1%

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

   1. lift-pow.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. pow-pow N/A

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

   4. lift-neg.f64 N/A

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

   5. lift-neg.f64 N/A

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

   6. sqr-neg-rev N/A

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

   7. lift-*.f64 N/A

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

   8. sqr-pow N/A

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

   9. pow-prod-down N/A

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

   10. lift-*.f64 N/A

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

   11. associate-*r/ N/A

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

   12. pow-unpow N/A

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

   13. lower-pow.f64 N/A

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

   14. lower-pow.f64 N/A

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

   15. lift-exp.f64 N/A

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

   16. lift-exp.f64 N/A

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

   17. exp-lft-sqr-rev N/A

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

   18. metadata-eval N/A

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

   19. metadata-eval N/A

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

   20. lower-exp.f64 N/A

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

   21. metadata-eval N/A

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

   22. lower-/.f64 99.3

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

6. Applied rewrites 99.3%

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

   1. lift-pow.f64 N/A

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

   2. sqr-pow N/A

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

   3. pow-prod-down N/A

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

   4. lower-pow.f64 N/A

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

   5. lift-pow.f64 N/A

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

   6. lift-pow.f64 N/A

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

   7. pow-prod-down N/A

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

   8. lower-pow.f64 N/A

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

   9. lift-exp.f64 N/A

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

   10. lift-exp.f64 N/A

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

   11. exp-lft-sqr-rev N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval N/A

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

   14. lower-exp.f64 N/A

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

   15. metadata-eval N/A

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

   16. lift-/.f64 N/A

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

   17. associate-/l/ N/A

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

   18. metadata-eval N/A

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

   19. lift-/.f64 99.2

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

8. Applied rewrites 99.2%

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

   1. lift-/.f64 N/A

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

   2. frac-2neg N/A

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

   3. mul-1-neg N/A

      \[\leadsto \cos x \cdot {\left({\left(e^{40}\right)}^{x}\right)}^{\left(\frac{\color{blue}{-1 \cdot x}}{\mathsf{neg}\left(4\right)}\right)} \]

   4. *-commutative N/A

      \[\leadsto \cos x \cdot {\left({\left(e^{40}\right)}^{x}\right)}^{\left(\frac{\color{blue}{x \cdot -1}}{\mathsf{neg}\left(4\right)}\right)} \]

   5. associate-/l* N/A

      \[\leadsto \cos x \cdot {\left({\left(e^{40}\right)}^{x}\right)}^{\color{blue}{\left(x \cdot \frac{-1}{\mathsf{neg}\left(4\right)}\right)}} \]

   6. metadata-eval N/A

      \[\leadsto \cos x \cdot {\left({\left(e^{40}\right)}^{x}\right)}^{\left(x \cdot \frac{-1}{\color{blue}{-4}}\right)} \]

   7. metadata-eval N/A

      \[\leadsto \cos x \cdot {\left({\left(e^{40}\right)}^{x}\right)}^{\left(x \cdot \color{blue}{\frac{1}{4}}\right)} \]

   8. metadata-eval N/A

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

   9. lower-*.f64 N/A

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

   10. metadata-eval 99.2

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

10. Applied rewrites 99.2%

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

Alternative 5: 98.1% accurate, 0.5× speedup

Math

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

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) ^ Float64(-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 94.3%

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

   1. lift-exp.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. exp-prod N/A

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

   4. lift-*.f64 N/A

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

   5. sqr-neg-rev N/A

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

   6. pow-unpow N/A

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

   7. lower-pow.f64 N/A

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

   8. lower-pow.f64 N/A

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

   9. lower-exp.f64 N/A

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

   10. lower-neg.f64 N/A

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

   11. lower-neg.f64 98.1

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

4. Applied rewrites 98.1%

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

   1. lift-pow.f64 N/A

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

   2. lift-exp.f64 N/A

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

   3. pow-exp N/A

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

   4. lift-neg.f64 N/A

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

   5. distribute-rgt-neg-out N/A

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

   6. distribute-lft-neg-in N/A

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

   7. metadata-eval N/A

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

   8. pow-exp N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-exp.f64 98.1

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

6. Applied rewrites 98.1%

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

Alternative 6: 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 94.3%

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

   1. lift-exp.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. exp-prod N/A

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

   4. lift-*.f64 N/A

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

   5. pow-unpow N/A

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

   6. lower-pow.f64 N/A

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

   7. lower-pow.f64 N/A

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

   8. lower-exp.f64 98.0

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

4. Applied rewrites 98.0%

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

Alternative 7: 96.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.3%

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

3. Taylor expanded in x around inf

   \[\leadsto \cos x \cdot \color{blue}{e^{10 \cdot {x}^{2}}} \]
4. Step-by-step derivation

   1. unpow2 N/A

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

   2. associate-*r* N/A

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

   3. exp-prod N/A

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

   4. lower-pow.f64 N/A

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

   5. *-commutative N/A

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

   6. exp-prod N/A

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

   7. lower-pow.f64 N/A

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

   8. lower-exp.f64 96.8

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

5. Applied rewrites 96.8%

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

6. Final simplification 96.8%

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

Alternative 8: 94.5% accurate, 0.7× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (* (cos x) (pow (exp (* -10.0 (* x x))) -1.0)))

C

double code(double x) {
	return cos(x) * pow(exp((-10.0 * (x * x))), -1.0);
}

Fortran

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

Java

public static double code(double x) {
	return Math.cos(x) * Math.pow(Math.exp((-10.0 * (x * x))), -1.0);
}

Python

def code(x):
	return math.cos(x) * math.pow(math.exp((-10.0 * (x * x))), -1.0)

Julia

function code(x)
	return Float64(cos(x) * (exp(Float64(-10.0 * Float64(x * x))) ^ -1.0))
end

MATLAB

function tmp = code(x)
	tmp = cos(x) * (exp((-10.0 * (x * x))) ^ -1.0);
end

Wolfram

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

Derivation

1. Initial program 94.3%

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

   1. lift-exp.f64 N/A

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

   2. sinh-+-cosh-rev N/A

      \[\leadsto \cos x \cdot \color{blue}{\left(\cosh \left(10 \cdot \left(x \cdot x\right)\right) + \sinh \left(10 \cdot \left(x \cdot x\right)\right)\right)} \]

   3. flip-+ N/A

      \[\leadsto \cos x \cdot \color{blue}{\frac{\cosh \left(10 \cdot \left(x \cdot x\right)\right) \cdot \cosh \left(10 \cdot \left(x \cdot x\right)\right) - \sinh \left(10 \cdot \left(x \cdot x\right)\right) \cdot \sinh \left(10 \cdot \left(x \cdot x\right)\right)}{\cosh \left(10 \cdot \left(x \cdot x\right)\right) - \sinh \left(10 \cdot \left(x \cdot x\right)\right)}} \]

   4. sinh-cosh N/A

      \[\leadsto \cos x \cdot \frac{\color{blue}{1}}{\cosh \left(10 \cdot \left(x \cdot x\right)\right) - \sinh \left(10 \cdot \left(x \cdot x\right)\right)} \]

   5. sinh---cosh-rev N/A

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

   6. lower-/.f64 N/A

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

   7. lower-exp.f64 N/A

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

   8. lift-*.f64 N/A

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

   9. distribute-lft-neg-in N/A

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

   10. lower-*.f64 N/A

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

   11. metadata-eval 94.3

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

4. Applied rewrites 94.3%

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

5. Final simplification 94.3%

   \[\leadsto \cos x \cdot {\left(e^{-10 \cdot \left(x \cdot x\right)}\right)}^{-1} \]
6. Add Preprocessing

Alternative 9: 95.3% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.3%

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

   1. lift-exp.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. exp-prod N/A

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

   4. lower-pow.f64 N/A

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

   5. lower-exp.f64 95.3

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

4. Applied rewrites 95.3%

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

Alternative 10: 94.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \sin \left(\frac{\mathsf{PI}\left(\right)}{2} + x\right) \cdot e^{10 \cdot \left(x \cdot x\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (sin (+ (/ (PI) 2.0) x)) (exp (* 10.0 (* x x)))))

Derivation

1. Initial program 94.3%

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

   1. lift-cos.f64 N/A

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

   2. sin-+PI/2-rev N/A

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

   3. lower-sin.f64 N/A

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

   4. +-commutative N/A

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

   5. lower-+.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lower-PI.f64 94.4

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

4. Applied rewrites 94.4%

   \[\leadsto \color{blue}{\sin \left(\frac{\mathsf{PI}\left(\right)}{2} + x\right)} \cdot e^{10 \cdot \left(x \cdot x\right)} \]
5. Add Preprocessing

Alternative 11: 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 94.3%

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

Alternative 12: 21.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right) \cdot e^{10 \cdot \left(x \cdot x\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (fma (- (* 0.041666666666666664 (* x x)) 0.5) (* x x) 1.0)
  (exp (* 10.0 (* x x)))))

C

double code(double x) {
	return fma(((0.041666666666666664 * (x * x)) - 0.5), (x * x), 1.0) * exp((10.0 * (x * x)));
}

Julia

function code(x)
	return Float64(fma(Float64(Float64(0.041666666666666664 * Float64(x * x)) - 0.5), Float64(x * x), 1.0) * exp(Float64(10.0 * Float64(x * x))))
end

Wolfram

code[x_] := N[(N[(N[(N[(0.041666666666666664 * N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * N[Exp[N[(10.0 * N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.3%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right)} \cdot e^{10 \cdot \left(x \cdot x\right)} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right) + 1\right)} \cdot e^{10 \cdot \left(x \cdot x\right)} \]

   2. *-commutative N/A

      \[\leadsto \left(\color{blue}{\left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right) \cdot {x}^{2}} + 1\right) \cdot e^{10 \cdot \left(x \cdot x\right)} \]

   3. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}, {x}^{2}, 1\right)} \cdot e^{10 \cdot \left(x \cdot x\right)} \]

   4. lower--.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}}, {x}^{2}, 1\right) \cdot e^{10 \cdot \left(x \cdot x\right)} \]

   5. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{24} \cdot {x}^{2}} - \frac{1}{2}, {x}^{2}, 1\right) \cdot e^{10 \cdot \left(x \cdot x\right)} \]

   6. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{24} \cdot \color{blue}{\left(x \cdot x\right)} - \frac{1}{2}, {x}^{2}, 1\right) \cdot e^{10 \cdot \left(x \cdot x\right)} \]

   7. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{24} \cdot \color{blue}{\left(x \cdot x\right)} - \frac{1}{2}, {x}^{2}, 1\right) \cdot e^{10 \cdot \left(x \cdot x\right)} \]

   8. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(\frac{1}{24} \cdot \left(x \cdot x\right) - \frac{1}{2}, \color{blue}{x \cdot x}, 1\right) \cdot e^{10 \cdot \left(x \cdot x\right)} \]

   9. lower-*.f64 21.3

      \[\leadsto \mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right) - 0.5, \color{blue}{x \cdot x}, 1\right) \cdot e^{10 \cdot \left(x \cdot x\right)} \]

5. Applied rewrites 21.3%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.041666666666666664 \cdot \left(x \cdot x\right) - 0.5, x \cdot x, 1\right)} \cdot e^{10 \cdot \left(x \cdot x\right)} \]
6. Add Preprocessing

Alternative 13: 18.2% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 94.3%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 18.2

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

5. Applied rewrites 18.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, x \cdot x, 1\right)} \cdot e^{10 \cdot \left(x \cdot x\right)} \]
6. Add Preprocessing

Alternative 14: 9.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \cos x \cdot \mathsf{fma}\left(10, x \cdot x, 1\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (cos x) (fma 10.0 (* x x) 1.0)))

C

double code(double x) {
	return cos(x) * fma(10.0, (x * x), 1.0);
}

Julia

function code(x)
	return Float64(cos(x) * fma(10.0, Float64(x * x), 1.0))
end

Wolfram

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

Derivation

1. Initial program 94.3%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

      \[\leadsto \cos x \cdot \color{blue}{\mathsf{fma}\left(10, {x}^{2}, 1\right)} \]

   3. unpow2 N/A

      \[\leadsto \cos x \cdot \mathsf{fma}\left(10, \color{blue}{x \cdot x}, 1\right) \]

   4. lower-*.f64 9.8

      \[\leadsto \cos x \cdot \mathsf{fma}\left(10, \color{blue}{x \cdot x}, 1\right) \]

5. Applied rewrites 9.8%

   \[\leadsto \cos x \cdot \color{blue}{\mathsf{fma}\left(10, x \cdot x, 1\right)} \]

6. Final simplification 9.8%

   \[\leadsto \cos x \cdot \mathsf{fma}\left(10, x \cdot x, 1\right) \]
7. Add Preprocessing

Alternative 15: 9.7% accurate, 13.5× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (* (* (* -0.5 x) x) 1.0))

C

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

Fortran

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

Java

public static double code(double x) {
	return ((-0.5 * x) * x) * 1.0;
}

Python

def code(x):
	return ((-0.5 * x) * x) * 1.0

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.3%

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

   1. lift-exp.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. exp-prod N/A

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

   4. lift-*.f64 N/A

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

   5. sqr-neg-rev N/A

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

   6. pow-unpow N/A

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

   7. lower-pow.f64 N/A

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

   8. lower-pow.f64 N/A

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

   9. lower-exp.f64 N/A

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

   10. lower-neg.f64 N/A

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

   11. lower-neg.f64 98.1

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

4. Applied rewrites 98.1%

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

5. Taylor expanded in x around 0

   \[\leadsto \cos x \cdot \color{blue}{1} \]
6. Step-by-step derivation

7. Applied rewrites 9.6%

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

8. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1}{2}, {x}^{2}, 1\right)} \cdot 1 \]

   3. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{x \cdot x}, 1\right) \cdot 1 \]

   4. lower-*.f64 9.7

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

10. Applied rewrites 9.7%

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

11. Taylor expanded in x around inf

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

13. Applied rewrites 9.7%

    \[\leadsto \left(\left(-0.5 \cdot x\right) \cdot \color{blue}{x}\right) \cdot 1 \]
14. Add Preprocessing

Alternative 16: 1.5% accurate, 216.0× 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 94.3%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1} \]
4. Step-by-step derivation

5. Applied rewrites 1.5%

   \[\leadsto \color{blue}{1} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024337 
(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)))))