ENA, Section 1.4, Exercise 1

Percentage Accurate: 94.5% → 99.3%

Time: 11.0s

Alternatives: 20

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: 99.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{1}{{\left({\left(e^{20}\right)}^{x}\right)}^{\left(-0.5 \cdot x\right)}} \cdot \cos x \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (/ 1.0 (pow (pow (exp 20.0) x) (* -0.5 x))) (cos x)))

C

double code(double x) {
	return (1.0 / pow(pow(exp(20.0), x), (-0.5 * x))) * cos(x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (1.0d0 / ((exp(20.0d0) ** x) ** ((-0.5d0) * x))) * cos(x)
end function

Java

public static double code(double x) {
	return (1.0 / Math.pow(Math.pow(Math.exp(20.0), x), (-0.5 * x))) * Math.cos(x);
}

Python

def code(x):
	return (1.0 / math.pow(math.pow(math.exp(20.0), x), (-0.5 * x))) * math.cos(x)

Julia

function code(x)
	return Float64(Float64(1.0 / ((exp(20.0) ^ x) ^ Float64(-0.5 * x))) * cos(x))
end

MATLAB

function tmp = code(x)
	tmp = (1.0 / ((exp(20.0) ^ x) ^ (-0.5 * x))) * cos(x);
end

Wolfram

code[x_] := N[(N[(1.0 / N[Power[N[Power[N[Exp[20.0], $MachinePrecision], x], $MachinePrecision], N[(-0.5 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.6%

   \[\cos x \cdot e^{10 \cdot \left(x \cdot x\right)} \]
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. sqr-pow N/A

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

   5. pow-prod-up N/A

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

   6. flip-+ N/A

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

   7. +-inverses N/A

      \[\leadsto \cos x \cdot {\left(e^{10}\right)}^{\left(\frac{\color{blue}{0}}{\frac{x \cdot x}{2} - \frac{x \cdot x}{2}}\right)} \]

   8. +-inverses N/A

      \[\leadsto \cos x \cdot {\left(e^{10}\right)}^{\left(\frac{0}{\color{blue}{0}}\right)} \]

   9. metadata-eval N/A

      \[\leadsto \cos x \cdot {\left(e^{10}\right)}^{\left(\frac{0}{\color{blue}{2 \cdot 0}}\right)} \]

   10. associate-/l/ N/A

       \[\leadsto \cos x \cdot {\left(e^{10}\right)}^{\color{blue}{\left(\frac{\frac{0}{0}}{2}\right)}} \]

   11. +-inverses N/A

       \[\leadsto \cos x \cdot {\left(e^{10}\right)}^{\left(\frac{\frac{\color{blue}{\frac{x \cdot x}{2} \cdot \frac{x \cdot x}{2} - \frac{x \cdot x}{2} \cdot \frac{x \cdot x}{2}}}{0}}{2}\right)} \]

   12. +-inverses N/A

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

   13. flip-+ N/A

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

   14. count-2 N/A

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

   15. *-commutative N/A

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

   16. associate-*l/ N/A

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

   17. associate-/l* N/A

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

   18. metadata-eval N/A

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

   19. *-rgt-identity N/A

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

   20. frac-2neg N/A

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

   21. distribute-frac-neg N/A

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

4. Applied rewrites 95.2%

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

   1. lift-pow.f64 N/A

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

   2. sqr-pow N/A

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

   3. pow-prod-down N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-/l* N/A

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

   7. pow-unpow N/A

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

   8. lift-neg.f64 N/A

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

   9. neg-mul-1 N/A

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

   10. *-commutative N/A

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

   11. associate-/l* N/A

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

   12. metadata-eval N/A

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

   13. lower-pow.f64 N/A

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

   14. lower-pow.f64 N/A

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

   15. lift-exp.f64 N/A

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

   16. lift-exp.f64 N/A

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

   17. prod-exp N/A

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

   18. lower-exp.f64 N/A

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

   19. metadata-eval N/A

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

   20. *-commutative N/A

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

   21. lower-*.f64 99.3

       \[\leadsto \cos x \cdot \frac{1}{{\left({\left(e^{20}\right)}^{x}\right)}^{\color{blue}{\left(-0.5 \cdot x\right)}}} \]

6. Applied rewrites 99.3%

   \[\leadsto \cos x \cdot \frac{1}{\color{blue}{{\left({\left(e^{20}\right)}^{x}\right)}^{\left(-0.5 \cdot x\right)}}} \]

7. Final simplification 99.3%

   \[\leadsto \frac{1}{{\left({\left(e^{20}\right)}^{x}\right)}^{\left(-0.5 \cdot x\right)}} \cdot \cos x \]
8. Add Preprocessing

Alternative 2: 98.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.5%

   \[\cos x \cdot e^{10 \cdot \left(x \cdot x\right)} \]
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. sqr-pow N/A

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

   5. pow-prod-up N/A

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

   6. flip-+ N/A

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

   7. +-inverses N/A

      \[\leadsto \cos x \cdot {\left(e^{10}\right)}^{\left(\frac{\color{blue}{0}}{\frac{x \cdot x}{2} - \frac{x \cdot x}{2}}\right)} \]

   8. +-inverses N/A

      \[\leadsto \cos x \cdot {\left(e^{10}\right)}^{\left(\frac{0}{\color{blue}{0}}\right)} \]

   9. metadata-eval N/A

      \[\leadsto \cos x \cdot {\left(e^{10}\right)}^{\left(\frac{0}{\color{blue}{2 \cdot 0}}\right)} \]

   10. associate-/l/ N/A

       \[\leadsto \cos x \cdot {\left(e^{10}\right)}^{\color{blue}{\left(\frac{\frac{0}{0}}{2}\right)}} \]

   11. +-inverses N/A

       \[\leadsto \cos x \cdot {\left(e^{10}\right)}^{\left(\frac{\frac{\color{blue}{\frac{x \cdot x}{2} \cdot \frac{x \cdot x}{2} - \frac{x \cdot x}{2} \cdot \frac{x \cdot x}{2}}}{0}}{2}\right)} \]

   12. +-inverses N/A

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

   13. flip-+ N/A

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

   14. count-2 N/A

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

   15. *-commutative N/A

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

   16. associate-*l/ N/A

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

   17. associate-/l* N/A

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

   18. metadata-eval N/A

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

   19. *-rgt-identity N/A

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

   20. frac-2neg N/A

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

   21. distribute-frac-neg N/A

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

4. Applied rewrites 95.2%

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

   1. lift-pow.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. *-commutative N/A

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

   4. pow-unpow N/A

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

   5. lift-exp.f64 N/A

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

   6. exp-prod N/A

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

   7. *-commutative N/A

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

   8. pow-exp N/A

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

   9. lift-exp.f64 N/A

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

   10. lower-pow.f64 N/A

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

   11. lift-exp.f64 N/A

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

   12. pow-exp N/A

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

   13. *-commutative N/A

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

   14. lift-*.f64 N/A

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

   15. lower-exp.f64 95.0

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

6. Applied rewrites 95.0%

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

   1. lift-exp.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. exp-prod N/A

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

   4. lower-pow.f64 N/A

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

   5. lower-exp.f64 98.0

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

8. Applied rewrites 98.0%

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

9. Final simplification 98.0%

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

Reproduce

herbie shell --seed 2024230 
(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)))))