ENA, Section 1.4, Exercise 1

Percentage Accurate: 94.5% → 99.5%

Time: 12.2s

Alternatives: 18

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: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{\cos x}{{\left({\left(e^{20}\right)}^{\left(x \cdot 2\right)}\right)}^{\left(x \cdot -0.25\right)}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (cos x) (pow (pow (exp 20.0) (* x 2.0)) (* x -0.25))))

C

double code(double x) {
	return cos(x) / pow(pow(exp(20.0), (x * 2.0)), (x * -0.25));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = cos(x) / ((exp(20.0d0) ** (x * 2.0d0)) ** (x * (-0.25d0)))
end function

Java

public static double code(double x) {
	return Math.cos(x) / Math.pow(Math.pow(Math.exp(20.0), (x * 2.0)), (x * -0.25));
}

Python

def code(x):
	return math.cos(x) / math.pow(math.pow(math.exp(20.0), (x * 2.0)), (x * -0.25))

Julia

function code(x)
	return Float64(cos(x) / ((exp(20.0) ^ Float64(x * 2.0)) ^ Float64(x * -0.25)))
end

MATLAB

function tmp = code(x)
	tmp = cos(x) / ((exp(20.0) ^ (x * 2.0)) ^ (x * -0.25));
end

Wolfram

code[x_] := N[(N[Cos[x], $MachinePrecision] / N[Power[N[Power[N[Exp[20.0], $MachinePrecision], N[(x * 2.0), $MachinePrecision]], $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. 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%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. un-div-inv N/A

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

   4. lower-/.f64 95.2

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

   5. lift-pow.f64 N/A

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

   6. lift-exp.f64 N/A

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

   7. pow-exp N/A

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

   8. lower-exp.f64 N/A

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

   9. *-commutative N/A

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

   10. lift-*.f64 N/A

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

   11. associate-*l* N/A

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

   12. lift-neg.f64 N/A

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

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

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

   14. lift-*.f64 N/A

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

   15. lower-*.f64 N/A

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

   16. lift-*.f64 N/A

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

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

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

   18. lower-*.f64 N/A

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

   19. metadata-eval 94.2

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

6. Applied rewrites 94.2%

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

   1. lift-exp.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. lift-*.f64 N/A

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

   6. metadata-eval N/A

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

   7. associate-*l* N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. pow-exp N/A

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

   11. lift-exp.f64 N/A

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

   12. sqr-pow N/A

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

   13. pow-sqr N/A

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

   14. lift-*.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. associate-*r* N/A

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

   17. lift-*.f64 N/A

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

   18. associate-/l* N/A

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

   19. associate-*r* N/A

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

   20. pow-unpow N/A

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

   21. lower-pow.f64 N/A

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

8. Applied rewrites 99.4%

   \[\leadsto \frac{\cos x}{\color{blue}{{\left({\left(e^{20}\right)}^{\left(2 \cdot x\right)}\right)}^{\left(x \cdot -0.25\right)}}} \]

9. Final simplification 99.4%

   \[\leadsto \frac{\cos x}{{\left({\left(e^{20}\right)}^{\left(x \cdot 2\right)}\right)}^{\left(x \cdot -0.25\right)}} \]
10. Add Preprocessing

Alternative 2: 99.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \cos x \cdot {\left({\left(e^{-20}\right)}^{\left(x \cdot 2\right)}\right)}^{\left(x \cdot -0.25\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (cos x) (pow (pow (exp -20.0) (* x 2.0)) (* x -0.25))))

C

double code(double x) {
	return cos(x) * pow(pow(exp(-20.0), (x * 2.0)), (x * -0.25));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = cos(x) * ((exp((-20.0d0)) ** (x * 2.0d0)) ** (x * (-0.25d0)))
end function

Java

public static double code(double x) {
	return Math.cos(x) * Math.pow(Math.pow(Math.exp(-20.0), (x * 2.0)), (x * -0.25));
}

Python

def code(x):
	return math.cos(x) * math.pow(math.pow(math.exp(-20.0), (x * 2.0)), (x * -0.25))

Julia

function code(x)
	return Float64(cos(x) * ((exp(-20.0) ^ Float64(x * 2.0)) ^ Float64(x * -0.25)))
end

MATLAB

function tmp = code(x)
	tmp = cos(x) * ((exp(-20.0) ^ (x * 2.0)) ^ (x * -0.25));
end

Wolfram

code[x_] := N[(N[Cos[x], $MachinePrecision] * N[Power[N[Power[N[Exp[-20.0], $MachinePrecision], N[(x * 2.0), $MachinePrecision]], $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. 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%

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

   1. lift-pow.f64 N/A

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

   2. sqr-pow N/A

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

   3. pow-prod-down N/A

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

   4. frac-2neg N/A

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

   5. lift-*.f64 N/A

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

   6. lift-neg.f64 N/A

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

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

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

   8. lift-*.f64 N/A

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

   9. remove-double-neg N/A

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

   10. div-inv N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. *-commutative N/A

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

   14. lower-pow.f64 N/A

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

   15. lift-exp.f64 N/A

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

   16. lift-exp.f64 N/A

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

   17. prod-exp N/A

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

   18. lower-exp.f64 N/A

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

   19. metadata-eval N/A

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

   20. *-commutative N/A

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

   21. lower-*.f64 95.2

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

6. Applied rewrites 95.2%

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

8. Applied rewrites 99.4%

   \[\leadsto \cos x \cdot \color{blue}{{\left({\left(e^{-20}\right)}^{\left(2 \cdot x\right)}\right)}^{\left(x \cdot -0.25\right)}} \]

9. Final simplification 99.4%

   \[\leadsto \cos x \cdot {\left({\left(e^{-20}\right)}^{\left(x \cdot 2\right)}\right)}^{\left(x \cdot -0.25\right)} \]
10. Add Preprocessing

Alternative 3: 99.3% accurate, 0.5× speedup

Math

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

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) ^ Float64(-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. 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%

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

   1. lift-pow.f64 N/A

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

   2. sqr-pow N/A

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

   3. pow-prod-down N/A

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

   4. frac-2neg N/A

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

   5. lift-*.f64 N/A

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

   6. lift-neg.f64 N/A

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

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

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

   8. lift-*.f64 N/A

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

   9. remove-double-neg N/A

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

   10. div-inv N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. *-commutative N/A

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

   14. lower-pow.f64 N/A

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

   15. lift-exp.f64 N/A

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

   16. lift-exp.f64 N/A

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

   17. prod-exp N/A

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

   18. lower-exp.f64 N/A

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

   19. metadata-eval N/A

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

   20. *-commutative N/A

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

   21. lower-*.f64 95.2

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

6. Applied rewrites 95.2%

   \[\leadsto \cos x \cdot \frac{1}{\color{blue}{{\left(e^{20}\right)}^{\left(\left(x \cdot x\right) \cdot -0.5\right)}}} \]
7. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. pow-flip N/A

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

   4. sqr-pow N/A

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

   5. pow-prod-down N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. associate-*r* N/A

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

   9. lift-*.f64 N/A

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

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

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

   11. associate-/l* N/A

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

   12. pow-unpow N/A

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

   13. lower-pow.f64 N/A

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

8. Applied rewrites 99.2%

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

Alternative 4: 96.7% 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. 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%

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

   1. lift-pow.f64 N/A

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

   2. sqr-pow N/A

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

   3. pow-prod-down N/A

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

   4. frac-2neg N/A

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

   5. lift-*.f64 N/A

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

   6. lift-neg.f64 N/A

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

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

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

   8. lift-*.f64 N/A

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

   9. remove-double-neg N/A

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

   10. div-inv N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. *-commutative N/A

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

   14. lower-pow.f64 N/A

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

   15. lift-exp.f64 N/A

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

   16. lift-exp.f64 N/A

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

   17. prod-exp N/A

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

   18. lower-exp.f64 N/A

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

   19. metadata-eval N/A

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

   20. *-commutative N/A

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

   21. lower-*.f64 95.2

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

6. Applied rewrites 95.2%

   \[\leadsto \cos x \cdot \frac{1}{\color{blue}{{\left(e^{20}\right)}^{\left(\left(x \cdot x\right) \cdot -0.5\right)}}} \]
7. Step-by-step derivation

   1. lift-/.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. pow-to-exp N/A

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

   4. rec-exp N/A

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

   5. lift-exp.f64 N/A

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

   6. rem-log-exp N/A

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

   7. *-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. associate-*l* N/A

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

   10. metadata-eval N/A

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

   11. lift-*.f64 N/A

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

   12. associate-*r* N/A

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

   13. lift-*.f64 N/A

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

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

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

   15. pow-exp N/A

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

   16. lift-*.f64 N/A

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

   17. *-commutative N/A

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

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

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

   19. metadata-eval N/A

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

   20. pow-unpow N/A

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

   21. lower-pow.f64 N/A

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

8. Applied rewrites 96.8%

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

Alternative 5: 95.2% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   1. lift-pow.f64 N/A

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

   2. sqr-pow N/A

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

   3. pow-prod-down N/A

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

   4. frac-2neg N/A

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

   5. lift-*.f64 N/A

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

   6. lift-neg.f64 N/A

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

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

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

   8. lift-*.f64 N/A

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

   9. remove-double-neg N/A

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

   10. div-inv N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. *-commutative N/A

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

   14. lower-pow.f64 N/A

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

   15. lift-exp.f64 N/A

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

   16. lift-exp.f64 N/A

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

   17. prod-exp N/A

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

   18. lower-exp.f64 N/A

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

   19. metadata-eval N/A

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

   20. *-commutative N/A

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

   21. lower-*.f64 95.2

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

6. Applied rewrites 95.2%

   \[\leadsto \cos x \cdot \frac{1}{\color{blue}{{\left(e^{20}\right)}^{\left(\left(x \cdot x\right) \cdot -0.5\right)}}} \]
7. Step-by-step derivation

   1. lift-pow.f64 N/A

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

   2. lift-exp.f64 N/A

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

   3. pow-exp N/A

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

   4. *-commutative N/A

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

   5. exp-prod N/A

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

   6. metadata-eval N/A

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

   7. pow-sqr N/A

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

   8. pow-prod-down N/A

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

   9. lower-pow.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. exp-prod N/A

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

   12. lift-exp.f64 N/A

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

   13. lift-*.f64 N/A

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

   14. exp-prod N/A

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

   15. lift-exp.f64 N/A

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

   16. pow-prod-up N/A

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

   17. metadata-eval N/A

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

   18. inv-pow N/A

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

   19. lower-/.f64 95.2

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

8. Applied rewrites 95.2%

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

Alternative 6: 95.2% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. un-div-inv N/A

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

   4. lower-/.f64 95.2

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

   5. lift-pow.f64 N/A

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

   6. lift-exp.f64 N/A

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

   7. pow-exp N/A

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

   8. lower-exp.f64 N/A

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

   9. *-commutative N/A

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

   10. lift-*.f64 N/A

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

   11. associate-*l* N/A

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

   12. lift-neg.f64 N/A

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

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

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

   14. lift-*.f64 N/A

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

   15. lower-*.f64 N/A

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

   16. lift-*.f64 N/A

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

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

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

   18. lower-*.f64 N/A

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

   19. metadata-eval 94.2

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

6. Applied rewrites 94.2%

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

   1. lift-exp.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. lift-*.f64 N/A

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

   6. metadata-eval N/A

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

   7. associate-*l* N/A

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

   8. lift-*.f64 N/A

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

   9. exp-prod N/A

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

   10. metadata-eval N/A

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

   11. pow-sqr N/A

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

   12. pow-prod-down N/A

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

   13. lower-pow.f64 N/A

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

8. Applied rewrites 95.2%

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

Alternative 7: 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 95.2

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

4. Applied rewrites 95.2%

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

Alternative 8: 94.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \frac{\cos x}{e^{\mathsf{fma}\left(x \cdot x, -5, x \cdot \left(x \cdot -5\right)\right)}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (/ (cos x) (exp (fma (* x x) -5.0 (* x (* x -5.0))))))

C

double code(double x) {
	return cos(x) / exp(fma((x * x), -5.0, (x * (x * -5.0))));
}

Julia

function code(x)
	return Float64(cos(x) / exp(fma(Float64(x * x), -5.0, Float64(x * Float64(x * -5.0)))))
end

Wolfram

code[x_] := N[(N[Cos[x], $MachinePrecision] / N[Exp[N[(N[(x * x), $MachinePrecision] * -5.0 + N[(x * N[(x * -5.0), $MachinePrecision]), $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. 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%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. un-div-inv N/A

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

   4. lower-/.f64 95.2

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

   5. lift-pow.f64 N/A

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

   6. lift-exp.f64 N/A

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

   7. pow-exp N/A

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

   8. lower-exp.f64 N/A

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

   9. *-commutative N/A

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

   10. lift-*.f64 N/A

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

   11. associate-*l* N/A

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

   12. lift-neg.f64 N/A

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

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

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

   14. lift-*.f64 N/A

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

   15. lower-*.f64 N/A

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

   16. lift-*.f64 N/A

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

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

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

   18. lower-*.f64 N/A

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

   19. metadata-eval 94.2

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

6. Applied rewrites 94.2%

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

   1. rem-log-exp N/A

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

   2. lift-*.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. associate-*r* N/A

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

   5. lift-*.f64 N/A

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

   6. pow-exp N/A

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

   7. lift-exp.f64 N/A

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

   8. sqr-pow N/A

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

   9. pow2 N/A

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

   10. lift-exp.f64 N/A

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

   11. pow-exp N/A

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

   12. metadata-eval N/A

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

   13. metadata-eval N/A

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

   14. associate-*l* N/A

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

   15. lift-*.f64 N/A

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

   16. *-commutative N/A

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

   17. lift-*.f64 N/A

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

   18. lift-exp.f64 N/A

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

   19. pow2 N/A

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

8. Applied rewrites 94.4%

   \[\leadsto \frac{\cos x}{e^{\color{blue}{\mathsf{fma}\left(x \cdot x, -5, x \cdot \left(x \cdot -5\right)\right)}}} \]
9. Add Preprocessing

Alternative 9: 94.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   1. lift-pow.f64 N/A

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

   2. lift-exp.f64 N/A

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

   3. pow-exp N/A

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

   4. lift-*.f64 N/A

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

   5. lift-neg.f64 N/A

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

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

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

   7. lift-*.f64 N/A

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

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

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

   9. *-commutative N/A

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

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

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

   11. rem-log-exp N/A

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

   12. lift-*.f64 N/A

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

   13. pow-to-exp N/A

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

   14. lower-pow.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. lower-exp.f64 N/A

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

   17. metadata-eval 95.1

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

6. Applied rewrites 95.1%

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

7. Taylor expanded in x around inf

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

   1. exp-prod N/A

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

   2. *-commutative N/A

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

   3. metadata-eval N/A

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

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

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

   5. rec-exp N/A

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

   6. remove-double-div N/A

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

   7. lower-exp.f64 N/A

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

   8. lower-*.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 94.3

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

9. Applied rewrites 94.3%

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

10. Final simplification 94.3%

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

Alternative 10: 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 11: 27.5% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ e^{10 \cdot \left(x \cdot x\right)} \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (exp (* 10.0 (* x x)))
  (fma
   (* x x)
   (fma x (* x (fma (* x x) -0.001388888888888889 0.041666666666666664)) -0.5)
   1.0)))

C

double code(double x) {
	return exp((10.0 * (x * x))) * fma((x * x), fma(x, (x * fma((x * x), -0.001388888888888889, 0.041666666666666664)), -0.5), 1.0);
}

Julia

function code(x)
	return Float64(exp(Float64(10.0 * Float64(x * x))) * fma(Float64(x * x), fma(x, Float64(x * fma(Float64(x * x), -0.001388888888888889, 0.041666666666666664)), -0.5), 1.0))
end

Wolfram

code[x_] := N[(N[Exp[N[(10.0 * N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * -0.001388888888888889 + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + -0.5), $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

   \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left(\frac{1}{24} \cdot {x}^{2} - \frac{1}{2}\right)\right)} \cdot e^{10 \cdot \left(x \cdot x\right)} \]
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. lower-fma.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. sub-neg N/A

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

   6. *-commutative N/A

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

   7. metadata-eval N/A

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

   8. lower-fma.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 21.3

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

5. Applied rewrites 21.3%

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

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)\right)} \cdot e^{10 \cdot \left(x \cdot x\right)} \]
7. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{\left({x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right) + 1\right)} \cdot e^{10 \cdot \left(x \cdot x\right)} \]

   2. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{2}, {x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}, 1\right)} \cdot e^{10 \cdot \left(x \cdot x\right)} \]

   3. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, {x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}, 1\right) \cdot e^{10 \cdot \left(x \cdot x\right)} \]

   4. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{x \cdot x}, {x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}, 1\right) \cdot e^{10 \cdot \left(x \cdot x\right)} \]

   5. sub-neg N/A

      \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right)}, 1\right) \cdot e^{10 \cdot \left(x \cdot x\right)} \]

   6. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right)} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), 1\right) \cdot e^{10 \cdot \left(x \cdot x\right)} \]

   7. associate-*l* N/A

      \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(x \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right)\right)} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), 1\right) \cdot e^{10 \cdot \left(x \cdot x\right)} \]

   8. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(x \cdot x, x \cdot \color{blue}{\left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) \cdot x\right)} + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right), 1\right) \cdot e^{10 \cdot \left(x \cdot x\right)} \]

   9. metadata-eval N/A

      \[\leadsto \mathsf{fma}\left(x \cdot x, x \cdot \left(\left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) \cdot x\right) + \color{blue}{\frac{-1}{2}}, 1\right) \cdot e^{10 \cdot \left(x \cdot x\right)} \]

   10. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) \cdot x, \frac{-1}{2}\right)}, 1\right) \cdot e^{10 \cdot \left(x \cdot x\right)} \]

   11. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right)}, \frac{-1}{2}\right), 1\right) \cdot e^{10 \cdot \left(x \cdot x\right)} \]

   12. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right)}, \frac{-1}{2}\right), 1\right) \cdot e^{10 \cdot \left(x \cdot x\right)} \]

   13. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\left(\frac{-1}{720} \cdot {x}^{2} + \frac{1}{24}\right)}, \frac{-1}{2}\right), 1\right) \cdot e^{10 \cdot \left(x \cdot x\right)} \]

   14. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{-1}{720}} + \frac{1}{24}\right), \frac{-1}{2}\right), 1\right) \cdot e^{10 \cdot \left(x \cdot x\right)} \]

   15. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{-1}{720}, \frac{1}{24}\right)}, \frac{-1}{2}\right), 1\right) \cdot e^{10 \cdot \left(x \cdot x\right)} \]

   16. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{-1}{720}, \frac{1}{24}\right), \frac{-1}{2}\right), 1\right) \cdot e^{10 \cdot \left(x \cdot x\right)} \]

   17. lower-*.f64 27.5

       \[\leadsto \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right) \cdot e^{10 \cdot \left(x \cdot x\right)} \]

8. Applied rewrites 27.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right)} \cdot e^{10 \cdot \left(x \cdot x\right)} \]

9. Final simplification 27.5%

   \[\leadsto e^{10 \cdot \left(x \cdot x\right)} \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right) \]
10. Add Preprocessing

Alternative 12: 21.3% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

   1. lift-*.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-*r* N/A

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

   4. lower-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 94.2

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

4. Applied rewrites 94.2%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*l* N/A

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

   4. lower-fma.f64 N/A

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

   5. lower-*.f64 N/A

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

   6. sub-neg N/A

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

   7. *-commutative N/A

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

   8. metadata-eval N/A

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

   9. lower-fma.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 21.3

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

7. Applied rewrites 21.3%

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

8. Final simplification 21.3%

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

Alternative 13: 18.2% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

   \[\leadsto \color{blue}{\left(1 + \frac{-1}{2} \cdot {x}^{2}\right)} \cdot e^{10 \cdot \left(x \cdot x\right)} \]
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. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 18.2

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

5. Applied rewrites 18.2%

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

6. Final simplification 18.2%

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

Alternative 14: 10.3% accurate, 4.3× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x, x \cdot -0.5, 1\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 166.66666666666666, 50\right), 10\right), 1\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (fma x (* x -0.5) 1.0)
  (fma (* x x) (fma x (* x (fma x (* x 166.66666666666666) 50.0)) 10.0) 1.0)))

C

double code(double x) {
	return fma(x, (x * -0.5), 1.0) * fma((x * x), fma(x, (x * fma(x, (x * 166.66666666666666), 50.0)), 10.0), 1.0);
}

Julia

function code(x)
	return Float64(fma(x, Float64(x * -0.5), 1.0) * fma(Float64(x * x), fma(x, Float64(x * fma(x, Float64(x * 166.66666666666666), 50.0)), 10.0), 1.0))
end

Wolfram

code[x_] := N[(N[(x * N[(x * -0.5), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(x * N[(x * 166.66666666666666), $MachinePrecision] + 50.0), $MachinePrecision]), $MachinePrecision] + 10.0), $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

4. Applied rewrites 95.1%

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

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. *-commutative N/A

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

   3. unpow2 N/A

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

   4. associate-*l* N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-*.f64 9.7

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

10. Applied rewrites 9.7%

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

11. Taylor expanded in x around 0

    \[\leadsto \mathsf{fma}\left(x, x \cdot \frac{-1}{2}, 1\right) \cdot \color{blue}{\left(1 + {x}^{2} \cdot \left(10 + {x}^{2} \cdot \left(50 + \frac{500}{3} \cdot {x}^{2}\right)\right)\right)} \]
12. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(x, x \cdot \frac{-1}{2}, 1\right) \cdot \color{blue}{\left({x}^{2} \cdot \left(10 + {x}^{2} \cdot \left(50 + \frac{500}{3} \cdot {x}^{2}\right)\right) + 1\right)} \]

    2. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(x, x \cdot \frac{-1}{2}, 1\right) \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, 10 + {x}^{2} \cdot \left(50 + \frac{500}{3} \cdot {x}^{2}\right), 1\right)} \]

    3. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(x, x \cdot \frac{-1}{2}, 1\right) \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, 10 + {x}^{2} \cdot \left(50 + \frac{500}{3} \cdot {x}^{2}\right), 1\right) \]

    4. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(x, x \cdot \frac{-1}{2}, 1\right) \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, 10 + {x}^{2} \cdot \left(50 + \frac{500}{3} \cdot {x}^{2}\right), 1\right) \]

    5. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(x, x \cdot \frac{-1}{2}, 1\right) \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{{x}^{2} \cdot \left(50 + \frac{500}{3} \cdot {x}^{2}\right) + 10}, 1\right) \]

    6. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(x, x \cdot \frac{-1}{2}, 1\right) \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\left(x \cdot x\right)} \cdot \left(50 + \frac{500}{3} \cdot {x}^{2}\right) + 10, 1\right) \]

    7. associate-*l* N/A

       \[\leadsto \mathsf{fma}\left(x, x \cdot \frac{-1}{2}, 1\right) \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(x \cdot \left(50 + \frac{500}{3} \cdot {x}^{2}\right)\right)} + 10, 1\right) \]

    8. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(x, x \cdot \frac{-1}{2}, 1\right) \cdot \mathsf{fma}\left(x \cdot x, x \cdot \color{blue}{\left(\left(50 + \frac{500}{3} \cdot {x}^{2}\right) \cdot x\right)} + 10, 1\right) \]

    9. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(x, x \cdot \frac{-1}{2}, 1\right) \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, \left(50 + \frac{500}{3} \cdot {x}^{2}\right) \cdot x, 10\right)}, 1\right) \]

    10. *-commutative N/A

        \[\leadsto \mathsf{fma}\left(x, x \cdot \frac{-1}{2}, 1\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \left(50 + \frac{500}{3} \cdot {x}^{2}\right)}, 10\right), 1\right) \]

    11. lower-*.f64 N/A

        \[\leadsto \mathsf{fma}\left(x, x \cdot \frac{-1}{2}, 1\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \left(50 + \frac{500}{3} \cdot {x}^{2}\right)}, 10\right), 1\right) \]

    12. +-commutative N/A

        \[\leadsto \mathsf{fma}\left(x, x \cdot \frac{-1}{2}, 1\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\left(\frac{500}{3} \cdot {x}^{2} + 50\right)}, 10\right), 1\right) \]

    13. *-commutative N/A

        \[\leadsto \mathsf{fma}\left(x, x \cdot \frac{-1}{2}, 1\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{{x}^{2} \cdot \frac{500}{3}} + 50\right), 10\right), 1\right) \]

    14. unpow2 N/A

        \[\leadsto \mathsf{fma}\left(x, x \cdot \frac{-1}{2}, 1\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \frac{500}{3} + 50\right), 10\right), 1\right) \]

    15. associate-*l* N/A

        \[\leadsto \mathsf{fma}\left(x, x \cdot \frac{-1}{2}, 1\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \left(\color{blue}{x \cdot \left(x \cdot \frac{500}{3}\right)} + 50\right), 10\right), 1\right) \]

    16. lower-fma.f64 N/A

        \[\leadsto \mathsf{fma}\left(x, x \cdot \frac{-1}{2}, 1\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{500}{3}, 50\right)}, 10\right), 1\right) \]

    17. lower-*.f64 10.3

        \[\leadsto \mathsf{fma}\left(x, x \cdot -0.5, 1\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot 166.66666666666666}, 50\right), 10\right), 1\right) \]

13. Applied rewrites 10.3%

    \[\leadsto \mathsf{fma}\left(x, x \cdot -0.5, 1\right) \cdot \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 166.66666666666666, 50\right), 10\right), 1\right)} \]
14. Add Preprocessing

Alternative 15: 10.1% accurate, 5.5× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x, x \cdot -0.5, 1\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 50, 10\right), 1\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (fma x (* x -0.5) 1.0) (fma (* x x) (fma x (* x 50.0) 10.0) 1.0)))

C

double code(double x) {
	return fma(x, (x * -0.5), 1.0) * fma((x * x), fma(x, (x * 50.0), 10.0), 1.0);
}

Julia

function code(x)
	return Float64(fma(x, Float64(x * -0.5), 1.0) * fma(Float64(x * x), fma(x, Float64(x * 50.0), 10.0), 1.0))
end

Wolfram

code[x_] := N[(N[(x * N[(x * -0.5), $MachinePrecision] + 1.0), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * 50.0), $MachinePrecision] + 10.0), $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

4. Applied rewrites 95.1%

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

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. *-commutative N/A

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

   3. unpow2 N/A

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

   4. associate-*l* N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-*.f64 9.7

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

10. Applied rewrites 9.7%

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

11. Taylor expanded in x around 0

    \[\leadsto \mathsf{fma}\left(x, x \cdot \frac{-1}{2}, 1\right) \cdot \color{blue}{\left(1 + {x}^{2} \cdot \left(10 + 50 \cdot {x}^{2}\right)\right)} \]
12. Step-by-step derivation

    1. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(x, x \cdot \frac{-1}{2}, 1\right) \cdot \color{blue}{\left({x}^{2} \cdot \left(10 + 50 \cdot {x}^{2}\right) + 1\right)} \]

    2. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(x, x \cdot \frac{-1}{2}, 1\right) \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, 10 + 50 \cdot {x}^{2}, 1\right)} \]

    3. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(x, x \cdot \frac{-1}{2}, 1\right) \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, 10 + 50 \cdot {x}^{2}, 1\right) \]

    4. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(x, x \cdot \frac{-1}{2}, 1\right) \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, 10 + 50 \cdot {x}^{2}, 1\right) \]

    5. +-commutative N/A

       \[\leadsto \mathsf{fma}\left(x, x \cdot \frac{-1}{2}, 1\right) \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{50 \cdot {x}^{2} + 10}, 1\right) \]

    6. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(x, x \cdot \frac{-1}{2}, 1\right) \cdot \mathsf{fma}\left(x \cdot x, 50 \cdot \color{blue}{\left(x \cdot x\right)} + 10, 1\right) \]

    7. associate-*r* N/A

       \[\leadsto \mathsf{fma}\left(x, x \cdot \frac{-1}{2}, 1\right) \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\left(50 \cdot x\right) \cdot x} + 10, 1\right) \]

    8. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(x, x \cdot \frac{-1}{2}, 1\right) \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(50 \cdot x\right)} + 10, 1\right) \]

    9. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(x, x \cdot \frac{-1}{2}, 1\right) \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, 50 \cdot x, 10\right)}, 1\right) \]

    10. *-commutative N/A

        \[\leadsto \mathsf{fma}\left(x, x \cdot \frac{-1}{2}, 1\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot 50}, 10\right), 1\right) \]

    11. lower-*.f64 10.1

        \[\leadsto \mathsf{fma}\left(x, x \cdot -0.5, 1\right) \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot 50}, 10\right), 1\right) \]

13. Applied rewrites 10.1%

    \[\leadsto \mathsf{fma}\left(x, x \cdot -0.5, 1\right) \cdot \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot 50, 10\right), 1\right)} \]
14. Add Preprocessing

Alternative 16: 9.9% accurate, 7.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

4. Applied rewrites 95.1%

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

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. *-commutative N/A

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

   3. unpow2 N/A

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

   4. associate-*l* N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-*.f64 9.7

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

10. Applied rewrites 9.7%

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

11. Taylor expanded in x around 0

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

    1. +-commutative N/A

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

    2. *-commutative N/A

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

    3. unpow2 N/A

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

    4. associate-*l* N/A

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

    5. lower-fma.f64 N/A

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

    6. lower-*.f64 9.9

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

13. Applied rewrites 9.9%

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

Alternative 17: 9.7% accurate, 13.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.3%

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

4. Applied rewrites 95.1%

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

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. *-commutative N/A

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

   3. unpow2 N/A

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

   4. associate-*l* N/A

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

   5. lower-fma.f64 N/A

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

   6. lower-*.f64 9.7

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

10. Applied rewrites 9.7%

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

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%

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

Alternative 18: 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 2024238 
(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)))))