ENA, Section 1.4, Exercise 1

Percentage Accurate: 94.5% → 99.2%

Time: 10.6s

Alternatives: 21

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: 99.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \left(\cos x \cdot {\left({\left(e^{40}\right)}^{x}\right)}^{\left(0.125 \cdot x\right)}\right) \cdot {\left({\left(e^{20}\right)}^{x}\right)}^{\left(0.25 \cdot x\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (* (cos x) (pow (pow (exp 40.0) x) (* 0.125 x)))
  (pow (pow (exp 20.0) x) (* 0.25 x))))

C

double code(double x) {
	return (cos(x) * pow(pow(exp(40.0), x), (0.125 * x))) * pow(pow(exp(20.0), x), (0.25 * x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (cos(x) * ((exp(40.0d0) ** x) ** (0.125d0 * x))) * ((exp(20.0d0) ** x) ** (0.25d0 * x))
end function

Java

public static double code(double x) {
	return (Math.cos(x) * Math.pow(Math.pow(Math.exp(40.0), x), (0.125 * x))) * Math.pow(Math.pow(Math.exp(20.0), x), (0.25 * x));
}

Python

def code(x):
	return (math.cos(x) * math.pow(math.pow(math.exp(40.0), x), (0.125 * x))) * math.pow(math.pow(math.exp(20.0), x), (0.25 * x))

Julia

function code(x)
	return Float64(Float64(cos(x) * ((exp(40.0) ^ x) ^ Float64(0.125 * x))) * ((exp(20.0) ^ x) ^ Float64(0.25 * x)))
end

MATLAB

function tmp = code(x)
	tmp = (cos(x) * ((exp(40.0) ^ x) ^ (0.125 * x))) * ((exp(20.0) ^ x) ^ (0.25 * x));
end

Wolfram

code[x_] := N[(N[(N[Cos[x], $MachinePrecision] * N[Power[N[Power[N[Exp[40.0], $MachinePrecision], x], $MachinePrecision], N[(0.125 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Power[N[Power[N[Exp[20.0], $MachinePrecision], x], $MachinePrecision], N[(0.25 * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.5%

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

   1. lift-exp.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. exp-prod N/A

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

   4. lift-*.f64 N/A

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

   5. pow-unpow N/A

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

   6. sqr-pow N/A

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

   7. lower-*.f64 N/A

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

   8. lower-pow.f64 N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-exp.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower-pow.f64 N/A

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

   13. lower-pow.f64 N/A

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

   14. lower-exp.f64 N/A

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

   15. lower-/.f64 97.9

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

4. Applied rewrites 97.9%

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

   1. lift-*.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. lift-pow.f64 N/A

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

   4. pow-prod-down N/A

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

   5. lower-pow.f64 N/A

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

   6. lift-pow.f64 N/A

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

   7. lift-pow.f64 N/A

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

   8. pow-prod-down N/A

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

   9. lower-pow.f64 N/A

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

   10. lift-exp.f64 N/A

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

   11. lift-exp.f64 N/A

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

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

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. lower-exp.f64 N/A

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

   16. metadata-eval 99.3

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

   17. lift-/.f64 N/A

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

   18. clear-num N/A

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

   19. associate-/r/ N/A

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

   20. metadata-eval N/A

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

   21. lower-*.f64 99.3

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

6. Applied rewrites 99.3%

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

8. Applied rewrites 99.2%

   \[\leadsto \color{blue}{{\left({\left(e^{20}\right)}^{x}\right)}^{\left(x \cdot 0.25\right)} \cdot \left({\left({\left(e^{20}\right)}^{x}\right)}^{\left(x \cdot 0.25\right)} \cdot \cos x\right)} \]
9. Step-by-step derivation

   1. lift-pow.f64 N/A

      \[\leadsto {\left({\left(e^{20}\right)}^{x}\right)}^{\left(x \cdot \frac{1}{4}\right)} \cdot \left(\color{blue}{{\left({\left(e^{20}\right)}^{x}\right)}^{\left(x \cdot \frac{1}{4}\right)}} \cdot \cos x\right) \]

   2. sqr-pow N/A

      \[\leadsto {\left({\left(e^{20}\right)}^{x}\right)}^{\left(x \cdot \frac{1}{4}\right)} \cdot \left(\color{blue}{\left({\left({\left(e^{20}\right)}^{x}\right)}^{\left(\frac{x \cdot \frac{1}{4}}{2}\right)} \cdot {\left({\left(e^{20}\right)}^{x}\right)}^{\left(\frac{x \cdot \frac{1}{4}}{2}\right)}\right)} \cdot \cos x\right) \]

   3. pow-prod-down N/A

      \[\leadsto {\left({\left(e^{20}\right)}^{x}\right)}^{\left(x \cdot \frac{1}{4}\right)} \cdot \left(\color{blue}{{\left({\left(e^{20}\right)}^{x} \cdot {\left(e^{20}\right)}^{x}\right)}^{\left(\frac{x \cdot \frac{1}{4}}{2}\right)}} \cdot \cos x\right) \]

   4. lower-pow.f64 N/A

      \[\leadsto {\left({\left(e^{20}\right)}^{x}\right)}^{\left(x \cdot \frac{1}{4}\right)} \cdot \left(\color{blue}{{\left({\left(e^{20}\right)}^{x} \cdot {\left(e^{20}\right)}^{x}\right)}^{\left(\frac{x \cdot \frac{1}{4}}{2}\right)}} \cdot \cos x\right) \]

   5. lift-pow.f64 N/A

      \[\leadsto {\left({\left(e^{20}\right)}^{x}\right)}^{\left(x \cdot \frac{1}{4}\right)} \cdot \left({\left(\color{blue}{{\left(e^{20}\right)}^{x}} \cdot {\left(e^{20}\right)}^{x}\right)}^{\left(\frac{x \cdot \frac{1}{4}}{2}\right)} \cdot \cos x\right) \]

   6. lift-pow.f64 N/A

      \[\leadsto {\left({\left(e^{20}\right)}^{x}\right)}^{\left(x \cdot \frac{1}{4}\right)} \cdot \left({\left({\left(e^{20}\right)}^{x} \cdot \color{blue}{{\left(e^{20}\right)}^{x}}\right)}^{\left(\frac{x \cdot \frac{1}{4}}{2}\right)} \cdot \cos x\right) \]

   7. pow-prod-down N/A

      \[\leadsto {\left({\left(e^{20}\right)}^{x}\right)}^{\left(x \cdot \frac{1}{4}\right)} \cdot \left({\color{blue}{\left({\left(e^{20} \cdot e^{20}\right)}^{x}\right)}}^{\left(\frac{x \cdot \frac{1}{4}}{2}\right)} \cdot \cos x\right) \]

   8. lower-pow.f64 N/A

      \[\leadsto {\left({\left(e^{20}\right)}^{x}\right)}^{\left(x \cdot \frac{1}{4}\right)} \cdot \left({\color{blue}{\left({\left(e^{20} \cdot e^{20}\right)}^{x}\right)}}^{\left(\frac{x \cdot \frac{1}{4}}{2}\right)} \cdot \cos x\right) \]

   9. lift-exp.f64 N/A

      \[\leadsto {\left({\left(e^{20}\right)}^{x}\right)}^{\left(x \cdot \frac{1}{4}\right)} \cdot \left({\left({\left(\color{blue}{e^{20}} \cdot e^{20}\right)}^{x}\right)}^{\left(\frac{x \cdot \frac{1}{4}}{2}\right)} \cdot \cos x\right) \]

   10. lift-exp.f64 N/A

       \[\leadsto {\left({\left(e^{20}\right)}^{x}\right)}^{\left(x \cdot \frac{1}{4}\right)} \cdot \left({\left({\left(e^{20} \cdot \color{blue}{e^{20}}\right)}^{x}\right)}^{\left(\frac{x \cdot \frac{1}{4}}{2}\right)} \cdot \cos x\right) \]

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

       \[\leadsto {\left({\left(e^{20}\right)}^{x}\right)}^{\left(x \cdot \frac{1}{4}\right)} \cdot \left({\left({\color{blue}{\left(e^{20 \cdot 2}\right)}}^{x}\right)}^{\left(\frac{x \cdot \frac{1}{4}}{2}\right)} \cdot \cos x\right) \]

   12. lower-exp.f64 N/A

       \[\leadsto {\left({\left(e^{20}\right)}^{x}\right)}^{\left(x \cdot \frac{1}{4}\right)} \cdot \left({\left({\color{blue}{\left(e^{20 \cdot 2}\right)}}^{x}\right)}^{\left(\frac{x \cdot \frac{1}{4}}{2}\right)} \cdot \cos x\right) \]

   13. metadata-eval N/A

       \[\leadsto {\left({\left(e^{20}\right)}^{x}\right)}^{\left(x \cdot \frac{1}{4}\right)} \cdot \left({\left({\left(e^{\color{blue}{40}}\right)}^{x}\right)}^{\left(\frac{x \cdot \frac{1}{4}}{2}\right)} \cdot \cos x\right) \]

   14. lift-*.f64 N/A

       \[\leadsto {\left({\left(e^{20}\right)}^{x}\right)}^{\left(x \cdot \frac{1}{4}\right)} \cdot \left({\left({\left(e^{40}\right)}^{x}\right)}^{\left(\frac{\color{blue}{x \cdot \frac{1}{4}}}{2}\right)} \cdot \cos x\right) \]

   15. associate-/l* N/A

       \[\leadsto {\left({\left(e^{20}\right)}^{x}\right)}^{\left(x \cdot \frac{1}{4}\right)} \cdot \left({\left({\left(e^{40}\right)}^{x}\right)}^{\color{blue}{\left(x \cdot \frac{\frac{1}{4}}{2}\right)}} \cdot \cos x\right) \]

   16. metadata-eval N/A

       \[\leadsto {\left({\left(e^{20}\right)}^{x}\right)}^{\left(x \cdot \frac{1}{4}\right)} \cdot \left({\left({\left(e^{40}\right)}^{x}\right)}^{\left(x \cdot \color{blue}{\frac{1}{8}}\right)} \cdot \cos x\right) \]

   17. metadata-eval N/A

       \[\leadsto {\left({\left(e^{20}\right)}^{x}\right)}^{\left(x \cdot \frac{1}{4}\right)} \cdot \left({\left({\left(e^{40}\right)}^{x}\right)}^{\left(x \cdot \color{blue}{{\frac{1}{2}}^{3}}\right)} \cdot \cos x\right) \]

   18. lower-*.f64 N/A

       \[\leadsto {\left({\left(e^{20}\right)}^{x}\right)}^{\left(x \cdot \frac{1}{4}\right)} \cdot \left({\left({\left(e^{40}\right)}^{x}\right)}^{\color{blue}{\left(x \cdot {\frac{1}{2}}^{3}\right)}} \cdot \cos x\right) \]

   19. metadata-eval 99.3

       \[\leadsto {\left({\left(e^{20}\right)}^{x}\right)}^{\left(x \cdot 0.25\right)} \cdot \left({\left({\left(e^{40}\right)}^{x}\right)}^{\left(x \cdot \color{blue}{0.125}\right)} \cdot \cos x\right) \]

10. Applied rewrites 99.3%

    \[\leadsto {\left({\left(e^{20}\right)}^{x}\right)}^{\left(x \cdot 0.25\right)} \cdot \left(\color{blue}{{\left({\left(e^{40}\right)}^{x}\right)}^{\left(x \cdot 0.125\right)}} \cdot \cos x\right) \]

11. Final simplification 99.3%

    \[\leadsto \left(\cos x \cdot {\left({\left(e^{40}\right)}^{x}\right)}^{\left(0.125 \cdot x\right)}\right) \cdot {\left({\left(e^{20}\right)}^{x}\right)}^{\left(0.25 \cdot x\right)} \]
12. Add Preprocessing

Alternative 2: 99.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.5%

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

   1. lift-exp.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. exp-prod N/A

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

   4. lift-*.f64 N/A

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

   5. pow-unpow N/A

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

   6. sqr-pow N/A

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

   7. lower-*.f64 N/A

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

   8. lower-pow.f64 N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-exp.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower-pow.f64 N/A

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

   13. lower-pow.f64 N/A

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

   14. lower-exp.f64 N/A

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

   15. lower-/.f64 97.9

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

4. Applied rewrites 97.9%

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

   1. lift-*.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. lift-pow.f64 N/A

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

   4. pow-prod-down N/A

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

   5. lower-pow.f64 N/A

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

   6. lift-pow.f64 N/A

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

   7. lift-pow.f64 N/A

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

   8. pow-prod-down N/A

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

   9. lower-pow.f64 N/A

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

   10. lift-exp.f64 N/A

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

   11. lift-exp.f64 N/A

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

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

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

   13. metadata-eval N/A

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

   14. metadata-eval N/A

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

   15. lower-exp.f64 N/A

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

   16. metadata-eval 99.3

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

   17. lift-/.f64 N/A

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

   18. clear-num N/A

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

   19. associate-/r/ N/A

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

   20. metadata-eval N/A

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

   21. lower-*.f64 99.3

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

6. Applied rewrites 99.3%

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

7. Final simplification 99.3%

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

Alternative 3: 99.2% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.5%

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

3. Taylor expanded in x around inf

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

   1. unpow2 N/A

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

   2. associate-*r* N/A

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

   3. exp-prod N/A

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

   4. lower-pow.f64 N/A

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

   5. *-commutative N/A

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

   6. exp-prod N/A

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

   7. lower-pow.f64 N/A

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

   8. lower-exp.f64 96.6

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

5. Applied rewrites 96.6%

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

7. Applied rewrites 99.2%

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

8. Final simplification 99.2%

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

Alternative 4: 99.2% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.5%

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

3. Taylor expanded in x around inf

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

   1. unpow2 N/A

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

   2. associate-*r* N/A

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

   3. exp-prod N/A

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

   4. lower-pow.f64 N/A

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

   5. *-commutative N/A

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

   6. exp-prod N/A

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

   7. lower-pow.f64 N/A

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

   8. lower-exp.f64 96.6

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

5. Applied rewrites 96.6%

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

7. Applied rewrites 99.1%

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

8. Final simplification 99.1%

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

Alternative 5: 98.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.5%

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

   1. lift-exp.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. exp-prod N/A

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

   4. lift-*.f64 N/A

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

   5. pow-unpow N/A

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

   6. sqr-pow N/A

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

   7. lower-*.f64 N/A

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

   8. lower-pow.f64 N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-exp.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower-pow.f64 N/A

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

   13. lower-pow.f64 N/A

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

   14. lower-exp.f64 N/A

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

   15. lower-/.f64 97.9

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

4. Applied rewrites 97.9%

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

   1. lift-*.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. lift-pow.f64 N/A

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

   4. pow-pow N/A

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

   5. lift-/.f64 N/A

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

   6. associate-*r/ N/A

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

   7. lift-*.f64 N/A

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

   8. lift-pow.f64 N/A

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

   9. lift-pow.f64 N/A

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

   10. pow-pow N/A

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

   11. lift-/.f64 N/A

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

   12. associate-*r/ N/A

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

   13. lift-*.f64 N/A

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

   14. sqr-pow N/A

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

   15. lift-*.f64 N/A

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

   16. sqr-neg-rev N/A

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

   17. lift-neg.f64 N/A

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

   18. lift-neg.f64 N/A

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

   19. pow-unpow N/A

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

   20. lower-pow.f64 N/A

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

6. Applied rewrites 97.9%

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

7. Final simplification 97.9%

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

Alternative 6: 98.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.5%

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

   1. lift-exp.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. exp-prod N/A

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

   4. lift-*.f64 N/A

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

   5. pow-unpow N/A

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

   6. lower-pow.f64 N/A

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

   7. lower-pow.f64 N/A

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

   8. lower-exp.f64 97.9

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

4. Applied rewrites 97.9%

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

5. Final simplification 97.9%

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

Alternative 7: 96.7% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.5%

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

3. Taylor expanded in x around inf

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

   1. unpow2 N/A

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

   2. associate-*r* N/A

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

   3. exp-prod N/A

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

   4. lower-pow.f64 N/A

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

   5. *-commutative N/A

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

   6. exp-prod N/A

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

   7. lower-pow.f64 N/A

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

   8. lower-exp.f64 96.6

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

5. Applied rewrites 96.6%

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

6. Final simplification 96.6%

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

Alternative 8: 95.3% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.5%

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

   1. lift-exp.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. exp-prod N/A

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

   4. lift-*.f64 N/A

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

   5. pow-unpow N/A

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

   6. sqr-pow N/A

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

   7. lower-*.f64 N/A

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

   8. lower-pow.f64 N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-exp.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower-pow.f64 N/A

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

   13. lower-pow.f64 N/A

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

   14. lower-exp.f64 N/A

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

   15. lower-/.f64 97.9

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

4. Applied rewrites 97.9%

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

   1. lift-*.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. lift-pow.f64 N/A

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

   4. pow-prod-down N/A

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

   5. lift-/.f64 N/A

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

   6. frac-2neg N/A

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

   7. lift-neg.f64 N/A

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

   8. div-inv N/A

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

   9. pow-unpow N/A

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

   10. lower-pow.f64 N/A

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

6. Applied rewrites 99.3%

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

7. Taylor expanded in x around inf

   \[\leadsto \cos x \cdot \color{blue}{\sqrt{\frac{1}{e^{-20 \cdot {x}^{2}}}}} \]
8. Step-by-step derivation

   1. lower-sqrt.f64 N/A

      \[\leadsto \cos x \cdot \color{blue}{\sqrt{\frac{1}{e^{-20 \cdot {x}^{2}}}}} \]

   2. rec-exp N/A

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

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

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

   4. metadata-eval N/A

      \[\leadsto \cos x \cdot \sqrt{e^{\color{blue}{20} \cdot {x}^{2}}} \]

   5. exp-prod N/A

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

   6. lower-pow.f64 N/A

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

   7. lower-exp.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 95.3

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

9. Applied rewrites 95.3%

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

10. Final simplification 95.3%

    \[\leadsto \sqrt{{\left(e^{20}\right)}^{\left(x \cdot x\right)}} \cdot \cos x \]
11. Add Preprocessing

Alternative 9: 95.3% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.5%

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

3. Taylor expanded in x around inf

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

   1. unpow2 N/A

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

   2. associate-*r* N/A

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

   3. exp-prod N/A

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

   4. lower-pow.f64 N/A

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

   5. *-commutative N/A

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

   6. exp-prod N/A

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

   7. lower-pow.f64 N/A

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

   8. lower-exp.f64 96.6

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

5. Applied rewrites 96.6%

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

7. Applied rewrites 95.3%

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

8. Final simplification 95.3%

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

Alternative 10: 95.2% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.5%

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

   1. lift-exp.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. exp-prod N/A

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

   4. lower-pow.f64 N/A

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

   5. lower-exp.f64 95.3

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

4. Applied rewrites 95.3%

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

5. Final simplification 95.3%

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

Alternative 11: 95.1% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.5%

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

3. Taylor expanded in x around inf

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

   1. unpow2 N/A

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

   2. associate-*r* N/A

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

   3. exp-prod N/A

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

   4. lower-pow.f64 N/A

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

   5. *-commutative N/A

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

   6. exp-prod N/A

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

   7. lower-pow.f64 N/A

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

   8. lower-exp.f64 96.6

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

5. Applied rewrites 96.6%

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

7. Applied rewrites 95.0%

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

8. Final simplification 95.0%

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

Alternative 12: 94.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\\ e^{\left(x \cdot x\right) \cdot 10} \cdot \sin \left(\mathsf{fma}\left(\frac{x \cdot x}{\mathsf{fma}\left(-0.25, t\_0, x \cdot x\right)}, \mathsf{fma}\left(0.5, \mathsf{PI}\left(\right), x\right), \frac{t\_0 \cdot -0.25}{\mathsf{fma}\left(\mathsf{PI}\left(\right), -0.5, x\right)}\right)\right) \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* (PI) (PI))))
   (*
    (exp (* (* x x) 10.0))
    (sin
     (fma
      (/ (* x x) (fma -0.25 t_0 (* x x)))
      (fma 0.5 (PI) x)
      (/ (* t_0 -0.25) (fma (PI) -0.5 x)))))))

Derivation

1. Initial program 94.5%

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

   1. lift-cos.f64 N/A

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

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

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

   3. lower-sin.f64 N/A

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

   4. +-commutative N/A

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

   5. clear-num N/A

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

   6. associate-/r/ N/A

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

   7. lower-fma.f64 N/A

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

   8. metadata-eval N/A

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

   9. lower-PI.f64 94.5

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

4. Applied rewrites 94.5%

   \[\leadsto \color{blue}{\sin \left(\mathsf{fma}\left(0.5, \mathsf{PI}\left(\right), x\right)\right)} \cdot e^{10 \cdot \left(x \cdot x\right)} \]
5. Step-by-step derivation

   1. lift-fma.f64 N/A

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

   2. +-commutative N/A

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

   3. flip-+ N/A

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

   4. clear-num N/A

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

   5. lower-/.f64 N/A

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

   6. lower-/.f64 N/A

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

   7. lower--.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 N/A

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

   10. lift-*.f64 N/A

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

   11. lower--.f64 N/A

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

   12. swap-sqr N/A

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

   13. lift-PI.f64 N/A

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

   14. lift-PI.f64 N/A

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

   15. lower-*.f64 N/A

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

   16. metadata-eval N/A

       \[\leadsto \sin \left(\frac{1}{\frac{x - \mathsf{PI}\left(\right) \cdot \frac{1}{2}}{x \cdot x - \color{blue}{\frac{1}{4}} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}}\right) \cdot e^{10 \cdot \left(x \cdot x\right)} \]

   17. lift-PI.f64 N/A

       \[\leadsto \sin \left(\frac{1}{\frac{x - \mathsf{PI}\left(\right) \cdot \frac{1}{2}}{x \cdot x - \frac{1}{4} \cdot \left(\color{blue}{\mathsf{PI}\left(\right)} \cdot \mathsf{PI}\left(\right)\right)}}\right) \cdot e^{10 \cdot \left(x \cdot x\right)} \]

   18. lift-PI.f64 N/A

       \[\leadsto \sin \left(\frac{1}{\frac{x - \mathsf{PI}\left(\right) \cdot \frac{1}{2}}{x \cdot x - \frac{1}{4} \cdot \left(\mathsf{PI}\left(\right) \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)}}\right) \cdot e^{10 \cdot \left(x \cdot x\right)} \]

   19. lower-*.f64 94.1

       \[\leadsto \sin \left(\frac{1}{\frac{x - \mathsf{PI}\left(\right) \cdot 0.5}{x \cdot x - 0.25 \cdot \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}}}\right) \cdot e^{10 \cdot \left(x \cdot x\right)} \]

6. Applied rewrites 94.1%

   \[\leadsto \sin \color{blue}{\left(\frac{1}{\frac{x - \mathsf{PI}\left(\right) \cdot 0.5}{x \cdot x - 0.25 \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}}\right)} \cdot e^{10 \cdot \left(x \cdot x\right)} \]

8. Applied rewrites 94.6%

   \[\leadsto \sin \color{blue}{\left(\mathsf{fma}\left(\frac{x \cdot x}{\mathsf{fma}\left(-0.25, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), x \cdot x\right)}, \mathsf{fma}\left(0.5, \mathsf{PI}\left(\right), x\right), \frac{-0.25 \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}{\mathsf{fma}\left(\mathsf{PI}\left(\right), -0.5, x\right)}\right)\right)} \cdot e^{10 \cdot \left(x \cdot x\right)} \]

9. Final simplification 94.6%

   \[\leadsto e^{\left(x \cdot x\right) \cdot 10} \cdot \sin \left(\mathsf{fma}\left(\frac{x \cdot x}{\mathsf{fma}\left(-0.25, \mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right), x \cdot x\right)}, \mathsf{fma}\left(0.5, \mathsf{PI}\left(\right), x\right), \frac{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot -0.25}{\mathsf{fma}\left(\mathsf{PI}\left(\right), -0.5, x\right)}\right)\right) \]
10. Add Preprocessing

Alternative 13: 94.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \sin \left(\frac{\mathsf{fma}\left(x, x, \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot -0.25\right)}{\mathsf{fma}\left(-0.5, \mathsf{PI}\left(\right), x\right)}\right) \cdot e^{\left(x \cdot x\right) \cdot 10} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (sin (/ (fma x x (* (* (PI) (PI)) -0.25)) (fma -0.5 (PI) x)))
  (exp (* (* x x) 10.0))))

Derivation

1. Initial program 94.5%

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

   1. lift-cos.f64 N/A

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

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

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

   3. lower-sin.f64 N/A

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

   4. +-commutative N/A

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

   5. clear-num N/A

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

   6. associate-/r/ N/A

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

   7. lower-fma.f64 N/A

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

   8. metadata-eval N/A

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

   9. lower-PI.f64 94.5

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

4. Applied rewrites 94.5%

   \[\leadsto \color{blue}{\sin \left(\mathsf{fma}\left(0.5, \mathsf{PI}\left(\right), x\right)\right)} \cdot e^{10 \cdot \left(x \cdot x\right)} \]
5. Step-by-step derivation

   1. lift-fma.f64 N/A

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

   2. +-commutative N/A

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

   3. flip-+ N/A

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

   4. lift-*.f64 N/A

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

   5. div-sub N/A

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

   6. lower--.f64 N/A

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

   7. lower-/.f64 N/A

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

   8. lower--.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 N/A

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

6. Applied rewrites 93.8%

   \[\leadsto \sin \color{blue}{\left(\frac{x \cdot x}{x - \mathsf{PI}\left(\right) \cdot 0.5} - \frac{0.25 \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}{x - \mathsf{PI}\left(\right) \cdot 0.5}\right)} \cdot e^{10 \cdot \left(x \cdot x\right)} \]
7. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \sin \color{blue}{\left(\frac{x \cdot x}{x - \mathsf{PI}\left(\right) \cdot \frac{1}{2}} - \frac{\frac{1}{4} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}{x - \mathsf{PI}\left(\right) \cdot \frac{1}{2}}\right)} \cdot e^{10 \cdot \left(x \cdot x\right)} \]

   2. lift-/.f64 N/A

      \[\leadsto \sin \left(\color{blue}{\frac{x \cdot x}{x - \mathsf{PI}\left(\right) \cdot \frac{1}{2}}} - \frac{\frac{1}{4} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}{x - \mathsf{PI}\left(\right) \cdot \frac{1}{2}}\right) \cdot e^{10 \cdot \left(x \cdot x\right)} \]

   3. lift-/.f64 N/A

      \[\leadsto \sin \left(\frac{x \cdot x}{x - \mathsf{PI}\left(\right) \cdot \frac{1}{2}} - \color{blue}{\frac{\frac{1}{4} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}{x - \mathsf{PI}\left(\right) \cdot \frac{1}{2}}}\right) \cdot e^{10 \cdot \left(x \cdot x\right)} \]

   4. sub-div N/A

      \[\leadsto \sin \color{blue}{\left(\frac{x \cdot x - \frac{1}{4} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}{x - \mathsf{PI}\left(\right) \cdot \frac{1}{2}}\right)} \cdot e^{10 \cdot \left(x \cdot x\right)} \]

   5. lift-*.f64 N/A

      \[\leadsto \sin \left(\frac{x \cdot x - \color{blue}{\frac{1}{4} \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)}}{x - \mathsf{PI}\left(\right) \cdot \frac{1}{2}}\right) \cdot e^{10 \cdot \left(x \cdot x\right)} \]

   6. *-commutative N/A

      \[\leadsto \sin \left(\frac{x \cdot x - \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot \frac{1}{4}}}{x - \mathsf{PI}\left(\right) \cdot \frac{1}{2}}\right) \cdot e^{10 \cdot \left(x \cdot x\right)} \]

   7. lift-*.f64 N/A

      \[\leadsto \sin \left(\frac{x \cdot x - \color{blue}{\left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)} \cdot \frac{1}{4}}{x - \mathsf{PI}\left(\right) \cdot \frac{1}{2}}\right) \cdot e^{10 \cdot \left(x \cdot x\right)} \]

   8. metadata-eval N/A

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

   9. swap-sqr N/A

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

   10. lift-*.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. lift-*.f64 N/A

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

   13. lower-/.f64 N/A

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

8. Applied rewrites 94.6%

   \[\leadsto \sin \color{blue}{\left(\frac{\mathsf{fma}\left(x, x, -0.25 \cdot \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right)\right)}{\mathsf{fma}\left(-0.5, \mathsf{PI}\left(\right), x\right)}\right)} \cdot e^{10 \cdot \left(x \cdot x\right)} \]

9. Final simplification 94.6%

   \[\leadsto \sin \left(\frac{\mathsf{fma}\left(x, x, \left(\mathsf{PI}\left(\right) \cdot \mathsf{PI}\left(\right)\right) \cdot -0.25\right)}{\mathsf{fma}\left(-0.5, \mathsf{PI}\left(\right), x\right)}\right) \cdot e^{\left(x \cdot x\right) \cdot 10} \]
10. Add Preprocessing

Alternative 14: 94.5% accurate, 1.0× speedup

Math

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

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 94.5%

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

   1. lift-*.f64 N/A

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

   2. lift-exp.f64 N/A

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

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

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

   4. flip-+ N/A

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

   5. sinh-cosh N/A

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

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

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

   7. associate-*r/ N/A

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

   8. lift-cos.f64 N/A

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

   9. sin-PI/2 N/A

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

   10. lower-/.f64 N/A

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

   11. lift-cos.f64 N/A

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

   12. sin-PI/2 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 N/A

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

   15. lower-exp.f64 N/A

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

   16. lift-*.f64 N/A

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

4. Applied rewrites 94.5%

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

5. Final simplification 94.5%

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

Alternative 15: 94.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.5%

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

3. Final simplification 94.5%

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

Alternative 16: 27.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(N[(N[(N[(-0.001388888888888889 * N[(x * x), $MachinePrecision] + 0.041666666666666664), $MachinePrecision] * N[(x * x), $MachinePrecision] + -0.5), $MachinePrecision] * N[(x * x), $MachinePrecision] + 1.0), $MachinePrecision] * N[Exp[N[(N[(x * x), $MachinePrecision] * 10.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.5%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\left(1 + {x}^{2} \cdot \left({x}^{2} \cdot \left(\frac{1}{24} + \frac{-1}{720} \cdot {x}^{2}\right) - \frac{1}{2}\right)\right)} \cdot e^{10 \cdot \left(x \cdot x\right)} \]
4. 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. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. sub-neg N/A

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

   5. *-commutative N/A

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

   6. metadata-eval N/A

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

   7. lower-fma.f64 N/A

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

   8. +-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 N/A

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

   12. unpow2 N/A

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

   13. lower-*.f64 N/A

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

   14. unpow2 N/A

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

   15. lower-*.f64 27.5

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

5. Applied rewrites 27.5%

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

6. Final simplification 27.5%

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

Alternative 17: 21.3% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 94.5%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. sub-neg N/A

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

   5. metadata-eval N/A

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

   6. lower-fma.f64 N/A

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

   7. unpow2 N/A

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

   8. lower-*.f64 N/A

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

   9. unpow2 N/A

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

   10. lower-*.f64 21.3

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

5. Applied rewrites 21.3%

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

6. Final simplification 21.3%

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

Alternative 18: 18.2% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 94.5%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 18.2

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

5. Applied rewrites 18.2%

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

6. Final simplification 18.2%

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

Alternative 19: 9.8% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 94.5%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 9.8

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

5. Applied rewrites 9.8%

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

6. Final simplification 9.8%

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

Alternative 20: 9.7% accurate, 13.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.5%

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

   1. lift-exp.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. exp-prod N/A

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

   4. lift-*.f64 N/A

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

   5. pow-unpow N/A

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

   6. sqr-pow N/A

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

   7. lower-*.f64 N/A

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

   8. lower-pow.f64 N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-exp.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower-pow.f64 N/A

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

   13. lower-pow.f64 N/A

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

   14. lower-exp.f64 N/A

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

   15. lower-/.f64 97.9

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

4. Applied rewrites 97.9%

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

5. Taylor expanded in x around 0

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

7. Applied rewrites 9.6%

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

8. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. metadata-eval N/A

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

   5. sin-PI/2 N/A

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

   6. *-rgt-identity-rev N/A

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

   7. associate-/l* N/A

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

   8. metadata-eval N/A

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

   9. *-commutative N/A

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

   10. metadata-eval N/A

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

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

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

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

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

   13. metadata-eval N/A

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

   14. lower-fma.f64 N/A

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

10. Applied rewrites 9.7%

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

11. Taylor expanded in x around inf

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

13. Applied rewrites 9.7%

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

14. Final simplification 9.7%

    \[\leadsto 1 \cdot \left(\left(-0.5 \cdot x\right) \cdot x\right) \]
15. Add Preprocessing

Alternative 21: 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 94.5%

   \[\cos x \cdot e^{10 \cdot \left(x \cdot x\right)} \]
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 2024298 
(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)))))