ENA, Section 1.4, Exercise 1

Percentage Accurate: 94.5% → 99.4%

Time: 9.4s

Alternatives: 19

Speedup: 1.0×

Specification

\[1.99 \leq x \land x \leq 2.01\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 94.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.5× speedup

Math

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

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 94.4%

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

   1. lift-exp.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. exp-prod N/A

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

   4. sqr-pow N/A

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

   5. pow-prod-down N/A

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

   6. lift-*.f64 N/A

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

   7. associate-/l* N/A

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

   8. pow-unpow N/A

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

   9. lower-pow.f64 N/A

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

4. Applied rewrites 96.7%

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

   1. lift-pow.f64 N/A

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

   2. lift-exp.f64 N/A

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

   3. pow-exp N/A

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

   4. *-commutative N/A

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

   5. pow-exp N/A

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

   6. lift-exp.f64 N/A

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

   7. lift-pow.f64 99.4

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

6. Applied rewrites 99.4%

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

7. Final simplification 99.4%

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

Alternative 2: 99.3% 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 94.4%

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

   1. lift-exp.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. exp-prod N/A

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

   4. sqr-pow N/A

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

   5. pow-prod-down N/A

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

   6. lift-*.f64 N/A

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

   7. associate-/l* N/A

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

   8. pow-unpow N/A

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

   9. lower-pow.f64 N/A

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

4. Applied rewrites 96.7%

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

   1. lift-pow.f64 N/A

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

   2. lift-exp.f64 N/A

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

   3. pow-exp N/A

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

   4. *-commutative N/A

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

   5. pow-exp N/A

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

   6. lift-exp.f64 N/A

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

   7. lift-pow.f64 99.4

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

6. Applied rewrites 99.4%

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

   1. lift-pow.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. pow-pow N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. associate-*r* N/A

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

   7. lift-*.f64 N/A

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

   8. metadata-eval N/A

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

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

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

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

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

   11. lift-*.f64 N/A

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

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

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

   13. lift-neg.f64 N/A

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

   14. lift-*.f64 N/A

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

   15. lift-*.f64 N/A

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

   16. associate-*l* N/A

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

   17. *-commutative N/A

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

   18. lift-*.f64 N/A

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

   19. pow-pow N/A

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

   20. lower-pow.f64 N/A

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

8. Applied rewrites 99.3%

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

9. Final simplification 99.3%

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

Alternative 3: 98.1% accurate, 0.5× speedup

Math

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

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) ^ 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 94.4%

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

   1. lift-exp.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. exp-prod N/A

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

   4. sqr-pow N/A

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

   5. pow-prod-up N/A

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

   6. flip-+ N/A

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

   7. +-inverses N/A

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

   8. +-inverses N/A

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

   9. metadata-eval N/A

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

   10. associate-/l/ N/A

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

   11. +-inverses N/A

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

   12. +-inverses N/A

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

   13. flip-+ N/A

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

   14. count-2 N/A

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

   15. *-commutative N/A

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

   16. associate-*l/ N/A

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

   17. associate-/l* N/A

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

   18. metadata-eval N/A

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

   19. *-rgt-identity N/A

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

   20. frac-2neg N/A

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

   21. distribute-frac-neg N/A

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

4. Applied rewrites 95.3%

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

   1. lift-/.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. pow-unpow N/A

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

   5. pow-flip N/A

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

   6. lift-neg.f64 N/A

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

   7. lower-pow.f64 N/A

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

   8. lift-neg.f64 N/A

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

   9. neg-mul-1 N/A

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

   10. pow-unpow N/A

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

   11. lower-pow.f64 N/A

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

   12. pow-to-exp N/A

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

   13. lower-exp.f64 N/A

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

   14. lift-exp.f64 N/A

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

   15. rem-log-exp N/A

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

   16. metadata-eval 98.1

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

6. Applied rewrites 98.1%

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

7. Final simplification 98.1%

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

Alternative 4: 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 94.4%

   \[\cos x \cdot e^{10 \cdot \left(x \cdot x\right)} \]
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 97.8

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

4. Applied rewrites 97.8%

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

5. Final simplification 97.8%

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

Alternative 5: 96.8% 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 94.4%

   \[\cos x \cdot e^{10 \cdot \left(x \cdot x\right)} \]
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 96.7

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

5. Applied rewrites 96.7%

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

6. Final simplification 96.7%

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

Alternative 6: 95.0% 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 94.4%

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

   1. lift-exp.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. exp-prod N/A

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

   4. sqr-pow N/A

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

   5. pow-prod-up N/A

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

   6. flip-+ N/A

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

   7. +-inverses N/A

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

   8. +-inverses N/A

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

   9. metadata-eval N/A

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

   10. associate-/l/ N/A

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

   11. +-inverses N/A

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

   12. +-inverses N/A

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

   13. flip-+ N/A

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

   14. count-2 N/A

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

   15. *-commutative N/A

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

   16. associate-*l/ N/A

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

   17. associate-/l* N/A

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

   18. metadata-eval N/A

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

   19. *-rgt-identity N/A

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

   20. frac-2neg N/A

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

   21. distribute-frac-neg N/A

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

4. Applied rewrites 95.3%

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

   1. lift-/.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. pow-flip N/A

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

   4. lift-*.f64 N/A

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

   5. lift-neg.f64 N/A

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

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

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

   7. lift-*.f64 N/A

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

   8. remove-double-neg N/A

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

   9. lift-exp.f64 N/A

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

   10. exp-prod N/A

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

   11. lift-*.f64 N/A

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

   12. associate-*r* N/A

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

   13. *-commutative N/A

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

   14. pow-exp N/A

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

   15. lift-exp.f64 N/A

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

   16. pow-pow N/A

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

   17. metadata-eval N/A

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

   18. lower-pow.f64 N/A

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

   19. metadata-eval N/A

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

   20. lower-pow.f64 96.7

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

6. Applied rewrites 96.7%

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

   1. rem-exp-log N/A

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

   2. lift-pow.f64 N/A

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

   3. log-pow N/A

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

   4. metadata-eval N/A

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

   5. rem-log-exp N/A

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

   6. lift-exp.f64 N/A

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

   7. lift-exp.f64 N/A

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

   8. rem-log-exp N/A

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

   9. associate-*r* N/A

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

   10. *-commutative N/A

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

   11. metadata-eval N/A

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

   12. div-inv N/A

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

   13. pow-to-exp N/A

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

   14. sqrt-pow1 N/A

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

   15. lift-pow.f64 N/A

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

   16. unpow1/2 N/A

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

   17. pow-to-exp N/A

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

   18. lower-exp.f64 N/A

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

   19. lift-pow.f64 N/A

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

   20. pow-to-exp N/A

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

   21. rem-log-exp N/A

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

   22. associate-*r* N/A

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

   23. *-commutative N/A

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

   24. associate-*r* N/A

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

8. Applied rewrites 95.4%

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

9. Final simplification 95.4%

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

Alternative 7: 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 94.4%

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

   1. lift-exp.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. exp-prod N/A

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

   4. lower-pow.f64 N/A

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

   5. lower-exp.f64 95.2

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

4. Applied rewrites 95.2%

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

5. Final simplification 95.2%

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

Alternative 8: 94.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \sqrt{e^{2 \cdot \left(\left(x \cdot x\right) \cdot 10\right)}} \cdot \cos x \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (sqrt (exp (* 2.0 (* (* x x) 10.0)))) (cos x)))

C

double code(double x) {
	return sqrt(exp((2.0 * ((x * x) * 10.0)))) * cos(x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = sqrt(exp((2.0d0 * ((x * x) * 10.0d0)))) * cos(x)
end function

Java

public static double code(double x) {
	return Math.sqrt(Math.exp((2.0 * ((x * x) * 10.0)))) * Math.cos(x);
}

Python

def code(x):
	return math.sqrt(math.exp((2.0 * ((x * x) * 10.0)))) * math.cos(x)

Julia

function code(x)
	return Float64(sqrt(exp(Float64(2.0 * Float64(Float64(x * x) * 10.0)))) * cos(x))
end

MATLAB

function tmp = code(x)
	tmp = sqrt(exp((2.0 * ((x * x) * 10.0)))) * cos(x);
end

Wolfram

code[x_] := N[(N[Sqrt[N[Exp[N[(2.0 * N[(N[(x * x), $MachinePrecision] * 10.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] * N[Cos[x], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.4%

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

   1. lift-exp.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. exp-prod N/A

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

   4. sqr-pow N/A

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

   5. pow-prod-down N/A

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

   6. frac-2neg N/A

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

   7. div-inv N/A

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

   8. pow-unpow N/A

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

   9. lower-pow.f64 N/A

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

4. Applied rewrites 95.2%

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

5. Taylor expanded in x around inf

   \[\leadsto \cos x \cdot \color{blue}{\sqrt{\frac{1}{e^{-20 \cdot {x}^{2}}}}} \]
6. 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. unpow2 N/A

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

   4. associate-*r* N/A

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

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

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

   6. exp-prod N/A

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

   7. lower-pow.f64 N/A

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

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

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

   9. metadata-eval N/A

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

   10. exp-prod N/A

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

   11. lower-pow.f64 N/A

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

   12. lower-exp.f64 99.4

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

7. Applied rewrites 99.4%

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

9. Applied rewrites 94.4%

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

10. Final simplification 94.4%

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

Alternative 9: 94.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.4%

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

   1. lift-exp.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. exp-prod N/A

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

   4. sqr-pow N/A

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

   5. pow-prod-up N/A

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

   6. flip-+ N/A

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

   7. +-inverses N/A

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

   8. +-inverses N/A

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

   9. metadata-eval N/A

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

   10. associate-/l/ N/A

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

   11. +-inverses N/A

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

   12. +-inverses N/A

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

   13. flip-+ N/A

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

   14. count-2 N/A

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

   15. *-commutative N/A

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

   16. associate-*l/ N/A

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

   17. associate-/l* N/A

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

   18. metadata-eval N/A

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

   19. *-rgt-identity N/A

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

   20. frac-2neg N/A

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

   21. distribute-frac-neg N/A

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

4. Applied rewrites 95.3%

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

   1. lift-pow.f64 N/A

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

   2. lift-exp.f64 N/A

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

   3. pow-exp N/A

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

   4. lower-exp.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-neg.f64 N/A

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

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

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

   8. lift-*.f64 N/A

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

   9. neg-mul-1 N/A

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

   10. associate-*r* N/A

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

   11. rem-log-exp N/A

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

   12. lift-exp.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. lift-exp.f64 N/A

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

   15. rem-log-exp N/A

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

   16. metadata-eval 94.4

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

6. Applied rewrites 94.4%

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

7. Final simplification 94.4%

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

Alternative 10: 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 94.4%

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

3. Final simplification 94.4%

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

Alternative 11: 27.5% accurate, 1.4× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (*
  (+
   (*
    (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(-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(Float64(Float64(fma(Float64(fma(-0.001388888888888889, Float64(x * x), 0.041666666666666664) * x), x, -0.5) * Float64(x * x)) + 1.0) * exp(Float64(Float64(x * x) * 10.0)))
end

Wolfram

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

Derivation

1. Initial program 94.4%

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

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

   6. *-commutative N/A

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

   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. Step-by-step derivation

7. Applied rewrites 27.5%

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

9. Applied rewrites 27.5%

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

10. Final simplification 27.5%

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

Alternative 12: 27.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 94.4%

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

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

   6. *-commutative N/A

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

   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. Step-by-step derivation

7. Applied rewrites 27.5%

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

9. Applied rewrites 27.5%

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

10. Final simplification 27.5%

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

Alternative 13: 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 94.4%

   \[\cos x \cdot e^{10 \cdot \left(x \cdot x\right)} \]
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 14: 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 94.4%

   \[\cos x \cdot e^{10 \cdot \left(x \cdot x\right)} \]
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 15: 10.3% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 94.4%

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

   1. lift-exp.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. exp-prod N/A

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

   4. sqr-pow N/A

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

   5. pow-prod-down N/A

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

   6. frac-2neg N/A

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

   7. div-inv N/A

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

   8. pow-unpow N/A

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

   9. lower-pow.f64 N/A

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

4. Applied rewrites 95.2%

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

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. metadata-eval N/A

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

   3. metadata-eval N/A

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

   4. *-inverses N/A

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

   5. distribute-frac-neg2 N/A

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

   6. associate-/l* N/A

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

   7. associate-*l/ N/A

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

   8. distribute-neg-frac2 N/A

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

   9. pow-sqr N/A

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

   10. metadata-eval N/A

       \[\leadsto \left(\left(\left(\mathsf{neg}\left(\frac{\frac{1}{2}}{{x}^{\color{blue}{4}}}\right)\right) \cdot \left({x}^{2} \cdot {x}^{2}\right)\right) \cdot {x}^{2} + 1\right) \cdot 1 \]

   11. *-commutative N/A

       \[\leadsto \left(\color{blue}{\left(\left({x}^{2} \cdot {x}^{2}\right) \cdot \left(\mathsf{neg}\left(\frac{\frac{1}{2}}{{x}^{4}}\right)\right)\right)} \cdot {x}^{2} + 1\right) \cdot 1 \]

   12. lower-fma.f64 N/A

       \[\leadsto \color{blue}{\mathsf{fma}\left(\left({x}^{2} \cdot {x}^{2}\right) \cdot \left(\mathsf{neg}\left(\frac{\frac{1}{2}}{{x}^{4}}\right)\right), {x}^{2}, 1\right)} \cdot 1 \]

10. Applied rewrites 9.7%

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

11. Taylor expanded in x around 0

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

    1. +-commutative N/A

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

    2. *-commutative N/A

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

    3. lower-fma.f64 N/A

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

    4. +-commutative N/A

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

    5. *-commutative N/A

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

    6. lower-fma.f64 N/A

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

    7. +-commutative N/A

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

    8. lower-fma.f64 N/A

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

    9. unpow2 N/A

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

    10. lower-*.f64 N/A

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

    11. unpow2 N/A

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

    12. lower-*.f64 N/A

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

    13. unpow2 N/A

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

    14. lower-*.f64 10.3

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

13. Applied rewrites 10.3%

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

14. Final simplification 10.3%

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

Alternative 16: 10.1% accurate, 5.5× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 94.4%

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

   1. lift-exp.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. exp-prod N/A

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

   4. sqr-pow N/A

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

   5. pow-prod-down N/A

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

   6. frac-2neg N/A

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

   7. div-inv N/A

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

   8. pow-unpow N/A

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

   9. lower-pow.f64 N/A

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

4. Applied rewrites 95.2%

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

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. metadata-eval N/A

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

   3. metadata-eval N/A

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

   4. *-inverses N/A

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

   5. distribute-frac-neg2 N/A

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

   6. associate-/l* N/A

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

   7. associate-*l/ N/A

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

   8. distribute-neg-frac2 N/A

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

   9. pow-sqr N/A

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

   10. metadata-eval N/A

       \[\leadsto \left(\left(\left(\mathsf{neg}\left(\frac{\frac{1}{2}}{{x}^{\color{blue}{4}}}\right)\right) \cdot \left({x}^{2} \cdot {x}^{2}\right)\right) \cdot {x}^{2} + 1\right) \cdot 1 \]

   11. *-commutative N/A

       \[\leadsto \left(\color{blue}{\left(\left({x}^{2} \cdot {x}^{2}\right) \cdot \left(\mathsf{neg}\left(\frac{\frac{1}{2}}{{x}^{4}}\right)\right)\right)} \cdot {x}^{2} + 1\right) \cdot 1 \]

   12. lower-fma.f64 N/A

       \[\leadsto \color{blue}{\mathsf{fma}\left(\left({x}^{2} \cdot {x}^{2}\right) \cdot \left(\mathsf{neg}\left(\frac{\frac{1}{2}}{{x}^{4}}\right)\right), {x}^{2}, 1\right)} \cdot 1 \]

10. Applied rewrites 9.7%

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

11. Taylor expanded in x around 0

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

    1. +-commutative N/A

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

    2. *-commutative N/A

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

    3. lower-fma.f64 N/A

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

    4. +-commutative N/A

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

    5. lower-fma.f64 N/A

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

    6. unpow2 N/A

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

    7. lower-*.f64 N/A

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

    8. unpow2 N/A

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

    9. lower-*.f64 10.1

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

13. Applied rewrites 10.1%

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

14. Final simplification 10.1%

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

Alternative 17: 9.9% accurate, 7.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 94.4%

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

   1. lift-exp.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. exp-prod N/A

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

   4. sqr-pow N/A

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

   5. pow-prod-down N/A

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

   6. frac-2neg N/A

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

   7. div-inv N/A

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

   8. pow-unpow N/A

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

   9. lower-pow.f64 N/A

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

4. Applied rewrites 95.2%

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

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. metadata-eval N/A

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

   3. metadata-eval N/A

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

   4. *-inverses N/A

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

   5. distribute-frac-neg2 N/A

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

   6. associate-/l* N/A

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

   7. associate-*l/ N/A

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

   8. distribute-neg-frac2 N/A

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

   9. pow-sqr N/A

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

   10. metadata-eval N/A

       \[\leadsto \left(\left(\left(\mathsf{neg}\left(\frac{\frac{1}{2}}{{x}^{\color{blue}{4}}}\right)\right) \cdot \left({x}^{2} \cdot {x}^{2}\right)\right) \cdot {x}^{2} + 1\right) \cdot 1 \]

   11. *-commutative N/A

       \[\leadsto \left(\color{blue}{\left(\left({x}^{2} \cdot {x}^{2}\right) \cdot \left(\mathsf{neg}\left(\frac{\frac{1}{2}}{{x}^{4}}\right)\right)\right)} \cdot {x}^{2} + 1\right) \cdot 1 \]

   12. lower-fma.f64 N/A

       \[\leadsto \color{blue}{\mathsf{fma}\left(\left({x}^{2} \cdot {x}^{2}\right) \cdot \left(\mathsf{neg}\left(\frac{\frac{1}{2}}{{x}^{4}}\right)\right), {x}^{2}, 1\right)} \cdot 1 \]

10. Applied rewrites 9.7%

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

11. Taylor expanded in x around 0

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

    1. +-commutative N/A

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

    2. lower-fma.f64 N/A

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

    3. unpow2 N/A

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

    4. lower-*.f64 9.9

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

13. Applied rewrites 9.9%

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

14. Final simplification 9.9%

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

Alternative 18: 9.7% accurate, 13.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.4%

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

   1. lift-exp.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. exp-prod N/A

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

   4. sqr-pow N/A

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

   5. pow-prod-down N/A

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

   6. frac-2neg N/A

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

   7. div-inv N/A

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

   8. pow-unpow N/A

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

   9. lower-pow.f64 N/A

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

4. Applied rewrites 95.2%

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

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. metadata-eval N/A

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

   3. metadata-eval N/A

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

   4. *-inverses N/A

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

   5. distribute-frac-neg2 N/A

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

   6. associate-/l* N/A

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

   7. associate-*l/ N/A

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

   8. distribute-neg-frac2 N/A

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

   9. pow-sqr N/A

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

   10. metadata-eval N/A

       \[\leadsto \left(\left(\left(\mathsf{neg}\left(\frac{\frac{1}{2}}{{x}^{\color{blue}{4}}}\right)\right) \cdot \left({x}^{2} \cdot {x}^{2}\right)\right) \cdot {x}^{2} + 1\right) \cdot 1 \]

   11. *-commutative N/A

       \[\leadsto \left(\color{blue}{\left(\left({x}^{2} \cdot {x}^{2}\right) \cdot \left(\mathsf{neg}\left(\frac{\frac{1}{2}}{{x}^{4}}\right)\right)\right)} \cdot {x}^{2} + 1\right) \cdot 1 \]

   12. lower-fma.f64 N/A

       \[\leadsto \color{blue}{\mathsf{fma}\left(\left({x}^{2} \cdot {x}^{2}\right) \cdot \left(\mathsf{neg}\left(\frac{\frac{1}{2}}{{x}^{4}}\right)\right), {x}^{2}, 1\right)} \cdot 1 \]

10. Applied rewrites 9.7%

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

11. Taylor expanded in x around inf

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

13. Applied rewrites 9.7%

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

14. Final simplification 9.7%

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

Alternative 19: 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 94.4%

   \[\cos x \cdot e^{10 \cdot \left(x \cdot x\right)} \]
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 2024271 
(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)))))