ENA, Section 1.4, Exercise 1

Percentage Accurate: 94.5% → 99.4%

Time: 9.7s

Alternatives: 16

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 94.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.2%

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

   1. lift-exp.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. exp-prod N/A

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

   4. lift-*.f64 N/A

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

   5. pow-unpow N/A

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

   6. lower-pow.f64 N/A

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

   7. lower-pow.f64 N/A

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

   8. lower-exp.f64 98.0

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

4. Applied rewrites 98.0%

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

   1. lift-pow.f64 N/A

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

   2. sqr-pow N/A

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

   3. pow-prod-down N/A

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

   4. lower-pow.f64 N/A

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

6. Applied rewrites 99.4%

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

Alternative 2: 98.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \cos x \cdot {\left({\left(e^{5}\right)}^{\left(x + x\right)}\right)}^{x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (cos x) (pow (pow (exp 5.0) (+ x x)) x)))

C

double code(double x) {
	return cos(x) * pow(pow(exp(5.0), (x + x)), x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = cos(x) * ((exp(5.0d0) ** (x + x)) ** x)
end function

Java

public static double code(double x) {
	return Math.cos(x) * Math.pow(Math.pow(Math.exp(5.0), (x + x)), x);
}

Python

def code(x):
	return math.cos(x) * math.pow(math.pow(math.exp(5.0), (x + x)), x)

Julia

function code(x)
	return Float64(cos(x) * ((exp(5.0) ^ Float64(x + x)) ^ x))
end

MATLAB

function tmp = code(x)
	tmp = cos(x) * ((exp(5.0) ^ (x + x)) ^ x);
end

Wolfram

code[x_] := N[(N[Cos[x], $MachinePrecision] * N[Power[N[Power[N[Exp[5.0], $MachinePrecision], N[(x + x), $MachinePrecision]], $MachinePrecision], x], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.2%

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

   1. lift-exp.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. exp-prod N/A

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

   4. lift-*.f64 N/A

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

   5. pow-unpow N/A

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

   6. lower-pow.f64 N/A

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

   7. lower-pow.f64 N/A

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

   8. lower-exp.f64 98.0

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

4. Applied rewrites 98.0%

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

   1. lift-pow.f64 N/A

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

   2. sqr-pow N/A

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

   3. pow-prod-down N/A

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

   4. lower-pow.f64 N/A

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

   5. lift-exp.f64 N/A

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

   6. lift-exp.f64 N/A

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

   7. prod-exp N/A

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

   8. rem-log-exp N/A

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

   9. lift-exp.f64 N/A

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

   10. rem-log-exp N/A

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

   11. lift-exp.f64 N/A

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

   12. log-prod N/A

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

   13. lower-exp.f64 N/A

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

   14. pow2 N/A

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

   15. log-pow N/A

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

   16. lift-exp.f64 N/A

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

   17. rem-log-exp N/A

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

   18. metadata-eval N/A

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

   19. clear-num N/A

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

   20. associate-/r/ N/A

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

   21. metadata-eval N/A

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

   22. lower-*.f64 99.3

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

6. Applied rewrites 99.3%

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

   1. lift-pow.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. pow-unpow N/A

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

   4. sqr-pow N/A

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

   5. unpow-prod-down N/A

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

   6. pow-prod-up N/A

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

   7. lower-pow.f64 N/A

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

   8. pow-to-exp N/A

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

   9. lift-exp.f64 N/A

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

   10. rem-log-exp N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

       \[\leadsto \cos x \cdot {\left({\left(e^{\color{blue}{5}}\right)}^{\left(x + x\right)}\right)}^{x} \]

   13. metadata-eval N/A

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

   14. lower-exp.f64 N/A

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

   15. metadata-eval N/A

       \[\leadsto \cos x \cdot {\left({\left(e^{\color{blue}{5}}\right)}^{\left(x + x\right)}\right)}^{x} \]

   16. lower-+.f64 98.4

       \[\leadsto \cos x \cdot {\left({\left(e^{5}\right)}^{\color{blue}{\left(x + x\right)}}\right)}^{x} \]

8. Applied rewrites 98.4%

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

Alternative 3: 98.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \cos x \cdot {\left({\left(e^{5}\right)}^{x}\right)}^{\left(x + x\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (cos x) (pow (pow (exp 5.0) x) (+ x x))))

C

double code(double x) {
	return cos(x) * pow(pow(exp(5.0), x), (x + x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = cos(x) * ((exp(5.0d0) ** x) ** (x + x))
end function

Java

public static double code(double x) {
	return Math.cos(x) * Math.pow(Math.pow(Math.exp(5.0), x), (x + x));
}

Python

def code(x):
	return math.cos(x) * math.pow(math.pow(math.exp(5.0), x), (x + x))

Julia

function code(x)
	return Float64(cos(x) * ((exp(5.0) ^ x) ^ Float64(x + x)))
end

MATLAB

function tmp = code(x)
	tmp = cos(x) * ((exp(5.0) ^ x) ^ (x + x));
end

Wolfram

code[x_] := N[(N[Cos[x], $MachinePrecision] * N[Power[N[Power[N[Exp[5.0], $MachinePrecision], x], $MachinePrecision], N[(x + x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.2%

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

   1. lift-exp.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. exp-prod N/A

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

   4. lift-*.f64 N/A

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

   5. pow-unpow N/A

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

   6. lower-pow.f64 N/A

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

   7. lower-pow.f64 N/A

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

   8. lower-exp.f64 98.0

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

4. Applied rewrites 98.0%

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

6. Applied rewrites 99.4%

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

   1. lift-sqrt.f64 N/A

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

   2. pow1/2 N/A

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

   3. lift-pow.f64 N/A

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

   4. pow-pow N/A

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

   5. *-commutative N/A

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

   6. pow-pow N/A

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

   7. lift-pow.f64 N/A

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

   8. pow-unpow N/A

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

   9. *-commutative N/A

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

   10. lift-*.f64 N/A

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

   11. sqr-pow N/A

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

   12. unpow-prod-down N/A

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

   13. pow-prod-up N/A

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

   14. lower-pow.f64 N/A

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

8. Applied rewrites 98.1%

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

Alternative 4: 98.1% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.2%

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

   1. lift-exp.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. exp-prod N/A

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

   4. lift-*.f64 N/A

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

   5. pow-unpow N/A

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

   6. lower-pow.f64 N/A

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

   7. lower-pow.f64 N/A

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

   8. lower-exp.f64 98.0

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

4. Applied rewrites 98.0%

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

6. Applied rewrites 99.4%

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

8. Applied rewrites 98.0%

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

Alternative 5: 98.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.2%

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

   1. lift-exp.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. exp-prod N/A

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

   4. lift-*.f64 N/A

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

   5. pow-unpow N/A

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

   6. lower-pow.f64 N/A

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

   7. lower-pow.f64 N/A

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

   8. lower-exp.f64 98.0

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

4. Applied rewrites 98.0%

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

Alternative 6: 96.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.2%

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

Alternative 7: 95.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \cos x \cdot {\left(e^{\left(2 \cdot x\right) \cdot x}\right)}^{5} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (cos x) (pow (exp (* (* 2.0 x) x)) 5.0)))

C

double code(double x) {
	return cos(x) * pow(exp(((2.0 * x) * x)), 5.0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = cos(x) * (exp(((2.0d0 * x) * x)) ** 5.0d0)
end function

Java

public static double code(double x) {
	return Math.cos(x) * Math.pow(Math.exp(((2.0 * x) * x)), 5.0);
}

Python

def code(x):
	return math.cos(x) * math.pow(math.exp(((2.0 * x) * x)), 5.0)

Julia

function code(x)
	return Float64(cos(x) * (exp(Float64(Float64(2.0 * x) * x)) ^ 5.0))
end

MATLAB

function tmp = code(x)
	tmp = cos(x) * (exp(((2.0 * x) * x)) ^ 5.0);
end

Wolfram

code[x_] := N[(N[Cos[x], $MachinePrecision] * N[Power[N[Exp[N[(N[(2.0 * x), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision], 5.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.2%

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

   1. lift-exp.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. exp-prod N/A

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

   4. lift-*.f64 N/A

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

   5. pow-unpow N/A

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

   6. lower-pow.f64 N/A

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

   7. lower-pow.f64 N/A

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

   8. lower-exp.f64 98.0

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

4. Applied rewrites 98.0%

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

   1. lift-pow.f64 N/A

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

   2. pow-to-exp N/A

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

   3. *-commutative N/A

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

   4. pow-exp N/A

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

   5. lift-pow.f64 N/A

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

   6. lift-exp.f64 N/A

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

   7. pow-exp N/A

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

   8. rem-log-exp N/A

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

   9. metadata-eval N/A

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

   10. associate-*r* N/A

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

   11. lift-*.f64 N/A

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

   12. lift-exp.f64 N/A

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

   13. pow-pow N/A

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

   14. lift-pow.f64 N/A

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

   15. pow-to-exp N/A

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

   16. *-commutative N/A

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

   17. lift-pow.f64 N/A

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

   18. lift-exp.f64 N/A

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

   19. pow-exp N/A

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

   20. rem-log-exp N/A

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

   21. associate-*r* N/A

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

   22. exp-prod N/A

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

6. Applied rewrites 95.4%

   \[\leadsto \cos x \cdot \color{blue}{{\left(e^{\left(2 \cdot x\right) \cdot x}\right)}^{5}} \]
7. Add Preprocessing

Alternative 8: 95.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \cos x \cdot {\left(e^{5}\right)}^{\left(\left(x + x\right) \cdot x\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (cos x) (pow (exp 5.0) (* (+ x x) x))))

C

double code(double x) {
	return cos(x) * pow(exp(5.0), ((x + x) * x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = cos(x) * (exp(5.0d0) ** ((x + x) * x))
end function

Java

public static double code(double x) {
	return Math.cos(x) * Math.pow(Math.exp(5.0), ((x + x) * x));
}

Python

def code(x):
	return math.cos(x) * math.pow(math.exp(5.0), ((x + x) * x))

Julia

function code(x)
	return Float64(cos(x) * (exp(5.0) ^ Float64(Float64(x + x) * x)))
end

MATLAB

function tmp = code(x)
	tmp = cos(x) * (exp(5.0) ^ ((x + x) * x));
end

Wolfram

code[x_] := N[(N[Cos[x], $MachinePrecision] * N[Power[N[Exp[5.0], $MachinePrecision], N[(N[(x + x), $MachinePrecision] * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.2%

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

   1. lift-exp.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. exp-prod N/A

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

   4. lift-*.f64 N/A

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

   5. pow-unpow N/A

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

   6. lower-pow.f64 N/A

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

   7. lower-pow.f64 N/A

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

   8. lower-exp.f64 98.0

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

4. Applied rewrites 98.0%

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

6. Applied rewrites 99.4%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-pow.f64 N/A

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

   3. lift-pow.f64 N/A

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

   4. pow-pow N/A

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

   5. lift-*.f64 N/A

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

   6. sqrt-pow1 N/A

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

   7. sqrt-pow2 N/A

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

   8. unpow1/2 N/A

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

   9. sqr-pow N/A

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

   10. unpow-prod-down N/A

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

   11. pow-prod-up N/A

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

   12. lower-pow.f64 N/A

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

   13. pow-to-exp N/A

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

   14. lift-exp.f64 N/A

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

   15. rem-log-exp N/A

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

   16. metadata-eval N/A

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

   17. metadata-eval N/A

       \[\leadsto \cos x \cdot {\left(e^{\color{blue}{5}}\right)}^{\left(x \cdot x + x \cdot x\right)} \]

   18. metadata-eval N/A

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

   19. lower-exp.f64 N/A

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

   20. metadata-eval N/A

       \[\leadsto \cos x \cdot {\left(e^{\color{blue}{5}}\right)}^{\left(x \cdot x + x \cdot x\right)} \]

   21. rem-log-exp N/A

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

   22. lift-exp.f64 N/A

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

   23. rem-log-exp N/A

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

   24. lift-exp.f64 N/A

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

8. Applied rewrites 95.3%

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

Alternative 9: 95.3% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.2%

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

   1. lift-exp.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. exp-prod N/A

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

   4. lower-pow.f64 N/A

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

   5. lower-exp.f64 95.3

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

4. Applied rewrites 95.3%

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

Alternative 10: 94.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.2%

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

   1. lift-exp.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. exp-prod N/A

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

   4. lift-*.f64 N/A

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

   5. pow-unpow N/A

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

   6. lower-pow.f64 N/A

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

   7. lower-pow.f64 N/A

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

   8. lower-exp.f64 98.0

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

4. Applied rewrites 98.0%

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

   1. lift-pow.f64 N/A

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

   2. sqr-pow N/A

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

   3. pow-prod-down N/A

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

   4. lower-pow.f64 N/A

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

6. Applied rewrites 99.4%

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

8. Applied rewrites 94.2%

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

Alternative 11: 94.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.2%

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

Alternative 12: 27.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 94.2%

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

Alternative 13: 21.3% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 94.2%

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

Alternative 14: 18.2% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 94.2%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 18.2

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

5. Applied rewrites 18.2%

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

Alternative 15: 9.9% accurate, 8.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 94.2%

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

   1. lift-exp.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. exp-prod N/A

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

   4. lift-*.f64 N/A

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

   5. pow-unpow N/A

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

   6. lower-pow.f64 N/A

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

   7. lower-pow.f64 N/A

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

   8. lower-exp.f64 98.0

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

4. Applied rewrites 98.0%

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

5. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. unpow2 N/A

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

   5. lower-*.f64 9.8

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

7. Applied rewrites 9.8%

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

8. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 9.9

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

10. Applied rewrites 9.9%

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

11. Taylor expanded in x around inf

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

13. Applied rewrites 9.9%

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

14. Final simplification 9.9%

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

Alternative 16: 1.5% accurate, 216.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x) :precision binary64 1.0)

C

double code(double x) {
	return 1.0;
}

Fortran

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

Java

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

Python

def code(x):
	return 1.0

Julia

function code(x)
	return 1.0
end

MATLAB

function tmp = code(x)
	tmp = 1.0;
end

Wolfram

code[x_] := 1.0

Derivation

1. Initial program 94.2%

   \[\cos x \cdot e^{10 \cdot \left(x \cdot x\right)} \]
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 2024318 
(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)))))