ENA, Section 1.4, Exercise 1

Percentage Accurate: 94.5% → 99.4%

Time: 10.9s

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

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

FPCore

(FPCore (x)
 :precision binary64
 (/ (cos x) (pow (pow (exp -40.0) (+ x x)) (* x 0.125))))

C

double code(double x) {
	return cos(x) / pow(pow(exp(-40.0), (x + x)), (x * 0.125));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = cos(x) / ((exp((-40.0d0)) ** (x + x)) ** (x * 0.125d0))
end function

Java

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

Python

def code(x):
	return math.cos(x) / math.pow(math.pow(math.exp(-40.0), (x + x)), (x * 0.125))

Julia

function code(x)
	return Float64(cos(x) / ((exp(-40.0) ^ Float64(x + x)) ^ Float64(x * 0.125)))
end

MATLAB

function tmp = code(x)
	tmp = cos(x) / ((exp(-40.0) ^ (x + x)) ^ (x * 0.125));
end

Wolfram

code[x_] := N[(N[Cos[x], $MachinePrecision] / N[Power[N[Power[N[Exp[-40.0], $MachinePrecision], N[(x + x), $MachinePrecision]], $MachinePrecision], N[(x * 0.125), $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 inf

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

   1. *-commutative N/A

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

   2. *-lft-identity N/A

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

   3. *-lowering-*.f64 N/A

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

   4. exp-lowering-exp.f64 N/A

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

   5. *-commutative N/A

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

   6. unpow2 N/A

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

   7. associate-*l* N/A

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

   8. *-lowering-*.f64 N/A

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

   9. *-lowering-*.f64 N/A

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

   10. *-lft-identity N/A

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

   11. cos-lowering-cos.f64 94.3

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

5. Simplified 94.3%

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

   1. *-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

   4. exp-prod N/A

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

   5. remove-double-neg N/A

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

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

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

   7. pow-neg N/A

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

   8. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{{\left({\left(e^{20}\right)}^{x}\right)}^{0}}}{{\left(e^{10}\right)}^{\left(x \cdot \left(\mathsf{neg}\left(x\right)\right)\right)}} \cdot \cos x \]

   9. /-lowering-/.f64 N/A

      \[\leadsto \color{blue}{\frac{{\left({\left(e^{20}\right)}^{x}\right)}^{0}}{{\left(e^{10}\right)}^{\left(x \cdot \left(\mathsf{neg}\left(x\right)\right)\right)}}} \cdot \cos x \]

   10. metadata-eval N/A

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

   11. pow-lowering-pow.f64 N/A

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

   12. exp-lowering-exp.f64 N/A

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

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

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

   14. neg-lowering-neg.f64 N/A

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

   15. *-lowering-*.f64 95.3

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

7. Applied egg-rr 95.3%

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

9. Applied egg-rr 99.3%

   \[\leadsto \frac{1}{\color{blue}{{\left({\left(e^{-40}\right)}^{\left(x + x\right)}\right)}^{\left(\left(x \cdot -0.5\right) \cdot -0.25\right)}}} \cdot \cos x \]

10. Taylor expanded in x around inf

    \[\leadsto \color{blue}{\frac{\cos x}{e^{\frac{1}{8} \cdot \left(x \cdot \log \left({\left(e^{-40}\right)}^{\left(2 \cdot x\right)}\right)\right)}}} \]
11. Step-by-step derivation

    1. /-lowering-/.f64 N/A

       \[\leadsto \color{blue}{\frac{\cos x}{e^{\frac{1}{8} \cdot \left(x \cdot \log \left({\left(e^{-40}\right)}^{\left(2 \cdot x\right)}\right)\right)}}} \]

    2. cos-lowering-cos.f64 N/A

       \[\leadsto \frac{\color{blue}{\cos x}}{e^{\frac{1}{8} \cdot \left(x \cdot \log \left({\left(e^{-40}\right)}^{\left(2 \cdot x\right)}\right)\right)}} \]

    3. associate-*r* N/A

       \[\leadsto \frac{\cos x}{e^{\color{blue}{\left(\frac{1}{8} \cdot x\right) \cdot \log \left({\left(e^{-40}\right)}^{\left(2 \cdot x\right)}\right)}}} \]

    4. *-commutative N/A

       \[\leadsto \frac{\cos x}{e^{\color{blue}{\log \left({\left(e^{-40}\right)}^{\left(2 \cdot x\right)}\right) \cdot \left(\frac{1}{8} \cdot x\right)}}} \]

    5. exp-to-pow N/A

       \[\leadsto \frac{\cos x}{\color{blue}{{\left({\left(e^{-40}\right)}^{\left(2 \cdot x\right)}\right)}^{\left(\frac{1}{8} \cdot x\right)}}} \]

    6. pow-lowering-pow.f64 N/A

       \[\leadsto \frac{\cos x}{\color{blue}{{\left({\left(e^{-40}\right)}^{\left(2 \cdot x\right)}\right)}^{\left(\frac{1}{8} \cdot x\right)}}} \]

    7. pow-lowering-pow.f64 N/A

       \[\leadsto \frac{\cos x}{{\color{blue}{\left({\left(e^{-40}\right)}^{\left(2 \cdot x\right)}\right)}}^{\left(\frac{1}{8} \cdot x\right)}} \]

    8. exp-lowering-exp.f64 N/A

       \[\leadsto \frac{\cos x}{{\left({\color{blue}{\left(e^{-40}\right)}}^{\left(2 \cdot x\right)}\right)}^{\left(\frac{1}{8} \cdot x\right)}} \]

    9. count-2 N/A

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

    10. +-lowering-+.f64 N/A

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

    11. *-commutative N/A

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

    12. *-lowering-*.f64 99.4

        \[\leadsto \frac{\cos x}{{\left({\left(e^{-40}\right)}^{\left(x + x\right)}\right)}^{\color{blue}{\left(x \cdot 0.125\right)}}} \]

12. Simplified 99.4%

    \[\leadsto \color{blue}{\frac{\cos x}{{\left({\left(e^{-40}\right)}^{\left(x + x\right)}\right)}^{\left(x \cdot 0.125\right)}}} \]
13. Add Preprocessing

Alternative 2: 99.4% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (* (cos x) (pow (pow (exp -40.0) (+ x x)) (* x -0.125))))

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x)
	return Float64(cos(x) * ((exp(-40.0) ^ Float64(x + x)) ^ Float64(x * -0.125)))
end

MATLAB

function tmp = code(x)
	tmp = cos(x) * ((exp(-40.0) ^ (x + x)) ^ (x * -0.125));
end

Wolfram

code[x_] := N[(N[Cos[x], $MachinePrecision] * N[Power[N[Power[N[Exp[-40.0], $MachinePrecision], N[(x + x), $MachinePrecision]], $MachinePrecision], N[(x * -0.125), $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 inf

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

   1. *-commutative N/A

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

   2. *-lft-identity N/A

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

   3. *-lowering-*.f64 N/A

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

   4. exp-lowering-exp.f64 N/A

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

   5. *-commutative N/A

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

   6. unpow2 N/A

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

   7. associate-*l* N/A

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

   8. *-lowering-*.f64 N/A

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

   9. *-lowering-*.f64 N/A

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

   10. *-lft-identity N/A

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

   11. cos-lowering-cos.f64 94.3

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

5. Simplified 94.3%

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

   1. *-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

   4. exp-prod N/A

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

   5. remove-double-neg N/A

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

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

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

   7. pow-neg N/A

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

   8. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{{\left({\left(e^{20}\right)}^{x}\right)}^{0}}}{{\left(e^{10}\right)}^{\left(x \cdot \left(\mathsf{neg}\left(x\right)\right)\right)}} \cdot \cos x \]

   9. /-lowering-/.f64 N/A

      \[\leadsto \color{blue}{\frac{{\left({\left(e^{20}\right)}^{x}\right)}^{0}}{{\left(e^{10}\right)}^{\left(x \cdot \left(\mathsf{neg}\left(x\right)\right)\right)}}} \cdot \cos x \]

   10. metadata-eval N/A

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

   11. pow-lowering-pow.f64 N/A

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

   12. exp-lowering-exp.f64 N/A

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

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

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

   14. neg-lowering-neg.f64 N/A

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

   15. *-lowering-*.f64 95.3

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

7. Applied egg-rr 95.3%

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

9. Applied egg-rr 99.3%

   \[\leadsto \frac{1}{\color{blue}{{\left({\left(e^{-40}\right)}^{\left(x + x\right)}\right)}^{\left(\left(x \cdot -0.5\right) \cdot -0.25\right)}}} \cdot \cos x \]

10. Taylor expanded in x around inf

    \[\leadsto \color{blue}{\frac{1}{e^{\frac{1}{8} \cdot \left(x \cdot \log \left({\left(e^{-40}\right)}^{\left(2 \cdot x\right)}\right)\right)}}} \cdot \cos x \]
11. Step-by-step derivation

    1. rec-exp N/A

       \[\leadsto \color{blue}{e^{\mathsf{neg}\left(\frac{1}{8} \cdot \left(x \cdot \log \left({\left(e^{-40}\right)}^{\left(2 \cdot x\right)}\right)\right)\right)}} \cdot \cos x \]

    2. metadata-eval N/A

       \[\leadsto e^{\mathsf{neg}\left(\frac{1}{8} \cdot \left(x \cdot \log \left({\left(e^{-40}\right)}^{\left(\color{blue}{\left(\mathsf{neg}\left(-2\right)\right)} \cdot x\right)}\right)\right)\right)} \cdot \cos x \]

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

       \[\leadsto e^{\mathsf{neg}\left(\frac{1}{8} \cdot \left(x \cdot \log \left({\left(e^{-40}\right)}^{\color{blue}{\left(\mathsf{neg}\left(-2 \cdot x\right)\right)}}\right)\right)\right)} \cdot \cos x \]

    4. associate-*r* N/A

       \[\leadsto e^{\mathsf{neg}\left(\color{blue}{\left(\frac{1}{8} \cdot x\right) \cdot \log \left({\left(e^{-40}\right)}^{\left(\mathsf{neg}\left(-2 \cdot x\right)\right)}\right)}\right)} \cdot \cos x \]

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

       \[\leadsto e^{\mathsf{neg}\left(\left(\frac{1}{8} \cdot x\right) \cdot \log \left({\left(e^{-40}\right)}^{\color{blue}{\left(\left(\mathsf{neg}\left(-2\right)\right) \cdot x\right)}}\right)\right)} \cdot \cos x \]

    6. metadata-eval N/A

       \[\leadsto e^{\mathsf{neg}\left(\left(\frac{1}{8} \cdot x\right) \cdot \log \left({\left(e^{-40}\right)}^{\left(\color{blue}{2} \cdot x\right)}\right)\right)} \cdot \cos x \]

    7. *-commutative N/A

       \[\leadsto e^{\mathsf{neg}\left(\color{blue}{\log \left({\left(e^{-40}\right)}^{\left(2 \cdot x\right)}\right) \cdot \left(\frac{1}{8} \cdot x\right)}\right)} \cdot \cos x \]

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

       \[\leadsto e^{\color{blue}{\log \left({\left(e^{-40}\right)}^{\left(2 \cdot x\right)}\right) \cdot \left(\mathsf{neg}\left(\frac{1}{8} \cdot x\right)\right)}} \cdot \cos x \]

    9. exp-to-pow N/A

       \[\leadsto \color{blue}{{\left({\left(e^{-40}\right)}^{\left(2 \cdot x\right)}\right)}^{\left(\mathsf{neg}\left(\frac{1}{8} \cdot x\right)\right)}} \cdot \cos x \]

    10. pow-lowering-pow.f64 N/A

        \[\leadsto \color{blue}{{\left({\left(e^{-40}\right)}^{\left(2 \cdot x\right)}\right)}^{\left(\mathsf{neg}\left(\frac{1}{8} \cdot x\right)\right)}} \cdot \cos x \]

12. Simplified 99.4%

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

13. Final simplification 99.4%

    \[\leadsto \cos x \cdot {\left({\left(e^{-40}\right)}^{\left(x + x\right)}\right)}^{\left(x \cdot -0.125\right)} \]
14. Add Preprocessing

Alternative 3: 99.2% accurate, 0.5× speedup

Math

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

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) ^ Float64(x * 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], N[(x * 0.5), $MachinePrecision]], $MachinePrecision], 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. exp-prod N/A

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

   2. pow-unpow N/A

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

   3. pow-lowering-pow.f64 N/A

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

   4. pow-exp N/A

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

   5. exp-lowering-exp.f64 N/A

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

   6. *-commutative N/A

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

   7. *-lowering-*.f64 95.2

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

4. Applied egg-rr 95.2%

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

   1. *-commutative N/A

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

   2. exp-prod N/A

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

   3. sqr-pow N/A

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

   4. pow-prod-down N/A

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

   5. prod-exp N/A

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

   6. metadata-eval N/A

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

   7. pow-lowering-pow.f64 N/A

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

   8. rem-log-exp N/A

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

   9. exp-lowering-exp.f64 N/A

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

   10. rem-log-exp N/A

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

   11. div-inv N/A

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

   12. metadata-eval N/A

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

   13. *-lowering-*.f64 99.3

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

6. Applied egg-rr 99.3%

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

Alternative 4: 98.1% accurate, 0.5× speedup

Math

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

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) ^ 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[Power[N[Exp[-10.0], $MachinePrecision], (-x)], $MachinePrecision], 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. exp-prod N/A

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

   2. pow-unpow N/A

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

   3. pow-lowering-pow.f64 N/A

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

   4. pow-exp N/A

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

   5. exp-lowering-exp.f64 N/A

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

   6. *-commutative N/A

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

   7. *-lowering-*.f64 95.2

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

4. Applied egg-rr 95.2%

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

   1. *-commutative N/A

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

   2. exp-prod N/A

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

   3. sqr-pow N/A

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

   4. pow-prod-down N/A

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

   5. prod-exp N/A

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

   6. metadata-eval N/A

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

   7. frac-2neg N/A

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

   8. div-inv N/A

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

   9. metadata-eval N/A

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

   10. metadata-eval N/A

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

   11. *-commutative N/A

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

   12. pow-unpow N/A

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

   13. pow-lowering-pow.f64 N/A

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

   14. pow-to-exp N/A

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

   15. rem-log-exp N/A

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

   16. metadata-eval N/A

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

   17. metadata-eval N/A

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

   18. rem-log-exp N/A

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

   19. exp-lowering-exp.f64 N/A

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

   20. rem-log-exp N/A

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

   21. metadata-eval N/A

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

   22. neg-lowering-neg.f64 98.1

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

6. Applied egg-rr 98.1%

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

Alternative 5: 97.9% 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 94.4%

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

   1. exp-prod N/A

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

   2. pow-unpow N/A

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

   3. pow-lowering-pow.f64 N/A

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

   4. pow-exp N/A

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

   5. exp-lowering-exp.f64 N/A

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

   6. *-commutative N/A

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

   7. *-lowering-*.f64 95.2

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

4. Applied egg-rr 95.2%

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

   1. *-commutative N/A

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

   2. exp-prod N/A

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

   3. pow-lowering-pow.f64 N/A

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

   4. exp-lowering-exp.f64 97.9

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

6. Applied egg-rr 97.9%

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

Alternative 6: 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(-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 94.4%

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

3. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. *-lft-identity N/A

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

   3. *-lowering-*.f64 N/A

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

   4. exp-lowering-exp.f64 N/A

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

   5. *-commutative N/A

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

   6. unpow2 N/A

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

   7. associate-*l* N/A

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

   8. *-lowering-*.f64 N/A

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

   9. *-lowering-*.f64 N/A

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

   10. *-lft-identity N/A

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

   11. cos-lowering-cos.f64 94.3

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

5. Simplified 94.3%

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

   1. *-commutative N/A

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

   2. pow-exp N/A

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

   3. *-commutative N/A

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

   4. exp-prod N/A

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

   5. sqr-pow N/A

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

   6. pow-prod-down N/A

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

   7. prod-exp N/A

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

   8. metadata-eval N/A

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

   9. frac-2neg N/A

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

   10. div-inv N/A

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

   11. metadata-eval N/A

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

   12. metadata-eval N/A

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

   13. *-commutative N/A

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

   14. pow-unpow N/A

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

   15. pow-unpow N/A

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

   16. *-commutative N/A

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

   17. pow-lowering-pow.f64 N/A

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

7. Applied egg-rr 95.4%

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

8. Final simplification 95.4%

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

Alternative 7: 95.3% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.4%

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

   1. exp-prod N/A

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

   2. pow-lowering-pow.f64 N/A

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

   3. exp-lowering-exp.f64 N/A

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

   4. *-lowering-*.f64 95.4

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

4. Applied egg-rr 95.4%

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

Alternative 8: 94.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   1. *-commutative N/A

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

   2. *-lft-identity N/A

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

   3. *-lowering-*.f64 N/A

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

   4. exp-lowering-exp.f64 N/A

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

   5. *-commutative N/A

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

   6. unpow2 N/A

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

   7. associate-*l* N/A

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

   8. *-lowering-*.f64 N/A

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

   9. *-lowering-*.f64 N/A

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

   10. *-lft-identity N/A

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

   11. cos-lowering-cos.f64 94.3

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

5. Simplified 94.3%

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

   1. *-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-*l* N/A

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

   4. exp-prod N/A

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

   5. remove-double-neg N/A

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

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

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

   7. pow-neg N/A

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

   8. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{{\left({\left(e^{20}\right)}^{x}\right)}^{0}}}{{\left(e^{10}\right)}^{\left(x \cdot \left(\mathsf{neg}\left(x\right)\right)\right)}} \cdot \cos x \]

   9. /-lowering-/.f64 N/A

      \[\leadsto \color{blue}{\frac{{\left({\left(e^{20}\right)}^{x}\right)}^{0}}{{\left(e^{10}\right)}^{\left(x \cdot \left(\mathsf{neg}\left(x\right)\right)\right)}}} \cdot \cos x \]

   10. metadata-eval N/A

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

   11. pow-lowering-pow.f64 N/A

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

   12. exp-lowering-exp.f64 N/A

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

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

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

   14. neg-lowering-neg.f64 N/A

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

   15. *-lowering-*.f64 95.3

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

7. Applied egg-rr 95.3%

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

8. Taylor expanded in x around inf

   \[\leadsto \frac{1}{\color{blue}{e^{-10 \cdot {x}^{2}}}} \cdot \cos x \]
9. Step-by-step derivation

   1. exp-lowering-exp.f64 N/A

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

   2. *-commutative N/A

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

   3. *-lowering-*.f64 N/A

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

   4. unpow2 N/A

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

   5. *-lowering-*.f64 94.4

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

10. Simplified 94.4%

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

11. Final simplification 94.4%

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

Alternative 9: 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 94.4%

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

Alternative 10: 27.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(N[Exp[N[(10.0 * N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * -0.001388888888888889 + 0.041666666666666664), $MachinePrecision] + -0.5), $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. 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. accelerator-lowering-fma.f64 N/A

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

   3. unpow2 N/A

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

   4. *-lowering-*.f64 N/A

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

   5. sub-neg N/A

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

   6. metadata-eval N/A

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

   7. accelerator-lowering-fma.f64 N/A

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

   8. unpow2 N/A

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

   9. *-lowering-*.f64 N/A

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

   10. +-commutative N/A

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

   11. *-commutative N/A

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

   12. accelerator-lowering-fma.f64 N/A

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

   13. unpow2 N/A

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

   14. *-lowering-*.f64 27.6

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

5. Simplified 27.6%

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

6. Final simplification 27.6%

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

Alternative 11: 27.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(N[Exp[N[(x * N[(x * 10.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * N[(x * N[(x * -0.001388888888888889), $MachinePrecision] + 0.041666666666666664), $MachinePrecision]), $MachinePrecision] + -0.5), $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. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. *-lft-identity N/A

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

   3. *-lowering-*.f64 N/A

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

   4. exp-lowering-exp.f64 N/A

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

   5. *-commutative N/A

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

   6. unpow2 N/A

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

   7. associate-*l* N/A

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

   8. *-lowering-*.f64 N/A

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

   9. *-lowering-*.f64 N/A

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

   10. *-lft-identity N/A

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

   11. cos-lowering-cos.f64 94.3

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

5. Simplified 94.3%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. accelerator-lowering-fma.f64 N/A

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

   3. unpow2 N/A

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

   4. *-lowering-*.f64 N/A

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

   5. sub-neg N/A

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

   6. unpow2 N/A

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

   7. associate-*l* N/A

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

   8. *-commutative N/A

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

   9. metadata-eval N/A

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

   10. accelerator-lowering-fma.f64 N/A

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

   11. *-commutative N/A

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

   12. *-lowering-*.f64 N/A

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

   13. +-commutative N/A

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

   14. *-commutative N/A

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

   15. unpow2 N/A

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

   16. associate-*l* N/A

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

   17. accelerator-lowering-fma.f64 N/A

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

   18. *-lowering-*.f64 27.6

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

8. Simplified 27.6%

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

Alternative 12: 21.3% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(N[Exp[N[(x * N[(x * 10.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * 0.041666666666666664 + -0.5), $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. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. *-lft-identity N/A

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

   3. *-lowering-*.f64 N/A

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

   4. exp-lowering-exp.f64 N/A

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

   5. *-commutative N/A

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

   6. unpow2 N/A

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

   7. associate-*l* N/A

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

   8. *-lowering-*.f64 N/A

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

   9. *-lowering-*.f64 N/A

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

   10. *-lft-identity N/A

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

   11. cos-lowering-cos.f64 94.3

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

5. Simplified 94.3%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. accelerator-lowering-fma.f64 N/A

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

   3. unpow2 N/A

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

   4. *-lowering-*.f64 N/A

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

   5. sub-neg N/A

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

   6. *-commutative N/A

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

   7. metadata-eval N/A

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

   8. accelerator-lowering-fma.f64 N/A

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

   9. unpow2 N/A

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

   10. *-lowering-*.f64 21.3

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

8. Simplified 21.3%

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

Alternative 13: 18.2% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(N[Exp[N[(x * N[(x * 10.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(x * N[(x * -0.5), $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. Taylor expanded in x around inf

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

   1. *-commutative N/A

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

   2. *-lft-identity N/A

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

   3. *-lowering-*.f64 N/A

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

   4. exp-lowering-exp.f64 N/A

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

   5. *-commutative N/A

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

   6. unpow2 N/A

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

   7. associate-*l* N/A

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

   8. *-lowering-*.f64 N/A

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

   9. *-lowering-*.f64 N/A

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

   10. *-lft-identity N/A

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

   11. cos-lowering-cos.f64 94.3

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

5. Simplified 94.3%

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

6. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. unpow2 N/A

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

   4. associate-*l* N/A

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

   5. accelerator-lowering-fma.f64 N/A

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

   6. *-lowering-*.f64 18.2

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

8. Simplified 18.2%

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

Alternative 14: 9.8% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(N[Cos[x], $MachinePrecision] * N[(x * N[(x * 10.0), $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. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. unpow2 N/A

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

   4. associate-*l* N/A

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

   5. accelerator-lowering-fma.f64 N/A

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

   6. *-lowering-*.f64 9.8

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

5. Simplified 9.8%

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

Alternative 15: 9.7% accurate, 19.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(x * N[(x * -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. Taylor expanded in x around 0

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

5. Simplified 9.6%

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

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1 + \frac{-1}{2} \cdot {x}^{2}} \]
7. Step-by-step derivation

   1. +-commutative N/A

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

   2. unpow2 N/A

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

   3. associate-*r* N/A

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

   4. *-commutative N/A

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

   5. accelerator-lowering-fma.f64 N/A

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

   6. *-commutative N/A

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

   7. *-lowering-*.f64 9.7

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

8. Simplified 9.7%

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

9. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{-1}{2} \cdot {x}^{2}} \]
10. Step-by-step derivation

    1. unpow2 N/A

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

    2. associate-*r* N/A

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

    3. *-commutative N/A

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

    4. *-lowering-*.f64 N/A

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

    5. *-commutative N/A

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

    6. *-lowering-*.f64 9.7

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

11. Simplified 9.7%

    \[\leadsto \color{blue}{x \cdot \left(x \cdot -0.5\right)} \]
12. 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 94.4%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{1} \]
4. Step-by-step derivation

5. Simplified 1.5%

   \[\leadsto \color{blue}{1} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024205 
(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)))))