ENA, Section 1.4, Exercise 1

Percentage Accurate: 94.4% → 98.4%

Time: 9.1s

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: 94.4% 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: 98.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ {\left({\left({\left(e^{10}\right)}^{0.5}\right)}^{\left(x + x\right)}\right)}^{x} \cdot \cos x \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (pow (pow (pow (exp 10.0) 0.5) (+ x x)) x) (cos x)))

C

double code(double x) {
	return pow(pow(pow(exp(10.0), 0.5), (x + x)), x) * cos(x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (((exp(10.0d0) ** 0.5d0) ** (x + x)) ** x) * cos(x)
end function

Java

public static double code(double x) {
	return Math.pow(Math.pow(Math.pow(Math.exp(10.0), 0.5), (x + x)), x) * Math.cos(x);
}

Python

def code(x):
	return math.pow(math.pow(math.pow(math.exp(10.0), 0.5), (x + x)), x) * math.cos(x)

Julia

function code(x)
	return Float64((((exp(10.0) ^ 0.5) ^ Float64(x + x)) ^ x) * cos(x))
end

MATLAB

function tmp = code(x)
	tmp = (((exp(10.0) ^ 0.5) ^ (x + x)) ^ x) * cos(x);
end

Wolfram

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

Derivation

1. Initial program 94.4%

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. exp-prod N/A

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

   4. *-commutative N/A

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

   5. unpow1 N/A

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

   6. metadata-eval N/A

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

   7. sqr-pow N/A

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

   8. associate-*l* N/A

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

   9. pow-unpow N/A

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

   10. lower-pow.f64 N/A

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

   11. pow-exp N/A

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

   12. lower-exp.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. metadata-eval N/A

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

   15. metadata-eval N/A

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

   16. unpow1/2 N/A

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

   17. lower-sqrt.f64 N/A

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

   18. *-commutative N/A

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

   19. lower-*.f64 N/A

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

   20. metadata-eval N/A

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

   21. metadata-eval N/A

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

4. Applied egg-rr 93.2%

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

   1. lift-cos.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. rem-log-exp N/A

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

   5. lift-exp.f64 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. exp-prod N/A

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

   9. pow-to-exp N/A

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

   10. lift-pow.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 93.2

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

6. Applied egg-rr 93.3%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. exp-prod N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. pow-unpow N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-pow.f64 N/A

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

   11. lower-exp.f64 93.4

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 93.4

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

8. Applied egg-rr 93.4%

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

   1. lift-sqrt.f64 N/A

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

   2. rem-log-exp N/A

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

   3. lift-exp.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-exp.f64 N/A

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

   7. rem-log-exp N/A

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

   8. pow-exp N/A

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

   9. *-commutative N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. associate-*l* N/A

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

   13. exp-prod N/A

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

   14. lift-sqrt.f64 N/A

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

   15. lift-sqrt.f64 N/A

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

   16. rem-square-sqrt N/A

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

   17. *-rgt-identity N/A

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

   18. metadata-eval N/A

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

   19. distribute-lft-out N/A

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

   20. distribute-rgt-out N/A

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

   21. pow-unpow N/A

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

   22. lower-pow.f64 N/A

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

10. Applied egg-rr 98.4%

    \[\leadsto {\color{blue}{\left({\left({\left(e^{10}\right)}^{0.5}\right)}^{\left(x + x\right)}\right)}}^{x} \cdot \cos x \]
11. Add Preprocessing

Alternative 2: 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 94.4%

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

   1. associate-*r* N/A

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

   2. exp-prod N/A

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

   3. pow-exp N/A

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

   4. sqr-pow N/A

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

   5. unpow-prod-down N/A

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

   6. pow-prod-up N/A

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

   7. lower-pow.f64 N/A

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

   8. pow-to-exp N/A

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

   9. lower-exp.f64 N/A

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

   10. rem-log-exp N/A

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

   11. associate-*r/ N/A

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

   12. *-commutative N/A

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

   13. associate-/l* N/A

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

   14. lower-*.f64 N/A

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

   15. metadata-eval N/A

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

   16. lower-+.f64 94.9

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

4. Applied egg-rr 94.9%

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

   1. *-commutative N/A

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

   2. exp-prod N/A

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

   3. lower-pow.f64 N/A

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

   4. lower-exp.f64 98.3

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

6. Applied egg-rr 98.3%

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

Alternative 3: 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 94.4%

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. exp-prod N/A

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

   4. *-commutative N/A

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

   5. unpow1 N/A

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

   6. metadata-eval N/A

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

   7. sqr-pow N/A

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

   8. associate-*l* N/A

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

   9. pow-unpow N/A

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

   10. lower-pow.f64 N/A

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

   11. pow-exp N/A

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

   12. lower-exp.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. metadata-eval N/A

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

   15. metadata-eval N/A

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

   16. unpow1/2 N/A

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

   17. lower-sqrt.f64 N/A

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

   18. *-commutative N/A

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

   19. lower-*.f64 N/A

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

   20. metadata-eval N/A

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

   21. metadata-eval N/A

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

4. Applied egg-rr 93.2%

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

   1. lift-cos.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. rem-log-exp N/A

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

   5. lift-exp.f64 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. exp-prod N/A

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

   9. pow-to-exp N/A

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

   10. lift-pow.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 93.2

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

6. Applied egg-rr 93.3%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. exp-prod N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. pow-unpow N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-pow.f64 N/A

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

   11. lower-exp.f64 93.4

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 93.4

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

8. Applied egg-rr 93.4%

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

   1. lift-sqrt.f64 N/A

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

   2. rem-log-exp N/A

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

   3. lift-exp.f64 N/A

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

   4. lift-sqrt.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. lift-exp.f64 N/A

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

   7. rem-log-exp N/A

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

   8. pow-exp N/A

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

   9. *-commutative N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. associate-*l* N/A

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

   13. exp-prod N/A

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

   14. lift-sqrt.f64 N/A

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

   15. lift-sqrt.f64 N/A

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

   16. rem-square-sqrt N/A

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

   17. lower-pow.f64 N/A

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

   18. lower-exp.f64 97.9

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

10. Applied egg-rr 97.9%

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

11. Final simplification 97.9%

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

Alternative 4: 96.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.4%

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

4. Applied egg-rr 95.2%

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

   1. exp-prod N/A

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

   2. lower-pow.f64 N/A

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

   3. lower-exp.f64 96.7

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

6. Applied egg-rr 96.7%

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

Alternative 5: 95.2% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.4%

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

4. Applied egg-rr 95.2%

   \[\leadsto \cos x \cdot \color{blue}{{\left(e^{x \cdot x}\right)}^{10}} \]
5. 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(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. lift-*.f64 N/A

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

   2. exp-prod N/A

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

   3. lower-pow.f64 N/A

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

   4. lower-exp.f64 95.2

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

4. Applied egg-rr 95.2%

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

Alternative 7: 94.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 94.4%

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

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. exp-prod N/A

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

   4. *-commutative N/A

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

   5. unpow1 N/A

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

   6. metadata-eval N/A

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

   7. sqr-pow N/A

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

   8. associate-*l* N/A

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

   9. pow-unpow N/A

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

   10. lower-pow.f64 N/A

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

   11. pow-exp N/A

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

   12. lower-exp.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. metadata-eval N/A

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

   15. metadata-eval N/A

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

   16. unpow1/2 N/A

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

   17. lower-sqrt.f64 N/A

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

   18. *-commutative N/A

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

   19. lower-*.f64 N/A

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

   20. metadata-eval N/A

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

   21. metadata-eval N/A

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

4. Applied egg-rr 93.2%

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

   1. lift-cos.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. rem-log-exp N/A

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

   5. lift-exp.f64 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. exp-prod N/A

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

   9. pow-to-exp N/A

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

   10. lift-pow.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 93.2

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

6. Applied egg-rr 93.3%

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

   1. lift-sqrt.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. *-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. associate-*r* N/A

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

   9. *-commutative N/A

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

   10. associate-*r* N/A

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

   11. lower-*.f64 N/A

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

   12. lower-*.f64 93.3

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

   13. lift-*.f64 N/A

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

   14. *-commutative N/A

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

   15. lower-*.f64 93.3

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

8. Applied egg-rr 93.3%

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

10. Applied egg-rr 94.5%

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

11. Final simplification 94.5%

    \[\leadsto \cos x \cdot e^{\mathsf{fma}\left(x \cdot x, 5, x \cdot \left(x \cdot 5\right)\right)} \]
12. Add Preprocessing

Alternative 8: 94.4% 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 9: 27.5% accurate, 1.2× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (*
  (exp (* (sqrt x) (* x (* 10.0 (sqrt 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((sqrt(x) * (x * (10.0 * sqrt(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(sqrt(x) * Float64(x * Float64(10.0 * sqrt(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[(N[Sqrt[x], $MachinePrecision] * N[(x * N[(10.0 * N[Sqrt[x], $MachinePrecision]), $MachinePrecision]), $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. Step-by-step derivation

   1. associate-*r* N/A

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

   2. *-commutative N/A

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

   3. exp-prod N/A

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

   4. *-commutative N/A

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

   5. unpow1 N/A

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

   6. metadata-eval N/A

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

   7. sqr-pow N/A

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

   8. associate-*l* N/A

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

   9. pow-unpow N/A

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

   10. lower-pow.f64 N/A

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

   11. pow-exp N/A

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

   12. lower-exp.f64 N/A

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

   13. lower-*.f64 N/A

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

   14. metadata-eval N/A

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

   15. metadata-eval N/A

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

   16. unpow1/2 N/A

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

   17. lower-sqrt.f64 N/A

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

   18. *-commutative N/A

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

   19. lower-*.f64 N/A

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

   20. metadata-eval N/A

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

   21. metadata-eval N/A

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

4. Applied egg-rr 93.2%

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

   1. lift-cos.f64 N/A

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

   2. lift-sqrt.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. rem-log-exp N/A

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

   5. lift-exp.f64 N/A

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

   6. lift-sqrt.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. exp-prod N/A

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

   9. pow-to-exp N/A

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

   10. lift-pow.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 93.2

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

6. Applied egg-rr 93.3%

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

7. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. lower-fma.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. sub-neg N/A

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

   6. metadata-eval N/A

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

   7. lower-fma.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. +-commutative N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 27.5

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

9. Simplified 27.5%

   \[\leadsto e^{\sqrt{x} \cdot \left(x \cdot \left(10 \cdot \sqrt{x}\right)\right)} \cdot \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, -0.001388888888888889, 0.041666666666666664\right), -0.5\right), 1\right)} \]
10. 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. lower-fma.f64 N/A

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

   3. unpow2 N/A

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

   4. lower-*.f64 N/A

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

   5. sub-neg N/A

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

   6. 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. lower-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. lower-*.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. lower-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. lower-*.f64 27.5

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

5. Simplified 27.5%

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

6. Final simplification 27.5%

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

Alternative 11: 27.5% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(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] * N[Exp[N[(x * N[(10.0 * 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. 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. lower-*.f64 N/A

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

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

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

   9. lower-*.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. lower-cos.f64 94.4

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

5. Simplified 94.4%

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

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. lower-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. lower-*.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. metadata-eval N/A

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

   7. lower-fma.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. +-commutative N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f64 N/A

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

   13. unpow2 N/A

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

   14. lower-*.f64 27.5

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

8. Simplified 27.5%

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

9. Final simplification 27.5%

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

Alternative 12: 21.3% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(N[Exp[N[(x * N[(10.0 * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(N[(x * x), $MachinePrecision] * N[(x * N[(x * 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 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. lower-*.f64 N/A

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

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

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

   9. lower-*.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. lower-cos.f64 94.4

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

5. Simplified 94.4%

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

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. lower-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. lower-*.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. unpow2 N/A

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

   7. associate-*r* N/A

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

   8. *-commutative N/A

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

   9. metadata-eval N/A

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

   10. lower-fma.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-*.f64 21.3

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

8. Simplified 21.3%

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

9. Final simplification 21.3%

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

Alternative 13: 18.2% accurate, 1.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_] := N[(N[Exp[N[(x * N[(10.0 * x), $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. lower-*.f64 N/A

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

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

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

   9. lower-*.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. lower-cos.f64 94.4

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

5. Simplified 94.4%

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

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. lower-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. lower-*.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. Final simplification 18.2%

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

Alternative 14: 9.8% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 94.4%

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

3. 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. lower-fma.f64 N/A

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

   6. lower-*.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. Final simplification 9.8%

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

Alternative 15: 9.7% accurate, 19.6× speedup

Math

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

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(Float64(x * x) * -0.5)
end

MATLAB

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

Wolfram

code[x_] := N[(N[(x * x), $MachinePrecision] * -0.5), $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. lower-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. lower-*.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. *-commutative N/A

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

    2. lower-*.f64 N/A

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

    3. unpow2 N/A

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

    4. lower-*.f64 9.7

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

11. Simplified 9.7%

    \[\leadsto \color{blue}{\left(x \cdot x\right) \cdot -0.5} \]
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 2024207 
(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)))))