ENA, Section 1.4, Exercise 1

Percentage Accurate: 94.4% → 99.3%

Time: 13.0s

Alternatives: 14

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: 99.3% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.3%

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

   1. cos-neg 94.3%

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

   2. *-commutative 94.3%

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

   3. associate-*r* 94.1%

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

   4. exp-prod 94.9%

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

   5. sqr-pow 94.8%

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

   6. sqr-pow 94.9%

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

   7. exp-prod 98.0%

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

   8. cos-neg 98.0%

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

3. Simplified 98.0%

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

   1. add-sqr-sqrt_binary64 98.0%

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

5. Applied rewrite-once 98.0%

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

   1. sqrt-unprod 98.0%

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

   2. pow1/2 98.0%

      \[\leadsto {\color{blue}{\left({\left({\left(e^{10}\right)}^{x} \cdot {\left(e^{10}\right)}^{x}\right)}^{0.5}\right)}}^{x} \cdot \cos x \]

   3. pow-prod-down 98.0%

      \[\leadsto {\left({\color{blue}{\left({\left(e^{10} \cdot e^{10}\right)}^{x}\right)}}^{0.5}\right)}^{x} \cdot \cos x \]

   4. pow-pow 98.0%

      \[\leadsto {\color{blue}{\left({\left(e^{10} \cdot e^{10}\right)}^{\left(x \cdot 0.5\right)}\right)}}^{x} \cdot \cos x \]

   5. prod-exp 99.2%

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

   6. metadata-eval 99.2%

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

7. Applied egg-rr 99.2%

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

   1. sqr-pow 99.1%

      \[\leadsto \color{blue}{\left({\left({\left(e^{20}\right)}^{\left(x \cdot 0.5\right)}\right)}^{\left(\frac{x}{2}\right)} \cdot {\left({\left(e^{20}\right)}^{\left(x \cdot 0.5\right)}\right)}^{\left(\frac{x}{2}\right)}\right)} \cdot \cos x \]

   2. pow-unpow 99.1%

      \[\leadsto \left({\color{blue}{\left({\left({\left(e^{20}\right)}^{x}\right)}^{0.5}\right)}}^{\left(\frac{x}{2}\right)} \cdot {\left({\left(e^{20}\right)}^{\left(x \cdot 0.5\right)}\right)}^{\left(\frac{x}{2}\right)}\right) \cdot \cos x \]

   3. pow-pow 99.2%

      \[\leadsto \left(\color{blue}{{\left({\left(e^{20}\right)}^{x}\right)}^{\left(0.5 \cdot \frac{x}{2}\right)}} \cdot {\left({\left(e^{20}\right)}^{\left(x \cdot 0.5\right)}\right)}^{\left(\frac{x}{2}\right)}\right) \cdot \cos x \]

   4. pow-unpow 99.2%

      \[\leadsto \left({\left({\left(e^{20}\right)}^{x}\right)}^{\left(0.5 \cdot \frac{x}{2}\right)} \cdot {\color{blue}{\left({\left({\left(e^{20}\right)}^{x}\right)}^{0.5}\right)}}^{\left(\frac{x}{2}\right)}\right) \cdot \cos x \]

   5. pow-pow 99.1%

      \[\leadsto \left({\left({\left(e^{20}\right)}^{x}\right)}^{\left(0.5 \cdot \frac{x}{2}\right)} \cdot \color{blue}{{\left({\left(e^{20}\right)}^{x}\right)}^{\left(0.5 \cdot \frac{x}{2}\right)}}\right) \cdot \cos x \]

   6. pow-prod-down 99.4%

      \[\leadsto \color{blue}{{\left({\left(e^{20}\right)}^{x} \cdot {\left(e^{20}\right)}^{x}\right)}^{\left(0.5 \cdot \frac{x}{2}\right)}} \cdot \cos x \]

   7. metadata-eval 99.4%

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

   8. associate-/r/ 96.9%

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

   9. clear-num 99.4%

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

   10. frac-2neg 99.4%

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

   11. distribute-frac-neg 99.4%

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

   12. pow-neg 99.2%

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

   13. pow-prod-down 99.2%

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

   14. prod-exp 99.2%

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

   15. metadata-eval 99.2%

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

   16. frac-2neg 99.2%

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

   17. metadata-eval 99.2%

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

   18. metadata-eval 99.2%

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

9. Applied egg-rr 99.2%

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

10. Taylor expanded in x around inf 94.1%

    \[\leadsto \color{blue}{\frac{\cos x}{e^{-0.25 \cdot \left(x \cdot \log \left({\left(e^{40}\right)}^{x}\right)\right)}}} \]
11. Step-by-step derivation

    1. associate-*r* 94.1%

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

    2. *-commutative 94.1%

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

    3. log-pow 94.6%

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

    4. rem-exp-log 99.3%

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

12. Simplified 99.3%

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

13. Final simplification 99.3%

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

Alternative 2: 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 94.3%

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

   1. cos-neg 94.3%

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

   2. *-commutative 94.3%

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

   3. associate-*r* 94.1%

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

   4. exp-prod 94.9%

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

   5. sqr-pow 94.8%

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

   6. sqr-pow 94.9%

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

   7. exp-prod 98.0%

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

   8. cos-neg 98.0%

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

3. Simplified 98.0%

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

   1. add-sqr-sqrt_binary64 98.0%

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

5. Applied rewrite-once 98.0%

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

   1. sqrt-unprod 98.0%

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

   2. pow1/2 98.0%

      \[\leadsto {\color{blue}{\left({\left({\left(e^{10}\right)}^{x} \cdot {\left(e^{10}\right)}^{x}\right)}^{0.5}\right)}}^{x} \cdot \cos x \]

   3. pow-prod-down 98.0%

      \[\leadsto {\left({\color{blue}{\left({\left(e^{10} \cdot e^{10}\right)}^{x}\right)}}^{0.5}\right)}^{x} \cdot \cos x \]

   4. pow-pow 98.0%

      \[\leadsto {\color{blue}{\left({\left(e^{10} \cdot e^{10}\right)}^{\left(x \cdot 0.5\right)}\right)}}^{x} \cdot \cos x \]

   5. prod-exp 99.2%

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

   6. metadata-eval 99.2%

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

7. Applied egg-rr 99.2%

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

8. Final simplification 99.2%

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

Alternative 3: 99.3% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.3%

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

   1. cos-neg 94.3%

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

   2. *-commutative 94.3%

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

   3. associate-*r* 94.1%

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

   4. exp-prod 94.9%

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

   5. sqr-pow 94.8%

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

   6. sqr-pow 94.9%

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

   7. exp-prod 98.0%

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

   8. cos-neg 98.0%

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

3. Simplified 98.0%

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

   1. add-sqr-sqrt_binary64 98.0%

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

5. Applied rewrite-once 98.0%

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

   1. sqrt-unprod 98.0%

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

   2. pow1/2 98.0%

      \[\leadsto {\color{blue}{\left({\left({\left(e^{10}\right)}^{x} \cdot {\left(e^{10}\right)}^{x}\right)}^{0.5}\right)}}^{x} \cdot \cos x \]

   3. pow-prod-down 98.0%

      \[\leadsto {\left({\color{blue}{\left({\left(e^{10} \cdot e^{10}\right)}^{x}\right)}}^{0.5}\right)}^{x} \cdot \cos x \]

   4. pow-pow 98.0%

      \[\leadsto {\color{blue}{\left({\left(e^{10} \cdot e^{10}\right)}^{\left(x \cdot 0.5\right)}\right)}}^{x} \cdot \cos x \]

   5. prod-exp 99.2%

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

   6. metadata-eval 99.2%

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

7. Applied egg-rr 99.2%

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

   1. sqr-pow 99.1%

      \[\leadsto \color{blue}{\left({\left({\left(e^{20}\right)}^{\left(x \cdot 0.5\right)}\right)}^{\left(\frac{x}{2}\right)} \cdot {\left({\left(e^{20}\right)}^{\left(x \cdot 0.5\right)}\right)}^{\left(\frac{x}{2}\right)}\right)} \cdot \cos x \]

   2. pow-unpow 99.1%

      \[\leadsto \left({\color{blue}{\left({\left({\left(e^{20}\right)}^{x}\right)}^{0.5}\right)}}^{\left(\frac{x}{2}\right)} \cdot {\left({\left(e^{20}\right)}^{\left(x \cdot 0.5\right)}\right)}^{\left(\frac{x}{2}\right)}\right) \cdot \cos x \]

   3. pow-pow 99.2%

      \[\leadsto \left(\color{blue}{{\left({\left(e^{20}\right)}^{x}\right)}^{\left(0.5 \cdot \frac{x}{2}\right)}} \cdot {\left({\left(e^{20}\right)}^{\left(x \cdot 0.5\right)}\right)}^{\left(\frac{x}{2}\right)}\right) \cdot \cos x \]

   4. pow-unpow 99.2%

      \[\leadsto \left({\left({\left(e^{20}\right)}^{x}\right)}^{\left(0.5 \cdot \frac{x}{2}\right)} \cdot {\color{blue}{\left({\left({\left(e^{20}\right)}^{x}\right)}^{0.5}\right)}}^{\left(\frac{x}{2}\right)}\right) \cdot \cos x \]

   5. pow-pow 99.1%

      \[\leadsto \left({\left({\left(e^{20}\right)}^{x}\right)}^{\left(0.5 \cdot \frac{x}{2}\right)} \cdot \color{blue}{{\left({\left(e^{20}\right)}^{x}\right)}^{\left(0.5 \cdot \frac{x}{2}\right)}}\right) \cdot \cos x \]

   6. pow-prod-down 99.4%

      \[\leadsto \color{blue}{{\left({\left(e^{20}\right)}^{x} \cdot {\left(e^{20}\right)}^{x}\right)}^{\left(0.5 \cdot \frac{x}{2}\right)}} \cdot \cos x \]

   7. *-commutative 99.4%

      \[\leadsto {\left({\left(e^{20}\right)}^{x} \cdot {\left(e^{20}\right)}^{x}\right)}^{\color{blue}{\left(\frac{x}{2} \cdot 0.5\right)}} \cdot \cos x \]

   8. associate-*l/ 99.4%

      \[\leadsto {\left({\left(e^{20}\right)}^{x} \cdot {\left(e^{20}\right)}^{x}\right)}^{\color{blue}{\left(\frac{x \cdot 0.5}{2}\right)}} \cdot \cos x \]

   9. pow-to-exp 94.1%

      \[\leadsto \color{blue}{e^{\log \left({\left(e^{20}\right)}^{x} \cdot {\left(e^{20}\right)}^{x}\right) \cdot \frac{x \cdot 0.5}{2}}} \cdot \cos x \]

   10. pow-prod-down 94.1%

       \[\leadsto e^{\log \color{blue}{\left({\left(e^{20} \cdot e^{20}\right)}^{x}\right)} \cdot \frac{x \cdot 0.5}{2}} \cdot \cos x \]

   11. prod-exp 94.1%

       \[\leadsto e^{\log \left({\color{blue}{\left(e^{20 + 20}\right)}}^{x}\right) \cdot \frac{x \cdot 0.5}{2}} \cdot \cos x \]

   12. metadata-eval 94.1%

       \[\leadsto e^{\log \left({\left(e^{\color{blue}{40}}\right)}^{x}\right) \cdot \frac{x \cdot 0.5}{2}} \cdot \cos x \]

   13. associate-/l* 94.1%

       \[\leadsto e^{\log \left({\left(e^{40}\right)}^{x}\right) \cdot \color{blue}{\frac{x}{\frac{2}{0.5}}}} \cdot \cos x \]

   14. div-inv 94.1%

       \[\leadsto e^{\log \left({\left(e^{40}\right)}^{x}\right) \cdot \color{blue}{\left(x \cdot \frac{1}{\frac{2}{0.5}}\right)}} \cdot \cos x \]

   15. metadata-eval 94.1%

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

   16. metadata-eval 94.1%

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

9. Applied egg-rr 94.1%

   \[\leadsto \color{blue}{e^{\log \left({\left(e^{40}\right)}^{x}\right) \cdot \left(x \cdot 0.25\right)}} \cdot \cos x \]
10. Step-by-step derivation

    1. exp-to-pow 99.2%

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

11. Simplified 99.2%

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

12. Final simplification 99.2%

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

Alternative 4: 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 94.3%

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

   1. cos-neg 94.3%

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

   2. *-commutative 94.3%

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

   3. associate-*r* 94.1%

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

   4. exp-prod 94.9%

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

   5. sqr-pow 94.8%

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

   6. sqr-pow 94.9%

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

   7. exp-prod 98.0%

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

   8. cos-neg 98.0%

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

3. Simplified 98.0%

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

4. Final simplification 98.0%

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

Alternative 5: 95.3% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x)
	return Float64(cos(x) * (exp(5.0) ^ Float64(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[Exp[5.0], $MachinePrecision], N[(x * N[(x + x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.3%

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

   1. cos-neg 94.3%

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

   2. *-commutative 94.3%

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

   3. associate-*r* 94.1%

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

   4. exp-prod 94.9%

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

   5. sqr-pow 94.8%

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

   6. sqr-pow 94.9%

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

   7. exp-prod 98.0%

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

   8. cos-neg 98.0%

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

3. Simplified 98.0%

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

   1. add-sqr-sqrt_binary64 98.0%

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

5. Applied rewrite-once 98.0%

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

   1. rem-square-sqrt 98.0%

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

   2. pow-pow 95.4%

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

   3. exp-prod 94.3%

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

   4. *-commutative 94.3%

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

   5. pow-exp 95.4%

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

   6. sqr-pow 95.4%

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

   7. pow-prod-down 95.4%

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

   8. exp-prod 96.4%

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

   9. exp-prod 96.7%

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

   10. pow-prod-up 96.8%

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

   11. metadata-eval 96.8%

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

7. Applied egg-rr 96.8%

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

8. Taylor expanded in x around inf 95.5%

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

   1. exp-prod 94.3%

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

   2. unpow2 94.3%

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

   3. associate-*r* 94.3%

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

   4. count-2 94.3%

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

   5. rem-log-exp 94.3%

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

   6. log-pow 94.3%

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

   7. *-commutative 94.3%

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

   8. exp-prod 95.3%

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

   9. log-pow 95.4%

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

   10. rem-log-exp 95.4%

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

   11. *-commutative 95.4%

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

10. Simplified 95.4%

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

11. Final simplification 95.4%

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

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 94.3%

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

   1. cos-neg 94.3%

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

   2. *-commutative 94.3%

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

   3. sqr-neg 94.3%

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

   4. exp-prod 95.4%

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

   5. sqr-pow 95.4%

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

   6. sqr-pow 95.4%

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

   7. sqr-neg 95.4%

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

   8. cos-neg 95.4%

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

3. Simplified 95.4%

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

4. Final simplification 95.4%

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

Alternative 7: 95.3% 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 94.3%

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

   1. cos-neg 94.3%

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

   2. *-commutative 94.3%

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

   3. sqr-neg 94.3%

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

   4. *-commutative 94.3%

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

   5. exp-prod 95.4%

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

   6. sqr-neg 95.4%

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

   7. cos-neg 95.4%

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

3. Simplified 95.4%

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

4. Final simplification 95.4%

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

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 94.3%

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

2. Final simplification 94.3%

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

Alternative 9: 9.8% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ -20 - \cos -20 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- -20.0 (cos -20.0)))

C

double code(double x) {
	return -20.0 - cos(-20.0);
}

Fortran

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

Java

public static double code(double x) {
	return -20.0 - Math.cos(-20.0);
}

Python

def code(x):
	return -20.0 - math.cos(-20.0)

Julia

function code(x)
	return Float64(-20.0 - cos(-20.0))
end

MATLAB

function tmp = code(x)
	tmp = -20.0 - cos(-20.0);
end

Wolfram

code[x_] := N[(-20.0 - N[Cos[-20.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.3%

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

   1. cos-neg 94.3%

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

   2. *-commutative 94.3%

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

   3. associate-*r* 94.1%

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

   4. exp-prod 94.9%

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

   5. sqr-pow 94.8%

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

   6. sqr-pow 94.9%

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

   7. exp-prod 98.0%

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

   8. cos-neg 98.0%

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

3. Simplified 98.0%

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

   1. add-sqr-sqrt_binary64 98.0%

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

5. Applied rewrite-once 98.0%

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

   1. sqrt-unprod 98.0%

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

   2. pow1/2 98.0%

      \[\leadsto {\color{blue}{\left({\left({\left(e^{10}\right)}^{x} \cdot {\left(e^{10}\right)}^{x}\right)}^{0.5}\right)}}^{x} \cdot \cos x \]

   3. pow-prod-down 98.0%

      \[\leadsto {\left({\color{blue}{\left({\left(e^{10} \cdot e^{10}\right)}^{x}\right)}}^{0.5}\right)}^{x} \cdot \cos x \]

   4. pow-pow 98.0%

      \[\leadsto {\color{blue}{\left({\left(e^{10} \cdot e^{10}\right)}^{\left(x \cdot 0.5\right)}\right)}}^{x} \cdot \cos x \]

   5. prod-exp 99.2%

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

   6. metadata-eval 99.2%

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

7. Applied egg-rr 99.2%

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

   1. sqr-pow 99.1%

      \[\leadsto \color{blue}{\left({\left({\left(e^{20}\right)}^{\left(x \cdot 0.5\right)}\right)}^{\left(\frac{x}{2}\right)} \cdot {\left({\left(e^{20}\right)}^{\left(x \cdot 0.5\right)}\right)}^{\left(\frac{x}{2}\right)}\right)} \cdot \cos x \]

   2. pow-unpow 99.1%

      \[\leadsto \left({\color{blue}{\left({\left({\left(e^{20}\right)}^{x}\right)}^{0.5}\right)}}^{\left(\frac{x}{2}\right)} \cdot {\left({\left(e^{20}\right)}^{\left(x \cdot 0.5\right)}\right)}^{\left(\frac{x}{2}\right)}\right) \cdot \cos x \]

   3. pow-pow 99.2%

      \[\leadsto \left(\color{blue}{{\left({\left(e^{20}\right)}^{x}\right)}^{\left(0.5 \cdot \frac{x}{2}\right)}} \cdot {\left({\left(e^{20}\right)}^{\left(x \cdot 0.5\right)}\right)}^{\left(\frac{x}{2}\right)}\right) \cdot \cos x \]

   4. pow-unpow 99.2%

      \[\leadsto \left({\left({\left(e^{20}\right)}^{x}\right)}^{\left(0.5 \cdot \frac{x}{2}\right)} \cdot {\color{blue}{\left({\left({\left(e^{20}\right)}^{x}\right)}^{0.5}\right)}}^{\left(\frac{x}{2}\right)}\right) \cdot \cos x \]

   5. pow-pow 99.1%

      \[\leadsto \left({\left({\left(e^{20}\right)}^{x}\right)}^{\left(0.5 \cdot \frac{x}{2}\right)} \cdot \color{blue}{{\left({\left(e^{20}\right)}^{x}\right)}^{\left(0.5 \cdot \frac{x}{2}\right)}}\right) \cdot \cos x \]

   6. pow-prod-down 99.4%

      \[\leadsto \color{blue}{{\left({\left(e^{20}\right)}^{x} \cdot {\left(e^{20}\right)}^{x}\right)}^{\left(0.5 \cdot \frac{x}{2}\right)}} \cdot \cos x \]

   7. metadata-eval 99.4%

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

   8. associate-/r/ 96.9%

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

   9. clear-num 99.4%

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

   10. frac-2neg 99.4%

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

   11. distribute-frac-neg 99.4%

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

   12. pow-neg 99.2%

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

   13. pow-prod-down 99.2%

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

   14. prod-exp 99.2%

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

   15. metadata-eval 99.2%

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

   16. frac-2neg 99.2%

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

   17. metadata-eval 99.2%

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

   18. metadata-eval 99.2%

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

9. Applied egg-rr 99.2%

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

11. Applied egg-rr 9.8%

    \[\leadsto \color{blue}{-20 - \cos -20} \]

12. Final simplification 9.8%

    \[\leadsto -20 - \cos -20 \]

Alternative 10: 9.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ -\cos -20 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (- (cos -20.0)))

C

double code(double x) {
	return -cos(-20.0);
}

Fortran

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

Java

public static double code(double x) {
	return -Math.cos(-20.0);
}

Python

def code(x):
	return -math.cos(-20.0)

Julia

function code(x)
	return Float64(-cos(-20.0))
end

MATLAB

function tmp = code(x)
	tmp = -cos(-20.0);
end

Wolfram

code[x_] := (-N[Cos[-20.0], $MachinePrecision])

Derivation

1. Initial program 94.3%

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

   1. cos-neg 94.3%

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

   2. *-commutative 94.3%

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

   3. associate-*r* 94.1%

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

   4. exp-prod 94.9%

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

   5. sqr-pow 94.8%

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

   6. sqr-pow 94.9%

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

   7. exp-prod 98.0%

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

   8. cos-neg 98.0%

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

3. Simplified 98.0%

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

   1. add-sqr-sqrt_binary64 98.0%

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

5. Applied rewrite-once 98.0%

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

   1. sqrt-unprod 98.0%

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

   2. pow1/2 98.0%

      \[\leadsto {\color{blue}{\left({\left({\left(e^{10}\right)}^{x} \cdot {\left(e^{10}\right)}^{x}\right)}^{0.5}\right)}}^{x} \cdot \cos x \]

   3. pow-prod-down 98.0%

      \[\leadsto {\left({\color{blue}{\left({\left(e^{10} \cdot e^{10}\right)}^{x}\right)}}^{0.5}\right)}^{x} \cdot \cos x \]

   4. pow-pow 98.0%

      \[\leadsto {\color{blue}{\left({\left(e^{10} \cdot e^{10}\right)}^{\left(x \cdot 0.5\right)}\right)}}^{x} \cdot \cos x \]

   5. prod-exp 99.2%

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

   6. metadata-eval 99.2%

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

7. Applied egg-rr 99.2%

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

   1. sqr-pow 99.1%

      \[\leadsto \color{blue}{\left({\left({\left(e^{20}\right)}^{\left(x \cdot 0.5\right)}\right)}^{\left(\frac{x}{2}\right)} \cdot {\left({\left(e^{20}\right)}^{\left(x \cdot 0.5\right)}\right)}^{\left(\frac{x}{2}\right)}\right)} \cdot \cos x \]

   2. pow-unpow 99.1%

      \[\leadsto \left({\color{blue}{\left({\left({\left(e^{20}\right)}^{x}\right)}^{0.5}\right)}}^{\left(\frac{x}{2}\right)} \cdot {\left({\left(e^{20}\right)}^{\left(x \cdot 0.5\right)}\right)}^{\left(\frac{x}{2}\right)}\right) \cdot \cos x \]

   3. pow-pow 99.2%

      \[\leadsto \left(\color{blue}{{\left({\left(e^{20}\right)}^{x}\right)}^{\left(0.5 \cdot \frac{x}{2}\right)}} \cdot {\left({\left(e^{20}\right)}^{\left(x \cdot 0.5\right)}\right)}^{\left(\frac{x}{2}\right)}\right) \cdot \cos x \]

   4. pow-unpow 99.2%

      \[\leadsto \left({\left({\left(e^{20}\right)}^{x}\right)}^{\left(0.5 \cdot \frac{x}{2}\right)} \cdot {\color{blue}{\left({\left({\left(e^{20}\right)}^{x}\right)}^{0.5}\right)}}^{\left(\frac{x}{2}\right)}\right) \cdot \cos x \]

   5. pow-pow 99.1%

      \[\leadsto \left({\left({\left(e^{20}\right)}^{x}\right)}^{\left(0.5 \cdot \frac{x}{2}\right)} \cdot \color{blue}{{\left({\left(e^{20}\right)}^{x}\right)}^{\left(0.5 \cdot \frac{x}{2}\right)}}\right) \cdot \cos x \]

   6. pow-prod-down 99.4%

      \[\leadsto \color{blue}{{\left({\left(e^{20}\right)}^{x} \cdot {\left(e^{20}\right)}^{x}\right)}^{\left(0.5 \cdot \frac{x}{2}\right)}} \cdot \cos x \]

   7. metadata-eval 99.4%

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

   8. associate-/r/ 96.9%

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

   9. clear-num 99.4%

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

   10. frac-2neg 99.4%

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

   11. distribute-frac-neg 99.4%

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

   12. pow-neg 99.2%

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

   13. pow-prod-down 99.2%

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

   14. prod-exp 99.2%

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

   15. metadata-eval 99.2%

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

   16. frac-2neg 99.2%

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

   17. metadata-eval 99.2%

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

   18. metadata-eval 99.2%

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

9. Applied egg-rr 99.2%

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

11. Applied egg-rr 9.6%

    \[\leadsto \color{blue}{\frac{-20}{-\frac{-20}{\cos -20}}} \]
12. Step-by-step derivation

    1. distribute-neg-frac 9.6%

       \[\leadsto \frac{-20}{\color{blue}{\frac{--20}{\cos -20}}} \]

    2. metadata-eval 9.6%

       \[\leadsto \frac{-20}{\frac{\color{blue}{20}}{\cos -20}} \]

    3. associate-/r/ 9.6%

       \[\leadsto \color{blue}{\frac{-20}{20} \cdot \cos -20} \]

    4. metadata-eval 9.6%

       \[\leadsto \color{blue}{-1} \cdot \cos -20 \]

    5. neg-mul-1 9.6%

       \[\leadsto \color{blue}{-\cos -20} \]

13. Simplified 9.6%

    \[\leadsto \color{blue}{-\cos -20} \]

14. Final simplification 9.6%

    \[\leadsto -\cos -20 \]

Alternative 11: 1.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \cos -20 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (cos -20.0))

C

double code(double x) {
	return cos(-20.0);
}

Fortran

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

Java

public static double code(double x) {
	return Math.cos(-20.0);
}

Python

def code(x):
	return math.cos(-20.0)

Julia

function code(x)
	return cos(-20.0)
end

MATLAB

function tmp = code(x)
	tmp = cos(-20.0);
end

Wolfram

code[x_] := N[Cos[-20.0], $MachinePrecision]

Derivation

1. Initial program 94.3%

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

   1. cos-neg 94.3%

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

   2. *-commutative 94.3%

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

   3. associate-*r* 94.1%

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

   4. exp-prod 94.9%

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

   5. sqr-pow 94.8%

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

   6. sqr-pow 94.9%

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

   7. exp-prod 98.0%

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

   8. cos-neg 98.0%

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

3. Simplified 98.0%

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

   1. add-sqr-sqrt_binary64 98.0%

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

5. Applied rewrite-once 98.0%

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

   1. sqrt-unprod 98.0%

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

   2. pow1/2 98.0%

      \[\leadsto {\color{blue}{\left({\left({\left(e^{10}\right)}^{x} \cdot {\left(e^{10}\right)}^{x}\right)}^{0.5}\right)}}^{x} \cdot \cos x \]

   3. pow-prod-down 98.0%

      \[\leadsto {\left({\color{blue}{\left({\left(e^{10} \cdot e^{10}\right)}^{x}\right)}}^{0.5}\right)}^{x} \cdot \cos x \]

   4. pow-pow 98.0%

      \[\leadsto {\color{blue}{\left({\left(e^{10} \cdot e^{10}\right)}^{\left(x \cdot 0.5\right)}\right)}}^{x} \cdot \cos x \]

   5. prod-exp 99.2%

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

   6. metadata-eval 99.2%

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

7. Applied egg-rr 99.2%

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

   1. sqr-pow 99.1%

      \[\leadsto \color{blue}{\left({\left({\left(e^{20}\right)}^{\left(x \cdot 0.5\right)}\right)}^{\left(\frac{x}{2}\right)} \cdot {\left({\left(e^{20}\right)}^{\left(x \cdot 0.5\right)}\right)}^{\left(\frac{x}{2}\right)}\right)} \cdot \cos x \]

   2. pow-unpow 99.1%

      \[\leadsto \left({\color{blue}{\left({\left({\left(e^{20}\right)}^{x}\right)}^{0.5}\right)}}^{\left(\frac{x}{2}\right)} \cdot {\left({\left(e^{20}\right)}^{\left(x \cdot 0.5\right)}\right)}^{\left(\frac{x}{2}\right)}\right) \cdot \cos x \]

   3. pow-pow 99.2%

      \[\leadsto \left(\color{blue}{{\left({\left(e^{20}\right)}^{x}\right)}^{\left(0.5 \cdot \frac{x}{2}\right)}} \cdot {\left({\left(e^{20}\right)}^{\left(x \cdot 0.5\right)}\right)}^{\left(\frac{x}{2}\right)}\right) \cdot \cos x \]

   4. pow-unpow 99.2%

      \[\leadsto \left({\left({\left(e^{20}\right)}^{x}\right)}^{\left(0.5 \cdot \frac{x}{2}\right)} \cdot {\color{blue}{\left({\left({\left(e^{20}\right)}^{x}\right)}^{0.5}\right)}}^{\left(\frac{x}{2}\right)}\right) \cdot \cos x \]

   5. pow-pow 99.1%

      \[\leadsto \left({\left({\left(e^{20}\right)}^{x}\right)}^{\left(0.5 \cdot \frac{x}{2}\right)} \cdot \color{blue}{{\left({\left(e^{20}\right)}^{x}\right)}^{\left(0.5 \cdot \frac{x}{2}\right)}}\right) \cdot \cos x \]

   6. pow-prod-down 99.4%

      \[\leadsto \color{blue}{{\left({\left(e^{20}\right)}^{x} \cdot {\left(e^{20}\right)}^{x}\right)}^{\left(0.5 \cdot \frac{x}{2}\right)}} \cdot \cos x \]

   7. metadata-eval 99.4%

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

   8. associate-/r/ 96.9%

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

   9. clear-num 99.4%

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

   10. frac-2neg 99.4%

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

   11. distribute-frac-neg 99.4%

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

   12. pow-neg 99.2%

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

   13. pow-prod-down 99.2%

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

   14. prod-exp 99.2%

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

   15. metadata-eval 99.2%

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

   16. frac-2neg 99.2%

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

   17. metadata-eval 99.2%

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

   18. metadata-eval 99.2%

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

9. Applied egg-rr 99.2%

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

11. Applied egg-rr 1.5%

    \[\leadsto \color{blue}{\frac{-20}{\frac{-20}{\cos -20}}} \]
12. Step-by-step derivation

    1. associate-/r/ 1.5%

       \[\leadsto \color{blue}{\frac{-20}{-20} \cdot \cos -20} \]

    2. metadata-eval 1.5%

       \[\leadsto \color{blue}{1} \cdot \cos -20 \]

    3. *-lft-identity 1.5%

       \[\leadsto \color{blue}{\cos -20} \]

13. Simplified 1.5%

    \[\leadsto \color{blue}{\cos -20} \]

14. Final simplification 1.5%

    \[\leadsto \cos -20 \]

Alternative 12: 9.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \cos x \end{array} \]

FPCore

(FPCore (x) :precision binary64 (cos x))

C

double code(double x) {
	return cos(x);
}

Fortran

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

Java

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

Python

def code(x):
	return math.cos(x)

Julia

function code(x)
	return cos(x)
end

MATLAB

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

Wolfram

code[x_] := N[Cos[x], $MachinePrecision]

Derivation

1. Initial program 94.3%

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

2. Taylor expanded in x around 0 9.6%

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

3. Final simplification 9.6%

   \[\leadsto \cos x \]

Alternative 13: 1.5% accurate, 18.8× speedup

Math

\[\begin{array}{l} \\ 1 + \left(\left(\left(x \cdot x\right) \cdot 0.6666666666666666\right) \cdot 9.5 + 3.1666666666666665\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (+ 1.0 (+ (* (* (* x x) 0.6666666666666666) 9.5) 3.1666666666666665)))

C

double code(double x) {
	return 1.0 + ((((x * x) * 0.6666666666666666) * 9.5) + 3.1666666666666665);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 + ((((x * x) * 0.6666666666666666d0) * 9.5d0) + 3.1666666666666665d0)
end function

Java

public static double code(double x) {
	return 1.0 + ((((x * x) * 0.6666666666666666) * 9.5) + 3.1666666666666665);
}

Python

def code(x):
	return 1.0 + ((((x * x) * 0.6666666666666666) * 9.5) + 3.1666666666666665)

Julia

function code(x)
	return Float64(1.0 + Float64(Float64(Float64(Float64(x * x) * 0.6666666666666666) * 9.5) + 3.1666666666666665))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 + ((((x * x) * 0.6666666666666666) * 9.5) + 3.1666666666666665);
end

Wolfram

code[x_] := N[(1.0 + N[(N[(N[(N[(x * x), $MachinePrecision] * 0.6666666666666666), $MachinePrecision] * 9.5), $MachinePrecision] + 3.1666666666666665), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.3%

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

   1. cos-neg 94.3%

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

   2. *-commutative 94.3%

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

   3. associate-*r* 94.1%

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

   4. exp-prod 94.9%

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

   5. sqr-pow 94.8%

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

   6. sqr-pow 94.9%

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

   7. exp-prod 98.0%

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

   8. cos-neg 98.0%

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

3. Simplified 98.0%

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

   1. add-sqr-sqrt_binary64 98.0%

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

5. Applied rewrite-once 98.0%

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

6. Taylor expanded in x around 0 1.5%

   \[\leadsto \color{blue}{1 + 9.5 \cdot {x}^{2}} \]
7. Step-by-step derivation

   1. *-commutative 1.5%

      \[\leadsto 1 + \color{blue}{{x}^{2} \cdot 9.5} \]

   2. unpow2 1.5%

      \[\leadsto 1 + \color{blue}{\left(x \cdot x\right)} \cdot 9.5 \]

   3. associate-*l* 1.5%

      \[\leadsto 1 + \color{blue}{x \cdot \left(x \cdot 9.5\right)} \]

8. Simplified 1.5%

   \[\leadsto \color{blue}{1 + x \cdot \left(x \cdot 9.5\right)} \]
9. Step-by-step derivation

   1. associate-*r* 1.5%

      \[\leadsto 1 + \color{blue}{\left(x \cdot x\right) \cdot 9.5} \]

   2. rem-log-exp 1.5%

      \[\leadsto 1 + \color{blue}{\log \left(e^{x \cdot x}\right)} \cdot 9.5 \]

   3. rem-3cbrt-lft 1.5%

      \[\leadsto 1 + \log \color{blue}{\left(\left(\sqrt[3]{e^{x \cdot x}} \cdot \sqrt[3]{e^{x \cdot x}}\right) \cdot \sqrt[3]{e^{x \cdot x}}\right)} \cdot 9.5 \]

   4. *-commutative 1.5%

      \[\leadsto 1 + \color{blue}{9.5 \cdot \log \left(\left(\sqrt[3]{e^{x \cdot x}} \cdot \sqrt[3]{e^{x \cdot x}}\right) \cdot \sqrt[3]{e^{x \cdot x}}\right)} \]

   5. log-prod 1.5%

      \[\leadsto 1 + 9.5 \cdot \color{blue}{\left(\log \left(\sqrt[3]{e^{x \cdot x}} \cdot \sqrt[3]{e^{x \cdot x}}\right) + \log \left(\sqrt[3]{e^{x \cdot x}}\right)\right)} \]

   6. distribute-rgt-in 1.5%

      \[\leadsto 1 + \color{blue}{\left(\log \left(\sqrt[3]{e^{x \cdot x}} \cdot \sqrt[3]{e^{x \cdot x}}\right) \cdot 9.5 + \log \left(\sqrt[3]{e^{x \cdot x}}\right) \cdot 9.5\right)} \]

10. Applied egg-rr 1.5%

    \[\leadsto 1 + \color{blue}{\left(\left(\left(x \cdot x\right) \cdot 0.6666666666666666\right) \cdot 9.5 + \left(x \cdot \left(x \cdot 0.3333333333333333\right)\right) \cdot 9.5\right)} \]

12. Applied egg-rr 1.5%

    \[\leadsto 1 + \left(\left(\left(x \cdot x\right) \cdot 0.6666666666666666\right) \cdot 9.5 + \color{blue}{3.1666666666666665}\right) \]

13. Final simplification 1.5%

    \[\leadsto 1 + \left(\left(\left(x \cdot x\right) \cdot 0.6666666666666666\right) \cdot 9.5 + 3.1666666666666665\right) \]

Alternative 14: 1.5% accurate, 29.6× speedup

Math

\[\begin{array}{l} \\ 1 + x \cdot \left(x \cdot 9.5\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (+ 1.0 (* x (* x 9.5))))

C

double code(double x) {
	return 1.0 + (x * (x * 9.5));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 + (x * (x * 9.5d0))
end function

Java

public static double code(double x) {
	return 1.0 + (x * (x * 9.5));
}

Python

def code(x):
	return 1.0 + (x * (x * 9.5))

Julia

function code(x)
	return Float64(1.0 + Float64(x * Float64(x * 9.5)))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 + (x * (x * 9.5));
end

Wolfram

code[x_] := N[(1.0 + N[(x * N[(x * 9.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.3%

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

   1. cos-neg 94.3%

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

   2. *-commutative 94.3%

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

   3. associate-*r* 94.1%

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

   4. exp-prod 94.9%

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

   5. sqr-pow 94.8%

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

   6. sqr-pow 94.9%

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

   7. exp-prod 98.0%

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

   8. cos-neg 98.0%

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

3. Simplified 98.0%

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

   1. add-sqr-sqrt_binary64 98.0%

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

5. Applied rewrite-once 98.0%

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

6. Taylor expanded in x around 0 1.5%

   \[\leadsto \color{blue}{1 + 9.5 \cdot {x}^{2}} \]
7. Step-by-step derivation

   1. *-commutative 1.5%

      \[\leadsto 1 + \color{blue}{{x}^{2} \cdot 9.5} \]

   2. unpow2 1.5%

      \[\leadsto 1 + \color{blue}{\left(x \cdot x\right)} \cdot 9.5 \]

   3. associate-*l* 1.5%

      \[\leadsto 1 + \color{blue}{x \cdot \left(x \cdot 9.5\right)} \]

8. Simplified 1.5%

   \[\leadsto \color{blue}{1 + x \cdot \left(x \cdot 9.5\right)} \]

9. Final simplification 1.5%

   \[\leadsto 1 + x \cdot \left(x \cdot 9.5\right) \]

Reproduce

herbie shell --seed 2023297 
(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)))))