ENA, Section 1.4, Exercise 1

Percentage Accurate: 94.4% → 99.3%

Time: 8.7s

Alternatives: 9

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, 0.3× speedup

Math

\[\begin{array}{l} \\ \cos x \cdot \sqrt{\sqrt[3]{{\left({\left(e^{60}\right)}^{x}\right)}^{x}}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (cos x) (sqrt (cbrt (pow (pow (exp 60.0) x) x)))))

C

double code(double x) {
	return cos(x) * sqrt(cbrt(pow(pow(exp(60.0), x), x)));
}

Java

public static double code(double x) {
	return Math.cos(x) * Math.sqrt(Math.cbrt(Math.pow(Math.pow(Math.exp(60.0), x), x)));
}

Julia

function code(x)
	return Float64(cos(x) * sqrt(cbrt(((exp(60.0) ^ x) ^ x))))
end

Wolfram

code[x_] := N[(N[Cos[x], $MachinePrecision] * N[Sqrt[N[Power[N[Power[N[Power[N[Exp[60.0], $MachinePrecision], x], $MachinePrecision], x], $MachinePrecision], 1/3], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.6%

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

   1. associate-*r* 94.6%

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

   2. add-log-exp 94.6%

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

   3. log-pow 94.5%

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

   4. pow-pow 94.9%

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

   5. add-exp-log 96.8%

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

   6. add-cbrt-cube 96.8%

      \[\leadsto \cos x \cdot \color{blue}{\sqrt[3]{\left({\left({\left(e^{x}\right)}^{10}\right)}^{x} \cdot {\left({\left(e^{x}\right)}^{10}\right)}^{x}\right) \cdot {\left({\left(e^{x}\right)}^{10}\right)}^{x}}} \]

   7. add-exp-log 96.5%

      \[\leadsto \cos x \cdot \sqrt[3]{\left({\left({\left(e^{x}\right)}^{10}\right)}^{x} \cdot {\left({\left(e^{x}\right)}^{10}\right)}^{x}\right) \cdot \color{blue}{e^{\log \left({\left({\left(e^{x}\right)}^{10}\right)}^{x}\right)}}} \]

   8. pow-pow 96.2%

      \[\leadsto \cos x \cdot \sqrt[3]{\left({\left({\left(e^{x}\right)}^{10}\right)}^{x} \cdot {\left({\left(e^{x}\right)}^{10}\right)}^{x}\right) \cdot e^{\log \color{blue}{\left({\left(e^{x}\right)}^{\left(10 \cdot x\right)}\right)}}} \]

   9. log-pow 96.4%

      \[\leadsto \cos x \cdot \sqrt[3]{\left({\left({\left(e^{x}\right)}^{10}\right)}^{x} \cdot {\left({\left(e^{x}\right)}^{10}\right)}^{x}\right) \cdot e^{\color{blue}{\left(10 \cdot x\right) \cdot \log \left(e^{x}\right)}}} \]

   10. add-log-exp 96.4%

       \[\leadsto \cos x \cdot \sqrt[3]{\left({\left({\left(e^{x}\right)}^{10}\right)}^{x} \cdot {\left({\left(e^{x}\right)}^{10}\right)}^{x}\right) \cdot e^{\left(10 \cdot x\right) \cdot \color{blue}{x}}} \]

   11. associate-*r* 96.4%

       \[\leadsto \cos x \cdot \sqrt[3]{\left({\left({\left(e^{x}\right)}^{10}\right)}^{x} \cdot {\left({\left(e^{x}\right)}^{10}\right)}^{x}\right) \cdot e^{\color{blue}{10 \cdot \left(x \cdot x\right)}}} \]

   12. pow-exp 96.8%

       \[\leadsto \cos x \cdot \sqrt[3]{\left({\left({\left(e^{x}\right)}^{10}\right)}^{x} \cdot {\left({\left(e^{x}\right)}^{10}\right)}^{x}\right) \cdot \color{blue}{{\left(e^{10}\right)}^{\left(x \cdot x\right)}}} \]

   13. *-commutative 96.8%

       \[\leadsto \cos x \cdot \sqrt[3]{\color{blue}{{\left(e^{10}\right)}^{\left(x \cdot x\right)} \cdot \left({\left({\left(e^{x}\right)}^{10}\right)}^{x} \cdot {\left({\left(e^{x}\right)}^{10}\right)}^{x}\right)}} \]

3. Applied egg-rr 98.8%

   \[\leadsto \cos x \cdot \color{blue}{\sqrt[3]{{\left({\left(e^{30}\right)}^{x}\right)}^{x}}} \]
4. Step-by-step derivation

   1. add-sqr-sqrt 98.7%

      \[\leadsto \cos x \cdot \sqrt[3]{{\color{blue}{\left(\sqrt{{\left(e^{30}\right)}^{x}} \cdot \sqrt{{\left(e^{30}\right)}^{x}}\right)}}^{x}} \]

   2. sqrt-unprod 98.8%

      \[\leadsto \cos x \cdot \sqrt[3]{{\color{blue}{\left(\sqrt{{\left(e^{30}\right)}^{x} \cdot {\left(e^{30}\right)}^{x}}\right)}}^{x}} \]

   3. pow-prod-down 98.8%

      \[\leadsto \cos x \cdot \sqrt[3]{{\left(\sqrt{\color{blue}{{\left(e^{30} \cdot e^{30}\right)}^{x}}}\right)}^{x}} \]

   4. prod-exp 99.1%

      \[\leadsto \cos x \cdot \sqrt[3]{{\left(\sqrt{{\color{blue}{\left(e^{30 + 30}\right)}}^{x}}\right)}^{x}} \]

   5. metadata-eval 99.1%

      \[\leadsto \cos x \cdot \sqrt[3]{{\left(\sqrt{{\left(e^{\color{blue}{60}}\right)}^{x}}\right)}^{x}} \]

5. Applied egg-rr 99.1%

   \[\leadsto \cos x \cdot \sqrt[3]{{\color{blue}{\left(\sqrt{{\left(e^{60}\right)}^{x}}\right)}}^{x}} \]
6. Step-by-step derivation

   1. add-sqr-sqrt 98.9%

      \[\leadsto \cos x \cdot \color{blue}{\left(\sqrt{\sqrt[3]{{\left(\sqrt{{\left(e^{60}\right)}^{x}}\right)}^{x}}} \cdot \sqrt{\sqrt[3]{{\left(\sqrt{{\left(e^{60}\right)}^{x}}\right)}^{x}}}\right)} \]

   2. sqrt-unprod 99.1%

      \[\leadsto \cos x \cdot \color{blue}{\sqrt{\sqrt[3]{{\left(\sqrt{{\left(e^{60}\right)}^{x}}\right)}^{x}} \cdot \sqrt[3]{{\left(\sqrt{{\left(e^{60}\right)}^{x}}\right)}^{x}}}} \]

   3. cbrt-unprod 99.2%

      \[\leadsto \cos x \cdot \sqrt{\color{blue}{\sqrt[3]{{\left(\sqrt{{\left(e^{60}\right)}^{x}}\right)}^{x} \cdot {\left(\sqrt{{\left(e^{60}\right)}^{x}}\right)}^{x}}}} \]

   4. pow-prod-down 99.3%

      \[\leadsto \cos x \cdot \sqrt{\sqrt[3]{\color{blue}{{\left(\sqrt{{\left(e^{60}\right)}^{x}} \cdot \sqrt{{\left(e^{60}\right)}^{x}}\right)}^{x}}}} \]

   5. add-sqr-sqrt 99.4%

      \[\leadsto \cos x \cdot \sqrt{\sqrt[3]{{\color{blue}{\left({\left(e^{60}\right)}^{x}\right)}}^{x}}} \]

7. Applied egg-rr 99.4%

   \[\leadsto \cos x \cdot \color{blue}{\sqrt{\sqrt[3]{{\left({\left(e^{60}\right)}^{x}\right)}^{x}}}} \]

8. Final simplification 99.4%

   \[\leadsto \cos x \cdot \sqrt{\sqrt[3]{{\left({\left(e^{60}\right)}^{x}\right)}^{x}}} \]

Alternative 2: 99.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \cos x \cdot \sqrt[3]{{\left(\sqrt{{\left(e^{60}\right)}^{x}}\right)}^{x}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (cos x) (cbrt (pow (sqrt (pow (exp 60.0) x)) x))))

C

double code(double x) {
	return cos(x) * cbrt(pow(sqrt(pow(exp(60.0), x)), x));
}

Java

public static double code(double x) {
	return Math.cos(x) * Math.cbrt(Math.pow(Math.sqrt(Math.pow(Math.exp(60.0), x)), x));
}

Julia

function code(x)
	return Float64(cos(x) * cbrt((sqrt((exp(60.0) ^ x)) ^ x)))
end

Wolfram

code[x_] := N[(N[Cos[x], $MachinePrecision] * N[Power[N[Power[N[Sqrt[N[Power[N[Exp[60.0], $MachinePrecision], x], $MachinePrecision]], $MachinePrecision], x], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.6%

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

   1. associate-*r* 94.6%

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

   2. add-log-exp 94.6%

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

   3. log-pow 94.5%

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

   4. pow-pow 94.9%

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

   5. add-exp-log 96.8%

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

   6. add-cbrt-cube 96.8%

      \[\leadsto \cos x \cdot \color{blue}{\sqrt[3]{\left({\left({\left(e^{x}\right)}^{10}\right)}^{x} \cdot {\left({\left(e^{x}\right)}^{10}\right)}^{x}\right) \cdot {\left({\left(e^{x}\right)}^{10}\right)}^{x}}} \]

   7. add-exp-log 96.5%

      \[\leadsto \cos x \cdot \sqrt[3]{\left({\left({\left(e^{x}\right)}^{10}\right)}^{x} \cdot {\left({\left(e^{x}\right)}^{10}\right)}^{x}\right) \cdot \color{blue}{e^{\log \left({\left({\left(e^{x}\right)}^{10}\right)}^{x}\right)}}} \]

   8. pow-pow 96.2%

      \[\leadsto \cos x \cdot \sqrt[3]{\left({\left({\left(e^{x}\right)}^{10}\right)}^{x} \cdot {\left({\left(e^{x}\right)}^{10}\right)}^{x}\right) \cdot e^{\log \color{blue}{\left({\left(e^{x}\right)}^{\left(10 \cdot x\right)}\right)}}} \]

   9. log-pow 96.4%

      \[\leadsto \cos x \cdot \sqrt[3]{\left({\left({\left(e^{x}\right)}^{10}\right)}^{x} \cdot {\left({\left(e^{x}\right)}^{10}\right)}^{x}\right) \cdot e^{\color{blue}{\left(10 \cdot x\right) \cdot \log \left(e^{x}\right)}}} \]

   10. add-log-exp 96.4%

       \[\leadsto \cos x \cdot \sqrt[3]{\left({\left({\left(e^{x}\right)}^{10}\right)}^{x} \cdot {\left({\left(e^{x}\right)}^{10}\right)}^{x}\right) \cdot e^{\left(10 \cdot x\right) \cdot \color{blue}{x}}} \]

   11. associate-*r* 96.4%

       \[\leadsto \cos x \cdot \sqrt[3]{\left({\left({\left(e^{x}\right)}^{10}\right)}^{x} \cdot {\left({\left(e^{x}\right)}^{10}\right)}^{x}\right) \cdot e^{\color{blue}{10 \cdot \left(x \cdot x\right)}}} \]

   12. pow-exp 96.8%

       \[\leadsto \cos x \cdot \sqrt[3]{\left({\left({\left(e^{x}\right)}^{10}\right)}^{x} \cdot {\left({\left(e^{x}\right)}^{10}\right)}^{x}\right) \cdot \color{blue}{{\left(e^{10}\right)}^{\left(x \cdot x\right)}}} \]

   13. *-commutative 96.8%

       \[\leadsto \cos x \cdot \sqrt[3]{\color{blue}{{\left(e^{10}\right)}^{\left(x \cdot x\right)} \cdot \left({\left({\left(e^{x}\right)}^{10}\right)}^{x} \cdot {\left({\left(e^{x}\right)}^{10}\right)}^{x}\right)}} \]

3. Applied egg-rr 98.8%

   \[\leadsto \cos x \cdot \color{blue}{\sqrt[3]{{\left({\left(e^{30}\right)}^{x}\right)}^{x}}} \]
4. Step-by-step derivation

   1. add-sqr-sqrt 98.7%

      \[\leadsto \cos x \cdot \sqrt[3]{{\color{blue}{\left(\sqrt{{\left(e^{30}\right)}^{x}} \cdot \sqrt{{\left(e^{30}\right)}^{x}}\right)}}^{x}} \]

   2. sqrt-unprod 98.8%

      \[\leadsto \cos x \cdot \sqrt[3]{{\color{blue}{\left(\sqrt{{\left(e^{30}\right)}^{x} \cdot {\left(e^{30}\right)}^{x}}\right)}}^{x}} \]

   3. pow-prod-down 98.8%

      \[\leadsto \cos x \cdot \sqrt[3]{{\left(\sqrt{\color{blue}{{\left(e^{30} \cdot e^{30}\right)}^{x}}}\right)}^{x}} \]

   4. prod-exp 99.1%

      \[\leadsto \cos x \cdot \sqrt[3]{{\left(\sqrt{{\color{blue}{\left(e^{30 + 30}\right)}}^{x}}\right)}^{x}} \]

   5. metadata-eval 99.1%

      \[\leadsto \cos x \cdot \sqrt[3]{{\left(\sqrt{{\left(e^{\color{blue}{60}}\right)}^{x}}\right)}^{x}} \]

5. Applied egg-rr 99.1%

   \[\leadsto \cos x \cdot \sqrt[3]{{\color{blue}{\left(\sqrt{{\left(e^{60}\right)}^{x}}\right)}}^{x}} \]

6. Final simplification 99.1%

   \[\leadsto \cos x \cdot \sqrt[3]{{\left(\sqrt{{\left(e^{60}\right)}^{x}}\right)}^{x}} \]

Alternative 3: 98.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \cos x \cdot \sqrt[3]{{\left({\left(e^{30}\right)}^{x}\right)}^{x}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (cos x) (cbrt (pow (pow (exp 30.0) x) x))))

C

double code(double x) {
	return cos(x) * cbrt(pow(pow(exp(30.0), x), x));
}

Java

public static double code(double x) {
	return Math.cos(x) * Math.cbrt(Math.pow(Math.pow(Math.exp(30.0), x), x));
}

Julia

function code(x)
	return Float64(cos(x) * cbrt(((exp(30.0) ^ x) ^ x)))
end

Wolfram

code[x_] := N[(N[Cos[x], $MachinePrecision] * N[Power[N[Power[N[Power[N[Exp[30.0], $MachinePrecision], x], $MachinePrecision], x], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.6%

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

   1. associate-*r* 94.6%

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

   2. add-log-exp 94.6%

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

   3. log-pow 94.5%

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

   4. pow-pow 94.9%

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

   5. add-exp-log 96.8%

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

   6. add-cbrt-cube 96.8%

      \[\leadsto \cos x \cdot \color{blue}{\sqrt[3]{\left({\left({\left(e^{x}\right)}^{10}\right)}^{x} \cdot {\left({\left(e^{x}\right)}^{10}\right)}^{x}\right) \cdot {\left({\left(e^{x}\right)}^{10}\right)}^{x}}} \]

   7. add-exp-log 96.5%

      \[\leadsto \cos x \cdot \sqrt[3]{\left({\left({\left(e^{x}\right)}^{10}\right)}^{x} \cdot {\left({\left(e^{x}\right)}^{10}\right)}^{x}\right) \cdot \color{blue}{e^{\log \left({\left({\left(e^{x}\right)}^{10}\right)}^{x}\right)}}} \]

   8. pow-pow 96.2%

      \[\leadsto \cos x \cdot \sqrt[3]{\left({\left({\left(e^{x}\right)}^{10}\right)}^{x} \cdot {\left({\left(e^{x}\right)}^{10}\right)}^{x}\right) \cdot e^{\log \color{blue}{\left({\left(e^{x}\right)}^{\left(10 \cdot x\right)}\right)}}} \]

   9. log-pow 96.4%

      \[\leadsto \cos x \cdot \sqrt[3]{\left({\left({\left(e^{x}\right)}^{10}\right)}^{x} \cdot {\left({\left(e^{x}\right)}^{10}\right)}^{x}\right) \cdot e^{\color{blue}{\left(10 \cdot x\right) \cdot \log \left(e^{x}\right)}}} \]

   10. add-log-exp 96.4%

       \[\leadsto \cos x \cdot \sqrt[3]{\left({\left({\left(e^{x}\right)}^{10}\right)}^{x} \cdot {\left({\left(e^{x}\right)}^{10}\right)}^{x}\right) \cdot e^{\left(10 \cdot x\right) \cdot \color{blue}{x}}} \]

   11. associate-*r* 96.4%

       \[\leadsto \cos x \cdot \sqrt[3]{\left({\left({\left(e^{x}\right)}^{10}\right)}^{x} \cdot {\left({\left(e^{x}\right)}^{10}\right)}^{x}\right) \cdot e^{\color{blue}{10 \cdot \left(x \cdot x\right)}}} \]

   12. pow-exp 96.8%

       \[\leadsto \cos x \cdot \sqrt[3]{\left({\left({\left(e^{x}\right)}^{10}\right)}^{x} \cdot {\left({\left(e^{x}\right)}^{10}\right)}^{x}\right) \cdot \color{blue}{{\left(e^{10}\right)}^{\left(x \cdot x\right)}}} \]

   13. *-commutative 96.8%

       \[\leadsto \cos x \cdot \sqrt[3]{\color{blue}{{\left(e^{10}\right)}^{\left(x \cdot x\right)} \cdot \left({\left({\left(e^{x}\right)}^{10}\right)}^{x} \cdot {\left({\left(e^{x}\right)}^{10}\right)}^{x}\right)}} \]

3. Applied egg-rr 98.8%

   \[\leadsto \cos x \cdot \color{blue}{\sqrt[3]{{\left({\left(e^{30}\right)}^{x}\right)}^{x}}} \]

4. Final simplification 98.8%

   \[\leadsto \cos x \cdot \sqrt[3]{{\left({\left(e^{30}\right)}^{x}\right)}^{x}} \]

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 94.6%

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

   1. associate-*r* 94.6%

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

   2. exp-prod 95.0%

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

   3. sqr-pow 95.0%

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

   4. sqr-pow 95.0%

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

   5. exp-prod 97.9%

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

3. Simplified 97.9%

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

4. Final simplification 97.9%

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

Alternative 5: 95.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \cos x \cdot \sqrt[3]{e^{x \cdot \left(x \cdot 30\right)}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (cos x) (cbrt (exp (* x (* x 30.0))))))

C

double code(double x) {
	return cos(x) * cbrt(exp((x * (x * 30.0))));
}

Java

public static double code(double x) {
	return Math.cos(x) * Math.cbrt(Math.exp((x * (x * 30.0))));
}

Julia

function code(x)
	return Float64(cos(x) * cbrt(exp(Float64(x * Float64(x * 30.0)))))
end

Wolfram

code[x_] := N[(N[Cos[x], $MachinePrecision] * N[Power[N[Exp[N[(x * N[(x * 30.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.6%

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

   1. associate-*r* 94.6%

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

   2. add-log-exp 94.6%

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

   3. log-pow 94.5%

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

   4. pow-pow 94.9%

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

   5. add-exp-log 96.8%

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

   6. add-cbrt-cube 96.8%

      \[\leadsto \cos x \cdot \color{blue}{\sqrt[3]{\left({\left({\left(e^{x}\right)}^{10}\right)}^{x} \cdot {\left({\left(e^{x}\right)}^{10}\right)}^{x}\right) \cdot {\left({\left(e^{x}\right)}^{10}\right)}^{x}}} \]

   7. add-exp-log 96.5%

      \[\leadsto \cos x \cdot \sqrt[3]{\left({\left({\left(e^{x}\right)}^{10}\right)}^{x} \cdot {\left({\left(e^{x}\right)}^{10}\right)}^{x}\right) \cdot \color{blue}{e^{\log \left({\left({\left(e^{x}\right)}^{10}\right)}^{x}\right)}}} \]

   8. pow-pow 96.2%

      \[\leadsto \cos x \cdot \sqrt[3]{\left({\left({\left(e^{x}\right)}^{10}\right)}^{x} \cdot {\left({\left(e^{x}\right)}^{10}\right)}^{x}\right) \cdot e^{\log \color{blue}{\left({\left(e^{x}\right)}^{\left(10 \cdot x\right)}\right)}}} \]

   9. log-pow 96.4%

      \[\leadsto \cos x \cdot \sqrt[3]{\left({\left({\left(e^{x}\right)}^{10}\right)}^{x} \cdot {\left({\left(e^{x}\right)}^{10}\right)}^{x}\right) \cdot e^{\color{blue}{\left(10 \cdot x\right) \cdot \log \left(e^{x}\right)}}} \]

   10. add-log-exp 96.4%

       \[\leadsto \cos x \cdot \sqrt[3]{\left({\left({\left(e^{x}\right)}^{10}\right)}^{x} \cdot {\left({\left(e^{x}\right)}^{10}\right)}^{x}\right) \cdot e^{\left(10 \cdot x\right) \cdot \color{blue}{x}}} \]

   11. associate-*r* 96.4%

       \[\leadsto \cos x \cdot \sqrt[3]{\left({\left({\left(e^{x}\right)}^{10}\right)}^{x} \cdot {\left({\left(e^{x}\right)}^{10}\right)}^{x}\right) \cdot e^{\color{blue}{10 \cdot \left(x \cdot x\right)}}} \]

   12. pow-exp 96.8%

       \[\leadsto \cos x \cdot \sqrt[3]{\left({\left({\left(e^{x}\right)}^{10}\right)}^{x} \cdot {\left({\left(e^{x}\right)}^{10}\right)}^{x}\right) \cdot \color{blue}{{\left(e^{10}\right)}^{\left(x \cdot x\right)}}} \]

   13. *-commutative 96.8%

       \[\leadsto \cos x \cdot \sqrt[3]{\color{blue}{{\left(e^{10}\right)}^{\left(x \cdot x\right)} \cdot \left({\left({\left(e^{x}\right)}^{10}\right)}^{x} \cdot {\left({\left(e^{x}\right)}^{10}\right)}^{x}\right)}} \]

3. Applied egg-rr 98.8%

   \[\leadsto \cos x \cdot \color{blue}{\sqrt[3]{{\left({\left(e^{30}\right)}^{x}\right)}^{x}}} \]
4. Step-by-step derivation

   1. add-exp-log 95.6%

      \[\leadsto \cos x \cdot \sqrt[3]{\color{blue}{e^{\log \left({\left({\left(e^{30}\right)}^{x}\right)}^{x}\right)}}} \]

   2. log-pow 95.3%

      \[\leadsto \cos x \cdot \sqrt[3]{e^{\color{blue}{x \cdot \log \left({\left(e^{30}\right)}^{x}\right)}}} \]

   3. pow-exp 95.4%

      \[\leadsto \cos x \cdot \sqrt[3]{e^{x \cdot \log \color{blue}{\left(e^{30 \cdot x}\right)}}} \]

   4. add-log-exp 95.4%

      \[\leadsto \cos x \cdot \sqrt[3]{e^{x \cdot \color{blue}{\left(30 \cdot x\right)}}} \]

5. Applied egg-rr 95.4%

   \[\leadsto \cos x \cdot \sqrt[3]{\color{blue}{e^{x \cdot \left(30 \cdot x\right)}}} \]

6. Final simplification 95.4%

   \[\leadsto \cos x \cdot \sqrt[3]{e^{x \cdot \left(x \cdot 30\right)}} \]

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 94.6%

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

   1. exp-prod 95.2%

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

3. Simplified 95.2%

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

4. Final simplification 95.2%

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

Alternative 7: 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 94.6%

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

2. Final simplification 94.6%

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

Alternative 8: 94.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

function code(x)
	return Float64(cos(x) * exp(Float64(x * Float64(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[Exp[N[(x * N[(x * 10.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.6%

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

   1. associate-*r* 94.6%

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

   2. add-log-exp 94.6%

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

   3. log-pow 94.5%

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

   4. pow-pow 94.9%

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

   5. add-exp-log 96.8%

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

   6. pow-pow 94.7%

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

   7. add-sqr-sqrt 94.4%

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

   8. unpow-prod-down 94.4%

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

   9. pow-prod-up 94.3%

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

   10. add-log-exp 94.3%

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

   11. log-pow 94.3%

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

   12. add-log-exp 94.3%

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

   13. log-pow 94.2%

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

   14. sum-log 94.2%

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

   15. pow-prod-down 94.3%

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

   16. prod-exp 94.4%

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

   17. pow-exp 94.3%

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

   18. add-log-exp 94.3%

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

   19. log-pow 94.3%

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

   20. pow-prod-up 94.3%

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

   21. add-exp-log 94.3%

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

3. Applied egg-rr 94.3%

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

   1. add-exp-log 94.2%

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

   2. sqrt-pow2 94.5%

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

   3. log-pow 94.6%

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

   4. associate-/l* 93.9%

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

   5. metadata-eval 93.9%

      \[\leadsto \cos x \cdot e^{\frac{x}{\color{blue}{0.1}} \cdot \log \left(e^{x}\right)} \]

   6. div-inv 94.6%

      \[\leadsto \cos x \cdot e^{\color{blue}{\left(x \cdot \frac{1}{0.1}\right)} \cdot \log \left(e^{x}\right)} \]

   7. metadata-eval 94.6%

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

   8. add-log-exp 94.6%

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

5. Applied egg-rr 94.6%

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

6. Final simplification 94.6%

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

Alternative 9: 9.7% accurate, 41.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.6%

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

   1. associate-*r* 94.6%

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

   2. add-log-exp 94.6%

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

   3. log-pow 94.5%

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

   4. pow-pow 94.9%

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

   5. add-exp-log 96.8%

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

   6. pow-pow 94.7%

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

   7. add-sqr-sqrt 93.0%

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

   8. pow-unpow 93.2%

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

3. Applied egg-rr 93.2%

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

   1. metadata-eval 93.2%

      \[\leadsto \cos x \cdot {\left({\left(e^{x}\right)}^{\left(\sqrt{\color{blue}{\left(2 \cdot 5\right)} \cdot x}\right)}\right)}^{\left(\sqrt{10 \cdot x}\right)} \]

   2. associate-*r* 93.2%

      \[\leadsto \cos x \cdot {\left({\left(e^{x}\right)}^{\left(\sqrt{\color{blue}{2 \cdot \left(5 \cdot x\right)}}\right)}\right)}^{\left(\sqrt{10 \cdot x}\right)} \]

   3. metadata-eval 93.2%

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

   4. associate-/r/ 93.6%

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

   5. associate-*r/ 93.6%

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

   6. metadata-eval 93.6%

      \[\leadsto \cos x \cdot {\left({\left(e^{x}\right)}^{\left(\sqrt{\frac{\color{blue}{20}}{\frac{2}{x}}}\right)}\right)}^{\left(\sqrt{10 \cdot x}\right)} \]

   7. associate-/l* 93.2%

      \[\leadsto \cos x \cdot {\left({\left(e^{x}\right)}^{\left(\sqrt{\color{blue}{\frac{20 \cdot x}{2}}}\right)}\right)}^{\left(\sqrt{10 \cdot x}\right)} \]

   8. *-commutative 93.2%

      \[\leadsto \cos x \cdot {\left({\left(e^{x}\right)}^{\left(\sqrt{\frac{\color{blue}{x \cdot 20}}{2}}\right)}\right)}^{\left(\sqrt{10 \cdot x}\right)} \]

   9. associate-/l* 93.8%

      \[\leadsto \cos x \cdot {\left({\left(e^{x}\right)}^{\left(\sqrt{\color{blue}{\frac{x}{\frac{2}{20}}}}\right)}\right)}^{\left(\sqrt{10 \cdot x}\right)} \]

   10. metadata-eval 93.8%

       \[\leadsto \cos x \cdot {\left({\left(e^{x}\right)}^{\left(\sqrt{\frac{x}{\color{blue}{0.1}}}\right)}\right)}^{\left(\sqrt{10 \cdot x}\right)} \]

   11. metadata-eval 93.8%

       \[\leadsto \cos x \cdot {\left({\left(e^{x}\right)}^{\left(\sqrt{\frac{x}{0.1}}\right)}\right)}^{\left(\sqrt{\color{blue}{\left(2 \cdot 5\right)} \cdot x}\right)} \]

   12. associate-*r* 93.8%

       \[\leadsto \cos x \cdot {\left({\left(e^{x}\right)}^{\left(\sqrt{\frac{x}{0.1}}\right)}\right)}^{\left(\sqrt{\color{blue}{2 \cdot \left(5 \cdot x\right)}}\right)} \]

   13. metadata-eval 93.8%

       \[\leadsto \cos x \cdot {\left({\left(e^{x}\right)}^{\left(\sqrt{\frac{x}{0.1}}\right)}\right)}^{\left(\sqrt{2 \cdot \left(\color{blue}{\frac{10}{2}} \cdot x\right)}\right)} \]

   14. associate-/r/ 93.6%

       \[\leadsto \cos x \cdot {\left({\left(e^{x}\right)}^{\left(\sqrt{\frac{x}{0.1}}\right)}\right)}^{\left(\sqrt{2 \cdot \color{blue}{\frac{10}{\frac{2}{x}}}}\right)} \]

   15. associate-*r/ 93.6%

       \[\leadsto \cos x \cdot {\left({\left(e^{x}\right)}^{\left(\sqrt{\frac{x}{0.1}}\right)}\right)}^{\left(\sqrt{\color{blue}{\frac{2 \cdot 10}{\frac{2}{x}}}}\right)} \]

   16. metadata-eval 93.6%

       \[\leadsto \cos x \cdot {\left({\left(e^{x}\right)}^{\left(\sqrt{\frac{x}{0.1}}\right)}\right)}^{\left(\sqrt{\frac{\color{blue}{20}}{\frac{2}{x}}}\right)} \]

   17. associate-/l* 93.8%

       \[\leadsto \cos x \cdot {\left({\left(e^{x}\right)}^{\left(\sqrt{\frac{x}{0.1}}\right)}\right)}^{\left(\sqrt{\color{blue}{\frac{20 \cdot x}{2}}}\right)} \]

   18. *-commutative 93.8%

       \[\leadsto \cos x \cdot {\left({\left(e^{x}\right)}^{\left(\sqrt{\frac{x}{0.1}}\right)}\right)}^{\left(\sqrt{\frac{\color{blue}{x \cdot 20}}{2}}\right)} \]

   19. associate-/l* 93.0%

       \[\leadsto \cos x \cdot {\left({\left(e^{x}\right)}^{\left(\sqrt{\frac{x}{0.1}}\right)}\right)}^{\left(\sqrt{\color{blue}{\frac{x}{\frac{2}{20}}}}\right)} \]

   20. metadata-eval 93.0%

       \[\leadsto \cos x \cdot {\left({\left(e^{x}\right)}^{\left(\sqrt{\frac{x}{0.1}}\right)}\right)}^{\left(\sqrt{\frac{x}{\color{blue}{0.1}}}\right)} \]

5. Simplified 93.0%

   \[\leadsto \cos x \cdot \color{blue}{{\left({\left(e^{x}\right)}^{\left(\sqrt{\frac{x}{0.1}}\right)}\right)}^{\left(\sqrt{\frac{x}{0.1}}\right)}} \]

6. Taylor expanded in x around 0 9.7%

   \[\leadsto \color{blue}{1 + -0.5 \cdot {x}^{2}} \]
7. Step-by-step derivation

   1. *-commutative 9.7%

      \[\leadsto 1 + \color{blue}{{x}^{2} \cdot -0.5} \]

   2. unpow2 9.7%

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

8. Simplified 9.7%

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

9. Taylor expanded in x around inf 9.7%

   \[\leadsto \color{blue}{-0.5 \cdot {x}^{2}} \]
10. Step-by-step derivation

    1. *-commutative 9.7%

       \[\leadsto \color{blue}{{x}^{2} \cdot -0.5} \]

    2. unpow2 9.7%

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

    3. associate-*r* 9.7%

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

11. Simplified 9.7%

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

12. Final simplification 9.7%

    \[\leadsto x \cdot \left(x \cdot -0.5\right) \]

Reproduce

herbie shell --seed 2023174 
(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)))))