ENA, Section 1.4, Exercise 1

Percentage Accurate: 94.5% → 99.1%

Time: 9.4s

Alternatives: 10

Speedup: 1.0×

Specification

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

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 94.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.1% accurate, 0.3× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (cbrt (* (sqrt (pow (pow (exp 60.0) x) x)) (pow (cos x) 3.0))))

C

double code(double x) {
	return cbrt((sqrt(pow(pow(exp(60.0), x), x)) * pow(cos(x), 3.0)));
}

Java

public static double code(double x) {
	return Math.cbrt((Math.sqrt(Math.pow(Math.pow(Math.exp(60.0), x), x)) * Math.pow(Math.cos(x), 3.0)));
}

Julia

function code(x)
	return cbrt(Float64(sqrt(((exp(60.0) ^ x) ^ x)) * (cos(x) ^ 3.0)))
end

Wolfram

code[x_] := N[Power[N[(N[Sqrt[N[Power[N[Power[N[Exp[60.0], $MachinePrecision], x], $MachinePrecision], x], $MachinePrecision]], $MachinePrecision] * N[Power[N[Cos[x], $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]

Derivation

1. Initial program 94.7%

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

3. Applied egg-rr 98.8%

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

   1. add-sqr-sqrt 98.7%

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

   2. sqrt-unprod 98.8%

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

   3. pow-prod-down 98.9%

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

   4. prod-exp 99.2%

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

   5. metadata-eval 99.2%

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

5. Applied egg-rr 99.2%

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

   1. add-sqr-sqrt 99.1%

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

   2. sqrt-unprod 99.2%

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

   3. pow-prod-down 99.2%

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

   4. add-sqr-sqrt 99.2%

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

7. Applied egg-rr 99.2%

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

8. Final simplification 99.2%

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

Alternative 2: 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 94.7%

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

3. Applied egg-rr 98.6%

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

4. Final simplification 98.6%

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

Alternative 3: 99.2% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.7%

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

   1. *-un-lft-identity 94.7%

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

   2. exp-prod 94.0%

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

   3. associate-*r* 93.9%

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

   4. pow-unpow 94.3%

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

3. Applied egg-rr 94.3%

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

   1. add-sqr-sqrt 94.4%

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

   2. sqrt-unprod 94.3%

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

   3. pow-exp 94.6%

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

   4. *-un-lft-identity 94.6%

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

   5. pow-exp 95.8%

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

   6. pow-exp 96.0%

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

   7. *-un-lft-identity 96.0%

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

   8. pow-exp 97.9%

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

   9. pow-prod-down 98.0%

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

   10. prod-exp 99.1%

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

   11. metadata-eval 99.1%

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

5. Applied egg-rr 99.1%

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

6. Final simplification 99.1%

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

Alternative 4: 98.2% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.7%

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

   1. *-un-lft-identity 94.7%

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

   2. exp-prod 94.0%

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

   3. associate-*r* 93.9%

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

   4. pow-unpow 94.3%

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

3. Applied egg-rr 94.3%

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

   1. add-sqr-sqrt 94.4%

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

   2. unpow-prod-down 94.3%

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

   3. pow-exp 94.6%

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

   4. *-un-lft-identity 94.6%

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

   5. pow-exp 95.7%

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

   6. pow-exp 96.0%

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

   7. *-un-lft-identity 96.0%

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

   8. pow-exp 97.8%

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

5. Applied egg-rr 97.8%

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

   1. pow-sqr 97.9%

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

   2. *-commutative 97.9%

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

7. Simplified 97.9%

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

8. Taylor expanded in x around inf 94.8%

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

   1. *-commutative 94.8%

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

   2. unpow1/2 94.8%

      \[\leadsto \cos x \cdot {\color{blue}{\left({\left(e^{x \cdot 10}\right)}^{0.5}\right)}}^{\left(x \cdot 2\right)} \]

   3. exp-prod 94.8%

      \[\leadsto \cos x \cdot {\color{blue}{\left(e^{\left(x \cdot 10\right) \cdot 0.5}\right)}}^{\left(x \cdot 2\right)} \]

   4. associate-*l* 94.8%

      \[\leadsto \cos x \cdot {\left(e^{\color{blue}{x \cdot \left(10 \cdot 0.5\right)}}\right)}^{\left(x \cdot 2\right)} \]

   5. metadata-eval 94.8%

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

   6. *-commutative 94.8%

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

   7. exp-prod 98.2%

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

10. Simplified 98.2%

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

11. Final simplification 98.2%

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

Alternative 5: 98.0% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.7%

   \[\cos x \cdot e^{10 \cdot \left(x \cdot x\right)} \]
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 94.8%

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

   3. sqr-pow 94.8%

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

   4. sqr-pow 94.8%

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

   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 6: 95.3% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.7%

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

   1. pow-exp 95.2%

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

   2. sqr-pow 95.2%

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

   3. pow-prod-down 95.2%

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

   4. prod-exp 95.3%

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

   5. metadata-eval 95.3%

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

   6. div-inv 95.3%

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

   7. metadata-eval 95.3%

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

3. Applied egg-rr 95.3%

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

4. Final simplification 95.3%

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

Alternative 7: 95.3% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.7%

   \[\cos x \cdot e^{10 \cdot \left(x \cdot x\right)} \]
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 8: 94.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.7%

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

2. Final simplification 94.7%

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

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 94.7%

   \[\cos x \cdot e^{10 \cdot \left(x \cdot x\right)} \]
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 95.0%

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

   5. add-exp-log 96.8%

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

   6. sqr-pow 96.8%

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

   7. pow-prod-down 96.9%

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

   8. add-sqr-sqrt 94.6%

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

   9. pow-unpow 94.1%

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

   10. pow-prod-up 94.1%

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

   11. metadata-eval 94.1%

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

   12. div-inv 94.1%

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

   13. metadata-eval 94.1%

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

   14. div-inv 94.1%

       \[\leadsto \cos x \cdot {\left({\left({\left(e^{x}\right)}^{20}\right)}^{\left(\sqrt{x \cdot 0.5}\right)}\right)}^{\left(\sqrt{\color{blue}{x \cdot \frac{1}{2}}}\right)} \]

   15. metadata-eval 94.1%

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

3. Applied egg-rr 94.1%

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

4. Taylor expanded in x around 0 9.7%

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

   1. unpow2 9.7%

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

6. Simplified 9.7%

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

7. Taylor expanded in x around inf 9.7%

   \[\leadsto \color{blue}{-0.5 \cdot {x}^{2}} \]
8. 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%

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

9. Simplified 9.7%

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

10. Final simplification 9.7%

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

Alternative 10: 1.5% accurate, 207.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 1.0)

C

double code(double x) {
	return 1.0;
}

Fortran

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

Java

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

Python

def code(x):
	return 1.0

Julia

function code(x)
	return 1.0
end

MATLAB

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

Wolfram

code[x_] := 1.0

Derivation

1. Initial program 94.7%

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

3. Applied egg-rr 98.8%

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

   1. add-sqr-sqrt 98.7%

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

   2. sqrt-unprod 98.8%

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

   3. pow-prod-down 98.9%

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

   4. prod-exp 99.2%

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

   5. metadata-eval 99.2%

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

5. Applied egg-rr 99.2%

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

   1. add-sqr-sqrt 99.1%

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

   2. sqrt-unprod 99.2%

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

   3. pow-prod-down 99.2%

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

   4. add-sqr-sqrt 99.2%

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

7. Applied egg-rr 99.2%

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

8. Taylor expanded in x around 0 1.5%

   \[\leadsto \color{blue}{1} \]

9. Final simplification 1.5%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023195 
(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)))))