ENA, Section 1.4, Exercise 1

Percentage Accurate: 94.4% → 99.1%

Time: 8.5s

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: 94.4% 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: 99.1% 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 94.3%

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

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 98.6%

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

   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 99.1%

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

   5. metadata-eval 99.1%

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

5. Applied egg-rr 99.1%

   \[\leadsto \sqrt[3]{{\color{blue}{\left(\sqrt{{\left(e^{60}\right)}^{x}}\right)}}^{x} \cdot {\cos x}^{3}} \]
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 99.1%

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

   3. pow-prod-down 99.1%

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

   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: 99.1% 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 94.3%

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

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 98.6%

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

   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 99.1%

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

   5. metadata-eval 99.1%

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

5. Applied egg-rr 99.1%

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

6. Taylor expanded in x around inf 94.1%

   \[\leadsto \color{blue}{\cos x \cdot {\left(1 \cdot e^{\log \left(\sqrt{{\left(e^{60}\right)}^{x}}\right) \cdot x}\right)}^{0.3333333333333333}} \]
7. Step-by-step derivation

   1. unpow1/3 95.0%

      \[\leadsto \cos x \cdot \color{blue}{\sqrt[3]{1 \cdot e^{\log \left(\sqrt{{\left(e^{60}\right)}^{x}}\right) \cdot x}}} \]

   2. *-lft-identity 95.0%

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

   3. exp-to-pow 99.0%

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

8. Simplified 99.0%

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

9. Final simplification 99.0%

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

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 94.3%

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

3. Applied egg-rr 98.7%

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

4. Final simplification 98.7%

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

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 94.3%

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

   1. associate-*r* 94.2%

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

   2. exp-prod 95.0%

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

   3. sqr-pow 94.9%

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

   4. sqr-pow 95.0%

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

   5. exp-prod 98.0%

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

3. Simplified 98.0%

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

4. Final simplification 98.0%

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

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 94.3%

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

3. Applied egg-rr 98.8%

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

4. Taylor expanded in x around inf 93.9%

   \[\leadsto \color{blue}{{\left({\left({\left(e^{30}\right)}^{x}\right)}^{x} \cdot 1\right)}^{0.3333333333333333} \cdot \cos x} \]
5. Step-by-step derivation

   1. unpow1/3 98.7%

      \[\leadsto \color{blue}{\sqrt[3]{{\left({\left(e^{30}\right)}^{x}\right)}^{x} \cdot 1}} \cdot \cos x \]

   2. *-rgt-identity 98.7%

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

   3. exp-prod 95.5%

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

   4. exp-prod 95.0%

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

   5. associate-*r* 94.6%

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

   6. unpow2 94.6%

      \[\leadsto \sqrt[3]{e^{30 \cdot \color{blue}{{x}^{2}}}} \cdot \cos x \]

   7. exp-prod 95.0%

      \[\leadsto \sqrt[3]{\color{blue}{{\left(e^{30}\right)}^{\left({x}^{2}\right)}}} \cdot \cos x \]

   8. unpow2 95.0%

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

   9. *-commutative 95.0%

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

6. Simplified 95.0%

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

   1. add-exp-log 94.7%

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

   2. pow-pow 95.5%

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

   3. log-pow 95.1%

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

   4. log-pow 95.0%

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

   5. add-log-exp 95.0%

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

8. Applied egg-rr 95.0%

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

9. Final simplification 95.0%

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

Alternative 6: 95.2% 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. exp-prod 94.9%

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

3. Simplified 94.9%

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

4. Final simplification 94.9%

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

Alternative 7: 95.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[Cos[x], $MachinePrecision] * N[Power[N[Exp[N[(x * 10.0), $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. associate-*r* 94.2%

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

   2. exp-prod 95.0%

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

   3. sqr-pow 94.9%

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

   4. sqr-pow 95.0%

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

   5. exp-prod 98.0%

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

3. Simplified 98.0%

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

4. Taylor expanded in x around inf 95.0%

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

5. Final simplification 95.0%

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

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: 9.6% 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. Step-by-step derivation

   1. associate-*r* 94.2%

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

   2. add-log-exp 94.2%

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

   3. log-pow 94.2%

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

   4. pow-pow 94.7%

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

   5. add-exp-log 96.8%

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

   6. pow-pow 94.6%

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

   7. sqr-pow 94.6%

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

   8. pow-prod-down 94.6%

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

   9. prod-exp 94.8%

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

3. Applied egg-rr 94.8%

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

   1. expm1-log1p-u 0.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos x \cdot {\left(e^{x + x}\right)}^{\left(\frac{10 \cdot x}{2}\right)}\right)\right)} \]

   2. expm1-udef 0.0%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\cos x \cdot {\left(e^{x + x}\right)}^{\left(\frac{10 \cdot x}{2}\right)}\right)} - 1} \]

   3. pow-exp 0.0%

      \[\leadsto e^{\mathsf{log1p}\left(\cos x \cdot \color{blue}{e^{\left(x + x\right) \cdot \frac{10 \cdot x}{2}}}\right)} - 1 \]

   4. flip-+ 0.0%

      \[\leadsto e^{\mathsf{log1p}\left(\cos x \cdot e^{\color{blue}{\frac{x \cdot x - x \cdot x}{x - x}} \cdot \frac{10 \cdot x}{2}}\right)} - 1 \]

   5. associate-/l* 0.0%

      \[\leadsto e^{\mathsf{log1p}\left(\cos x \cdot e^{\frac{x \cdot x - x \cdot x}{x - x} \cdot \color{blue}{\frac{10}{\frac{2}{x}}}}\right)} - 1 \]

   6. frac-times 0.0%

      \[\leadsto e^{\mathsf{log1p}\left(\cos x \cdot e^{\color{blue}{\frac{\left(x \cdot x - x \cdot x\right) \cdot 10}{\left(x - x\right) \cdot \frac{2}{x}}}}\right)} - 1 \]

   7. +-inverses 0.0%

      \[\leadsto e^{\mathsf{log1p}\left(\cos x \cdot e^{\frac{\color{blue}{0} \cdot 10}{\left(x - x\right) \cdot \frac{2}{x}}}\right)} - 1 \]

   8. metadata-eval 0.0%

      \[\leadsto e^{\mathsf{log1p}\left(\cos x \cdot e^{\frac{\color{blue}{0}}{\left(x - x\right) \cdot \frac{2}{x}}}\right)} - 1 \]

   9. +-inverses 0.0%

      \[\leadsto e^{\mathsf{log1p}\left(\cos x \cdot e^{\frac{0}{\color{blue}{0} \cdot \frac{2}{x}}}\right)} - 1 \]

5. Applied egg-rr 0.0%

   \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\cos x \cdot e^{\frac{0}{0 \cdot \frac{2}{x}}}\right)} - 1} \]
6. Step-by-step derivation

   1. expm1-def 0.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos x \cdot e^{\frac{0}{0 \cdot \frac{2}{x}}}\right)\right)} \]

   2. expm1-log1p 0.0%

      \[\leadsto \color{blue}{\cos x \cdot e^{\frac{0}{0 \cdot \frac{2}{x}}}} \]

   3. div0 9.6%

      \[\leadsto \cos x \cdot e^{\color{blue}{0}} \]

   4. 1-exp 9.6%

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

   5. *-rgt-identity 9.6%

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

7. Simplified 9.6%

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

8. Final simplification 9.6%

   \[\leadsto \cos x \]

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 94.3%

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

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 98.6%

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

   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 99.1%

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

   5. metadata-eval 99.1%

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

5. Applied egg-rr 99.1%

   \[\leadsto \sqrt[3]{{\color{blue}{\left(\sqrt{{\left(e^{60}\right)}^{x}}\right)}}^{x} \cdot {\cos x}^{3}} \]
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 99.1%

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

   3. pow-prod-down 99.1%

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

   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 2023185 
(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)))))