ENA, Section 1.4, Exercise 1

Percentage Accurate: 94.5% → 99.4%

Time: 8.3s

Alternatives: 8

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: 99.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

function code(x)
	return Float64(cos(x) * ((exp(20.0) ^ x) ^ Float64(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[Power[N[Exp[20.0], $MachinePrecision], x], $MachinePrecision], N[(x * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.5%

   \[\cos x \cdot e^{10 \cdot \left(x \cdot x\right)} \]
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. *-commutative 95.2%

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

   3. pow-exp 94.5%

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

   4. associate-*r* 94.3%

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

   5. add-log-exp 94.3%

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

   6. log-pow 94.3%

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

   7. pow-pow 94.8%

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

   8. add-exp-log 96.9%

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

   9. add-cbrt-cube 96.8%

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

   10. add-cbrt-cube 96.8%

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

   11. cbrt-unprod 96.8%

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

3. Applied egg-rr 98.9%

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

   1. add-cube-cbrt 98.9%

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

   2. rem-cbrt-cube 98.9%

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

   3. rem-cbrt-cube 98.9%

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

   4. pow2 98.9%

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

   5. rem-cbrt-cube 98.9%

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

5. Applied egg-rr 98.9%

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

6. Taylor expanded in x around inf 94.0%

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

   1. *-commutative 94.0%

      \[\leadsto \color{blue}{\cos x \cdot {\left({\left({\left(e^{30}\right)}^{x}\right)}^{x} \cdot 1\right)}^{0.3333333333333333}} \]

   2. unpow1/3 98.8%

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

   3. *-rgt-identity 98.8%

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

   4. exp-prod 95.4%

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

   5. *-commutative 95.4%

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

   6. exp-prod 95.2%

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

   7. *-commutative 95.2%

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

   8. exp-prod 95.5%

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

8. Simplified 95.5%

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

   1. add-exp-log 94.6%

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

   2. pow1/3 94.1%

      \[\leadsto \cos x \cdot e^{\log \color{blue}{\left({\left({\left(e^{x}\right)}^{\left(x \cdot 30\right)}\right)}^{0.3333333333333333}\right)}} \]

   3. pow-pow 93.7%

      \[\leadsto \cos x \cdot e^{\log \color{blue}{\left({\left(e^{x}\right)}^{\left(\left(x \cdot 30\right) \cdot 0.3333333333333333\right)}\right)}} \]

   4. log-pow 93.9%

      \[\leadsto \cos x \cdot e^{\color{blue}{\left(\left(x \cdot 30\right) \cdot 0.3333333333333333\right) \cdot \log \left(e^{x}\right)}} \]

   5. add-log-exp 93.9%

      \[\leadsto \cos x \cdot e^{\left(\left(x \cdot 30\right) \cdot 0.3333333333333333\right) \cdot \color{blue}{x}} \]

   6. pow-exp 94.0%

      \[\leadsto \cos x \cdot \color{blue}{{\left(e^{\left(x \cdot 30\right) \cdot 0.3333333333333333}\right)}^{x}} \]

   7. pow-exp 94.5%

      \[\leadsto \cos x \cdot {\color{blue}{\left({\left(e^{x \cdot 30}\right)}^{0.3333333333333333}\right)}}^{x} \]

   8. *-commutative 94.5%

      \[\leadsto \cos x \cdot {\left({\left(e^{\color{blue}{30 \cdot x}}\right)}^{0.3333333333333333}\right)}^{x} \]

   9. pow-exp 94.0%

      \[\leadsto \cos x \cdot {\left({\color{blue}{\left({\left(e^{30}\right)}^{x}\right)}}^{0.3333333333333333}\right)}^{x} \]

10. Applied egg-rr 99.4%

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

11. Final simplification 99.4%

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

Alternative 2: 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 94.5%

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

   1. associate-*r* 94.3%

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

   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 3: 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 94.5%

   \[\cos x \cdot e^{10 \cdot \left(x \cdot x\right)} \]
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. *-commutative 95.2%

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

   3. pow-exp 94.5%

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

   4. associate-*r* 94.3%

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

   5. add-log-exp 94.3%

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

   6. log-pow 94.3%

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

   7. pow-pow 94.8%

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

   8. add-exp-log 96.9%

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

   9. add-cbrt-cube 96.8%

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

   10. add-cbrt-cube 96.8%

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

   11. cbrt-unprod 96.8%

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

3. Applied egg-rr 98.9%

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

   1. add-cube-cbrt 98.9%

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

   2. rem-cbrt-cube 98.9%

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

   3. rem-cbrt-cube 98.9%

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

   4. pow2 98.9%

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

   5. rem-cbrt-cube 98.9%

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

5. Applied egg-rr 98.9%

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

6. Taylor expanded in x around inf 94.0%

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

   1. *-commutative 94.0%

      \[\leadsto \color{blue}{\cos x \cdot {\left({\left({\left(e^{30}\right)}^{x}\right)}^{x} \cdot 1\right)}^{0.3333333333333333}} \]

   2. unpow1/3 98.8%

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

   3. *-rgt-identity 98.8%

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

   4. exp-prod 95.4%

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

   5. *-commutative 95.4%

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

   6. exp-prod 95.2%

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

   7. *-commutative 95.2%

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

   8. exp-prod 95.5%

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

8. Simplified 95.5%

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

   1. pow-exp 95.2%

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

10. Applied egg-rr 95.2%

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

11. Final simplification 95.2%

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

Alternative 4: 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 94.5%

   \[\cos x \cdot e^{10 \cdot \left(x \cdot x\right)} \]
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 5: 95.2% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.5%

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

   1. *-commutative 94.5%

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

   2. exp-prod 95.2%

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

   3. exp-prod 96.7%

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

3. Applied egg-rr 96.7%

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

4. Taylor expanded in x around inf 95.2%

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

   1. unpow2 95.2%

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

6. Simplified 95.2%

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

7. Final simplification 95.2%

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

Alternative 6: 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 94.5%

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

2. Final simplification 94.5%

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

Alternative 7: 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 94.5%

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

2. Taylor expanded in x around 0 9.6%

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

3. Final simplification 9.6%

   \[\leadsto \cos x \]

Alternative 8: 1.5% accurate, 29.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.5%

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

2. Taylor expanded in x around 0 1.5%

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

   1. unpow2 1.5%

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

4. Simplified 1.5%

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

5. Final simplification 1.5%

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

Reproduce

herbie shell --seed 2023178 
(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)))))