ENA, Section 1.4, Exercise 1

Percentage Accurate: 94.5% → 99.4%

Time: 8.1s

Alternatives: 11

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, 0.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[Sqrt[N[Power[N[Power[N[Exp[20.0], $MachinePrecision], x], $MachinePrecision], x], $MachinePrecision]], $MachinePrecision] * N[Cos[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. cos-neg 94.6%

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

   2. *-commutative 94.6%

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

   3. cos-neg 94.6%

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

3. Simplified 94.6%

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

   1. pow-exp 95.1%

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

   2. add-sqr-sqrt 95.1%

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

   3. sqrt-unprod 95.1%

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

   4. pow-prod-down 95.1%

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

   5. pow-pow 98.1%

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

   6. prod-exp 99.4%

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

   7. metadata-eval 99.4%

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

5. Applied egg-rr 99.4%

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

6. Final simplification 99.4%

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

Alternative 2: 98.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[Cos[x], $MachinePrecision] * N[Power[N[Power[N[Exp[5.0], $MachinePrecision], N[(x + x), $MachinePrecision]], $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. cos-neg 94.6%

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

   2. *-commutative 94.6%

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

   3. cos-neg 94.6%

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

3. Simplified 94.6%

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

   1. pow-exp 95.1%

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

   2. add-sqr-sqrt 95.1%

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

   3. sqrt-unprod 95.1%

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

   4. pow-prod-down 95.1%

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

   5. pow-pow 98.1%

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

   6. prod-exp 99.4%

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

   7. metadata-eval 99.4%

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

5. Applied egg-rr 99.4%

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

   1. sqrt-pow1 99.4%

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

   2. add-sqr-sqrt 99.1%

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

   3. unpow-prod-down 99.0%

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

   4. metadata-eval 99.0%

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

   5. prod-exp 98.7%

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

   6. pow-prod-down 98.7%

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

   7. exp-prod 97.1%

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

   8. exp-prod 96.0%

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

   9. sqrt-unprod 95.9%

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

   10. add-sqr-sqrt 96.0%

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

   11. sqrt-pow2 95.9%

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

7. Applied egg-rr 98.4%

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

8. Final simplification 98.4%

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

Alternative 3: 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 94.6%

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

   1. cos-neg 94.6%

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

   2. *-commutative 94.6%

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

   3. cos-neg 94.6%

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

3. Simplified 94.6%

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

   1. pow-exp 95.1%

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

   2. pow-pow 97.9%

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

   3. sqr-pow 98.0%

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

   4. pow-prod-down 98.0%

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

   5. pow-prod-down 98.0%

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

   6. prod-exp 99.4%

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

   7. metadata-eval 99.4%

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

   8. div-inv 99.4%

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

   9. metadata-eval 99.4%

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

5. Applied egg-rr 99.4%

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

6. Final simplification 99.4%

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

Alternative 4: 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 94.6%

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

   1. cos-neg 94.6%

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

   2. *-commutative 94.6%

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

   3. associate-*r* 94.6%

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

   4. exp-prod 94.8%

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

   5. exp-prod 97.9%

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

   6. cos-neg 97.9%

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

3. Simplified 97.9%

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

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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[Cos[x], $MachinePrecision] * N[Power[N[Exp[5.0], $MachinePrecision], N[(2.0 * 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. Step-by-step derivation

   1. cos-neg 94.6%

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

   2. *-commutative 94.6%

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

   3. cos-neg 94.6%

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

3. Simplified 94.6%

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

   1. pow-exp 95.1%

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

   2. add-sqr-sqrt 95.2%

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

   3. pow2 95.2%

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

   4. pow-pow 95.1%

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

   5. pow1/2 95.1%

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

   6. pow-exp 95.1%

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

   7. metadata-eval 95.1%

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

5. Applied egg-rr 95.1%

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

6. Final simplification 95.1%

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

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 94.6%

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

   1. cos-neg 94.6%

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

   2. *-commutative 94.6%

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

   3. exp-prod 95.1%

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

   4. cos-neg 95.1%

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

3. Simplified 95.1%

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

4. Final simplification 95.1%

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

Alternative 7: 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 94.6%

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

2. Final simplification 94.6%

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

Alternative 8: 18.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ e^{10 \cdot \left(x \cdot x\right)} \cdot \left(1 + \left(x \cdot x\right) \cdot -0.5\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (exp (* 10.0 (* x x))) (+ 1.0 (* (* x x) -0.5))))

C

double code(double x) {
	return exp((10.0 * (x * x))) * (1.0 + ((x * x) * -0.5));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = exp((10.0d0 * (x * x))) * (1.0d0 + ((x * x) * (-0.5d0)))
end function

Java

public static double code(double x) {
	return Math.exp((10.0 * (x * x))) * (1.0 + ((x * x) * -0.5));
}

Python

def code(x):
	return math.exp((10.0 * (x * x))) * (1.0 + ((x * x) * -0.5))

Julia

function code(x)
	return Float64(exp(Float64(10.0 * Float64(x * x))) * Float64(1.0 + Float64(Float64(x * x) * -0.5)))
end

MATLAB

function tmp = code(x)
	tmp = exp((10.0 * (x * x))) * (1.0 + ((x * x) * -0.5));
end

Wolfram

code[x_] := N[(N[Exp[N[(10.0 * N[(x * x), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(N[(x * x), $MachinePrecision] * -0.5), $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. cos-neg 94.6%

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

   2. *-commutative 94.6%

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

   3. cos-neg 94.6%

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

3. Simplified 94.6%

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

4. Taylor expanded in x around 0 18.2%

   \[\leadsto e^{10 \cdot \left(x \cdot x\right)} \cdot \color{blue}{\left(1 + -0.5 \cdot {x}^{2}\right)} \]
5. Step-by-step derivation

   1. unpow2 9.7%

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

6. Simplified 18.2%

   \[\leadsto e^{10 \cdot \left(x \cdot x\right)} \cdot \color{blue}{\left(1 + -0.5 \cdot \left(x \cdot x\right)\right)} \]

7. Final simplification 18.2%

   \[\leadsto e^{10 \cdot \left(x \cdot x\right)} \cdot \left(1 + \left(x \cdot x\right) \cdot -0.5\right) \]

Alternative 9: 9.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \cos x \cdot \left(10 \cdot \left(x \cdot x\right) + 1\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (cos x) (+ (* 10.0 (* x x)) 1.0)))

C

double code(double x) {
	return cos(x) * ((10.0 * (x * x)) + 1.0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = cos(x) * ((10.0d0 * (x * x)) + 1.0d0)
end function

Java

public static double code(double x) {
	return Math.cos(x) * ((10.0 * (x * x)) + 1.0);
}

Python

def code(x):
	return math.cos(x) * ((10.0 * (x * x)) + 1.0)

Julia

function code(x)
	return Float64(cos(x) * Float64(Float64(10.0 * Float64(x * x)) + 1.0))
end

MATLAB

function tmp = code(x)
	tmp = cos(x) * ((10.0 * (x * x)) + 1.0);
end

Wolfram

code[x_] := N[(N[Cos[x], $MachinePrecision] * N[(N[(10.0 * N[(x * x), $MachinePrecision]), $MachinePrecision] + 1.0), $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. cos-neg 94.6%

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

   2. *-commutative 94.6%

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

   3. cos-neg 94.6%

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

3. Simplified 94.6%

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

   1. pow-exp 95.1%

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

   2. add-sqr-sqrt 95.1%

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

   3. sqrt-unprod 95.1%

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

   4. pow-prod-down 95.1%

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

   5. pow-pow 98.1%

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

   6. prod-exp 99.4%

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

   7. metadata-eval 99.4%

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

5. Applied egg-rr 99.4%

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

6. Taylor expanded in x around 0 9.8%

   \[\leadsto \color{blue}{\left(1 + 10 \cdot {x}^{2}\right)} \cdot \cos x \]
7. Step-by-step derivation

   1. unpow2 9.8%

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

8. Simplified 9.8%

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

9. Final simplification 9.8%

   \[\leadsto \cos x \cdot \left(10 \cdot \left(x \cdot x\right) + 1\right) \]

Alternative 10: 9.7% accurate, 29.6× speedup

Math

\[\begin{array}{l} \\ 1 + \left(x \cdot x\right) \cdot -0.5 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (+ 1.0 (* (* x x) -0.5)))

C

double code(double x) {
	return 1.0 + ((x * x) * -0.5);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 + ((x * x) * (-0.5d0))
end function

Java

public static double code(double x) {
	return 1.0 + ((x * x) * -0.5);
}

Python

def code(x):
	return 1.0 + ((x * x) * -0.5)

Julia

function code(x)
	return Float64(1.0 + Float64(Float64(x * x) * -0.5))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 + ((x * x) * -0.5);
end

Wolfram

code[x_] := N[(1.0 + N[(N[(x * x), $MachinePrecision] * -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. cos-neg 94.6%

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

   2. *-commutative 94.6%

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

   3. cos-neg 94.6%

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

3. Simplified 94.6%

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

4. Taylor expanded in x around 0 9.6%

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

5. Taylor expanded in x around 0 9.7%

   \[\leadsto 1 \cdot \color{blue}{\left(1 + -0.5 \cdot {x}^{2}\right)} \]
6. Step-by-step derivation

   1. unpow2 9.7%

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

7. Simplified 9.7%

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

8. Final simplification 9.7%

   \[\leadsto 1 + \left(x \cdot x\right) \cdot -0.5 \]

Alternative 11: 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 94.6%

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

   1. cos-neg 94.6%

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

   2. *-commutative 94.6%

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

   3. cos-neg 94.6%

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

3. Simplified 94.6%

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

   1. pow-exp 95.1%

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

   2. add-sqr-sqrt 95.1%

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

   3. sqrt-unprod 95.1%

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

   4. pow-prod-down 95.1%

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

   5. pow-pow 98.1%

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

   6. prod-exp 99.4%

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

   7. metadata-eval 99.4%

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

5. Applied egg-rr 99.4%

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

   1. sqrt-pow1 99.4%

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

7. Applied egg-rr 99.4%

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

8. Taylor expanded in x around 0 1.5%

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

9. Final simplification 1.5%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023290 
(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)))))