ENA, Section 1.4, Exercise 1

Percentage Accurate: 94.4% → 99.2%

Time: 11.8s

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: 94.4% 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: 99.2% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.1%

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

   1. pow-exp 95.1%

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

   2. pow-unpow 97.9%

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

4. Applied egg-rr 97.9%

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

   1. add-sqr-sqrt 97.7%

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

   2. sqrt-unprod 97.9%

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

   3. pow-prod-down 97.9%

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

   4. prod-exp 99.2%

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

   5. metadata-eval 99.2%

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

6. Applied egg-rr 99.2%

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

   1. pow1/2 99.2%

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

   2. pow-pow 99.2%

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

8. Applied egg-rr 99.2%

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

9. Final simplification 99.2%

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

Alternative 2: 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 94.1%

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

   1. pow-exp 95.1%

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

   2. pow-unpow 97.9%

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

4. Applied egg-rr 97.9%

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

5. Final simplification 97.9%

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

Alternative 3: 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 94.1%

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

   1. exp-prod 95.1%

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

3. Simplified 95.1%

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

5. Final simplification 95.1%

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

Alternative 4: 94.4% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x)
 :precision binary64
 (* (/ -1.0 (/ -1.0 (cos x))) (exp (* x (* x 10.0)))))

C

double code(double x) {
	return (-1.0 / (-1.0 / cos(x))) * exp((x * (x * 10.0)));
}

Fortran

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

Java

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

Python

def code(x):
	return (-1.0 / (-1.0 / math.cos(x))) * math.exp((x * (x * 10.0)))

Julia

function code(x)
	return Float64(Float64(-1.0 / Float64(-1.0 / cos(x))) * exp(Float64(x * Float64(x * 10.0))))
end

MATLAB

function tmp = code(x)
	tmp = (-1.0 / (-1.0 / cos(x))) * exp((x * (x * 10.0)));
end

Wolfram

code[x_] := N[(N[(-1.0 / N[(-1.0 / N[Cos[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Exp[N[(x * N[(x * 10.0), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.1%

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

   1. associate-*r* 94.1%

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

3. Simplified 94.1%

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

   1. add-log-exp 94.1%

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

   2. add-cube-cbrt 94.2%

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

   3. log-prod 94.1%

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

   4. cos-sum 94.1%

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

6. Applied egg-rr 94.1%

   \[\leadsto \color{blue}{\left(\cos \left(2 \cdot \left(0.3333333333333333 \cdot x\right)\right) \cdot \cos \left(0.3333333333333333 \cdot x\right) - \sin \left(2 \cdot \left(0.3333333333333333 \cdot x\right)\right) \cdot \sin \left(0.3333333333333333 \cdot x\right)\right)} \cdot e^{\left(10 \cdot x\right) \cdot x} \]
7. Step-by-step derivation

   1. associate-*r* 94.1%

      \[\leadsto \left(\cos \color{blue}{\left(\left(2 \cdot 0.3333333333333333\right) \cdot x\right)} \cdot \cos \left(0.3333333333333333 \cdot x\right) - \sin \left(2 \cdot \left(0.3333333333333333 \cdot x\right)\right) \cdot \sin \left(0.3333333333333333 \cdot x\right)\right) \cdot e^{\left(10 \cdot x\right) \cdot x} \]

   2. *-commutative 94.1%

      \[\leadsto \left(\cos \color{blue}{\left(x \cdot \left(2 \cdot 0.3333333333333333\right)\right)} \cdot \cos \left(0.3333333333333333 \cdot x\right) - \sin \left(2 \cdot \left(0.3333333333333333 \cdot x\right)\right) \cdot \sin \left(0.3333333333333333 \cdot x\right)\right) \cdot e^{\left(10 \cdot x\right) \cdot x} \]

   3. metadata-eval 94.1%

      \[\leadsto \left(\cos \left(x \cdot \color{blue}{0.6666666666666666}\right) \cdot \cos \left(0.3333333333333333 \cdot x\right) - \sin \left(2 \cdot \left(0.3333333333333333 \cdot x\right)\right) \cdot \sin \left(0.3333333333333333 \cdot x\right)\right) \cdot e^{\left(10 \cdot x\right) \cdot x} \]

   4. *-commutative 94.1%

      \[\leadsto \left(\cos \left(x \cdot 0.6666666666666666\right) \cdot \cos \color{blue}{\left(x \cdot 0.3333333333333333\right)} - \sin \left(2 \cdot \left(0.3333333333333333 \cdot x\right)\right) \cdot \sin \left(0.3333333333333333 \cdot x\right)\right) \cdot e^{\left(10 \cdot x\right) \cdot x} \]

   5. associate-*r* 94.1%

      \[\leadsto \left(\cos \left(x \cdot 0.6666666666666666\right) \cdot \cos \left(x \cdot 0.3333333333333333\right) - \sin \color{blue}{\left(\left(2 \cdot 0.3333333333333333\right) \cdot x\right)} \cdot \sin \left(0.3333333333333333 \cdot x\right)\right) \cdot e^{\left(10 \cdot x\right) \cdot x} \]

   6. *-commutative 94.1%

      \[\leadsto \left(\cos \left(x \cdot 0.6666666666666666\right) \cdot \cos \left(x \cdot 0.3333333333333333\right) - \sin \color{blue}{\left(x \cdot \left(2 \cdot 0.3333333333333333\right)\right)} \cdot \sin \left(0.3333333333333333 \cdot x\right)\right) \cdot e^{\left(10 \cdot x\right) \cdot x} \]

   7. metadata-eval 94.1%

      \[\leadsto \left(\cos \left(x \cdot 0.6666666666666666\right) \cdot \cos \left(x \cdot 0.3333333333333333\right) - \sin \left(x \cdot \color{blue}{0.6666666666666666}\right) \cdot \sin \left(0.3333333333333333 \cdot x\right)\right) \cdot e^{\left(10 \cdot x\right) \cdot x} \]

   8. *-commutative 94.1%

      \[\leadsto \left(\cos \left(x \cdot 0.6666666666666666\right) \cdot \cos \left(x \cdot 0.3333333333333333\right) - \sin \left(x \cdot 0.6666666666666666\right) \cdot \sin \color{blue}{\left(x \cdot 0.3333333333333333\right)}\right) \cdot e^{\left(10 \cdot x\right) \cdot x} \]

8. Simplified 94.1%

   \[\leadsto \color{blue}{\left(\cos \left(x \cdot 0.6666666666666666\right) \cdot \cos \left(x \cdot 0.3333333333333333\right) - \sin \left(x \cdot 0.6666666666666666\right) \cdot \sin \left(x \cdot 0.3333333333333333\right)\right)} \cdot e^{\left(10 \cdot x\right) \cdot x} \]
9. Step-by-step derivation

   1. flip-- 94.1%

      \[\leadsto \color{blue}{\frac{\left(\cos \left(x \cdot 0.6666666666666666\right) \cdot \cos \left(x \cdot 0.3333333333333333\right)\right) \cdot \left(\cos \left(x \cdot 0.6666666666666666\right) \cdot \cos \left(x \cdot 0.3333333333333333\right)\right) - \left(\sin \left(x \cdot 0.6666666666666666\right) \cdot \sin \left(x \cdot 0.3333333333333333\right)\right) \cdot \left(\sin \left(x \cdot 0.6666666666666666\right) \cdot \sin \left(x \cdot 0.3333333333333333\right)\right)}{\cos \left(x \cdot 0.6666666666666666\right) \cdot \cos \left(x \cdot 0.3333333333333333\right) + \sin \left(x \cdot 0.6666666666666666\right) \cdot \sin \left(x \cdot 0.3333333333333333\right)}} \cdot e^{\left(10 \cdot x\right) \cdot x} \]

   2. cos-diff 94.1%

      \[\leadsto \frac{\left(\cos \left(x \cdot 0.6666666666666666\right) \cdot \cos \left(x \cdot 0.3333333333333333\right)\right) \cdot \left(\cos \left(x \cdot 0.6666666666666666\right) \cdot \cos \left(x \cdot 0.3333333333333333\right)\right) - \left(\sin \left(x \cdot 0.6666666666666666\right) \cdot \sin \left(x \cdot 0.3333333333333333\right)\right) \cdot \left(\sin \left(x \cdot 0.6666666666666666\right) \cdot \sin \left(x \cdot 0.3333333333333333\right)\right)}{\color{blue}{\cos \left(x \cdot 0.6666666666666666 - x \cdot 0.3333333333333333\right)}} \cdot e^{\left(10 \cdot x\right) \cdot x} \]

   3. clear-num 94.1%

      \[\leadsto \color{blue}{\frac{1}{\frac{\cos \left(x \cdot 0.6666666666666666 - x \cdot 0.3333333333333333\right)}{\left(\cos \left(x \cdot 0.6666666666666666\right) \cdot \cos \left(x \cdot 0.3333333333333333\right)\right) \cdot \left(\cos \left(x \cdot 0.6666666666666666\right) \cdot \cos \left(x \cdot 0.3333333333333333\right)\right) - \left(\sin \left(x \cdot 0.6666666666666666\right) \cdot \sin \left(x \cdot 0.3333333333333333\right)\right) \cdot \left(\sin \left(x \cdot 0.6666666666666666\right) \cdot \sin \left(x \cdot 0.3333333333333333\right)\right)}}} \cdot e^{\left(10 \cdot x\right) \cdot x} \]

   4. distribute-lft-out-- 94.1%

      \[\leadsto \frac{1}{\frac{\cos \color{blue}{\left(x \cdot \left(0.6666666666666666 - 0.3333333333333333\right)\right)}}{\left(\cos \left(x \cdot 0.6666666666666666\right) \cdot \cos \left(x \cdot 0.3333333333333333\right)\right) \cdot \left(\cos \left(x \cdot 0.6666666666666666\right) \cdot \cos \left(x \cdot 0.3333333333333333\right)\right) - \left(\sin \left(x \cdot 0.6666666666666666\right) \cdot \sin \left(x \cdot 0.3333333333333333\right)\right) \cdot \left(\sin \left(x \cdot 0.6666666666666666\right) \cdot \sin \left(x \cdot 0.3333333333333333\right)\right)}} \cdot e^{\left(10 \cdot x\right) \cdot x} \]

   5. metadata-eval 94.1%

      \[\leadsto \frac{1}{\frac{\cos \left(x \cdot \color{blue}{0.3333333333333333}\right)}{\left(\cos \left(x \cdot 0.6666666666666666\right) \cdot \cos \left(x \cdot 0.3333333333333333\right)\right) \cdot \left(\cos \left(x \cdot 0.6666666666666666\right) \cdot \cos \left(x \cdot 0.3333333333333333\right)\right) - \left(\sin \left(x \cdot 0.6666666666666666\right) \cdot \sin \left(x \cdot 0.3333333333333333\right)\right) \cdot \left(\sin \left(x \cdot 0.6666666666666666\right) \cdot \sin \left(x \cdot 0.3333333333333333\right)\right)}} \cdot e^{\left(10 \cdot x\right) \cdot x} \]

10. Applied egg-rr 94.1%

    \[\leadsto \color{blue}{\frac{1}{\frac{\cos \left(x \cdot 0.3333333333333333\right)}{\cos \left(x \cdot 0.3333333333333333\right) \cdot \cos x}}} \cdot e^{\left(10 \cdot x\right) \cdot x} \]
11. Step-by-step derivation

    1. frac-2neg 94.1%

       \[\leadsto \color{blue}{\frac{-1}{-\frac{\cos \left(x \cdot 0.3333333333333333\right)}{\cos \left(x \cdot 0.3333333333333333\right) \cdot \cos x}}} \cdot e^{\left(10 \cdot x\right) \cdot x} \]

    2. metadata-eval 94.1%

       \[\leadsto \frac{\color{blue}{-1}}{-\frac{\cos \left(x \cdot 0.3333333333333333\right)}{\cos \left(x \cdot 0.3333333333333333\right) \cdot \cos x}} \cdot e^{\left(10 \cdot x\right) \cdot x} \]

    3. div-inv 94.1%

       \[\leadsto \color{blue}{\left(-1 \cdot \frac{1}{-\frac{\cos \left(x \cdot 0.3333333333333333\right)}{\cos \left(x \cdot 0.3333333333333333\right) \cdot \cos x}}\right)} \cdot e^{\left(10 \cdot x\right) \cdot x} \]

    4. /-rgt-identity 94.1%

       \[\leadsto \left(-1 \cdot \frac{1}{-\color{blue}{\frac{\frac{\cos \left(x \cdot 0.3333333333333333\right)}{\cos \left(x \cdot 0.3333333333333333\right) \cdot \cos x}}{1}}}\right) \cdot e^{\left(10 \cdot x\right) \cdot x} \]

    5. /-rgt-identity 94.1%

       \[\leadsto \left(-1 \cdot \frac{1}{-\color{blue}{\frac{\cos \left(x \cdot 0.3333333333333333\right)}{\cos \left(x \cdot 0.3333333333333333\right) \cdot \cos x}}}\right) \cdot e^{\left(10 \cdot x\right) \cdot x} \]

    6. associate-/r* 94.1%

       \[\leadsto \left(-1 \cdot \frac{1}{-\color{blue}{\frac{\frac{\cos \left(x \cdot 0.3333333333333333\right)}{\cos \left(x \cdot 0.3333333333333333\right)}}{\cos x}}}\right) \cdot e^{\left(10 \cdot x\right) \cdot x} \]

    7. *-inverses 94.1%

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

    8. distribute-neg-frac 94.1%

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

    9. metadata-eval 94.1%

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

12. Applied egg-rr 94.1%

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

13. Final simplification 94.1%

    \[\leadsto \frac{-1}{\frac{-1}{\cos x}} \cdot e^{x \cdot \left(x \cdot 10\right)} \]
14. Add Preprocessing

Alternative 5: 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 94.1%

   \[\cos x \cdot e^{10 \cdot \left(x \cdot x\right)} \]
2. Add Preprocessing

3. Final simplification 94.1%

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

Alternative 6: 94.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.1%

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

   1. associate-*r* 94.1%

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

3. Simplified 94.1%

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

5. Final simplification 94.1%

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

Alternative 7: 9.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ {x}^{2} \cdot -0.5 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (pow x 2.0) -0.5))

C

double code(double x) {
	return pow(x, 2.0) * -0.5;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x ** 2.0d0) * (-0.5d0)
end function

Java

public static double code(double x) {
	return Math.pow(x, 2.0) * -0.5;
}

Python

def code(x):
	return math.pow(x, 2.0) * -0.5

Julia

function code(x)
	return Float64((x ^ 2.0) * -0.5)
end

MATLAB

function tmp = code(x)
	tmp = (x ^ 2.0) * -0.5;
end

Wolfram

code[x_] := N[(N[Power[x, 2.0], $MachinePrecision] * -0.5), $MachinePrecision]

Derivation

1. Initial program 94.1%

   \[\cos x \cdot e^{10 \cdot \left(x \cdot x\right)} \]
2. Add Preprocessing

3. Taylor expanded in x around 0 9.6%

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

4. Taylor expanded in x around 0 9.7%

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

   1. *-commutative 9.7%

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

6. Simplified 9.7%

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

7. Taylor expanded in x around inf 9.7%

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

   1. *-commutative 9.7%

      \[\leadsto \color{blue}{\left({x}^{2} \cdot -0.5\right)} \cdot 1 \]

9. Simplified 9.7%

   \[\leadsto \color{blue}{\left({x}^{2} \cdot -0.5\right)} \cdot 1 \]

10. Final simplification 9.7%

    \[\leadsto {x}^{2} \cdot -0.5 \]
11. Add Preprocessing

Alternative 8: 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 94.1%

   \[\cos x \cdot e^{10 \cdot \left(x \cdot x\right)} \]
2. Add Preprocessing

3. Taylor expanded in x around 0 9.6%

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

4. Taylor expanded in x around 0 9.7%

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

   1. *-commutative 9.7%

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

6. Simplified 9.7%

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

   1. pow2 9.7%

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

8. Applied egg-rr 9.7%

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

9. Final simplification 9.7%

   \[\leadsto 1 + \left(x \cdot x\right) \cdot -0.5 \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024021 
(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)))))