ENA, Section 1.4, Exercise 1

Percentage Accurate: 94.5% → 99.4%

Time: 10.2s

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

\[\begin{array}{l} \\ \cos x \cdot {\left({\left(e^{20}\right)}^{x}\right)}^{\left(\frac{x}{2}\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (cos x) (pow (pow (exp 20.0) x) (/ x 2.0))))

C

double code(double x) {
	return cos(x) * pow(pow(exp(20.0), x), (x / 2.0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = cos(x) * ((exp(20.0d0) ** x) ** (x / 2.0d0))
end function

Java

public static double code(double x) {
	return Math.cos(x) * Math.pow(Math.pow(Math.exp(20.0), x), (x / 2.0));
}

Python

def code(x):
	return math.cos(x) * math.pow(math.pow(math.exp(20.0), x), (x / 2.0))

Julia

function code(x)
	return Float64(cos(x) * ((exp(20.0) ^ x) ^ Float64(x / 2.0)))
end

MATLAB

function tmp = code(x)
	tmp = cos(x) * ((exp(20.0) ^ x) ^ (x / 2.0));
end

Wolfram

code[x_] := N[(N[Cos[x], $MachinePrecision] * N[Power[N[Power[N[Exp[20.0], $MachinePrecision], x], $MachinePrecision], N[(x / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.5%

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

   1. pow-exp 95.2%

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

   2. sqr-pow 95.2%

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

   3. pow-prod-down 95.2%

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

   4. associate-/l* 95.2%

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

   5. pow-unpow 98.0%

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

   6. prod-exp 99.4%

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

   7. metadata-eval 99.4%

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

4. Applied egg-rr 99.4%

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

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. Add Preprocessing
3. 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. pow-unpow 98.0%

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

4. Applied egg-rr 98.0%

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

Alternative 3: 95.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \cos x \cdot {\left(e^{40}\right)}^{\left(x \cdot \left(x \cdot 0.25\right)\right)} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (cos x) (pow (exp 40.0) (* x (* x 0.25)))))

C

double code(double x) {
	return cos(x) * pow(exp(40.0), (x * (x * 0.25)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = cos(x) * (exp(40.0d0) ** (x * (x * 0.25d0)))
end function

Java

public static double code(double x) {
	return Math.cos(x) * Math.pow(Math.exp(40.0), (x * (x * 0.25)));
}

Python

def code(x):
	return math.cos(x) * math.pow(math.exp(40.0), (x * (x * 0.25)))

Julia

function code(x)
	return Float64(cos(x) * (exp(40.0) ^ Float64(x * Float64(x * 0.25))))
end

MATLAB

function tmp = code(x)
	tmp = cos(x) * (exp(40.0) ^ (x * (x * 0.25)));
end

Wolfram

code[x_] := N[(N[Cos[x], $MachinePrecision] * N[Power[N[Exp[40.0], $MachinePrecision], N[(x * N[(x * 0.25), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.5%

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

   1. pow-exp 95.2%

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

   2. sqr-pow 95.2%

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

   3. pow-prod-down 95.2%

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

   4. associate-/l* 95.2%

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

   5. pow-unpow 98.0%

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

   6. prod-exp 99.4%

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

   7. metadata-eval 99.4%

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

4. Applied egg-rr 99.4%

   \[\leadsto \cos x \cdot \color{blue}{{\left({\left(e^{20}\right)}^{x}\right)}^{\left(\frac{x}{2}\right)}} \]
5. Step-by-step derivation

   1. expm1-log1p-u 99.3%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos x\right)\right)} \cdot {\left({\left(e^{20}\right)}^{x}\right)}^{\left(\frac{x}{2}\right)} \]

6. Applied egg-rr 99.3%

   \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cos x\right)\right)} \cdot {\left({\left(e^{20}\right)}^{x}\right)}^{\left(\frac{x}{2}\right)} \]

7. Taylor expanded in x around inf 94.4%

   \[\leadsto \color{blue}{\cos x \cdot e^{0.5 \cdot \left(x \cdot \log \left({\left(e^{20}\right)}^{x}\right)\right)}} \]
8. Step-by-step derivation

   1. *-commutative 94.4%

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

   2. exp-prod 94.4%

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

   3. unpow1/2 94.4%

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

   4. *-commutative 94.4%

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

   5. exp-to-pow 99.4%

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

9. Simplified 99.4%

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

    1. sqrt-pow1 99.4%

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

    2. add-sqr-sqrt 99.1%

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

    3. add-sqr-sqrt 99.4%

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

    4. sqr-pow 99.1%

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

    5. pow-prod-down 99.2%

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

    6. prod-exp 99.2%

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

    7. metadata-eval 99.2%

       \[\leadsto \cos x \cdot {\left({\left(e^{\color{blue}{40}}\right)}^{\left(\frac{x}{2}\right)}\right)}^{\left(\frac{x}{2}\right)} \]

    8. sqrt-pow1 99.2%

       \[\leadsto \cos x \cdot {\color{blue}{\left(\sqrt{{\left(e^{40}\right)}^{x}}\right)}}^{\left(\frac{x}{2}\right)} \]

    9. sqrt-pow2 99.3%

       \[\leadsto \cos x \cdot \color{blue}{{\left({\left(e^{40}\right)}^{x}\right)}^{\left(\frac{\frac{x}{2}}{2}\right)}} \]

    10. pow-pow 95.3%

        \[\leadsto \cos x \cdot \color{blue}{{\left(e^{40}\right)}^{\left(x \cdot \frac{\frac{x}{2}}{2}\right)}} \]

    11. div-inv 95.3%

        \[\leadsto \cos x \cdot {\left(e^{40}\right)}^{\left(x \cdot \frac{\color{blue}{x \cdot \frac{1}{2}}}{2}\right)} \]

    12. metadata-eval 95.3%

        \[\leadsto \cos x \cdot {\left(e^{40}\right)}^{\left(x \cdot \frac{x \cdot \color{blue}{0.5}}{2}\right)} \]

    13. associate-/l* 95.3%

        \[\leadsto \cos x \cdot {\left(e^{40}\right)}^{\left(x \cdot \color{blue}{\left(x \cdot \frac{0.5}{2}\right)}\right)} \]

    14. metadata-eval 95.3%

        \[\leadsto \cos x \cdot {\left(e^{40}\right)}^{\left(x \cdot \left(x \cdot \color{blue}{0.25}\right)\right)} \]

11. Applied egg-rr 95.3%

    \[\leadsto \cos x \cdot \color{blue}{{\left(e^{40}\right)}^{\left(x \cdot \left(x \cdot 0.25\right)\right)}} \]
12. Add Preprocessing

Alternative 4: 95.3% 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 94.5%

   \[\cos x \cdot e^{10 \cdot \left(x \cdot x\right)} \]
2. Add Preprocessing
3. 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. add-sqr-sqrt 95.3%

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

   3. pow2 95.3%

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

   4. pow-pow 95.3%

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

   5. pow1/2 95.3%

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

   6. pow-to-exp 95.3%

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

   7. rem-log-exp 95.3%

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

   8. metadata-eval 95.3%

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

   9. pow2 95.3%

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

4. Applied egg-rr 95.3%

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

   1. pow2 95.3%

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

6. Applied egg-rr 95.3%

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

Alternative 5: 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. Add Preprocessing
5. Add Preprocessing

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. Add Preprocessing
3. Add Preprocessing

Alternative 7: 1.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{1 + {x}^{2} \cdot 28.5} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (cbrt (+ 1.0 (* (pow x 2.0) 28.5))))

C

double code(double x) {
	return cbrt((1.0 + (pow(x, 2.0) * 28.5)));
}

Java

public static double code(double x) {
	return Math.cbrt((1.0 + (Math.pow(x, 2.0) * 28.5)));
}

Julia

function code(x)
	return cbrt(Float64(1.0 + Float64((x ^ 2.0) * 28.5)))
end

Wolfram

code[x_] := N[Power[N[(1.0 + N[(N[Power[x, 2.0], $MachinePrecision] * 28.5), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]

Derivation

1. Initial program 94.5%

   \[\cos x \cdot e^{10 \cdot \left(x \cdot x\right)} \]
2. Add Preprocessing
3. 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. add-cbrt-cube 95.2%

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

   4. add-cbrt-cube 95.2%

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

   5. cbrt-unprod 95.2%

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

   6. pow3 95.2%

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

   7. add-exp-log 94.5%

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

   8. log-pow 94.6%

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

   9. pow-exp 95.2%

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

   10. pow-pow 95.2%

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

   11. pow2 95.2%

       \[\leadsto \sqrt[3]{{\left(e^{\color{blue}{{x}^{2}}}\right)}^{\left(\log \left(e^{10}\right) \cdot 3\right)} \cdot \left(\left(\cos x \cdot \cos x\right) \cdot \cos x\right)} \]

   12. rem-log-exp 95.2%

       \[\leadsto \sqrt[3]{{\left(e^{{x}^{2}}\right)}^{\left(\color{blue}{10} \cdot 3\right)} \cdot \left(\left(\cos x \cdot \cos x\right) \cdot \cos x\right)} \]

   13. metadata-eval 95.2%

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

   14. pow3 95.2%

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

4. Applied egg-rr 95.2%

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

5. Taylor expanded in x around 0 1.5%

   \[\leadsto \sqrt[3]{\color{blue}{1 + 28.5 \cdot {x}^{2}}} \]
6. Step-by-step derivation

   1. *-commutative 1.5%

      \[\leadsto \sqrt[3]{1 + \color{blue}{{x}^{2} \cdot 28.5}} \]

7. Simplified 1.5%

   \[\leadsto \sqrt[3]{\color{blue}{1 + {x}^{2} \cdot 28.5}} \]
8. Add Preprocessing

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. Add Preprocessing

3. Taylor expanded in x around 0 1.5%

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

   1. *-commutative 1.5%

      \[\leadsto 1 + \color{blue}{{x}^{2} \cdot 9.5} \]

5. Simplified 1.5%

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

   1. pow2 95.3%

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

7. Applied egg-rr 1.5%

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

Reproduce

herbie shell --seed 2024179 
(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)))))