ENA, Section 1.4, Exercise 1

Percentage Accurate: 94.4% → 99.4%

Time: 11.5s

Alternatives: 12

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: 99.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \cos x \cdot \sqrt{{\left({\left(e^{80}\right)}^{x}\right)}^{\left(x \cdot 0.25\right)}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (cos x) (sqrt (pow (pow (exp 80.0) x) (* x 0.25)))))

C

double code(double x) {
	return cos(x) * sqrt(pow(pow(exp(80.0), x), (x * 0.25)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = cos(x) * sqrt(((exp(80.0d0) ** x) ** (x * 0.25d0)))
end function

Java

public static double code(double x) {
	return Math.cos(x) * Math.sqrt(Math.pow(Math.pow(Math.exp(80.0), x), (x * 0.25)));
}

Python

def code(x):
	return math.cos(x) * math.sqrt(math.pow(math.pow(math.exp(80.0), x), (x * 0.25)))

Julia

function code(x)
	return Float64(cos(x) * sqrt(((exp(80.0) ^ x) ^ Float64(x * 0.25))))
end

MATLAB

function tmp = code(x)
	tmp = cos(x) * sqrt(((exp(80.0) ^ x) ^ (x * 0.25)));
end

Wolfram

code[x_] := N[(N[Cos[x], $MachinePrecision] * N[Sqrt[N[Power[N[Power[N[Exp[80.0], $MachinePrecision], x], $MachinePrecision], N[(x * 0.25), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.4%

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

   1. pow-exp 95.3%

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

   2. sqr-pow 95.3%

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

   3. pow-prod-down 95.3%

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

   4. associate-/l* 95.3%

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

   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. add-sqr-sqrt 99.2%

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

   2. sqrt-unprod 99.4%

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

   3. pow-prod-down 99.2%

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

   4. prod-exp 99.2%

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

   5. metadata-eval 99.2%

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

6. Applied egg-rr 99.2%

   \[\leadsto \cos x \cdot {\color{blue}{\left(\sqrt{{\left(e^{40}\right)}^{x}}\right)}}^{\left(\frac{x}{2}\right)} \]
7. Step-by-step derivation

   1. add-sqr-sqrt 99.2%

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

   2. sqrt-unprod 99.2%

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

   3. pow-prod-down 99.2%

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

   4. prod-exp 99.4%

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

   5. metadata-eval 99.4%

      \[\leadsto \cos x \cdot {\left(\sqrt{\sqrt{{\left(e^{\color{blue}{80}}\right)}^{x}}}\right)}^{\left(\frac{x}{2}\right)} \]

8. Applied egg-rr 99.4%

   \[\leadsto \cos x \cdot {\left(\sqrt{\color{blue}{\sqrt{{\left(e^{80}\right)}^{x}}}}\right)}^{\left(\frac{x}{2}\right)} \]
9. Step-by-step derivation

   1. add-sqr-sqrt 99.0%

      \[\leadsto \cos x \cdot \color{blue}{\left(\sqrt{{\left(\sqrt{\sqrt{{\left(e^{80}\right)}^{x}}}\right)}^{\left(\frac{x}{2}\right)}} \cdot \sqrt{{\left(\sqrt{\sqrt{{\left(e^{80}\right)}^{x}}}\right)}^{\left(\frac{x}{2}\right)}}\right)} \]

   2. sqrt-unprod 99.4%

      \[\leadsto \cos x \cdot \color{blue}{\sqrt{{\left(\sqrt{\sqrt{{\left(e^{80}\right)}^{x}}}\right)}^{\left(\frac{x}{2}\right)} \cdot {\left(\sqrt{\sqrt{{\left(e^{80}\right)}^{x}}}\right)}^{\left(\frac{x}{2}\right)}}} \]

   3. sqrt-pow2 99.4%

      \[\leadsto \cos x \cdot \sqrt{\color{blue}{{\left(\sqrt{{\left(e^{80}\right)}^{x}}\right)}^{\left(\frac{\frac{x}{2}}{2}\right)}} \cdot {\left(\sqrt{\sqrt{{\left(e^{80}\right)}^{x}}}\right)}^{\left(\frac{x}{2}\right)}} \]

   4. sqrt-pow2 99.4%

      \[\leadsto \cos x \cdot \sqrt{{\left(\sqrt{{\left(e^{80}\right)}^{x}}\right)}^{\left(\frac{\frac{x}{2}}{2}\right)} \cdot \color{blue}{{\left(\sqrt{{\left(e^{80}\right)}^{x}}\right)}^{\left(\frac{\frac{x}{2}}{2}\right)}}} \]

   5. pow-prod-down 99.4%

      \[\leadsto \cos x \cdot \sqrt{\color{blue}{{\left(\sqrt{{\left(e^{80}\right)}^{x}} \cdot \sqrt{{\left(e^{80}\right)}^{x}}\right)}^{\left(\frac{\frac{x}{2}}{2}\right)}}} \]

   6. add-sqr-sqrt 99.4%

      \[\leadsto \cos x \cdot \sqrt{{\color{blue}{\left({\left(e^{80}\right)}^{x}\right)}}^{\left(\frac{\frac{x}{2}}{2}\right)}} \]

   7. div-inv 99.4%

      \[\leadsto \cos x \cdot \sqrt{{\left({\left(e^{80}\right)}^{x}\right)}^{\left(\frac{\color{blue}{x \cdot \frac{1}{2}}}{2}\right)}} \]

   8. metadata-eval 99.4%

      \[\leadsto \cos x \cdot \sqrt{{\left({\left(e^{80}\right)}^{x}\right)}^{\left(\frac{x \cdot \color{blue}{0.5}}{2}\right)}} \]

   9. associate-/l* 99.4%

      \[\leadsto \cos x \cdot \sqrt{{\left({\left(e^{80}\right)}^{x}\right)}^{\color{blue}{\left(x \cdot \frac{0.5}{2}\right)}}} \]

   10. metadata-eval 99.4%

       \[\leadsto \cos x \cdot \sqrt{{\left({\left(e^{80}\right)}^{x}\right)}^{\left(x \cdot \color{blue}{0.25}\right)}} \]

10. Applied egg-rr 99.4%

    \[\leadsto \cos x \cdot \color{blue}{\sqrt{{\left({\left(e^{80}\right)}^{x}\right)}^{\left(x \cdot 0.25\right)}}} \]

11. Final simplification 99.4%

    \[\leadsto \cos x \cdot \sqrt{{\left({\left(e^{80}\right)}^{x}\right)}^{\left(x \cdot 0.25\right)}} \]
12. Add Preprocessing

Alternative 2: 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 94.4%

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

   1. pow-exp 95.3%

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

   2. sqr-pow 95.3%

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

   3. pow-prod-down 95.3%

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

   4. associate-/l* 95.3%

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

   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. Final simplification 99.4%

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

Alternative 3: 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 94.4%

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

   1. pow-exp 95.3%

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

   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. Final simplification 98.0%

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

Alternative 4: 95.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \cos x \cdot {\left(1 + \mathsf{expm1}\left(x \cdot 20\right)\right)}^{\left(\frac{x}{2}\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (cos x) (pow (+ 1.0 (expm1 (* x 20.0))) (/ x 2.0))))

C

double code(double x) {
	return cos(x) * pow((1.0 + expm1((x * 20.0))), (x / 2.0));
}

Java

public static double code(double x) {
	return Math.cos(x) * Math.pow((1.0 + Math.expm1((x * 20.0))), (x / 2.0));
}

Python

def code(x):
	return math.cos(x) * math.pow((1.0 + math.expm1((x * 20.0))), (x / 2.0))

Julia

function code(x)
	return Float64(cos(x) * (Float64(1.0 + expm1(Float64(x * 20.0))) ^ Float64(x / 2.0)))
end

Wolfram

code[x_] := N[(N[Cos[x], $MachinePrecision] * N[Power[N[(1.0 + N[(Exp[N[(x * 20.0), $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision], N[(x / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.4%

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

   1. pow-exp 95.3%

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

   2. sqr-pow 95.3%

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

   3. pow-prod-down 95.3%

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

   4. associate-/l* 95.3%

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

   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. add-sqr-sqrt 99.2%

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

   2. sqrt-unprod 99.4%

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

   3. pow-prod-down 99.2%

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

   4. prod-exp 99.2%

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

   5. metadata-eval 99.2%

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

6. Applied egg-rr 99.2%

   \[\leadsto \cos x \cdot {\color{blue}{\left(\sqrt{{\left(e^{40}\right)}^{x}}\right)}}^{\left(\frac{x}{2}\right)} \]
7. Step-by-step derivation

   1. sqrt-pow1 99.2%

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

   2. metadata-eval 99.2%

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

   3. prod-exp 99.2%

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

   4. pow-prod-down 99.2%

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

   5. expm1-log1p-u 95.4%

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

   6. sqr-pow 95.4%

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

   7. expm1-undefine 95.4%

      \[\leadsto \cos x \cdot {\color{blue}{\left(e^{\mathsf{log1p}\left({\left(e^{20}\right)}^{x}\right)} - 1\right)}}^{\left(\frac{x}{2}\right)} \]

8. Applied egg-rr 95.4%

   \[\leadsto \cos x \cdot {\color{blue}{\left(e^{\mathsf{log1p}\left({\left(e^{20}\right)}^{x}\right)} - 1\right)}}^{\left(\frac{x}{2}\right)} \]
9. Step-by-step derivation

   1. log1p-undefine 95.4%

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

   2. rem-exp-log 99.4%

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

   3. associate--l+ 99.4%

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

   4. exp-prod 95.4%

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

   5. *-commutative 95.4%

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

   6. expm1-define 95.4%

      \[\leadsto \cos x \cdot {\left(1 + \color{blue}{\mathsf{expm1}\left(x \cdot 20\right)}\right)}^{\left(\frac{x}{2}\right)} \]

   7. *-commutative 95.4%

      \[\leadsto \cos x \cdot {\left(1 + \mathsf{expm1}\left(\color{blue}{20 \cdot x}\right)\right)}^{\left(\frac{x}{2}\right)} \]

10. Simplified 95.4%

    \[\leadsto \cos x \cdot {\color{blue}{\left(1 + \mathsf{expm1}\left(20 \cdot x\right)\right)}}^{\left(\frac{x}{2}\right)} \]

11. Final simplification 95.4%

    \[\leadsto \cos x \cdot {\left(1 + \mathsf{expm1}\left(x \cdot 20\right)\right)}^{\left(\frac{x}{2}\right)} \]
12. Add Preprocessing

Alternative 5: 95.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.4%

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

   1. pow-exp 95.3%

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

   2. sqr-pow 95.3%

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

   3. pow-prod-down 95.3%

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

   4. associate-/l* 95.3%

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

   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. add-exp-log 95.4%

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

   2. log-pow 95.4%

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

   3. rem-log-exp 95.4%

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

6. Applied egg-rr 95.4%

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

7. Final simplification 95.4%

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

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 94.4%

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

   1. exp-prod 95.3%

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

3. Simplified 95.3%

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

5. Final simplification 95.3%

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

Alternative 7: 95.0% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.4%

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

   1. pow-exp 95.3%

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

   2. sqr-pow 95.3%

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

   3. pow-prod-down 95.3%

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

   4. associate-/l* 95.3%

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

   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. Taylor expanded in x around inf 94.4%

   \[\leadsto \color{blue}{\cos x \cdot e^{0.5 \cdot \left(x \cdot \log \left({\left(e^{20}\right)}^{x}\right)\right)}} \]
6. 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 99.3%

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

7. Simplified 99.3%

   \[\leadsto \color{blue}{\cos x \cdot \sqrt{{\left({\left(e^{20}\right)}^{x}\right)}^{x}}} \]
8. 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. sqr-pow 99.2%

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

   3. pow-prod-down 99.2%

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

   4. prod-exp 99.2%

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

   5. metadata-eval 99.2%

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

   6. sqrt-pow1 99.2%

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

   7. add-sqr-sqrt 99.0%

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

   8. unpow-prod-down 98.9%

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

   9. sqrt-pow1 98.9%

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

   10. metadata-eval 98.9%

       \[\leadsto \cos x \cdot \left({\left(\sqrt{{\left(e^{\color{blue}{20 + 20}}\right)}^{\left(\frac{x}{2}\right)}}\right)}^{\left(\frac{x}{2}\right)} \cdot {\left(\sqrt{\sqrt{{\left(e^{40}\right)}^{x}}}\right)}^{\left(\frac{x}{2}\right)}\right) \]

   11. prod-exp 98.9%

       \[\leadsto \cos x \cdot \left({\left(\sqrt{{\color{blue}{\left(e^{20} \cdot e^{20}\right)}}^{\left(\frac{x}{2}\right)}}\right)}^{\left(\frac{x}{2}\right)} \cdot {\left(\sqrt{\sqrt{{\left(e^{40}\right)}^{x}}}\right)}^{\left(\frac{x}{2}\right)}\right) \]

   12. pow-prod-down 99.0%

       \[\leadsto \cos x \cdot \left({\left(\sqrt{\color{blue}{{\left(e^{20}\right)}^{\left(\frac{x}{2}\right)} \cdot {\left(e^{20}\right)}^{\left(\frac{x}{2}\right)}}}\right)}^{\left(\frac{x}{2}\right)} \cdot {\left(\sqrt{\sqrt{{\left(e^{40}\right)}^{x}}}\right)}^{\left(\frac{x}{2}\right)}\right) \]

   13. sqr-pow 99.0%

       \[\leadsto \cos x \cdot \left({\left(\sqrt{\color{blue}{{\left(e^{20}\right)}^{x}}}\right)}^{\left(\frac{x}{2}\right)} \cdot {\left(\sqrt{\sqrt{{\left(e^{40}\right)}^{x}}}\right)}^{\left(\frac{x}{2}\right)}\right) \]

   14. div-inv 99.0%

       \[\leadsto \cos x \cdot \left({\left(\sqrt{{\left(e^{20}\right)}^{x}}\right)}^{\color{blue}{\left(x \cdot \frac{1}{2}\right)}} \cdot {\left(\sqrt{\sqrt{{\left(e^{40}\right)}^{x}}}\right)}^{\left(\frac{x}{2}\right)}\right) \]

   15. metadata-eval 99.0%

       \[\leadsto \cos x \cdot \left({\left(\sqrt{{\left(e^{20}\right)}^{x}}\right)}^{\left(x \cdot \color{blue}{0.5}\right)} \cdot {\left(\sqrt{\sqrt{{\left(e^{40}\right)}^{x}}}\right)}^{\left(\frac{x}{2}\right)}\right) \]

9. Applied egg-rr 99.0%

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

    1. pow-sqr 99.2%

       \[\leadsto \cos x \cdot \color{blue}{{\left(\sqrt{{\left(e^{20}\right)}^{x}}\right)}^{\left(2 \cdot \left(x \cdot 0.5\right)\right)}} \]

    2. count-2 99.2%

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

    3. distribute-lft-out 99.2%

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

    4. metadata-eval 99.2%

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

    5. *-rgt-identity 99.2%

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

11. Simplified 99.2%

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

12. Taylor expanded in x around inf 95.3%

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

    1. exp-prod 99.2%

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

    2. unpow1/2 99.2%

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

    3. exp-prod 95.3%

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

    4. *-commutative 95.3%

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

    5. exp-prod 95.3%

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

    6. associate-*l* 95.3%

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

    7. metadata-eval 95.3%

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

14. Simplified 95.3%

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

15. Final simplification 95.3%

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

Alternative 8: 94.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.4%

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

3. Final simplification 94.4%

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

Alternative 9: 94.3% 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 94.4%

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

   1. associate-*r* 94.4%

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

3. Simplified 94.4%

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

5. Final simplification 94.4%

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

Alternative 10: 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 94.4%

   \[\cos x \cdot e^{10 \cdot \left(x \cdot x\right)} \]
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 11: 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 94.4%

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

3. Taylor expanded in x around 0 9.6%

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

4. Final simplification 9.6%

   \[\leadsto \cos x \]
5. Add Preprocessing

Alternative 12: 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 94.4%

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

   1. pow-exp 95.3%

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

   2. sqr-pow 95.3%

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

   3. pow-prod-down 95.3%

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

   4. associate-/l* 95.3%

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

   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. Taylor expanded in x around 0 1.5%

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

6. Final simplification 1.5%

   \[\leadsto 1 \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2024078 
(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)))))