ENA, Section 1.4, Exercise 1

Percentage Accurate: 94.5% → 99.4%

Time: 7.8s

Alternatives: 9

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

\[\begin{array}{l} \\ \cos x \cdot {\left({\left(e^{80}\right)}^{\left(x \cdot 0.25\right)}\right)}^{\left(\frac{x}{2}\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (cos x) (pow (pow (exp 80.0) (* x 0.25)) (/ x 2.0))))

C

double code(double x) {
	return cos(x) * pow(pow(exp(80.0), (x * 0.25)), (x / 2.0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = cos(x) * ((exp(80.0d0) ** (x * 0.25d0)) ** (x / 2.0d0))
end function

Java

public static double code(double x) {
	return Math.cos(x) * Math.pow(Math.pow(Math.exp(80.0), (x * 0.25)), (x / 2.0));
}

Python

def code(x):
	return math.cos(x) * math.pow(math.pow(math.exp(80.0), (x * 0.25)), (x / 2.0))

Julia

function code(x)
	return Float64(cos(x) * ((exp(80.0) ^ Float64(x * 0.25)) ^ Float64(x / 2.0)))
end

MATLAB

function tmp = code(x)
	tmp = cos(x) * ((exp(80.0) ^ (x * 0.25)) ^ (x / 2.0));
end

Wolfram

code[x_] := N[(N[Cos[x], $MachinePrecision] * N[Power[N[Power[N[Exp[80.0], $MachinePrecision], N[(x * 0.25), $MachinePrecision]], $MachinePrecision], N[(x / 2.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.7%

   \[\cos x \cdot e^{10 \cdot \left(x \cdot x\right)} \]
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-cbrt-cube 99.0%

      \[\leadsto \cos x \cdot {\color{blue}{\left(\sqrt[3]{\left({\left(e^{20}\right)}^{x} \cdot {\left(e^{20}\right)}^{x}\right) \cdot {\left(e^{20}\right)}^{x}}\right)}}^{\left(\frac{x}{2}\right)} \]

   2. pow3 99.0%

      \[\leadsto \cos x \cdot {\left(\sqrt[3]{\color{blue}{{\left({\left(e^{20}\right)}^{x}\right)}^{3}}}\right)}^{\left(\frac{x}{2}\right)} \]

6. Applied egg-rr 99.0%

   \[\leadsto \cos x \cdot {\color{blue}{\left(\sqrt[3]{{\left({\left(e^{20}\right)}^{x}\right)}^{3}}\right)}}^{\left(\frac{x}{2}\right)} \]

7. Taylor expanded in x around inf 94.9%

   \[\leadsto \cos x \cdot {\left(\sqrt[3]{\color{blue}{{\left(e^{20 \cdot x}\right)}^{3}}}\right)}^{\left(\frac{x}{2}\right)} \]
8. Step-by-step derivation

   1. pow-exp 99.0%

      \[\leadsto \cos x \cdot {\left(\sqrt[3]{{\color{blue}{\left({\left(e^{20}\right)}^{x}\right)}}^{3}}\right)}^{\left(\frac{x}{2}\right)} \]

   2. rem-cbrt-cube 99.4%

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

   3. sqr-pow 99.1%

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

   4. pow-prod-down 99.3%

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

   5. prod-exp 99.3%

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

   6. metadata-eval 99.3%

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

   7. sqr-pow 99.1%

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

   8. pow-prod-down 99.3%

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

   9. prod-exp 99.4%

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

   10. metadata-eval 99.4%

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

   11. div-inv 99.4%

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

   12. metadata-eval 99.4%

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

   13. associate-/l* 99.4%

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

   14. metadata-eval 99.4%

       \[\leadsto \cos x \cdot {\left({\left(e^{80}\right)}^{\left(x \cdot \color{blue}{0.25}\right)}\right)}^{\left(\frac{x}{2}\right)} \]

9. Applied egg-rr 99.4%

   \[\leadsto \cos x \cdot {\color{blue}{\left({\left(e^{80}\right)}^{\left(x \cdot 0.25\right)}\right)}}^{\left(\frac{x}{2}\right)} \]
10. 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 94.7%

   \[\cos x \cdot e^{10 \cdot \left(x \cdot x\right)} \]
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. 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 94.7%

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

Alternative 4: 95.2% accurate, 0.7× speedup

Math

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

FPCore

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

C

double code(double x) {
	return ((cos(x) + 1.0) - 1.0) * pow(exp(10.0), (x * x));
}

Fortran

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

Java

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

Python

def code(x):
	return ((math.cos(x) + 1.0) - 1.0) * math.pow(math.exp(10.0), (x * x))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.7%

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

   1. log1p-expm1-u 95.3%

      \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos x\right)\right)} \cdot {\left(e^{10}\right)}^{\left(x \cdot x\right)} \]

6. Applied egg-rr 95.3%

   \[\leadsto \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left(\cos x\right)\right)} \cdot {\left(e^{10}\right)}^{\left(x \cdot x\right)} \]
7. Step-by-step derivation

   1. expm1-log1p-u 95.3%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\mathsf{log1p}\left(\mathsf{expm1}\left(\cos x\right)\right)\right)\right)} \cdot {\left(e^{10}\right)}^{\left(x \cdot x\right)} \]

   2. log1p-expm1-u 95.3%

      \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\cos x}\right)\right) \cdot {\left(e^{10}\right)}^{\left(x \cdot x\right)} \]

   3. expm1-undefine 95.3%

      \[\leadsto \color{blue}{\left(e^{\mathsf{log1p}\left(\cos x\right)} - 1\right)} \cdot {\left(e^{10}\right)}^{\left(x \cdot x\right)} \]

   4. log1p-undefine 95.3%

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

   5. rem-exp-log 95.3%

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

   6. +-commutative 95.3%

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

8. Applied egg-rr 95.3%

   \[\leadsto \color{blue}{\left(\left(\cos x + 1\right) - 1\right)} \cdot {\left(e^{10}\right)}^{\left(x \cdot x\right)} \]
9. Add Preprocessing

Alternative 5: 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 94.7%

   \[\cos x \cdot e^{10 \cdot \left(x \cdot x\right)} \]
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. 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 94.7%

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

Alternative 7: 9.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ 1 + -0.5 \cdot {x}^{2} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (+ 1.0 (* -0.5 (pow x 2.0))))

C

double code(double x) {
	return 1.0 + (-0.5 * pow(x, 2.0));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0 + ((-0.5d0) * (x ** 2.0d0))
end function

Java

public static double code(double x) {
	return 1.0 + (-0.5 * Math.pow(x, 2.0));
}

Python

def code(x):
	return 1.0 + (-0.5 * math.pow(x, 2.0))

Julia

function code(x)
	return Float64(1.0 + Float64(-0.5 * (x ^ 2.0)))
end

MATLAB

function tmp = code(x)
	tmp = 1.0 + (-0.5 * (x ^ 2.0));
end

Wolfram

code[x_] := N[(1.0 + N[(-0.5 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.7%

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

   \[\leadsto \color{blue}{1 + -0.5 \cdot {x}^{2}} \]
5. Add Preprocessing

Alternative 8: 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 94.7%

   \[\cos x \cdot e^{10 \cdot \left(x \cdot x\right)} \]
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 inf 9.6%

   \[\leadsto \color{blue}{\cos x} \]
5. Add Preprocessing

Alternative 9: 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 94.7%

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

3. Taylor expanded in x around 0 1.5%

   \[\leadsto \color{blue}{1} \]
4. Add Preprocessing

Reproduce

herbie shell --seed 2024116 -o generate:simplify
(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)))))