ENA, Section 1.4, Exercise 1

Percentage Accurate: 94.5% → 99.4%

Time: 7.7s

Alternatives: 11

Speedup: 1.0×

Specification

\[1.99 \leq x \land x \leq 2.01\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 94.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.4% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[Power[N[Power[N[Exp[20.0], $MachinePrecision], x], $MachinePrecision], N[(x * 0.5), $MachinePrecision]], $MachinePrecision] * N[Cos[x], $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. cos-neg 94.5%

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

   2. *-commutative 94.5%

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

   3. cos-neg 94.5%

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

3. Simplified 94.5%

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

   1. pow-exp 95.4%

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

   2. pow-pow 98.0%

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

   3. sqr-pow 98.0%

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

   4. pow-prod-down 98.0%

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

   5. pow-prod-down 98.0%

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

   6. prod-exp 99.4%

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

   7. metadata-eval 99.4%

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

   8. div-inv 99.4%

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

   9. metadata-eval 99.4%

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

5. Applied egg-rr 99.4%

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

6. Final simplification 99.4%

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

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. Step-by-step derivation

   1. cos-neg 94.5%

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

   2. *-commutative 94.5%

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

   3. associate-*r* 94.5%

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

   4. exp-prod 95.2%

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

   5. exp-prod 98.0%

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

   6. cos-neg 98.0%

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

3. Simplified 98.0%

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

4. Final simplification 98.0%

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

Alternative 3: 95.3% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[Cos[x], $MachinePrecision] * N[Power[N[Exp[20.0], $MachinePrecision], N[(0.5 * 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. Step-by-step derivation

   1. cos-neg 94.5%

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

   2. *-commutative 94.5%

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

   3. cos-neg 94.5%

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

3. Simplified 94.5%

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

   1. pow-exp 95.4%

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

   2. sqr-pow 95.4%

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

   3. pow-prod-down 95.4%

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

   4. prod-exp 95.4%

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

   5. metadata-eval 95.4%

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

   6. div-inv 95.4%

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

   7. metadata-eval 95.4%

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

5. Applied egg-rr 95.4%

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

6. Final simplification 95.4%

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

Alternative 4: 95.3% accurate, 0.7× speedup

Math

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

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(Float64(x * 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[(N[(x * x), $MachinePrecision] * 0.25), $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. cos-neg 94.5%

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

   2. *-commutative 94.5%

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

   3. cos-neg 94.5%

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

3. Simplified 94.5%

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

   1. pow-exp 95.4%

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

   2. add-sqr-sqrt 95.4%

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

   3. sqrt-unprod 95.4%

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

   4. pow-prod-down 95.4%

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

   5. pow-pow 98.0%

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

   6. prod-exp 99.4%

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

   7. metadata-eval 99.4%

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

5. Applied egg-rr 99.4%

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

   1. pow1/2 99.4%

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

   2. pow-exp 95.2%

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

   3. *-commutative 95.2%

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

   4. pow-pow 95.2%

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

   5. *-commutative 95.2%

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

   6. pow-exp 99.4%

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

   7. sqr-pow 99.2%

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

   8. pow-prod-down 99.2%

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

   9. prod-exp 99.2%

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

   10. metadata-eval 99.2%

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

   11. sqrt-pow1 99.2%

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

   12. pow1/2 99.2%

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

   13. pow1/2 99.2%

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

   14. sqrt-pow1 99.2%

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

   15. pow-pow 95.4%

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

   16. div-inv 95.4%

       \[\leadsto {\left(e^{40}\right)}^{\left(\color{blue}{\left(x \cdot \frac{1}{2}\right)} \cdot \left(x \cdot 0.5\right)\right)} \cdot \cos x \]

   17. metadata-eval 95.4%

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

   18. swap-sqr 95.4%

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

   19. metadata-eval 95.4%

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

7. Applied egg-rr 95.4%

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

8. Final simplification 95.4%

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

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. cos-neg 94.5%

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

   2. *-commutative 94.5%

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

   3. exp-prod 95.4%

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

   4. cos-neg 95.4%

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

3. Simplified 95.4%

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

4. Final simplification 95.4%

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

Alternative 6: 95.2% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

function code(x)
	return Float64(cos(x) * (exp(Float64(x * 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[Power[N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision], 10.0], $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. cos-neg 94.5%

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

   2. *-commutative 94.5%

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

   3. cos-neg 94.5%

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

3. Simplified 94.5%

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

4. Taylor expanded in x around inf 94.5%

   \[\leadsto \color{blue}{e^{10 \cdot {x}^{2}}} \cdot \cos x \]
5. Step-by-step derivation

   1. unpow2 94.5%

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

   2. *-commutative 94.5%

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

   3. exp-prod 95.4%

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

6. Simplified 95.4%

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

7. Final simplification 95.4%

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

Alternative 7: 94.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.5%

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

2. Final simplification 94.5%

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

Alternative 8: 9.8% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.5%

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

   1. cos-neg 94.5%

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

   2. *-commutative 94.5%

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

   3. cos-neg 94.5%

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

3. Simplified 94.5%

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

4. Taylor expanded in x around 0 9.8%

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

   1. unpow2 9.8%

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

6. Simplified 9.8%

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

7. Final simplification 9.8%

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

Alternative 9: 9.8% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(x * N[(N[Cos[x], $MachinePrecision] * N[(x * 10.0), $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. cos-neg 94.5%

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

   2. *-commutative 94.5%

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

   3. cos-neg 94.5%

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

3. Simplified 94.5%

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

4. Taylor expanded in x around 0 9.8%

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

   1. unpow2 9.8%

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

6. Simplified 9.8%

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

7. Taylor expanded in x around inf 9.8%

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

   1. associate-*r* 9.8%

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

   2. *-commutative 9.8%

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

   3. unpow2 9.8%

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

   4. associate-*r* 9.8%

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

   5. associate-*l* 9.8%

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

9. Simplified 9.8%

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

10. Final simplification 9.8%

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

Alternative 10: 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 94.5%

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

   1. cos-neg 94.5%

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

   2. *-commutative 94.5%

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

   3. cos-neg 94.5%

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

3. Simplified 94.5%

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

4. Taylor expanded in x around 0 9.6%

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

5. Final simplification 9.6%

   \[\leadsto \cos x \]

Alternative 11: 1.5% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x) :precision binary64 1.0)

C

double code(double x) {
	return 1.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1.0d0
end function

Java

public static double code(double x) {
	return 1.0;
}

Python

def code(x):
	return 1.0

Julia

function code(x)
	return 1.0
end

MATLAB

function tmp = code(x)
	tmp = 1.0;
end

Wolfram

code[x_] := 1.0

Derivation

1. Initial program 94.5%

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

   1. cos-neg 94.5%

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

   2. *-commutative 94.5%

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

   3. cos-neg 94.5%

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

3. Simplified 94.5%

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

   1. pow-exp 95.4%

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

   2. pow-pow 98.0%

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

   3. sqr-pow 98.0%

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

   4. pow-prod-down 98.0%

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

   5. pow-prod-down 98.0%

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

   6. prod-exp 99.4%

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

   7. metadata-eval 99.4%

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

   8. div-inv 99.4%

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

   9. metadata-eval 99.4%

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

5. Applied egg-rr 99.4%

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

6. Taylor expanded in x around 0 1.5%

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

7. Final simplification 1.5%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023271 
(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)))))