ENA, Section 1.4, Exercise 1

Percentage Accurate: 94.5% → 99.1%

Time: 8.6s

Alternatives: 10

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: 99.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{\sqrt{{\left({\left(e^{60}\right)}^{x}\right)}^{x}} \cdot \mathsf{log1p}\left(\mathsf{expm1}\left({\cos x}^{3}\right)\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (cbrt
  (* (sqrt (pow (pow (exp 60.0) x) x)) (log1p (expm1 (pow (cos x) 3.0))))))

C

double code(double x) {
	return cbrt((sqrt(pow(pow(exp(60.0), x), x)) * log1p(expm1(pow(cos(x), 3.0)))));
}

Java

public static double code(double x) {
	return Math.cbrt((Math.sqrt(Math.pow(Math.pow(Math.exp(60.0), x), x)) * Math.log1p(Math.expm1(Math.pow(Math.cos(x), 3.0)))));
}

Julia

function code(x)
	return cbrt(Float64(sqrt(((exp(60.0) ^ x) ^ x)) * log1p(expm1((cos(x) ^ 3.0)))))
end

Wolfram

code[x_] := N[Power[N[(N[Sqrt[N[Power[N[Power[N[Exp[60.0], $MachinePrecision], x], $MachinePrecision], x], $MachinePrecision]], $MachinePrecision] * N[Log[1 + N[(Exp[N[Power[N[Cos[x], $MachinePrecision], 3.0], $MachinePrecision]] - 1), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]

Derivation

1. Initial program 94.7%

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

3. Applied egg-rr 98.8%

   \[\leadsto \color{blue}{\sqrt[3]{{\left({\left(e^{30}\right)}^{x}\right)}^{x} \cdot {\cos x}^{3}}} \]
4. Step-by-step derivation

   1. add-sqr-sqrt 98.7%

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

   2. sqrt-unprod 98.8%

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

   3. pow-prod-down 98.8%

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

   4. pow-prod-down 98.9%

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

   5. prod-exp 99.0%

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

   6. metadata-eval 99.0%

      \[\leadsto \sqrt[3]{\sqrt{{\left({\left(e^{\color{blue}{60}}\right)}^{x}\right)}^{x}} \cdot {\cos x}^{3}} \]

5. Applied egg-rr 99.0%

   \[\leadsto \sqrt[3]{\color{blue}{\sqrt{{\left({\left(e^{60}\right)}^{x}\right)}^{x}}} \cdot {\cos x}^{3}} \]
6. Step-by-step derivation

   1. log1p-expm1-u 99.0%

      \[\leadsto \sqrt[3]{\sqrt{{\left({\left(e^{60}\right)}^{x}\right)}^{x}} \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left({\cos x}^{3}\right)\right)}} \]

7. Applied egg-rr 99.0%

   \[\leadsto \sqrt[3]{\sqrt{{\left({\left(e^{60}\right)}^{x}\right)}^{x}} \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left({\cos x}^{3}\right)\right)}} \]

8. Final simplification 99.0%

   \[\leadsto \sqrt[3]{\sqrt{{\left({\left(e^{60}\right)}^{x}\right)}^{x}} \cdot \mathsf{log1p}\left(\mathsf{expm1}\left({\cos x}^{3}\right)\right)} \]

Alternative 2: 99.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{{\cos x}^{3} \cdot {\left({\left(e^{60}\right)}^{x}\right)}^{\left(x \cdot 0.5\right)}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (cbrt (* (pow (cos x) 3.0) (pow (pow (exp 60.0) x) (* x 0.5)))))

C

double code(double x) {
	return cbrt((pow(cos(x), 3.0) * pow(pow(exp(60.0), x), (x * 0.5))));
}

Java

public static double code(double x) {
	return Math.cbrt((Math.pow(Math.cos(x), 3.0) * Math.pow(Math.pow(Math.exp(60.0), x), (x * 0.5))));
}

Julia

function code(x)
	return cbrt(Float64((cos(x) ^ 3.0) * ((exp(60.0) ^ x) ^ Float64(x * 0.5))))
end

Wolfram

code[x_] := N[Power[N[(N[Power[N[Cos[x], $MachinePrecision], 3.0], $MachinePrecision] * N[Power[N[Power[N[Exp[60.0], $MachinePrecision], x], $MachinePrecision], N[(x * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]

Derivation

1. Initial program 94.7%

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

3. Applied egg-rr 98.8%

   \[\leadsto \color{blue}{\sqrt[3]{{\left({\left(e^{30}\right)}^{x}\right)}^{x} \cdot {\cos x}^{3}}} \]
4. Step-by-step derivation

   1. add-sqr-sqrt 98.7%

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

   2. sqrt-unprod 98.8%

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

   3. pow-prod-down 98.8%

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

   4. pow-prod-down 98.9%

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

   5. prod-exp 99.0%

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

   6. metadata-eval 99.0%

      \[\leadsto \sqrt[3]{\sqrt{{\left({\left(e^{\color{blue}{60}}\right)}^{x}\right)}^{x}} \cdot {\cos x}^{3}} \]

5. Applied egg-rr 99.0%

   \[\leadsto \sqrt[3]{\color{blue}{\sqrt{{\left({\left(e^{60}\right)}^{x}\right)}^{x}}} \cdot {\cos x}^{3}} \]
6. Step-by-step derivation

   1. log1p-expm1-u 99.0%

      \[\leadsto \sqrt[3]{\sqrt{{\left({\left(e^{60}\right)}^{x}\right)}^{x}} \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left({\cos x}^{3}\right)\right)}} \]

7. Applied egg-rr 99.0%

   \[\leadsto \sqrt[3]{\sqrt{{\left({\left(e^{60}\right)}^{x}\right)}^{x}} \cdot \color{blue}{\mathsf{log1p}\left(\mathsf{expm1}\left({\cos x}^{3}\right)\right)}} \]
8. Step-by-step derivation

   1. pow1/3 0.0%

      \[\leadsto \color{blue}{{\left(\sqrt{{\left({\left(e^{60}\right)}^{x}\right)}^{x}} \cdot \mathsf{log1p}\left(\mathsf{expm1}\left({\cos x}^{3}\right)\right)\right)}^{0.3333333333333333}} \]

   2. pow-to-exp 0.0%

      \[\leadsto \color{blue}{e^{\log \left(\sqrt{{\left({\left(e^{60}\right)}^{x}\right)}^{x}} \cdot \mathsf{log1p}\left(\mathsf{expm1}\left({\cos x}^{3}\right)\right)\right) \cdot 0.3333333333333333}} \]

   3. log-prod 0.0%

      \[\leadsto e^{\color{blue}{\left(\log \left(\sqrt{{\left({\left(e^{60}\right)}^{x}\right)}^{x}}\right) + \log \left(\mathsf{log1p}\left(\mathsf{expm1}\left({\cos x}^{3}\right)\right)\right)\right)} \cdot 0.3333333333333333} \]

   4. pow1/2 0.0%

      \[\leadsto e^{\left(\log \color{blue}{\left({\left({\left({\left(e^{60}\right)}^{x}\right)}^{x}\right)}^{0.5}\right)} + \log \left(\mathsf{log1p}\left(\mathsf{expm1}\left({\cos x}^{3}\right)\right)\right)\right) \cdot 0.3333333333333333} \]

   5. log-pow 0.0%

      \[\leadsto e^{\left(\color{blue}{0.5 \cdot \log \left({\left({\left(e^{60}\right)}^{x}\right)}^{x}\right)} + \log \left(\mathsf{log1p}\left(\mathsf{expm1}\left({\cos x}^{3}\right)\right)\right)\right) \cdot 0.3333333333333333} \]

   6. log-pow 0.0%

      \[\leadsto e^{\left(0.5 \cdot \color{blue}{\left(x \cdot \log \left({\left(e^{60}\right)}^{x}\right)\right)} + \log \left(\mathsf{log1p}\left(\mathsf{expm1}\left({\cos x}^{3}\right)\right)\right)\right) \cdot 0.3333333333333333} \]

   7. pow-exp 0.0%

      \[\leadsto e^{\left(0.5 \cdot \left(x \cdot \log \color{blue}{\left(e^{60 \cdot x}\right)}\right) + \log \left(\mathsf{log1p}\left(\mathsf{expm1}\left({\cos x}^{3}\right)\right)\right)\right) \cdot 0.3333333333333333} \]

   8. add-log-exp 0.0%

      \[\leadsto e^{\left(0.5 \cdot \left(x \cdot \color{blue}{\left(60 \cdot x\right)}\right) + \log \left(\mathsf{log1p}\left(\mathsf{expm1}\left({\cos x}^{3}\right)\right)\right)\right) \cdot 0.3333333333333333} \]

   9. log1p-expm1-u 0.0%

      \[\leadsto e^{\left(0.5 \cdot \left(x \cdot \left(60 \cdot x\right)\right) + \log \color{blue}{\left({\cos x}^{3}\right)}\right) \cdot 0.3333333333333333} \]

   10. log-pow 0.0%

       \[\leadsto e^{\left(0.5 \cdot \left(x \cdot \left(60 \cdot x\right)\right) + \color{blue}{3 \cdot \log \cos x}\right) \cdot 0.3333333333333333} \]

9. Applied egg-rr 0.0%

   \[\leadsto \color{blue}{e^{\left(0.5 \cdot \left(x \cdot \left(60 \cdot x\right)\right) + 3 \cdot \log \cos x\right) \cdot 0.3333333333333333}} \]
10. Step-by-step derivation

    1. exp-prod 0.0%

       \[\leadsto \color{blue}{{\left(e^{0.5 \cdot \left(x \cdot \left(60 \cdot x\right)\right) + 3 \cdot \log \cos x}\right)}^{0.3333333333333333}} \]

    2. unpow1/3 0.0%

       \[\leadsto \color{blue}{\sqrt[3]{e^{0.5 \cdot \left(x \cdot \left(60 \cdot x\right)\right) + 3 \cdot \log \cos x}}} \]

    3. +-commutative 0.0%

       \[\leadsto \sqrt[3]{e^{\color{blue}{3 \cdot \log \cos x + 0.5 \cdot \left(x \cdot \left(60 \cdot x\right)\right)}}} \]

    4. exp-sum 0.0%

       \[\leadsto \sqrt[3]{\color{blue}{e^{3 \cdot \log \cos x} \cdot e^{0.5 \cdot \left(x \cdot \left(60 \cdot x\right)\right)}}} \]

    5. *-commutative 0.0%

       \[\leadsto \sqrt[3]{e^{\color{blue}{\log \cos x \cdot 3}} \cdot e^{0.5 \cdot \left(x \cdot \left(60 \cdot x\right)\right)}} \]

    6. exp-to-pow 95.1%

       \[\leadsto \sqrt[3]{\color{blue}{{\cos x}^{3}} \cdot e^{0.5 \cdot \left(x \cdot \left(60 \cdot x\right)\right)}} \]

    7. associate-*r* 95.1%

       \[\leadsto \sqrt[3]{{\cos x}^{3} \cdot e^{\color{blue}{\left(0.5 \cdot x\right) \cdot \left(60 \cdot x\right)}}} \]

    8. *-commutative 95.1%

       \[\leadsto \sqrt[3]{{\cos x}^{3} \cdot e^{\color{blue}{\left(60 \cdot x\right) \cdot \left(0.5 \cdot x\right)}}} \]

    9. exp-prod 95.7%

       \[\leadsto \sqrt[3]{{\cos x}^{3} \cdot \color{blue}{{\left(e^{60 \cdot x}\right)}^{\left(0.5 \cdot x\right)}}} \]

    10. exp-prod 99.0%

        \[\leadsto \sqrt[3]{{\cos x}^{3} \cdot {\color{blue}{\left({\left(e^{60}\right)}^{x}\right)}}^{\left(0.5 \cdot x\right)}} \]

    11. *-commutative 99.0%

        \[\leadsto \sqrt[3]{{\cos x}^{3} \cdot {\left({\left(e^{60}\right)}^{x}\right)}^{\color{blue}{\left(x \cdot 0.5\right)}}} \]

11. Simplified 99.0%

    \[\leadsto \color{blue}{\sqrt[3]{{\cos x}^{3} \cdot {\left({\left(e^{60}\right)}^{x}\right)}^{\left(x \cdot 0.5\right)}}} \]

12. Final simplification 99.0%

    \[\leadsto \sqrt[3]{{\cos x}^{3} \cdot {\left({\left(e^{60}\right)}^{x}\right)}^{\left(x \cdot 0.5\right)}} \]

Alternative 3: 98.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \sqrt[3]{{\cos x}^{3} \cdot {\left({\left(e^{30}\right)}^{x}\right)}^{x}} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (cbrt (* (pow (cos x) 3.0) (pow (pow (exp 30.0) x) x))))

C

double code(double x) {
	return cbrt((pow(cos(x), 3.0) * pow(pow(exp(30.0), x), x)));
}

Java

public static double code(double x) {
	return Math.cbrt((Math.pow(Math.cos(x), 3.0) * Math.pow(Math.pow(Math.exp(30.0), x), x)));
}

Julia

function code(x)
	return cbrt(Float64((cos(x) ^ 3.0) * ((exp(30.0) ^ x) ^ x)))
end

Wolfram

code[x_] := N[Power[N[(N[Power[N[Cos[x], $MachinePrecision], 3.0], $MachinePrecision] * N[Power[N[Power[N[Exp[30.0], $MachinePrecision], x], $MachinePrecision], x], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]

Derivation

1. Initial program 94.7%

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

3. Applied egg-rr 98.8%

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

4. Final simplification 98.8%

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

Alternative 4: 98.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \cos x \cdot \sqrt[3]{{\left({\left(e^{30}\right)}^{x}\right)}^{x}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (cos x) (cbrt (pow (pow (exp 30.0) x) x))))

C

double code(double x) {
	return cos(x) * cbrt(pow(pow(exp(30.0), x), x));
}

Java

public static double code(double x) {
	return Math.cos(x) * Math.cbrt(Math.pow(Math.pow(Math.exp(30.0), x), x));
}

Julia

function code(x)
	return Float64(cos(x) * cbrt(((exp(30.0) ^ x) ^ x)))
end

Wolfram

code[x_] := N[(N[Cos[x], $MachinePrecision] * N[Power[N[Power[N[Power[N[Exp[30.0], $MachinePrecision], x], $MachinePrecision], x], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.7%

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

3. Applied egg-rr 98.7%

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

4. Final simplification 98.7%

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

Alternative 5: 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 94.7%

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

   1. associate-*r* 94.5%

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

   2. exp-prod 94.9%

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

   3. sqr-pow 94.9%

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

   4. sqr-pow 94.9%

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

   5. exp-prod 98.0%

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

3. Simplified 98.0%

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

4. Final simplification 98.0%

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

Alternative 6: 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 94.7%

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

   1. pow-exp 95.4%

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

   2. sqr-pow 95.4%

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

   3. pow-prod-down 95.4%

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

   4. prod-exp 95.4%

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

   5. metadata-eval 95.4%

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

   6. div-inv 95.4%

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

   7. metadata-eval 95.4%

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

3. Applied egg-rr 95.4%

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

4. Final simplification 95.4%

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

Alternative 7: 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 94.7%

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

   1. exp-prod 95.4%

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

3. Simplified 95.4%

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

4. Final simplification 95.4%

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

Alternative 8: 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. Final simplification 94.7%

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

Alternative 9: 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. Taylor expanded in x around 0 9.6%

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

3. Final simplification 9.6%

   \[\leadsto \cos x \]

Alternative 10: 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%

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

3. Applied egg-rr 98.8%

   \[\leadsto \color{blue}{\sqrt[3]{{\left({\left(e^{30}\right)}^{x}\right)}^{x} \cdot {\cos x}^{3}}} \]
4. Step-by-step derivation

   1. add-sqr-sqrt 98.7%

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

   2. sqrt-unprod 98.8%

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

   3. pow-prod-down 98.8%

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

   4. pow-prod-down 98.9%

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

   5. prod-exp 99.0%

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

   6. metadata-eval 99.0%

      \[\leadsto \sqrt[3]{\sqrt{{\left({\left(e^{\color{blue}{60}}\right)}^{x}\right)}^{x}} \cdot {\cos x}^{3}} \]

5. Applied egg-rr 99.0%

   \[\leadsto \sqrt[3]{\color{blue}{\sqrt{{\left({\left(e^{60}\right)}^{x}\right)}^{x}}} \cdot {\cos x}^{3}} \]

6. Taylor expanded in x around 0 1.5%

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

7. Final simplification 1.5%

   \[\leadsto 1 \]

Reproduce

herbie shell --seed 2023200 
(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)))))