ENA, Section 1.4, Exercise 1

Percentage Accurate: 94.4% → 99.4%

Time: 10.4s

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: 94.4% 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

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

FPCore

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

C

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

Python

def code(x):
	return math.cos(x) * math.pow(math.sqrt(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[Power[N[Sqrt[N[Power[N[Exp[80.0], $MachinePrecision], x], $MachinePrecision]], $MachinePrecision], N[(x * 0.25), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.6%

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

   1. exp-prod 95.6%

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

3. Simplified 95.6%

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

   1. add-cbrt-cube 95.6%

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

   2. pow3 95.6%

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

   3. add-exp-log 94.6%

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

   4. log-pow 94.7%

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

   5. pow-exp 95.6%

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

   6. pow-pow 95.6%

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

   7. pow2 95.6%

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

   8. rem-log-exp 95.6%

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

   9. metadata-eval 95.6%

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

6. Applied egg-rr 95.6%

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

8. Applied egg-rr 99.3%

   \[\leadsto \cos x \cdot \color{blue}{{\left({\left(e^{40}\right)}^{x}\right)}^{\left(x \cdot 0.25\right)}} \]
9. Step-by-step derivation

   1. add-sqr-sqrt 99.2%

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

   2. sqrt-unprod 99.3%

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

   3. pow-prod-down 99.3%

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

   4. prod-exp 99.4%

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

   5. metadata-eval 99.4%

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

10. Applied egg-rr 99.4%

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

Alternative 2: 99.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \cos x \cdot {\left({\left(e^{80}\right)}^{\left(x \cdot 0.5\right)}\right)}^{\left(x \cdot 0.25\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (cos x) (pow (pow (exp 80.0) (* x 0.5)) (* x 0.25))))

C

double code(double x) {
	return cos(x) * pow(pow(exp(80.0), (x * 0.5)), (x * 0.25));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = cos(x) * ((exp(80.0d0) ** (x * 0.5d0)) ** (x * 0.25d0))
end function

Java

public static double code(double x) {
	return Math.cos(x) * Math.pow(Math.pow(Math.exp(80.0), (x * 0.5)), (x * 0.25));
}

Python

def code(x):
	return math.cos(x) * math.pow(math.pow(math.exp(80.0), (x * 0.5)), (x * 0.25))

Julia

function code(x)
	return Float64(cos(x) * ((exp(80.0) ^ Float64(x * 0.5)) ^ Float64(x * 0.25)))
end

MATLAB

function tmp = code(x)
	tmp = cos(x) * ((exp(80.0) ^ (x * 0.5)) ^ (x * 0.25));
end

Wolfram

code[x_] := N[(N[Cos[x], $MachinePrecision] * N[Power[N[Power[N[Exp[80.0], $MachinePrecision], N[(x * 0.5), $MachinePrecision]], $MachinePrecision], N[(x * 0.25), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.6%

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

   1. exp-prod 95.6%

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

3. Simplified 95.6%

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

   1. add-cbrt-cube 95.6%

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

   2. pow3 95.6%

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

   3. add-exp-log 94.6%

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

   4. log-pow 94.7%

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

   5. pow-exp 95.6%

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

   6. pow-pow 95.6%

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

   7. pow2 95.6%

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

   8. rem-log-exp 95.6%

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

   9. metadata-eval 95.6%

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

6. Applied egg-rr 95.6%

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

8. Applied egg-rr 99.3%

   \[\leadsto \cos x \cdot \color{blue}{{\left({\left(e^{40}\right)}^{x}\right)}^{\left(x \cdot 0.25\right)}} \]
9. Step-by-step derivation

   1. add-sqr-sqrt 99.2%

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

   2. sqrt-unprod 99.3%

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

   3. pow-prod-down 99.3%

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

   4. prod-exp 99.4%

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

   5. metadata-eval 99.4%

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

10. Applied egg-rr 99.4%

    \[\leadsto \cos x \cdot {\color{blue}{\left(\sqrt{{\left(e^{80}\right)}^{x}}\right)}}^{\left(x \cdot 0.25\right)} \]
11. Step-by-step derivation

    1. pow1/2 99.4%

       \[\leadsto \cos x \cdot {\color{blue}{\left({\left({\left(e^{80}\right)}^{x}\right)}^{0.5}\right)}}^{\left(x \cdot 0.25\right)} \]

    2. pow-pow 99.4%

       \[\leadsto \cos x \cdot {\color{blue}{\left({\left(e^{80}\right)}^{\left(x \cdot 0.5\right)}\right)}}^{\left(x \cdot 0.25\right)} \]

12. Applied egg-rr 99.4%

    \[\leadsto \cos x \cdot {\color{blue}{\left({\left(e^{80}\right)}^{\left(x \cdot 0.5\right)}\right)}}^{\left(x \cdot 0.25\right)} \]
13. Add Preprocessing

Alternative 3: 99.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \cos x \cdot {\left({\left(e^{20}\right)}^{\left(x \cdot 2\right)}\right)}^{\left(x \cdot 0.25\right)} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (cos x) (pow (pow (exp 20.0) (* x 2.0)) (* x 0.25))))

C

double code(double x) {
	return cos(x) * pow(pow(exp(20.0), (x * 2.0)), (x * 0.25));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = cos(x) * ((exp(20.0d0) ** (x * 2.0d0)) ** (x * 0.25d0))
end function

Java

public static double code(double x) {
	return Math.cos(x) * Math.pow(Math.pow(Math.exp(20.0), (x * 2.0)), (x * 0.25));
}

Python

def code(x):
	return math.cos(x) * math.pow(math.pow(math.exp(20.0), (x * 2.0)), (x * 0.25))

Julia

function code(x)
	return Float64(cos(x) * ((exp(20.0) ^ Float64(x * 2.0)) ^ Float64(x * 0.25)))
end

MATLAB

function tmp = code(x)
	tmp = cos(x) * ((exp(20.0) ^ (x * 2.0)) ^ (x * 0.25));
end

Wolfram

code[x_] := N[(N[Cos[x], $MachinePrecision] * N[Power[N[Power[N[Exp[20.0], $MachinePrecision], N[(x * 2.0), $MachinePrecision]], $MachinePrecision], N[(x * 0.25), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.6%

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

   1. exp-prod 95.6%

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

3. Simplified 95.6%

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

   1. add-cbrt-cube 95.6%

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

   2. pow3 95.6%

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

   3. add-exp-log 94.6%

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

   4. log-pow 94.7%

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

   5. pow-exp 95.6%

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

   6. pow-pow 95.6%

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

   7. pow2 95.6%

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

   8. rem-log-exp 95.6%

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

   9. metadata-eval 95.6%

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

6. Applied egg-rr 95.6%

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

8. Applied egg-rr 99.3%

   \[\leadsto \cos x \cdot \color{blue}{{\left({\left(e^{40}\right)}^{x}\right)}^{\left(x \cdot 0.25\right)}} \]
9. Step-by-step derivation

   1. add-sqr-sqrt 99.3%

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

   2. unpow-prod-down 99.4%

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

   3. metadata-eval 99.4%

      \[\leadsto \cos x \cdot {\left({\left(\sqrt{e^{\color{blue}{20 + 20}}}\right)}^{x} \cdot {\left(\sqrt{e^{40}}\right)}^{x}\right)}^{\left(x \cdot 0.25\right)} \]

   4. prod-exp 99.4%

      \[\leadsto \cos x \cdot {\left({\left(\sqrt{\color{blue}{e^{20} \cdot e^{20}}}\right)}^{x} \cdot {\left(\sqrt{e^{40}}\right)}^{x}\right)}^{\left(x \cdot 0.25\right)} \]

   5. expm1-log1p-u 97.3%

      \[\leadsto \cos x \cdot {\left({\left(\sqrt{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{20}\right)\right)} \cdot e^{20}}\right)}^{x} \cdot {\left(\sqrt{e^{40}}\right)}^{x}\right)}^{\left(x \cdot 0.25\right)} \]

   6. expm1-log1p-u 96.5%

      \[\leadsto \cos x \cdot {\left({\left(\sqrt{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{20}\right)\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{20}\right)\right)}}\right)}^{x} \cdot {\left(\sqrt{e^{40}}\right)}^{x}\right)}^{\left(x \cdot 0.25\right)} \]

   7. sqrt-unprod 96.8%

      \[\leadsto \cos x \cdot {\left({\color{blue}{\left(\sqrt{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{20}\right)\right)} \cdot \sqrt{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{20}\right)\right)}\right)}}^{x} \cdot {\left(\sqrt{e^{40}}\right)}^{x}\right)}^{\left(x \cdot 0.25\right)} \]

   8. add-sqr-sqrt 96.5%

      \[\leadsto \cos x \cdot {\left({\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(e^{20}\right)\right)\right)}}^{x} \cdot {\left(\sqrt{e^{40}}\right)}^{x}\right)}^{\left(x \cdot 0.25\right)} \]

   9. expm1-log1p-u 99.4%

      \[\leadsto \cos x \cdot {\left({\color{blue}{\left(e^{20}\right)}}^{x} \cdot {\left(\sqrt{e^{40}}\right)}^{x}\right)}^{\left(x \cdot 0.25\right)} \]

   10. metadata-eval 99.4%

       \[\leadsto \cos x \cdot {\left({\left(e^{20}\right)}^{x} \cdot {\left(\sqrt{e^{\color{blue}{20 + 20}}}\right)}^{x}\right)}^{\left(x \cdot 0.25\right)} \]

   11. prod-exp 99.4%

       \[\leadsto \cos x \cdot {\left({\left(e^{20}\right)}^{x} \cdot {\left(\sqrt{\color{blue}{e^{20} \cdot e^{20}}}\right)}^{x}\right)}^{\left(x \cdot 0.25\right)} \]

   12. expm1-log1p-u 97.3%

       \[\leadsto \cos x \cdot {\left({\left(e^{20}\right)}^{x} \cdot {\left(\sqrt{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{20}\right)\right)} \cdot e^{20}}\right)}^{x}\right)}^{\left(x \cdot 0.25\right)} \]

   13. expm1-log1p-u 96.5%

       \[\leadsto \cos x \cdot {\left({\left(e^{20}\right)}^{x} \cdot {\left(\sqrt{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{20}\right)\right) \cdot \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{20}\right)\right)}}\right)}^{x}\right)}^{\left(x \cdot 0.25\right)} \]

   14. sqrt-unprod 96.8%

       \[\leadsto \cos x \cdot {\left({\left(e^{20}\right)}^{x} \cdot {\color{blue}{\left(\sqrt{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{20}\right)\right)} \cdot \sqrt{\mathsf{expm1}\left(\mathsf{log1p}\left(e^{20}\right)\right)}\right)}}^{x}\right)}^{\left(x \cdot 0.25\right)} \]

   15. add-sqr-sqrt 96.5%

       \[\leadsto \cos x \cdot {\left({\left(e^{20}\right)}^{x} \cdot {\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(e^{20}\right)\right)\right)}}^{x}\right)}^{\left(x \cdot 0.25\right)} \]

   16. expm1-log1p-u 99.4%

       \[\leadsto \cos x \cdot {\left({\left(e^{20}\right)}^{x} \cdot {\color{blue}{\left(e^{20}\right)}}^{x}\right)}^{\left(x \cdot 0.25\right)} \]

10. Applied egg-rr 99.4%

    \[\leadsto \cos x \cdot {\color{blue}{\left({\left(e^{20}\right)}^{x} \cdot {\left(e^{20}\right)}^{x}\right)}}^{\left(x \cdot 0.25\right)} \]
11. Step-by-step derivation

    1. pow-sqr 99.4%

       \[\leadsto \cos x \cdot {\color{blue}{\left({\left(e^{20}\right)}^{\left(2 \cdot x\right)}\right)}}^{\left(x \cdot 0.25\right)} \]

    2. *-commutative 99.4%

       \[\leadsto \cos x \cdot {\left({\left(e^{20}\right)}^{\color{blue}{\left(x \cdot 2\right)}}\right)}^{\left(x \cdot 0.25\right)} \]

12. Simplified 99.4%

    \[\leadsto \cos x \cdot {\color{blue}{\left({\left(e^{20}\right)}^{\left(x \cdot 2\right)}\right)}}^{\left(x \cdot 0.25\right)} \]
13. Add Preprocessing

Alternative 4: 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 94.6%

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

   1. pow-exp 95.6%

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

   2. sqr-pow 95.5%

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

   3. pow-prod-down 95.6%

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

   4. associate-/l* 95.6%

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

   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 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 94.6%

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

   1. pow-exp 95.6%

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

   2. pow-unpow 97.9%

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

4. Applied egg-rr 97.9%

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

Alternative 6: 95.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \cos x \cdot {\left(e^{{x}^{2}}\right)}^{10} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (cos x) (pow (exp (pow x 2.0)) 10.0)))

C

double code(double x) {
	return cos(x) * pow(exp(pow(x, 2.0)), 10.0);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = cos(x) * (exp((x ** 2.0d0)) ** 10.0d0)
end function

Java

public static double code(double x) {
	return Math.cos(x) * Math.pow(Math.exp(Math.pow(x, 2.0)), 10.0);
}

Python

def code(x):
	return math.cos(x) * math.pow(math.exp(math.pow(x, 2.0)), 10.0)

Julia

function code(x)
	return Float64(cos(x) * (exp((x ^ 2.0)) ^ 10.0))
end

MATLAB

function tmp = code(x)
	tmp = cos(x) * (exp((x ^ 2.0)) ^ 10.0);
end

Wolfram

code[x_] := N[(N[Cos[x], $MachinePrecision] * N[Power[N[Exp[N[Power[x, 2.0], $MachinePrecision]], $MachinePrecision], 10.0], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.6%

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

   1. *-commutative 94.6%

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

   2. exp-prod 95.6%

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

   3. pow2 95.6%

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

4. Applied egg-rr 95.6%

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

Alternative 7: 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 94.6%

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

   1. pow-exp 95.6%

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

   2. sqr-pow 95.5%

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

   3. pow-prod-down 95.6%

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

   4. prod-exp 95.6%

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

   5. metadata-eval 95.6%

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

   6. div-inv 95.6%

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

   7. pow2 95.6%

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

   8. metadata-eval 95.6%

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

4. Applied egg-rr 95.6%

   \[\leadsto \cos x \cdot \color{blue}{{\left(e^{20}\right)}^{\left({x}^{2} \cdot 0.5\right)}} \]
5. Step-by-step derivation

   1. unpow2 95.6%

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

6. Applied egg-rr 95.6%

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

7. Final simplification 95.6%

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

Alternative 8: 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 94.6%

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

   1. exp-prod 95.6%

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

3. Simplified 95.6%

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

Alternative 9: 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 94.6%

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

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 94.6%

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

   1. pow-exp 95.6%

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

   2. sqr-pow 95.5%

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

   3. pow-prod-down 95.6%

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

   4. associate-/l* 95.6%

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

   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. Add Preprocessing

Reproduce

herbie shell --seed 2024144 
(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)))))