ENA, Section 1.4, Exercise 1

Percentage Accurate: 94.5% → 99.4%

Time: 8.5s

Alternatives: 7

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

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

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 94.5%

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

   1. pow-exp 95.0%

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

   2. sqr-pow 95.1%

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

   3. pow-prod-down 95.0%

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

   4. associate-/l* 95.0%

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

   5. pow-unpow 97.9%

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

   6. prod-exp 99.5%

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

   7. metadata-eval 99.5%

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

4. Applied egg-rr 99.5%

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

Alternative 2: 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 94.5%

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

   1. pow-exp 95.0%

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

   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 3: 95.1% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.5%

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

   1. pow-exp 95.0%

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

   2. sqr-pow 95.1%

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

   3. pow-prod-down 95.0%

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

   4. associate-/l* 95.0%

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

   5. pow-unpow 97.9%

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

   6. prod-exp 99.5%

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

   7. metadata-eval 99.5%

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

4. Applied egg-rr 99.5%

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

   1. add-exp-log 95.1%

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

   2. log-pow 95.1%

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

   3. rem-log-exp 95.1%

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

6. Applied egg-rr 95.1%

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

Alternative 4: 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. exp-prod 95.0%

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

3. Simplified 95.0%

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

Alternative 5: 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. Add Preprocessing
3. Add Preprocessing

Alternative 6: 9.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \cos x \cdot \frac{e}{e} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (cos x) (/ E E)))

C

double code(double x) {
	return cos(x) * (((double) M_E) / ((double) M_E));
}

Java

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

Python

def code(x):
	return math.cos(x) * (math.e / math.e)

Julia

function code(x)
	return Float64(cos(x) * Float64(exp(1) / exp(1)))
end

MATLAB

function tmp = code(x)
	tmp = cos(x) * (2.71828182845904523536 / 2.71828182845904523536);
end

Wolfram

code[x_] := N[(N[Cos[x], $MachinePrecision] * N[(E / E), $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. exp-prod 95.0%

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

3. Simplified 95.0%

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

   1. pow-exp 94.5%

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

   2. *-un-lft-identity 94.5%

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

   3. exp-prod 94.3%

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

   4. expm1-log1p-u 92.5%

      \[\leadsto \cos x \cdot {\left(e^{1}\right)}^{\color{blue}{\left(\mathsf{expm1}\left(\mathsf{log1p}\left(10 \cdot \left(x \cdot x\right)\right)\right)\right)}} \]

   5. expm1-undefine 92.6%

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

   6. pow-sub 92.6%

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

   7. rem-log-exp 92.6%

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

   8. *-commutative 92.6%

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

   9. pow2 92.6%

      \[\leadsto \cos x \cdot \frac{{\left(e^{1}\right)}^{\left(e^{\mathsf{log1p}\left(\color{blue}{{x}^{2}} \cdot \log \left(e^{10}\right)\right)}\right)}}{{\left(e^{1}\right)}^{1}} \]

   10. rem-log-exp 92.6%

       \[\leadsto \cos x \cdot \frac{{\left(e^{1}\right)}^{\left(e^{\mathsf{log1p}\left({x}^{2} \cdot \color{blue}{10}\right)}\right)}}{{\left(e^{1}\right)}^{1}} \]

6. Applied egg-rr 92.6%

   \[\leadsto \cos x \cdot \color{blue}{\frac{{\left(e^{1}\right)}^{\left(e^{\mathsf{log1p}\left({x}^{2} \cdot 10\right)}\right)}}{{\left(e^{1}\right)}^{1}}} \]
7. Step-by-step derivation

   1. exp-1-e 92.6%

      \[\leadsto \cos x \cdot \frac{{\color{blue}{e}}^{\left(e^{\mathsf{log1p}\left({x}^{2} \cdot 10\right)}\right)}}{{\left(e^{1}\right)}^{1}} \]

   2. log1p-undefine 92.6%

      \[\leadsto \cos x \cdot \frac{{e}^{\left(e^{\color{blue}{\log \left(1 + {x}^{2} \cdot 10\right)}}\right)}}{{\left(e^{1}\right)}^{1}} \]

   3. rem-exp-log 94.3%

      \[\leadsto \cos x \cdot \frac{{e}^{\color{blue}{\left(1 + {x}^{2} \cdot 10\right)}}}{{\left(e^{1}\right)}^{1}} \]

   4. +-commutative 94.3%

      \[\leadsto \cos x \cdot \frac{{e}^{\color{blue}{\left({x}^{2} \cdot 10 + 1\right)}}}{{\left(e^{1}\right)}^{1}} \]

   5. fma-define 94.3%

      \[\leadsto \cos x \cdot \frac{{e}^{\color{blue}{\left(\mathsf{fma}\left({x}^{2}, 10, 1\right)\right)}}}{{\left(e^{1}\right)}^{1}} \]

   6. unpow1 94.3%

      \[\leadsto \cos x \cdot \frac{{e}^{\left(\mathsf{fma}\left({x}^{2}, 10, 1\right)\right)}}{\color{blue}{e^{1}}} \]

   7. exp-1-e 94.3%

      \[\leadsto \cos x \cdot \frac{{e}^{\left(\mathsf{fma}\left({x}^{2}, 10, 1\right)\right)}}{\color{blue}{e}} \]

8. Simplified 94.3%

   \[\leadsto \cos x \cdot \color{blue}{\frac{{e}^{\left(\mathsf{fma}\left({x}^{2}, 10, 1\right)\right)}}{e}} \]

9. Taylor expanded in x around 0 9.6%

   \[\leadsto \cos x \cdot \frac{\color{blue}{e}}{e} \]
10. Add Preprocessing

Alternative 7: 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. Add Preprocessing
3. Step-by-step derivation

   1. pow-exp 95.0%

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

   2. sqr-pow 95.1%

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

   3. pow-prod-down 95.0%

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

   4. associate-/l* 95.0%

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

   5. pow-unpow 97.9%

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

   6. prod-exp 99.5%

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

   7. metadata-eval 99.5%

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

4. Applied egg-rr 99.5%

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

5. Taylor expanded in x around 0 1.5%

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

Reproduce

herbie shell --seed 2024157 
(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)))))