ENA, Section 1.4, Exercise 1

Percentage Accurate: 94.5% → 99.4%

Time: 8.9s

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 94.2%

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

   1. pow-exp 95.3%

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

   2. sqr-pow 95.3%

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

   3. pow-prod-down 95.3%

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

   4. associate-/l* 95.3%

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

   5. pow-unpow 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 99.4%

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

   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 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 94.2%

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

   1. pow-exp 95.3%

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

   2. pow-unpow 98.0%

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

4. Applied egg-rr 98.0%

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

Alternative 3: 95.2% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.2%

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

   1. exp-prod 95.3%

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

3. Simplified 95.3%

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

   1. add-cbrt-cube 95.3%

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

   2. pow3 95.3%

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

6. Applied egg-rr 95.3%

   \[\leadsto \color{blue}{\sqrt[3]{{\cos x}^{3}}} \cdot {\left(e^{10}\right)}^{\left(x \cdot x\right)} \]
7. Step-by-step derivation

   1. rem-cbrt-cube 95.3%

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

   2. expm1-log1p-u 95.3%

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

   3. log1p-define 95.4%

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

   4. +-commutative 95.4%

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

   5. expm1-define 95.4%

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

   6. add-exp-log 95.4%

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

   7. +-commutative 95.4%

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

   8. associate--l+ 95.4%

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

   9. sub-neg 95.4%

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

   10. metadata-eval 95.4%

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

8. Applied egg-rr 95.4%

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

Alternative 4: 95.2% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

function code(x)
	return Float64(cos(x) * (exp(20.0) ^ Float64(Float64(x * x) / Float64(-(-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[Exp[20.0], $MachinePrecision], N[(N[(x * x), $MachinePrecision] / (--2.0)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 94.2%

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

   1. pow-exp 95.3%

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

   2. sqr-pow 95.3%

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

   3. pow-prod-down 95.3%

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

   4. 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. pow2 95.4%

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

   8. metadata-eval 95.4%

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

4. Applied egg-rr 95.4%

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

   1. unpow2 95.4%

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

   2. associate-*r* 95.4%

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

   3. metadata-eval 95.4%

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

   4. div-inv 95.4%

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

   5. frac-2neg 95.4%

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

   6. associate-*r/ 95.4%

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

   7. metadata-eval 95.4%

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

6. Applied egg-rr 95.4%

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

7. Final simplification 95.4%

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

Alternative 5: 95.2% accurate, 0.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.2%

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

   1. exp-prod 95.3%

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

3. Simplified 95.3%

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

Alternative 6: 94.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 94.2%

   \[\cos x \cdot e^{10 \cdot \left(x \cdot x\right)} \]
2. Add Preprocessing
3. 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 94.2%

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

   1. pow-exp 95.3%

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

   2. sqr-pow 95.3%

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

   3. pow-prod-down 95.3%

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

   4. associate-/l* 95.3%

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

   5. pow-unpow 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 99.4%

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

   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 2024137 
(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)))))