cos2 (problem 3.4.1)

Percentage Accurate: 51.0% → 50.3%

Time: 10.7s

Alternatives: 4

Speedup: N/A×

Specification

Math

\[\begin{array}{l} \\ \frac{1 - \cos x}{x \cdot x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (- 1.0 (cos x)) (* x x)))

C

double code(double x) {
	return (1.0 - cos(x)) / (x * x);
}

Fortran

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

Java

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

Python

def code(x):
	return (1.0 - math.cos(x)) / (x * x)

Julia

function code(x)
	return Float64(Float64(1.0 - cos(x)) / Float64(x * x))
end

MATLAB

function tmp = code(x)
	tmp = (1.0 - cos(x)) / (x * x);
end

Wolfram

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

Initial Program: 51.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1 - \cos x}{x \cdot x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (- 1.0 (cos x)) (* x x)))

C

double code(double x) {
	return (1.0 - cos(x)) / (x * x);
}

Fortran

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

Java

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

Python

def code(x):
	return (1.0 - math.cos(x)) / (x * x)

Julia

function code(x)
	return Float64(Float64(1.0 - cos(x)) / Float64(x * x))
end

MATLAB

function tmp = code(x)
	tmp = (1.0 - cos(x)) / (x * x);
end

Wolfram

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

Alternative 1: 50.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 0.5 + {x}^{2} \cdot -0.041666666666666664 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (+ 0.5 (* (pow x 2.0) -0.041666666666666664)))

C

double code(double x) {
	return 0.5 + (pow(x, 2.0) * -0.041666666666666664);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.5d0 + ((x ** 2.0d0) * (-0.041666666666666664d0))
end function

Java

public static double code(double x) {
	return 0.5 + (Math.pow(x, 2.0) * -0.041666666666666664);
}

Python

def code(x):
	return 0.5 + (math.pow(x, 2.0) * -0.041666666666666664)

Julia

function code(x)
	return Float64(0.5 + Float64((x ^ 2.0) * -0.041666666666666664))
end

MATLAB

function tmp = code(x)
	tmp = 0.5 + ((x ^ 2.0) * -0.041666666666666664);
end

Wolfram

code[x_] := N[(0.5 + N[(N[Power[x, 2.0], $MachinePrecision] * -0.041666666666666664), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 48.6%

   \[\frac{1 - \cos x}{x \cdot x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0 53.1%

   \[\leadsto \color{blue}{0.5 + -0.041666666666666664 \cdot {x}^{2}} \]
4. Step-by-step derivation

   1. *-commutative 53.1%

      \[\leadsto 0.5 + \color{blue}{{x}^{2} \cdot -0.041666666666666664} \]

5. Simplified 53.1%

   \[\leadsto \color{blue}{0.5 + {x}^{2} \cdot -0.041666666666666664} \]

6. Final simplification 53.1%

   \[\leadsto 0.5 + {x}^{2} \cdot -0.041666666666666664 \]
7. Add Preprocessing

Alternative 2: 32.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{2}{{x}^{2}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 2.0 (pow x 2.0)))

C

double code(double x) {
	return 2.0 / pow(x, 2.0);
}

Fortran

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

Java

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

Python

def code(x):
	return 2.0 / math.pow(x, 2.0)

Julia

function code(x)
	return Float64(2.0 / (x ^ 2.0))
end

MATLAB

function tmp = code(x)
	tmp = 2.0 / (x ^ 2.0);
end

Wolfram

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

Derivation

1. Initial program 48.6%

   \[\frac{1 - \cos x}{x \cdot x} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. associate-/r* 49.7%

      \[\leadsto \color{blue}{\frac{\frac{1 - \cos x}{x}}{x}} \]

   2. div-inv 49.7%

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

4. Applied egg-rr 49.7%

   \[\leadsto \color{blue}{\frac{1 - \cos x}{x} \cdot \frac{1}{x}} \]
5. Step-by-step derivation

   1. associate-*l/ 49.7%

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

   2. associate-*r/ 49.7%

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

   3. *-rgt-identity 49.7%

      \[\leadsto \frac{\frac{\color{blue}{1 - \cos x}}{x}}{x} \]

   4. sub-neg 49.7%

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

   5. add-sqr-sqrt 26.3%

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

   6. distribute-rgt-neg-in 26.3%

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

   7. add-sqr-sqrt 0.0%

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

   8. sqrt-unprod 18.8%

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

   9. sqr-neg 18.8%

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

   10. add-sqr-sqrt 18.8%

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

   11. add-sqr-sqrt 32.2%

       \[\leadsto \frac{\frac{1 + \color{blue}{\cos x}}{x}}{x} \]

6. Applied egg-rr 32.2%

   \[\leadsto \color{blue}{\frac{\frac{1 + \cos x}{x}}{x}} \]

7. Taylor expanded in x around 0 32.9%

   \[\leadsto \color{blue}{\frac{2}{{x}^{2}}} \]

8. Final simplification 32.9%

   \[\leadsto \frac{2}{{x}^{2}} \]
9. Add Preprocessing

Alternative 3: 32.1% accurate, 21.4× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{2}{x}}{x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (/ 2.0 x) x))

C

double code(double x) {
	return (2.0 / x) / x;
}

Fortran

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

Java

public static double code(double x) {
	return (2.0 / x) / x;
}

Python

def code(x):
	return (2.0 / x) / x

Julia

function code(x)
	return Float64(Float64(2.0 / x) / x)
end

MATLAB

function tmp = code(x)
	tmp = (2.0 / x) / x;
end

Wolfram

code[x_] := N[(N[(2.0 / x), $MachinePrecision] / x), $MachinePrecision]

Derivation

1. Initial program 48.6%

   \[\frac{1 - \cos x}{x \cdot x} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. associate-/r* 49.7%

      \[\leadsto \color{blue}{\frac{\frac{1 - \cos x}{x}}{x}} \]

   2. div-inv 49.7%

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

4. Applied egg-rr 49.7%

   \[\leadsto \color{blue}{\frac{1 - \cos x}{x} \cdot \frac{1}{x}} \]
5. Step-by-step derivation

   1. associate-*l/ 49.7%

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

   2. associate-*r/ 49.7%

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

   3. *-rgt-identity 49.7%

      \[\leadsto \frac{\frac{\color{blue}{1 - \cos x}}{x}}{x} \]

   4. sub-neg 49.7%

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

   5. add-sqr-sqrt 26.3%

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

   6. distribute-rgt-neg-in 26.3%

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

   7. add-sqr-sqrt 0.0%

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

   8. sqrt-unprod 18.8%

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

   9. sqr-neg 18.8%

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

   10. add-sqr-sqrt 18.8%

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

   11. add-sqr-sqrt 32.2%

       \[\leadsto \frac{\frac{1 + \color{blue}{\cos x}}{x}}{x} \]

6. Applied egg-rr 32.2%

   \[\leadsto \color{blue}{\frac{\frac{1 + \cos x}{x}}{x}} \]

7. Taylor expanded in x around 0 32.8%

   \[\leadsto \frac{\frac{\color{blue}{2}}{x}}{x} \]

8. Final simplification 32.8%

   \[\leadsto \frac{\frac{2}{x}}{x} \]
9. Add Preprocessing

Alternative 4: 27.9% accurate, 107.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.0)

C

double code(double x) {
	return 0.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.0d0
end function

Java

public static double code(double x) {
	return 0.0;
}

Python

def code(x):
	return 0.0

Julia

function code(x)
	return 0.0
end

MATLAB

function tmp = code(x)
	tmp = 0.0;
end

Wolfram

code[x_] := 0.0

Derivation

1. Initial program 48.6%

   \[\frac{1 - \cos x}{x \cdot x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0 28.3%

   \[\leadsto \frac{1 - \color{blue}{1}}{x \cdot x} \]

4. Taylor expanded in x around 0 29.1%

   \[\leadsto \color{blue}{0} \]

5. Final simplification 29.1%

   \[\leadsto 0 \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024033 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))