cos2 (problem 3.4.1)

Percentage Accurate: 50.9% → 99.8%

Time: 10.9s

Alternatives: 5

Speedup: 107.0×

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: 50.9% 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: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\tan \left(\frac{x}{2}\right)}{x}}{\frac{x}{\sin x}} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

public static double code(double x) {
	return (Math.tan((x / 2.0)) / x) / (x / Math.sin(x));
}

Python

def code(x):
	return (math.tan((x / 2.0)) / x) / (x / math.sin(x))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 47.4%

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

   1. associate-/r* 48.4%

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

   2. div-inv 48.4%

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

4. Applied egg-rr 48.4%

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

   1. frac-times 47.4%

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

   2. unpow2 47.4%

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

   3. clear-num 47.4%

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

   4. *-rgt-identity 47.4%

      \[\leadsto \frac{1}{\frac{{x}^{2}}{\color{blue}{1 - \cos x}}} \]

   5. flip-- 47.1%

      \[\leadsto \frac{1}{\frac{{x}^{2}}{\color{blue}{\frac{1 \cdot 1 - \cos x \cdot \cos x}{1 + \cos x}}}} \]

   6. metadata-eval 47.1%

      \[\leadsto \frac{1}{\frac{{x}^{2}}{\frac{\color{blue}{1} - \cos x \cdot \cos x}{1 + \cos x}}} \]

   7. 1-sub-cos 74.1%

      \[\leadsto \frac{1}{\frac{{x}^{2}}{\frac{\color{blue}{\sin x \cdot \sin x}}{1 + \cos x}}} \]

   8. associate-/r/ 74.6%

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

   9. associate-/l/ 74.6%

      \[\leadsto \color{blue}{\frac{\frac{1}{1 + \cos x}}{\frac{{x}^{2}}{\sin x \cdot \sin x}}} \]

   10. add-sqr-sqrt 74.6%

       \[\leadsto \frac{\frac{1}{1 + \cos x}}{\color{blue}{\sqrt{\frac{{x}^{2}}{\sin x \cdot \sin x}} \cdot \sqrt{\frac{{x}^{2}}{\sin x \cdot \sin x}}}} \]

   11. associate-/r* 74.6%

       \[\leadsto \color{blue}{\frac{\frac{\frac{1}{1 + \cos x}}{\sqrt{\frac{{x}^{2}}{\sin x \cdot \sin x}}}}{\sqrt{\frac{{x}^{2}}{\sin x \cdot \sin x}}}} \]

6. Applied egg-rr 99.5%

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

7. Taylor expanded in x around inf 99.5%

   \[\leadsto \frac{\color{blue}{\frac{\sin x}{x \cdot \left(1 + \cos x\right)}}}{\frac{x}{\sin x}} \]
8. Step-by-step derivation

   1. *-commutative 99.5%

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

   2. associate-/r* 99.5%

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

   3. hang-0p-tan 99.8%

      \[\leadsto \frac{\frac{\color{blue}{\tan \left(\frac{x}{2}\right)}}{x}}{\frac{x}{\sin x}} \]

9. Simplified 99.8%

   \[\leadsto \frac{\color{blue}{\frac{\tan \left(\frac{x}{2}\right)}{x}}}{\frac{x}{\sin x}} \]

10. Final simplification 99.8%

    \[\leadsto \frac{\frac{\tan \left(\frac{x}{2}\right)}{x}}{\frac{x}{\sin x}} \]
11. Add Preprocessing

Alternative 2: 74.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0042:\\ \;\;\;\;0.5 + {x}^{2} \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.0042)
   (+ 0.5 (* (pow x 2.0) -0.041666666666666664))
   (/ (- 1.0 (cos x)) (* x x))))

C

double code(double x) {
	double tmp;
	if (x <= 0.0042) {
		tmp = 0.5 + (pow(x, 2.0) * -0.041666666666666664);
	} else {
		tmp = (1.0 - cos(x)) / (x * x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.0042d0) then
        tmp = 0.5d0 + ((x ** 2.0d0) * (-0.041666666666666664d0))
    else
        tmp = (1.0d0 - cos(x)) / (x * x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.0042) {
		tmp = 0.5 + (Math.pow(x, 2.0) * -0.041666666666666664);
	} else {
		tmp = (1.0 - Math.cos(x)) / (x * x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.0042:
		tmp = 0.5 + (math.pow(x, 2.0) * -0.041666666666666664)
	else:
		tmp = (1.0 - math.cos(x)) / (x * x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.0042)
		tmp = Float64(0.5 + Float64((x ^ 2.0) * -0.041666666666666664));
	else
		tmp = Float64(Float64(1.0 - cos(x)) / Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.0042)
		tmp = 0.5 + ((x ^ 2.0) * -0.041666666666666664);
	else
		tmp = (1.0 - cos(x)) / (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.0042], N[(0.5 + N[(N[Power[x, 2.0], $MachinePrecision] * -0.041666666666666664), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.00419999999999999974

   1. Initial program 31.5%

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

   3. Taylor expanded in x around 0 70.8%

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

      1. *-commutative 70.8%

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

   5. Simplified 70.8%

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

   if 0.00419999999999999974 < x

   1. Initial program 98.2%

      \[\frac{1 - \cos x}{x \cdot x} \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0042:\\ \;\;\;\;0.5 + {x}^{2} \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0042:\\ \;\;\;\;0.5 + {x}^{2} \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.0042)
   (+ 0.5 (* (pow x 2.0) -0.041666666666666664))
   (/ (/ (- 1.0 (cos x)) x) x)))

C

double code(double x) {
	double tmp;
	if (x <= 0.0042) {
		tmp = 0.5 + (pow(x, 2.0) * -0.041666666666666664);
	} else {
		tmp = ((1.0 - cos(x)) / x) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.0042d0) then
        tmp = 0.5d0 + ((x ** 2.0d0) * (-0.041666666666666664d0))
    else
        tmp = ((1.0d0 - cos(x)) / x) / x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.0042) {
		tmp = 0.5 + (Math.pow(x, 2.0) * -0.041666666666666664);
	} else {
		tmp = ((1.0 - Math.cos(x)) / x) / x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.0042:
		tmp = 0.5 + (math.pow(x, 2.0) * -0.041666666666666664)
	else:
		tmp = ((1.0 - math.cos(x)) / x) / x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.0042)
		tmp = Float64(0.5 + Float64((x ^ 2.0) * -0.041666666666666664));
	else
		tmp = Float64(Float64(Float64(1.0 - cos(x)) / x) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.0042)
		tmp = 0.5 + ((x ^ 2.0) * -0.041666666666666664);
	else
		tmp = ((1.0 - cos(x)) / x) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.0042], N[(0.5 + N[(N[Power[x, 2.0], $MachinePrecision] * -0.041666666666666664), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.00419999999999999974

   1. Initial program 31.5%

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

   3. Taylor expanded in x around 0 70.8%

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

      1. *-commutative 70.8%

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

   5. Simplified 70.8%

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

   if 0.00419999999999999974 < x

   1. Initial program 98.2%

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

      1. associate-/r* 99.3%

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

      2. div-inv 99.3%

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

   4. Applied egg-rr 99.3%

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

      1. un-div-inv 99.3%

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

   6. Applied egg-rr 99.3%

      \[\leadsto \color{blue}{\frac{\frac{1 - \cos x}{x}}{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0042:\\ \;\;\;\;0.5 + {x}^{2} \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 52.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{\frac{x}{\sin x}} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ 0.5 (/ x (sin x))))

C

double code(double x) {
	return 0.5 / (x / sin(x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.5d0 / (x / sin(x))
end function

Java

public static double code(double x) {
	return 0.5 / (x / Math.sin(x));
}

Python

def code(x):
	return 0.5 / (x / math.sin(x))

Julia

function code(x)
	return Float64(0.5 / Float64(x / sin(x)))
end

MATLAB

function tmp = code(x)
	tmp = 0.5 / (x / sin(x));
end

Wolfram

code[x_] := N[(0.5 / N[(x / N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 47.4%

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

   1. associate-/r* 48.4%

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

   2. div-inv 48.4%

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

4. Applied egg-rr 48.4%

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

   1. frac-times 47.4%

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

   2. unpow2 47.4%

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

   3. clear-num 47.4%

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

   4. *-rgt-identity 47.4%

      \[\leadsto \frac{1}{\frac{{x}^{2}}{\color{blue}{1 - \cos x}}} \]

   5. flip-- 47.1%

      \[\leadsto \frac{1}{\frac{{x}^{2}}{\color{blue}{\frac{1 \cdot 1 - \cos x \cdot \cos x}{1 + \cos x}}}} \]

   6. metadata-eval 47.1%

      \[\leadsto \frac{1}{\frac{{x}^{2}}{\frac{\color{blue}{1} - \cos x \cdot \cos x}{1 + \cos x}}} \]

   7. 1-sub-cos 74.1%

      \[\leadsto \frac{1}{\frac{{x}^{2}}{\frac{\color{blue}{\sin x \cdot \sin x}}{1 + \cos x}}} \]

   8. associate-/r/ 74.6%

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

   9. associate-/l/ 74.6%

      \[\leadsto \color{blue}{\frac{\frac{1}{1 + \cos x}}{\frac{{x}^{2}}{\sin x \cdot \sin x}}} \]

   10. add-sqr-sqrt 74.6%

       \[\leadsto \frac{\frac{1}{1 + \cos x}}{\color{blue}{\sqrt{\frac{{x}^{2}}{\sin x \cdot \sin x}} \cdot \sqrt{\frac{{x}^{2}}{\sin x \cdot \sin x}}}} \]

   11. associate-/r* 74.6%

       \[\leadsto \color{blue}{\frac{\frac{\frac{1}{1 + \cos x}}{\sqrt{\frac{{x}^{2}}{\sin x \cdot \sin x}}}}{\sqrt{\frac{{x}^{2}}{\sin x \cdot \sin x}}}} \]

6. Applied egg-rr 99.5%

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

7. Taylor expanded in x around 0 56.0%

   \[\leadsto \frac{\color{blue}{0.5}}{\frac{x}{\sin x}} \]

8. Final simplification 56.0%

   \[\leadsto \frac{0.5}{\frac{x}{\sin x}} \]
9. Add Preprocessing

Alternative 5: 51.5% accurate, 107.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \end{array} \]

FPCore

(FPCore (x) :precision binary64 0.5)

C

double code(double x) {
	return 0.5;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.5d0
end function

Java

public static double code(double x) {
	return 0.5;
}

Python

def code(x):
	return 0.5

Julia

function code(x)
	return 0.5
end

MATLAB

function tmp = code(x)
	tmp = 0.5;
end

Wolfram

code[x_] := 0.5

Derivation

1. Initial program 47.4%

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

3. Taylor expanded in x around 0 55.3%

   \[\leadsto \color{blue}{0.5} \]

4. Final simplification 55.3%

   \[\leadsto 0.5 \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024055 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))