cos2 (problem 3.4.1)

Percentage Accurate: 50.5% → 99.4%

Time: 41.3s

Alternatives: 6

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: 50.5% 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: 99.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{\sin x}{x \cdot \frac{x}{\tan \left(x \cdot 0.5\right)}} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.2%

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

   1. flip-- 54.0%

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

   2. associate-/l/ 54.0%

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

   3. metadata-eval 54.0%

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

   4. 1-sub-cos 76.1%

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

   5. pow2 76.1%

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

4. Applied egg-rr 76.1%

   \[\leadsto \color{blue}{\frac{\sin x \cdot \sin x}{{x}^{2} \cdot \left(1 + \cos x\right)}} \]
5. Step-by-step derivation

   1. associate-/l* 76.3%

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

   2. *-commutative 76.3%

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

   3. associate-/r* 76.3%

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

   4. hang-0p-tan 76.5%

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

6. Simplified 76.5%

   \[\leadsto \color{blue}{\sin x \cdot \frac{\tan \left(\frac{x}{2}\right)}{{x}^{2}}} \]
7. Step-by-step derivation

   1. pow2 76.5%

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

   2. associate-*r/ 75.9%

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

   3. *-commutative 75.9%

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

   4. times-frac 99.8%

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

   5. div-inv 99.8%

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

   6. metadata-eval 99.8%

      \[\leadsto \frac{\tan \left(x \cdot \color{blue}{0.5}\right)}{x} \cdot \frac{\sin x}{x} \]

8. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{\frac{\tan \left(x \cdot 0.5\right)}{x} \cdot \frac{\sin x}{x}} \]
9. Step-by-step derivation

   1. clear-num 99.8%

      \[\leadsto \color{blue}{\frac{1}{\frac{x}{\tan \left(x \cdot 0.5\right)}}} \cdot \frac{\sin x}{x} \]

   2. frac-times 99.8%

      \[\leadsto \color{blue}{\frac{1 \cdot \sin x}{\frac{x}{\tan \left(x \cdot 0.5\right)} \cdot x}} \]

   3. *-un-lft-identity 99.8%

      \[\leadsto \frac{\color{blue}{\sin x}}{\frac{x}{\tan \left(x \cdot 0.5\right)} \cdot x} \]

10. Applied egg-rr 99.8%

    \[\leadsto \color{blue}{\frac{\sin x}{\frac{x}{\tan \left(x \cdot 0.5\right)} \cdot x}} \]

11. Final simplification 99.8%

    \[\leadsto \frac{\sin x}{x \cdot \frac{x}{\tan \left(x \cdot 0.5\right)}} \]
12. Add Preprocessing

Alternative 2: 99.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \frac{\tan \left(x \cdot 0.5\right)}{x} \cdot \frac{\sin x}{x} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.2%

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

   1. flip-- 54.0%

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

   2. associate-/l/ 54.0%

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

   3. metadata-eval 54.0%

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

   4. 1-sub-cos 76.1%

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

   5. pow2 76.1%

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

4. Applied egg-rr 76.1%

   \[\leadsto \color{blue}{\frac{\sin x \cdot \sin x}{{x}^{2} \cdot \left(1 + \cos x\right)}} \]
5. Step-by-step derivation

   1. associate-/l* 76.3%

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

   2. *-commutative 76.3%

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

   3. associate-/r* 76.3%

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

   4. hang-0p-tan 76.5%

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

6. Simplified 76.5%

   \[\leadsto \color{blue}{\sin x \cdot \frac{\tan \left(\frac{x}{2}\right)}{{x}^{2}}} \]
7. Step-by-step derivation

   1. pow2 76.5%

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

   2. associate-*r/ 75.9%

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

   3. *-commutative 75.9%

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

   4. times-frac 99.8%

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

   5. div-inv 99.8%

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

   6. metadata-eval 99.8%

      \[\leadsto \frac{\tan \left(x \cdot \color{blue}{0.5}\right)}{x} \cdot \frac{\sin x}{x} \]

8. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{\frac{\tan \left(x \cdot 0.5\right)}{x} \cdot \frac{\sin x}{x}} \]
9. Add Preprocessing

Alternative 3: 75.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.00014:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.00014) 0.5 (/ (/ (- 1.0 (cos x)) x) x)))

C

double code(double x) {
	double tmp;
	if (x <= 0.00014) {
		tmp = 0.5;
	} 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 <= 0.00014d0) then
        tmp = 0.5d0
    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 <= 0.00014) {
		tmp = 0.5;
	} else {
		tmp = ((1.0 - Math.cos(x)) / x) / x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.00014:
		tmp = 0.5
	else:
		tmp = ((1.0 - math.cos(x)) / x) / x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.00014)
		tmp = 0.5;
	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 <= 0.00014)
		tmp = 0.5;
	else
		tmp = ((1.0 - cos(x)) / x) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.00014], 0.5, N[(N[(N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.3999999999999999e-4

   1. Initial program 38.9%

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

   3. Taylor expanded in x around 0 63.6%

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

   if 1.3999999999999999e-4 < x

   1. Initial program 99.3%

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

      1. associate-/r* 99.3%

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

   4. Applied egg-rr 99.3%

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

Alternative 4: 75.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.00014:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.00014) 0.5 (/ (- 1.0 (cos x)) (* x x))))

C

double code(double x) {
	double tmp;
	if (x <= 0.00014) {
		tmp = 0.5;
	} 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 <= 0.00014d0) then
        tmp = 0.5d0
    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 <= 0.00014) {
		tmp = 0.5;
	} else {
		tmp = (1.0 - Math.cos(x)) / (x * x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.00014:
		tmp = 0.5
	else:
		tmp = (1.0 - math.cos(x)) / (x * x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.00014)
		tmp = 0.5;
	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 <= 0.00014)
		tmp = 0.5;
	else
		tmp = (1.0 - cos(x)) / (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.00014], 0.5, N[(N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.3999999999999999e-4

   1. Initial program 38.9%

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

   3. Taylor expanded in x around 0 63.6%

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

   if 1.3999999999999999e-4 < x

   1. Initial program 99.3%

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

Alternative 5: 52.7% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.2%

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

   1. flip-- 54.0%

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

   2. associate-/l/ 54.0%

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

   3. metadata-eval 54.0%

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

   4. 1-sub-cos 76.1%

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

   5. pow2 76.1%

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

4. Applied egg-rr 76.1%

   \[\leadsto \color{blue}{\frac{\sin x \cdot \sin x}{{x}^{2} \cdot \left(1 + \cos x\right)}} \]
5. Step-by-step derivation

   1. associate-/l* 76.3%

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

   2. *-commutative 76.3%

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

   3. associate-/r* 76.3%

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

   4. hang-0p-tan 76.5%

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

6. Simplified 76.5%

   \[\leadsto \color{blue}{\sin x \cdot \frac{\tan \left(\frac{x}{2}\right)}{{x}^{2}}} \]
7. Step-by-step derivation

   1. pow2 76.5%

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

   2. associate-*r/ 75.9%

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

   3. *-commutative 75.9%

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

   4. times-frac 99.8%

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

   5. div-inv 99.8%

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

   6. metadata-eval 99.8%

      \[\leadsto \frac{\tan \left(x \cdot \color{blue}{0.5}\right)}{x} \cdot \frac{\sin x}{x} \]

8. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{\frac{\tan \left(x \cdot 0.5\right)}{x} \cdot \frac{\sin x}{x}} \]

9. Taylor expanded in x around 0 49.6%

   \[\leadsto \color{blue}{0.5} \cdot \frac{\sin x}{x} \]
10. Add Preprocessing

Alternative 6: 51.8% 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 54.2%

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

3. Taylor expanded in x around 0 48.7%

   \[\leadsto \color{blue}{0.5} \]
4. Add Preprocessing

Reproduce

herbie shell --seed 2024096 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))