cos2 (problem 3.4.1)

Percentage Accurate: 50.2% → 99.8%

Time: 10.1s

Alternatives: 5

Speedup: 120.0×

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: 50.2% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := N[(N[(N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision] * N[Tan[N[(x * 0.5), $MachinePrecision]], $MachinePrecision]), $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. lift-cos.f64 N/A

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

   2. flip-- N/A

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

   3. lift-*.f64 N/A

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

   4. associate-/l/ N/A

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

   5. metadata-eval N/A

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

   6. lift-cos.f64 N/A

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

   7. lift-cos.f64 N/A

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

   8. 1-sub-cos N/A

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

   9. times-frac N/A

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

   10. lower-*.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower-sin.f64 N/A

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

   13. lift-cos.f64 N/A

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

   14. hang-0p-tan N/A

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

   15. lower-tan.f64 N/A

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

   16. lower-/.f64 75.1

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

4. Applied egg-rr 75.1%

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

   1. lift-sin.f64 N/A

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

   2. associate-/r* N/A

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

   3. lift-/.f64 N/A

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

   4. lift-tan.f64 N/A

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

   5. associate-*l/ N/A

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

   6. lower-/.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 99.9

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

   9. lift-/.f64 N/A

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

   10. div-inv N/A

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

   11. lower-*.f64 N/A

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

   12. metadata-eval 99.9

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

6. Applied egg-rr 99.9%

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

Alternative 2: 75.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0052:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot -0.041666666666666664, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.0052)
   (fma x (* x -0.041666666666666664) 0.5)
   (/ (/ (- 1.0 (cos x)) x) x)))

C

double code(double x) {
	double tmp;
	if (x <= 0.0052) {
		tmp = fma(x, (x * -0.041666666666666664), 0.5);
	} else {
		tmp = ((1.0 - cos(x)) / x) / x;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.0052)
		tmp = fma(x, Float64(x * -0.041666666666666664), 0.5);
	else
		tmp = Float64(Float64(Float64(1.0 - cos(x)) / x) / x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 0.0052], N[(x * N[(x * -0.041666666666666664), $MachinePrecision] + 0.5), $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 < 0.0051999999999999998

   1. Initial program 34.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} + \frac{-1}{24} \cdot {x}^{2}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{-1}{24} \cdot {x}^{2} + \frac{1}{2}} \]

      2. unpow2 N/A

         \[\leadsto \frac{-1}{24} \cdot \color{blue}{\left(x \cdot x\right)} + \frac{1}{2} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{24} \cdot x\right) \cdot x} + \frac{1}{2} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{x \cdot \left(\frac{-1}{24} \cdot x\right)} + \frac{1}{2} \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{-1}{24} \cdot x, \frac{1}{2}\right)} \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{-1}{24}}, \frac{1}{2}\right) \]

      7. lower-*.f64 67.4

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot -0.041666666666666664}, 0.5\right) \]

   5. Simplified 67.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot -0.041666666666666664, 0.5\right)} \]

   if 0.0051999999999999998 < x

   1. Initial program 97.1%

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

      1. lift-cos.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. div-sub N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 97.1

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

   4. Applied egg-rr 97.1%

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

      1. associate-/r* N/A

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

      2. lift-cos.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/r* N/A

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

      6. sub-div N/A

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

      7. lower-/.f64 N/A

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

      8. sub-div N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 99.1

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

   6. Applied egg-rr 99.1%

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

Alternative 3: 75.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0052:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot -0.041666666666666664, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.0052)
   (fma x (* x -0.041666666666666664) 0.5)
   (/ (- 1.0 (cos x)) (* x x))))

C

double code(double x) {
	double tmp;
	if (x <= 0.0052) {
		tmp = fma(x, (x * -0.041666666666666664), 0.5);
	} else {
		tmp = (1.0 - cos(x)) / (x * x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.0052)
		tmp = fma(x, Float64(x * -0.041666666666666664), 0.5);
	else
		tmp = Float64(Float64(1.0 - cos(x)) / Float64(x * x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 0.0052], N[(x * N[(x * -0.041666666666666664), $MachinePrecision] + 0.5), $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 < 0.0051999999999999998

   1. Initial program 34.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} + \frac{-1}{24} \cdot {x}^{2}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{-1}{24} \cdot {x}^{2} + \frac{1}{2}} \]

      2. unpow2 N/A

         \[\leadsto \frac{-1}{24} \cdot \color{blue}{\left(x \cdot x\right)} + \frac{1}{2} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{24} \cdot x\right) \cdot x} + \frac{1}{2} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{x \cdot \left(\frac{-1}{24} \cdot x\right)} + \frac{1}{2} \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{-1}{24} \cdot x, \frac{1}{2}\right)} \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{-1}{24}}, \frac{1}{2}\right) \]

      7. lower-*.f64 67.4

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot -0.041666666666666664}, 0.5\right) \]

   5. Simplified 67.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot -0.041666666666666664, 0.5\right)} \]

   if 0.0051999999999999998 < x

   1. Initial program 97.1%

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

Alternative 4: 65.1% accurate, 5.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.4:\\ \;\;\;\;\mathsf{fma}\left(x, x \cdot -0.041666666666666664, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.4) (fma x (* x -0.041666666666666664) 0.5) (/ 2.0 (* x x))))

C

double code(double x) {
	double tmp;
	if (x <= 2.4) {
		tmp = fma(x, (x * -0.041666666666666664), 0.5);
	} else {
		tmp = 2.0 / (x * x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.4)
		tmp = fma(x, Float64(x * -0.041666666666666664), 0.5);
	else
		tmp = Float64(2.0 / Float64(x * x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 2.4], N[(x * N[(x * -0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision], N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.39999999999999991

   1. Initial program 34.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} + \frac{-1}{24} \cdot {x}^{2}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{-1}{24} \cdot {x}^{2} + \frac{1}{2}} \]

      2. unpow2 N/A

         \[\leadsto \frac{-1}{24} \cdot \color{blue}{\left(x \cdot x\right)} + \frac{1}{2} \]

      3. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{24} \cdot x\right) \cdot x} + \frac{1}{2} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{x \cdot \left(\frac{-1}{24} \cdot x\right)} + \frac{1}{2} \]

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{-1}{24} \cdot x, \frac{1}{2}\right)} \]

      6. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{-1}{24}}, \frac{1}{2}\right) \]

      7. lower-*.f64 67.4

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot -0.041666666666666664}, 0.5\right) \]

   5. Simplified 67.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot -0.041666666666666664, 0.5\right)} \]

   if 2.39999999999999991 < x

   1. Initial program 97.1%

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

   4. Applied egg-rr 67.2%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{2}{{x}^{2}}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. unpow2 N/A

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

      3. lower-*.f64 67.9

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

   7. Simplified 67.9%

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

Alternative 5: 52.3% accurate, 120.0× 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 48.6%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{1}{2}} \]
4. Step-by-step derivation

5. Simplified 53.7%

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

Reproduce

herbie shell --seed 2024221 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))