cos2 (problem 3.4.1)

Percentage Accurate: 50.6% → 99.8%

Time: 9.2s

Alternatives: 6

Speedup: 17.8×

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: 50.6% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 49.7%

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

   1. flip-- 49.6%

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

   2. div-inv 49.6%

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

   3. metadata-eval 49.6%

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

   4. pow2 49.6%

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

4. Applied egg-rr 49.6%

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

   1. associate-*r/ 49.6%

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

   2. *-rgt-identity 49.6%

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

6. Simplified 49.6%

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

   1. unpow2 49.6%

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

   2. 1-sub-cos 73.9%

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

8. Applied egg-rr 73.9%

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

   1. associate-/l* 73.9%

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

   2. times-frac 99.6%

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

   3. hang-0p-tan 99.8%

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

10. Applied egg-rr 99.8%

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

Alternative 2: 74.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.035:\\ \;\;\;\;0.5 + {x}^{2} \cdot \left(0.001388888888888889 \cdot \left(x \cdot x\right) - 0.041666666666666664\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.035)
   (+
    0.5
    (* (pow x 2.0) (- (* 0.001388888888888889 (* x x)) 0.041666666666666664)))
   (/ (/ (- 1.0 (cos x)) x) x)))

C

double code(double x) {
	double tmp;
	if (x <= 0.035) {
		tmp = 0.5 + (pow(x, 2.0) * ((0.001388888888888889 * (x * x)) - 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 <= 0.035d0) then
        tmp = 0.5d0 + ((x ** 2.0d0) * ((0.001388888888888889d0 * (x * x)) - 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 <= 0.035) {
		tmp = 0.5 + (Math.pow(x, 2.0) * ((0.001388888888888889 * (x * x)) - 0.041666666666666664));
	} else {
		tmp = ((1.0 - Math.cos(x)) / x) / x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.035:
		tmp = 0.5 + (math.pow(x, 2.0) * ((0.001388888888888889 * (x * x)) - 0.041666666666666664))
	else:
		tmp = ((1.0 - math.cos(x)) / x) / x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.035)
		tmp = Float64(0.5 + Float64((x ^ 2.0) * Float64(Float64(0.001388888888888889 * Float64(x * x)) - 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 <= 0.035)
		tmp = 0.5 + ((x ^ 2.0) * ((0.001388888888888889 * (x * x)) - 0.041666666666666664));
	else
		tmp = ((1.0 - cos(x)) / x) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.035], N[(0.5 + N[(N[Power[x, 2.0], $MachinePrecision] * N[(N[(0.001388888888888889 * N[(x * x), $MachinePrecision]), $MachinePrecision] - 0.041666666666666664), $MachinePrecision]), $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 < 0.035000000000000003

   1. Initial program 35.2%

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

   3. Taylor expanded in x around 0 66.6%

      \[\leadsto \color{blue}{0.5 + {x}^{2} \cdot \left(0.001388888888888889 \cdot {x}^{2} - 0.041666666666666664\right)} \]
   4. Step-by-step derivation

      1. pow2 66.6%

         \[\leadsto 0.5 + {x}^{2} \cdot \left(0.001388888888888889 \cdot \color{blue}{\left(x \cdot x\right)} - 0.041666666666666664\right) \]

   5. Applied egg-rr 66.6%

      \[\leadsto 0.5 + {x}^{2} \cdot \left(0.001388888888888889 \cdot \color{blue}{\left(x \cdot x\right)} - 0.041666666666666664\right) \]

   if 0.035000000000000003 < x

   1. Initial program 96.1%

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

      1. associate-/r* 99.3%

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

      2. clear-num 96.2%

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

      3. inv-pow 96.2%

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

   4. Applied egg-rr 96.2%

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

      1. unpow-1 96.2%

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

      2. clear-num 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. Add Preprocessing

Alternative 3: 75.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.000175:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.000175) 0.5 (/ (/ (- 1.0 (cos x)) x) x)))

C

double code(double x) {
	double tmp;
	if (x <= 0.000175) {
		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 <= 0.000175d0) 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 <= 0.000175) {
		tmp = 0.5;
	} else {
		tmp = ((1.0 - Math.cos(x)) / x) / x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.000175:
		tmp = 0.5
	else:
		tmp = ((1.0 - math.cos(x)) / x) / x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.000175)
		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 <= 0.000175)
		tmp = 0.5;
	else
		tmp = ((1.0 - cos(x)) / x) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.000175], 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 < 1.74999999999999998e-4

   1. Initial program 34.9%

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

   3. Taylor expanded in x around 0 67.0%

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

   if 1.74999999999999998e-4 < x

   1. Initial program 95.9%

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

      1. associate-/r* 99.1%

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

      2. clear-num 96.0%

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

      3. inv-pow 96.0%

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

   4. Applied egg-rr 96.0%

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

      1. unpow-1 96.0%

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

      2. clear-num 99.1%

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

   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 4: 74.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.000175:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.000175) 0.5 (/ (- 1.0 (cos x)) (* x x))))

C

double code(double x) {
	double tmp;
	if (x <= 0.000175) {
		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 <= 0.000175d0) 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 <= 0.000175) {
		tmp = 0.5;
	} else {
		tmp = (1.0 - Math.cos(x)) / (x * x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.000175:
		tmp = 0.5
	else:
		tmp = (1.0 - math.cos(x)) / (x * x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.000175)
		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 <= 0.000175)
		tmp = 0.5;
	else
		tmp = (1.0 - cos(x)) / (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.000175], 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 < 1.74999999999999998e-4

   1. Initial program 34.9%

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

   3. Taylor expanded in x around 0 67.0%

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

   if 1.74999999999999998e-4 < x

   1. Initial program 95.9%

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

Alternative 5: 63.6% accurate, 17.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.15 \cdot 10^{+77}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 1.15e+77) 0.5 0.0))

C

double code(double x) {
	double tmp;
	if (x <= 1.15e+77) {
		tmp = 0.5;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.15d+77) then
        tmp = 0.5d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.15e+77) {
		tmp = 0.5;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.15e+77:
		tmp = 0.5
	else:
		tmp = 0.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.15e+77)
		tmp = 0.5;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.15e+77)
		tmp = 0.5;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.15e+77], 0.5, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 1.14999999999999997e77

   1. Initial program 40.8%

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

   3. Taylor expanded in x around 0 61.6%

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

   if 1.14999999999999997e77 < x

   1. Initial program 95.1%

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

   3. Taylor expanded in x around 0 60.4%

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

   4. Taylor expanded in x around 0 60.4%

      \[\leadsto \color{blue}{0} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 26.9% 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 49.7%

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

3. Taylor expanded in x around 0 26.9%

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

4. Taylor expanded in x around 0 27.7%

   \[\leadsto \color{blue}{0} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024123 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))