cos2 (problem 3.4.1)

Percentage Accurate: 51.3% → 99.4%

Time: 19.1s

Alternatives: 4

Speedup: 6.7×

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: 51.3% 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, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.000115:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x\_m}{x\_m}}{x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.000115) 0.5 (/ (/ (- 1.0 (cos x_m)) x_m) x_m)))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.000115) {
		tmp = 0.5;
	} else {
		tmp = ((1.0 - cos(x_m)) / x_m) / x_m;
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 0.000115d0) then
        tmp = 0.5d0
    else
        tmp = ((1.0d0 - cos(x_m)) / x_m) / x_m
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 0.000115) {
		tmp = 0.5;
	} else {
		tmp = ((1.0 - Math.cos(x_m)) / x_m) / x_m;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 0.000115:
		tmp = 0.5
	else:
		tmp = ((1.0 - math.cos(x_m)) / x_m) / x_m
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.000115)
		tmp = 0.5;
	else
		tmp = Float64(Float64(Float64(1.0 - cos(x_m)) / x_m) / x_m);
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 0.000115)
		tmp = 0.5;
	else
		tmp = ((1.0 - cos(x_m)) / x_m) / x_m;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.000115], 0.5, N[(N[(N[(1.0 - N[Cos[x$95$m], $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision] / x$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.15e-4

   1. Initial program 35.5%

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

   3. Taylor expanded in x around 0 66.8%

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

   if 1.15e-4 < x

   1. Initial program 98.8%

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

      1. associate-/r* 99.4%

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

      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. *-commutative 99.3%

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

      2. frac-2neg 99.3%

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

      3. associate-*r/ 99.3%

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

      4. sub-neg 99.3%

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

      5. distribute-neg-in 99.3%

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

      6. metadata-eval 99.3%

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

      7. add-sqr-sqrt 49.1%

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

      8. sqrt-unprod 71.7%

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

      9. sqr-neg 71.7%

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

      10. sqrt-unprod 22.6%

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

      11. add-sqr-sqrt 52.5%

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

      12. add-sqr-sqrt 29.9%

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

      13. sqrt-unprod 80.1%

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

      14. sqr-neg 80.1%

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

      15. sqrt-unprod 49.8%

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

      16. add-sqr-sqrt 99.3%

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

   6. Applied egg-rr 99.3%

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

   7. Taylor expanded in x around inf 99.4%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.000115:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.000115:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x\_m}{x\_m \cdot x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.000115) 0.5 (/ (- 1.0 (cos x_m)) (* x_m x_m))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.000115) {
		tmp = 0.5;
	} else {
		tmp = (1.0 - cos(x_m)) / (x_m * x_m);
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 0.000115d0) then
        tmp = 0.5d0
    else
        tmp = (1.0d0 - cos(x_m)) / (x_m * x_m)
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 0.000115) {
		tmp = 0.5;
	} else {
		tmp = (1.0 - Math.cos(x_m)) / (x_m * x_m);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 0.000115:
		tmp = 0.5
	else:
		tmp = (1.0 - math.cos(x_m)) / (x_m * x_m)
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.000115)
		tmp = 0.5;
	else
		tmp = Float64(Float64(1.0 - cos(x_m)) / Float64(x_m * x_m));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 0.000115)
		tmp = 0.5;
	else
		tmp = (1.0 - cos(x_m)) / (x_m * x_m);
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.000115], 0.5, N[(N[(1.0 - N[Cos[x$95$m], $MachinePrecision]), $MachinePrecision] / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.15e-4

   1. Initial program 35.5%

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

   3. Taylor expanded in x around 0 66.8%

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

   if 1.15e-4 < x

   1. Initial program 98.8%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.000115:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.4% accurate, 6.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.4 \cdot 10^{+77}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x\_m} \cdot \left(\frac{1}{x\_m} - \frac{1}{x\_m}\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.4e+77) 0.5 (* (/ 1.0 x_m) (- (/ 1.0 x_m) (/ 1.0 x_m)))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.4e+77) {
		tmp = 0.5;
	} else {
		tmp = (1.0 / x_m) * ((1.0 / x_m) - (1.0 / x_m));
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 1.4d+77) then
        tmp = 0.5d0
    else
        tmp = (1.0d0 / x_m) * ((1.0d0 / x_m) - (1.0d0 / x_m))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 1.4e+77) {
		tmp = 0.5;
	} else {
		tmp = (1.0 / x_m) * ((1.0 / x_m) - (1.0 / x_m));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 1.4e+77:
		tmp = 0.5
	else:
		tmp = (1.0 / x_m) * ((1.0 / x_m) - (1.0 / x_m))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1.4e+77)
		tmp = 0.5;
	else
		tmp = Float64(Float64(1.0 / x_m) * Float64(Float64(1.0 / x_m) - Float64(1.0 / x_m)));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 1.4e+77)
		tmp = 0.5;
	else
		tmp = (1.0 / x_m) * ((1.0 / x_m) - (1.0 / x_m));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1.4e+77], 0.5, N[(N[(1.0 / x$95$m), $MachinePrecision] * N[(N[(1.0 / x$95$m), $MachinePrecision] - N[(1.0 / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.4e77

   1. Initial program 42.2%

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

   3. Taylor expanded in x around 0 60.5%

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

   if 1.4e77 < x

   1. Initial program 98.7%

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

      1. associate-/r* 99.7%

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

      2. div-inv 99.6%

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

   4. Applied egg-rr 99.6%

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

      1. div-sub 99.4%

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

      2. sub-neg 99.4%

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

   6. Applied egg-rr 99.4%

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

      1. sub-neg 99.4%

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

   8. Simplified 99.4%

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

   9. Taylor expanded in x around 0 65.3%

      \[\leadsto \left(\frac{1}{x} - \color{blue}{\frac{1}{x}}\right) \cdot \frac{1}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 61.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.4 \cdot 10^{+77}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} \cdot \left(\frac{1}{x} - \frac{1}{x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 51.2% accurate, 107.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ 0.5 \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 0.5)

C

x_m = fabs(x);
double code(double x_m) {
	return 0.5;
}

Fortran

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

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return 0.5;
}

Python

x_m = math.fabs(x)
def code(x_m):
	return 0.5

Julia

x_m = abs(x)
function code(x_m)
	return 0.5
end

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := 0.5

Derivation

1. Initial program 52.6%

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

3. Taylor expanded in x around 0 50.0%

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

4. Final simplification 50.0%

   \[\leadsto 0.5 \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024076 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))