cos2 (problem 3.4.1)

Percentage Accurate: 50.4% → 99.4%

Time: 7.6s

Alternatives: 7

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: 50.4% 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, 0.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.000118:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \cos x\_m\right) \cdot {x\_m}^{-2}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.000118) 0.5 (* (- 1.0 (cos x_m)) (pow x_m -2.0))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.000118) {
		tmp = 0.5;
	} else {
		tmp = (1.0 - cos(x_m)) * pow(x_m, -2.0);
	}
	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 <= 0.000118d0) then
        tmp = 0.5d0
    else
        tmp = (1.0d0 - cos(x_m)) * (x_m ** (-2.0d0))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 0.000118) {
		tmp = 0.5;
	} else {
		tmp = (1.0 - Math.cos(x_m)) * Math.pow(x_m, -2.0);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 0.000118:
		tmp = 0.5
	else:
		tmp = (1.0 - math.cos(x_m)) * math.pow(x_m, -2.0)
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.000118)
		tmp = 0.5;
	else
		tmp = Float64(Float64(1.0 - cos(x_m)) * (x_m ^ -2.0));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 0.000118)
		tmp = 0.5;
	else
		tmp = (1.0 - cos(x_m)) * (x_m ^ -2.0);
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.000118], 0.5, N[(N[(1.0 - N[Cos[x$95$m], $MachinePrecision]), $MachinePrecision] * N[Power[x$95$m, -2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.18e-4

   1. Initial program 34.9%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 66.6%

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

   if 1.18e-4 < x

   1. Initial program 99.6%

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

   4. Applied rewrites 99.6%

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

4. Final simplification 75.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.000118:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\left(1 - \cos x\right) \cdot {x}^{-2}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.6% accurate, 0.5× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (* (tan (* 0.5 x_m)) (/ (/ (sin x_m) x_m) x_m)))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(N[Tan[N[(0.5 * x$95$m), $MachinePrecision]], $MachinePrecision] * N[(N[(N[Sin[x$95$m], $MachinePrecision] / x$95$m), $MachinePrecision] / x$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.6%

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

   1. lift-/.f64 N/A

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

   2. lift--.f64 N/A

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

   3. flip-- N/A

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

   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 73.9

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

4. Applied rewrites 73.9%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/r* N/A

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

   4. lower-/.f64 N/A

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

   5. lower-/.f64 99.6

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

6. Applied rewrites 99.6%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. metadata-eval N/A

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

   5. lower-*.f64 99.6

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

8. Applied rewrites 99.6%

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

9. Final simplification 99.6%

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

Alternative 3: 99.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.000118:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x\_m}{x\_m}}{x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.000118) 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 <= 0.000118) {
		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 <= 0.000118d0) 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 <= 0.000118) {
		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 <= 0.000118:
		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 <= 0.000118)
		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 <= 0.000118)
		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, 0.000118], 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 < 1.18e-4

   1. Initial program 34.9%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 66.6%

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

   if 1.18e-4 < x

   1. Initial program 99.6%

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

   4. Applied rewrites 99.6%

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

Alternative 4: 99.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 0.000118:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x\_m}{x\_m \cdot x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.000118) 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 <= 0.000118) {
		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 <= 0.000118d0) 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 <= 0.000118) {
		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 <= 0.000118:
		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 <= 0.000118)
		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 <= 0.000118)
		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, 0.000118], 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 < 1.18e-4

   1. Initial program 34.9%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 66.6%

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

   if 1.18e-4 < x

   1. Initial program 99.6%

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

Alternative 5: 75.7% accurate, 4.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x\_m \leq 1.2 \cdot 10^{+77}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - 1}{x\_m \cdot x\_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.2e+77) 0.5 (/ (- 1.0 1.0) (* x_m x_m))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.2e+77) {
		tmp = 0.5;
	} else {
		tmp = (1.0 - 1.0) / (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 <= 1.2d+77) then
        tmp = 0.5d0
    else
        tmp = (1.0d0 - 1.0d0) / (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 <= 1.2e+77) {
		tmp = 0.5;
	} else {
		tmp = (1.0 - 1.0) / (x_m * x_m);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 1.2e+77:
		tmp = 0.5
	else:
		tmp = (1.0 - 1.0) / (x_m * x_m)
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1.2e+77)
		tmp = 0.5;
	else
		tmp = Float64(Float64(1.0 - 1.0) / 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 <= 1.2e+77)
		tmp = 0.5;
	else
		tmp = (1.0 - 1.0) / (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, 1.2e+77], 0.5, N[(N[(1.0 - 1.0), $MachinePrecision] / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.1999999999999999e77

   1. Initial program 39.9%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 62.1%

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

   if 1.1999999999999999e77 < x

   1. Initial program 99.7%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 70.2%

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

Alternative 6: 78.3% accurate, 5.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \frac{1}{\mathsf{fma}\left(0.16666666666666666, x\_m \cdot x\_m, 2\right)} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (/ 1.0 (fma 0.16666666666666666 (* x_m x_m) 2.0)))

C

x_m = fabs(x);
double code(double x_m) {
	return 1.0 / fma(0.16666666666666666, (x_m * x_m), 2.0);
}

Julia

x_m = abs(x)
function code(x_m)
	return Float64(1.0 / fma(0.16666666666666666, Float64(x_m * x_m), 2.0))
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := N[(1.0 / N[(0.16666666666666666 * N[(x$95$m * x$95$m), $MachinePrecision] + 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 51.6%

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

   1. lift-/.f64 N/A

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

   2. lift--.f64 N/A

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

   3. div-sub N/A

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

   4. frac-2neg N/A

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

   5. metadata-eval N/A

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

   6. lift-*.f64 N/A

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

   7. distribute-rgt-neg-in N/A

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

   8. associate-/r* N/A

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

   9. lift-*.f64 N/A

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

   10. associate-/r* N/A

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

   11. frac-2neg N/A

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

   12. sub-div N/A

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

   13. lower-/.f64 N/A

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

   14. lower--.f64 N/A

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

   15. lower-/.f64 N/A

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

   16. lower-neg.f64 N/A

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

   17. lower-/.f64 N/A

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

   18. lower-neg.f64 52.6

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

4. Applied rewrites 52.6%

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

   1. lift-/.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. div-inv N/A

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

   5. lift-neg.f64 N/A

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

   6. neg-mul-1 N/A

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

   7. distribute-lft-out-- N/A

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

   8. lift-/.f64 N/A

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

   9. div-sub N/A

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

   10. lift-cos.f64 N/A

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

   11. lift-neg.f64 N/A

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

   12. neg-mul-1 N/A

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

   13. times-frac N/A

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

   14. metadata-eval N/A

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

   15. associate-/r* N/A

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

   16. lift-*.f64 N/A

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

   17. clear-num N/A

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

   18. div-inv N/A

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

   19. lower-/.f64 N/A

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

   20. lower-/.f64 N/A

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

6. Applied rewrites 51.6%

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

7. Taylor expanded in x around 0

   \[\leadsto \frac{1}{\color{blue}{2 + \frac{1}{6} \cdot {x}^{2}}} \]
8. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \frac{1}{\color{blue}{\frac{1}{6} \cdot {x}^{2} + 2}} \]

   2. lower-fma.f64 N/A

      \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(\frac{1}{6}, {x}^{2}, 2\right)}} \]

   3. unpow2 N/A

      \[\leadsto \frac{1}{\mathsf{fma}\left(\frac{1}{6}, \color{blue}{x \cdot x}, 2\right)} \]

   4. lower-*.f64 80.5

      \[\leadsto \frac{1}{\mathsf{fma}\left(0.16666666666666666, \color{blue}{x \cdot x}, 2\right)} \]

9. Applied rewrites 80.5%

   \[\leadsto \frac{1}{\color{blue}{\mathsf{fma}\left(0.16666666666666666, x \cdot x, 2\right)}} \]
10. Add Preprocessing

Alternative 7: 52.1% accurate, 120.0× 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 51.6%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 50.6%

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

Reproduce

herbie shell --seed 2024296 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))