cos2 (problem 3.4.1)

Percentage Accurate: 50.5% → 99.8%

Time: 8.5s

Alternatives: 10

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: 50.5% 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: 99.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 48.0%

   \[\frac{1 - \cos x}{x \cdot x} \]
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. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. associate-*r/ N/A

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

   5. lift-*.f64 N/A

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

   6. times-frac N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. clear-num N/A

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

   11. associate-/r/ N/A

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

   12. metadata-eval N/A

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

   13. lower-*.f64 N/A

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

   14. lower-/.f64 99.8

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

6. Applied rewrites 99.8%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. un-div-inv N/A

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

   5. lower-/.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 99.8

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

8. Applied rewrites 99.8%

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

Alternative 2: 99.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 48.0%

   \[\frac{1 - \cos x}{x \cdot x} \]
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. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. associate-*r/ N/A

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

   5. lift-*.f64 N/A

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

   6. times-frac N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. clear-num N/A

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

   11. associate-/r/ N/A

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

   12. metadata-eval N/A

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

   13. lower-*.f64 N/A

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

   14. lower-/.f64 99.8

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

6. Applied rewrites 99.8%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. clear-num N/A

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

   4. un-div-inv N/A

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

   5. lower-/.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower-/.f64 99.8

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

8. Applied rewrites 99.8%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/r/ N/A

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

   4. associate-*l/ N/A

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

   5. lower-/.f64 N/A

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

   6. lower-*.f64 99.8

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

   7. lift-*.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-*.f64 99.8

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

10. Applied rewrites 99.8%

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

Alternative 3: 99.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 48.0%

   \[\frac{1 - \cos x}{x \cdot x} \]
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. *-commutative N/A

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

   3. lift-/.f64 N/A

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

   4. associate-*r/ N/A

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

   5. lift-*.f64 N/A

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

   6. times-frac N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. clear-num N/A

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

   11. associate-/r/ N/A

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

   12. metadata-eval N/A

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

   13. lower-*.f64 N/A

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

   14. lower-/.f64 99.8

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

6. Applied rewrites 99.8%

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

Alternative 4: 75.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.033:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, -0.041666666666666664\right), x \cdot x, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-1}{x} \cdot \left(\cos x - 1\right)}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.033)
   (fma (fma 0.001388888888888889 (* x x) -0.041666666666666664) (* x x) 0.5)
   (/ (* (/ -1.0 x) (- (cos x) 1.0)) x)))

C

double code(double x) {
	double tmp;
	if (x <= 0.033) {
		tmp = fma(fma(0.001388888888888889, (x * x), -0.041666666666666664), (x * x), 0.5);
	} else {
		tmp = ((-1.0 / x) * (cos(x) - 1.0)) / x;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.033)
		tmp = fma(fma(0.001388888888888889, Float64(x * x), -0.041666666666666664), Float64(x * x), 0.5);
	else
		tmp = Float64(Float64(Float64(-1.0 / x) * Float64(cos(x) - 1.0)) / x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 0.033], N[(N[(0.001388888888888889 * N[(x * x), $MachinePrecision] + -0.041666666666666664), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(N[(-1.0 / x), $MachinePrecision] * N[(N[Cos[x], $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.033000000000000002

   1. Initial program 34.2%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{720} \cdot {x}^{2} - \frac{1}{24}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{720} \cdot {x}^{2} - \frac{1}{24}\right) + \frac{1}{2}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{720} \cdot {x}^{2} - \frac{1}{24}\right) \cdot {x}^{2}} + \frac{1}{2} \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{720} \cdot {x}^{2} - \frac{1}{24}, {x}^{2}, \frac{1}{2}\right)} \]

      4. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{720} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{24}\right)\right)}, {x}^{2}, \frac{1}{2}\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{720} \cdot {x}^{2} + \color{blue}{\frac{-1}{24}}, {x}^{2}, \frac{1}{2}\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{720}, {x}^{2}, \frac{-1}{24}\right)}, {x}^{2}, \frac{1}{2}\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, \color{blue}{x \cdot x}, \frac{-1}{24}\right), {x}^{2}, \frac{1}{2}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, \color{blue}{x \cdot x}, \frac{-1}{24}\right), {x}^{2}, \frac{1}{2}\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, x \cdot x, \frac{-1}{24}\right), \color{blue}{x \cdot x}, \frac{1}{2}\right) \]

      10. lower-*.f64 67.8

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, -0.041666666666666664\right), \color{blue}{x \cdot x}, 0.5\right) \]

   5. Applied rewrites 67.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, -0.041666666666666664\right), x \cdot x, 0.5\right)} \]

   if 0.033000000000000002 < x

   1. Initial program 97.2%

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

   4. Applied rewrites 99.5%

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

   6. Applied rewrites 99.5%

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

Alternative 5: 75.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.033:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, -0.041666666666666664\right), x \cdot x, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.033)
   (fma (fma 0.001388888888888889 (* x x) -0.041666666666666664) (* x x) 0.5)
   (/ (/ (- 1.0 (cos x)) x) x)))

C

double code(double x) {
	double tmp;
	if (x <= 0.033) {
		tmp = fma(fma(0.001388888888888889, (x * x), -0.041666666666666664), (x * x), 0.5);
	} else {
		tmp = ((1.0 - cos(x)) / x) / x;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.033)
		tmp = fma(fma(0.001388888888888889, Float64(x * x), -0.041666666666666664), Float64(x * x), 0.5);
	else
		tmp = Float64(Float64(Float64(1.0 - cos(x)) / x) / x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 0.033], N[(N[(0.001388888888888889 * N[(x * x), $MachinePrecision] + -0.041666666666666664), $MachinePrecision] * N[(x * x), $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 < 0.033000000000000002

   1. Initial program 34.2%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{720} \cdot {x}^{2} - \frac{1}{24}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{720} \cdot {x}^{2} - \frac{1}{24}\right) + \frac{1}{2}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{720} \cdot {x}^{2} - \frac{1}{24}\right) \cdot {x}^{2}} + \frac{1}{2} \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{720} \cdot {x}^{2} - \frac{1}{24}, {x}^{2}, \frac{1}{2}\right)} \]

      4. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{720} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{24}\right)\right)}, {x}^{2}, \frac{1}{2}\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{720} \cdot {x}^{2} + \color{blue}{\frac{-1}{24}}, {x}^{2}, \frac{1}{2}\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{720}, {x}^{2}, \frac{-1}{24}\right)}, {x}^{2}, \frac{1}{2}\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, \color{blue}{x \cdot x}, \frac{-1}{24}\right), {x}^{2}, \frac{1}{2}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, \color{blue}{x \cdot x}, \frac{-1}{24}\right), {x}^{2}, \frac{1}{2}\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, x \cdot x, \frac{-1}{24}\right), \color{blue}{x \cdot x}, \frac{1}{2}\right) \]

      10. lower-*.f64 67.8

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, -0.041666666666666664\right), \color{blue}{x \cdot x}, 0.5\right) \]

   5. Applied rewrites 67.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, -0.041666666666666664\right), x \cdot x, 0.5\right)} \]

   if 0.033000000000000002 < x

   1. Initial program 97.2%

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

   4. Applied rewrites 99.5%

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

Alternative 6: 75.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.033:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, -0.041666666666666664\right), x \cdot x, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.033)
   (fma (fma 0.001388888888888889 (* x x) -0.041666666666666664) (* x x) 0.5)
   (/ (- 1.0 (cos x)) (* x x))))

C

double code(double x) {
	double tmp;
	if (x <= 0.033) {
		tmp = fma(fma(0.001388888888888889, (x * x), -0.041666666666666664), (x * x), 0.5);
	} else {
		tmp = (1.0 - cos(x)) / (x * x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.033)
		tmp = fma(fma(0.001388888888888889, Float64(x * x), -0.041666666666666664), Float64(x * x), 0.5);
	else
		tmp = Float64(Float64(1.0 - cos(x)) / Float64(x * x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 0.033], N[(N[(0.001388888888888889 * N[(x * x), $MachinePrecision] + -0.041666666666666664), $MachinePrecision] * N[(x * x), $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 < 0.033000000000000002

   1. Initial program 34.2%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{720} \cdot {x}^{2} - \frac{1}{24}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{720} \cdot {x}^{2} - \frac{1}{24}\right) + \frac{1}{2}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{720} \cdot {x}^{2} - \frac{1}{24}\right) \cdot {x}^{2}} + \frac{1}{2} \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{720} \cdot {x}^{2} - \frac{1}{24}, {x}^{2}, \frac{1}{2}\right)} \]

      4. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{720} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{24}\right)\right)}, {x}^{2}, \frac{1}{2}\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{720} \cdot {x}^{2} + \color{blue}{\frac{-1}{24}}, {x}^{2}, \frac{1}{2}\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{720}, {x}^{2}, \frac{-1}{24}\right)}, {x}^{2}, \frac{1}{2}\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, \color{blue}{x \cdot x}, \frac{-1}{24}\right), {x}^{2}, \frac{1}{2}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, \color{blue}{x \cdot x}, \frac{-1}{24}\right), {x}^{2}, \frac{1}{2}\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, x \cdot x, \frac{-1}{24}\right), \color{blue}{x \cdot x}, \frac{1}{2}\right) \]

      10. lower-*.f64 67.8

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, -0.041666666666666664\right), \color{blue}{x \cdot x}, 0.5\right) \]

   5. Applied rewrites 67.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, -0.041666666666666664\right), x \cdot x, 0.5\right)} \]

   if 0.033000000000000002 < x

   1. Initial program 97.2%

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

Alternative 7: 78.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ {\left(\mathsf{fma}\left(0.16666666666666666, x \cdot x, 2\right)\right)}^{-1} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (pow (fma 0.16666666666666666 (* x x) 2.0) -1.0))

C

double code(double x) {
	return pow(fma(0.16666666666666666, (x * x), 2.0), -1.0);
}

Julia

function code(x)
	return fma(0.16666666666666666, Float64(x * x), 2.0) ^ -1.0
end

Wolfram

code[x_] := N[Power[N[(0.16666666666666666 * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision], -1.0], $MachinePrecision]

Derivation

1. Initial program 48.0%

   \[\frac{1 - \cos x}{x \cdot x} \]
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. frac-2neg N/A

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

   14. metadata-eval N/A

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

   15. inv-pow N/A

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

   16. lower-/.f64 N/A

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

4. Applied rewrites 49.4%

   \[\leadsto \color{blue}{\frac{\frac{-1}{x} - \left(-\frac{\cos x}{x}\right)}{-x}} \]
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-neg.f64 N/A

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

   3. lift--.f64 N/A

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

   4. lift-/.f64 N/A

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

   5. div-inv N/A

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

   6. lift-neg.f64 N/A

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

   7. neg-mul-1 N/A

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

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

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

   9. lift-/.f64 N/A

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

   10. div-sub N/A

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

   11. lift-cos.f64 N/A

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

   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. associate-/r* N/A

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

   15. lift-*.f64 N/A

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

   16. times-frac N/A

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

   17. neg-mul-1 N/A

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

   18. neg-mul-1 N/A

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

   19. frac-2neg N/A

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

   20. lift-*.f64 N/A

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

6. Applied rewrites 48.4%

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

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 78.7

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

9. Applied rewrites 78.7%

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

10. Final simplification 78.7%

    \[\leadsto {\left(\mathsf{fma}\left(0.16666666666666666, x \cdot x, 2\right)\right)}^{-1} \]
11. Add Preprocessing

Alternative 8: 63.7% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4.2 \cdot 10^{+38}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, -0.041666666666666664\right), x \cdot x, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - 1}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 4.2e+38)
   (fma (fma 0.001388888888888889 (* x x) -0.041666666666666664) (* x x) 0.5)
   (/ (- 1.0 1.0) (* x x))))

C

double code(double x) {
	double tmp;
	if (x <= 4.2e+38) {
		tmp = fma(fma(0.001388888888888889, (x * x), -0.041666666666666664), (x * x), 0.5);
	} else {
		tmp = (1.0 - 1.0) / (x * x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 4.2e+38)
		tmp = fma(fma(0.001388888888888889, Float64(x * x), -0.041666666666666664), Float64(x * x), 0.5);
	else
		tmp = Float64(Float64(1.0 - 1.0) / Float64(x * x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 4.2e+38], N[(N[(0.001388888888888889 * N[(x * x), $MachinePrecision] + -0.041666666666666664), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.5), $MachinePrecision], N[(N[(1.0 - 1.0), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.2e38

   1. Initial program 36.4%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{1}{2} + {x}^{2} \cdot \left(\frac{1}{720} \cdot {x}^{2} - \frac{1}{24}\right)} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{{x}^{2} \cdot \left(\frac{1}{720} \cdot {x}^{2} - \frac{1}{24}\right) + \frac{1}{2}} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{1}{720} \cdot {x}^{2} - \frac{1}{24}\right) \cdot {x}^{2}} + \frac{1}{2} \]

      3. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{720} \cdot {x}^{2} - \frac{1}{24}, {x}^{2}, \frac{1}{2}\right)} \]

      4. sub-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{720} \cdot {x}^{2} + \left(\mathsf{neg}\left(\frac{1}{24}\right)\right)}, {x}^{2}, \frac{1}{2}\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{fma}\left(\frac{1}{720} \cdot {x}^{2} + \color{blue}{\frac{-1}{24}}, {x}^{2}, \frac{1}{2}\right) \]

      6. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{720}, {x}^{2}, \frac{-1}{24}\right)}, {x}^{2}, \frac{1}{2}\right) \]

      7. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, \color{blue}{x \cdot x}, \frac{-1}{24}\right), {x}^{2}, \frac{1}{2}\right) \]

      8. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, \color{blue}{x \cdot x}, \frac{-1}{24}\right), {x}^{2}, \frac{1}{2}\right) \]

      9. unpow2 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{720}, x \cdot x, \frac{-1}{24}\right), \color{blue}{x \cdot x}, \frac{1}{2}\right) \]

      10. lower-*.f64 65.8

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, -0.041666666666666664\right), \color{blue}{x \cdot x}, 0.5\right) \]

   5. Applied rewrites 65.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(0.001388888888888889, x \cdot x, -0.041666666666666664\right), x \cdot x, 0.5\right)} \]

   if 4.2e38 < x

   1. Initial program 96.9%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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 57.4%

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

Alternative 9: 63.9% accurate, 4.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.3 \cdot 10^{+77}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - 1}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.3e+77) 0.5 (/ (- 1.0 1.0) (* x x))))

C

double code(double x) {
	double tmp;
	if (x <= 1.3e+77) {
		tmp = 0.5;
	} else {
		tmp = (1.0 - 1.0) / (x * x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.3d+77) then
        tmp = 0.5d0
    else
        tmp = (1.0d0 - 1.0d0) / (x * x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.3e+77) {
		tmp = 0.5;
	} else {
		tmp = (1.0 - 1.0) / (x * x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.3e+77:
		tmp = 0.5
	else:
		tmp = (1.0 - 1.0) / (x * x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.3e+77)
		tmp = 0.5;
	else
		tmp = Float64(Float64(1.0 - 1.0) / Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.3e+77)
		tmp = 0.5;
	else
		tmp = (1.0 - 1.0) / (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.3e+77], 0.5, N[(N[(1.0 - 1.0), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.3000000000000001e77

   1. Initial program 37.6%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 65.0%

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

   if 1.3000000000000001e77 < x

   1. Initial program 96.7%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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 62.2%

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

Alternative 10: 52.0% 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 48.0%

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

3. Taylor expanded in x around 0

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

5. Applied rewrites 54.2%

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

Reproduce

herbie shell --seed 2024313 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))