cos2 (problem 3.4.1)

Percentage Accurate: 51.4% → 99.8%

Time: 7.6s

Alternatives: 8

Speedup: 17.1×

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: 51.4% 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: 99.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 51.1%

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

   1. lift--.f64 N/A

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

   2. flip-- N/A

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

   3. lower-/.f64 N/A

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

   4. metadata-eval N/A

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

   5. lift-cos.f64 N/A

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

   6. lift-cos.f64 N/A

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

   7. 1-sub-cos N/A

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

   8. pow2 N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. +-commutative N/A

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

   12. lower-+.f64 75.8

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

4. Applied rewrites 75.8%

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

   1. lift-/.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-/l/ N/A

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

   4. lift-pow.f64 N/A

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

   5. unpow2 N/A

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

   6. times-frac N/A

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

   7. lower-*.f64 N/A

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

   8. lift-sin.f64 N/A

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

   9. lift-+.f64 N/A

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

   10. +-commutative N/A

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

   11. lift-cos.f64 N/A

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

   12. hang-0p-tan N/A

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

   13. lower-tan.f64 N/A

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

   14. lower-/.f64 N/A

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

   15. lower-/.f64 76.1

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

6. Applied rewrites 76.1%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. associate-*r/ N/A

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

   4. lift-*.f64 N/A

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

   5. times-frac N/A

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

   6. lower-*.f64 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 99.8

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

8. Applied rewrites 99.8%

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

Alternative 2: 74.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0002:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, -0.041666666666666664, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\tan \left(\frac{x}{2}\right) \cdot \frac{\sin x}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.0002)
   (fma (* x x) -0.041666666666666664 0.5)
   (* (tan (/ x 2.0)) (/ (sin x) (* x x)))))

C

double code(double x) {
	double tmp;
	if (x <= 0.0002) {
		tmp = fma((x * x), -0.041666666666666664, 0.5);
	} else {
		tmp = tan((x / 2.0)) * (sin(x) / (x * x));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.0002)
		tmp = fma(Float64(x * x), -0.041666666666666664, 0.5);
	else
		tmp = Float64(tan(Float64(x / 2.0)) * Float64(sin(x) / Float64(x * x)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 0.0002], N[(N[(x * x), $MachinePrecision] * -0.041666666666666664 + 0.5), $MachinePrecision], N[(N[Tan[N[(x / 2.0), $MachinePrecision]], $MachinePrecision] * N[(N[Sin[x], $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.0000000000000001e-4

   1. Initial program 34.7%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 67.0

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

   5. Applied rewrites 67.0%

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

   if 2.0000000000000001e-4 < x

   1. Initial program 99.3%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. metadata-eval N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. 1-sub-cos N/A

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

      8. pow2 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 99.2

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

   4. Applied rewrites 99.2%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-cos.f64 N/A

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

      12. hang-0p-tan N/A

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

      13. lower-tan.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 99.6

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

   6. Applied rewrites 99.6%

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

Alternative 3: 74.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0056:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, -0.041666666666666664, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.0056)
   (fma (* x x) -0.041666666666666664 0.5)
   (/ (/ (- 1.0 (cos x)) x) x)))

C

double code(double x) {
	double tmp;
	if (x <= 0.0056) {
		tmp = fma((x * x), -0.041666666666666664, 0.5);
	} else {
		tmp = ((1.0 - cos(x)) / x) / x;
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.0056)
		tmp = fma(Float64(x * x), -0.041666666666666664, 0.5);
	else
		tmp = Float64(Float64(Float64(1.0 - cos(x)) / x) / x);
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 0.0056], N[(N[(x * x), $MachinePrecision] * -0.041666666666666664 + 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 < 0.00559999999999999994

   1. Initial program 34.7%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 67.0

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

   5. Applied rewrites 67.0%

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

   if 0.00559999999999999994 < x

   1. Initial program 99.3%

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

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 99.3

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

   4. Applied rewrites 99.3%

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

Alternative 4: 74.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0056:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, -0.041666666666666664, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.0056)
   (fma (* x x) -0.041666666666666664 0.5)
   (/ (- 1.0 (cos x)) (* x x))))

C

double code(double x) {
	double tmp;
	if (x <= 0.0056) {
		tmp = fma((x * x), -0.041666666666666664, 0.5);
	} else {
		tmp = (1.0 - cos(x)) / (x * x);
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.0056)
		tmp = fma(Float64(x * x), -0.041666666666666664, 0.5);
	else
		tmp = Float64(Float64(1.0 - cos(x)) / Float64(x * x));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 0.0056], N[(N[(x * x), $MachinePrecision] * -0.041666666666666664 + 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 < 0.00559999999999999994

   1. Initial program 34.7%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

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

      5. lower-*.f64 67.0

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

   5. Applied rewrites 67.0%

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

   if 0.00559999999999999994 < x

   1. Initial program 99.3%

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

Alternative 5: 63.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x}{x \cdot x}\\ \mathbf{if}\;x \leq 290000000000:\\ \;\;\;\;\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(0.001388888888888889 \cdot x, x, -0.041666666666666664\right), 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_0, t\_0, \frac{-1}{x \cdot x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ x (* x x))))
   (if (<= x 290000000000.0)
     (fma (* x x) (fma (* 0.001388888888888889 x) x -0.041666666666666664) 0.5)
     (fma t_0 t_0 (/ -1.0 (* x x))))))

C

double code(double x) {
	double t_0 = x / (x * x);
	double tmp;
	if (x <= 290000000000.0) {
		tmp = fma((x * x), fma((0.001388888888888889 * x), x, -0.041666666666666664), 0.5);
	} else {
		tmp = fma(t_0, t_0, (-1.0 / (x * x)));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(x / Float64(x * x))
	tmp = 0.0
	if (x <= 290000000000.0)
		tmp = fma(Float64(x * x), fma(Float64(0.001388888888888889 * x), x, -0.041666666666666664), 0.5);
	else
		tmp = fma(t_0, t_0, Float64(-1.0 / Float64(x * x)));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x / N[(x * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 290000000000.0], N[(N[(x * x), $MachinePrecision] * N[(N[(0.001388888888888889 * x), $MachinePrecision] * x + -0.041666666666666664), $MachinePrecision] + 0.5), $MachinePrecision], N[(t$95$0 * t$95$0 + N[(-1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 2.9e11

   1. Initial program 35.4%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 66.8%

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

   if 2.9e11 < x

   1. Initial program 99.3%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. metadata-eval N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. 1-sub-cos N/A

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

      8. pow2 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 99.2

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

   4. Applied rewrites 99.2%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lift-pow.f64 N/A

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

      5. unpow2 N/A

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

      6. times-frac N/A

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

      7. lower-*.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. lift-+.f64 N/A

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

      10. +-commutative N/A

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

      11. lift-cos.f64 N/A

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

      12. hang-0p-tan N/A

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

      13. lower-tan.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower-/.f64 99.6

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

   6. Applied rewrites 99.6%

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

   8. Applied rewrites 98.7%

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

   9. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(\frac{x}{x \cdot x}, \frac{x}{x \cdot x}, \frac{\color{blue}{-1}}{x \cdot x}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 53.3%

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

Alternative 6: 63.3% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.15 \cdot 10^{+80}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{x}, \frac{-1}{x}, \frac{-1}{x \cdot x}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.15e+80) 0.5 (fma (/ -1.0 x) (/ -1.0 x) (/ -1.0 (* x x)))))

C

double code(double x) {
	double tmp;
	if (x <= 2.15e+80) {
		tmp = 0.5;
	} else {
		tmp = fma((-1.0 / x), (-1.0 / x), (-1.0 / (x * x)));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.15e+80)
		tmp = 0.5;
	else
		tmp = fma(Float64(-1.0 / x), Float64(-1.0 / x), Float64(-1.0 / Float64(x * x)));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[x, 2.15e+80], 0.5, N[(N[(-1.0 / x), $MachinePrecision] * N[(-1.0 / x), $MachinePrecision] + N[(-1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.15000000000000002e80

   1. Initial program 40.2%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 62.5%

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

   if 2.15000000000000002e80 < x

   1. Initial program 99.7%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. metadata-eval N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. 1-sub-cos N/A

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

      8. pow2 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 99.5

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

   4. Applied rewrites 99.5%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-pow.f64 N/A

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

      4. unpow2 N/A

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

      5. lift-sin.f64 N/A

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

      6. lift-sin.f64 N/A

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

      7. 1-sub-cos N/A

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

      8. metadata-eval N/A

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

      9. lift-cos.f64 N/A

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

      10. lift-cos.f64 N/A

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

      11. lift-+.f64 N/A

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

      12. +-commutative N/A

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

      13. flip-- N/A

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

      14. *-rgt-identity N/A

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

      15. fp-cancel-sub-sign-inv N/A

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

      16. div-add N/A

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

   6. Applied rewrites 99.4%

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

   7. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(\frac{-1}{x}, \frac{-1}{x}, \frac{\color{blue}{-1} \cdot 1}{x \cdot x}\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 69.5%

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

4. Final simplification 63.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.15 \cdot 10^{+80}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{x}, \frac{-1}{x}, \frac{-1}{x \cdot x}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 63.3% accurate, 17.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 7.6 \cdot 10^{+76}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 7.6e+76) 0.5 0.0))

C

double code(double x) {
	double tmp;
	if (x <= 7.6e+76) {
		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 <= 7.6d+76) 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 <= 7.6e+76) {
		tmp = 0.5;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 7.6e+76:
		tmp = 0.5
	else:
		tmp = 0.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 7.6e+76)
		tmp = 0.5;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 7.6e+76)
		tmp = 0.5;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 7.6e+76], 0.5, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 7.60000000000000049e76

   1. Initial program 39.6%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 63.1%

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

   if 7.60000000000000049e76 < x

   1. Initial program 99.7%

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

      1. lift--.f64 N/A

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

      2. flip-- N/A

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

      3. lower-/.f64 N/A

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

      4. metadata-eval N/A

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

      5. lift-cos.f64 N/A

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

      6. lift-cos.f64 N/A

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

      7. 1-sub-cos N/A

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

      8. pow2 N/A

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

      9. lower-pow.f64 N/A

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

      10. lower-sin.f64 N/A

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

      11. +-commutative N/A

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

      12. lower-+.f64 99.5

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

   4. Applied rewrites 99.5%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. lift-/.f64 N/A

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

      4. distribute-neg-frac N/A

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

      5. lift-pow.f64 N/A

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

      6. unpow2 N/A

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

      7. lift-sin.f64 N/A

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

      8. lift-sin.f64 N/A

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

      9. sub-1-cos N/A

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

      10. lift-cos.f64 N/A

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

      11. lift-cos.f64 N/A

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

      12. metadata-eval N/A

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

      13. lift-+.f64 N/A

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

      14. flip-- N/A

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

      15. mul-1-neg N/A

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

      16. lift-*.f64 N/A

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

      17. div-sub N/A

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

      18. metadata-eval N/A

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

      19. lift-*.f64 N/A

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

      20. mul-1-neg N/A

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

   6. Applied rewrites 99.6%

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

      1. lift-neg.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. cos-neg-rev N/A

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

      4. lift-neg.f64 N/A

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

      5. cos-+PI-rev N/A

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

      6. lower-cos.f64 N/A

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

      7. lower-+.f64 N/A

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

      8. lower-PI.f64 69.9

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

   8. Applied rewrites 69.9%

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

   9. Taylor expanded in x around 0

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

       1. cos-PI N/A

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

       2. metadata-eval N/A

          \[\leadsto \frac{\color{blue}{0}}{{x}^{2}} \]

       3. div0 66.6

          \[\leadsto \color{blue}{0} \]

   11. Applied rewrites 66.6%

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

Alternative 8: 28.0% accurate, 120.0× 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 51.1%

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

   1. lift--.f64 N/A

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

   2. flip-- N/A

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

   3. lower-/.f64 N/A

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

   4. metadata-eval N/A

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

   5. lift-cos.f64 N/A

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

   6. lift-cos.f64 N/A

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

   7. 1-sub-cos N/A

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

   8. pow2 N/A

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

   9. lower-pow.f64 N/A

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

   10. lower-sin.f64 N/A

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

   11. +-commutative N/A

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

   12. lower-+.f64 75.8

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

4. Applied rewrites 75.8%

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

   1. lift-/.f64 N/A

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

   2. frac-2neg N/A

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

   3. lift-/.f64 N/A

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

   4. distribute-neg-frac N/A

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

   5. lift-pow.f64 N/A

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

   6. unpow2 N/A

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

   7. lift-sin.f64 N/A

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

   8. lift-sin.f64 N/A

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

   9. sub-1-cos N/A

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

   10. lift-cos.f64 N/A

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

   11. lift-cos.f64 N/A

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

   12. metadata-eval N/A

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

   13. lift-+.f64 N/A

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

   14. flip-- N/A

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

   15. mul-1-neg N/A

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

   16. lift-*.f64 N/A

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

   17. div-sub N/A

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

   18. metadata-eval N/A

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

   19. lift-*.f64 N/A

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

   20. mul-1-neg N/A

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

6. Applied rewrites 50.9%

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

   1. lift-neg.f64 N/A

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

   2. lift-cos.f64 N/A

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

   3. cos-neg-rev N/A

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

   4. lift-neg.f64 N/A

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

   5. cos-+PI-rev N/A

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

   6. lower-cos.f64 N/A

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

   7. lower-+.f64 N/A

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

   8. lower-PI.f64 31.0

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

8. Applied rewrites 31.0%

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

9. Taylor expanded in x around 0

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

    1. cos-PI N/A

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

    2. metadata-eval N/A

       \[\leadsto \frac{\color{blue}{0}}{{x}^{2}} \]

    3. div0 27.6

       \[\leadsto \color{blue}{0} \]

11. Applied rewrites 27.6%

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

Reproduce

herbie shell --seed 2024340 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))