cos2 (problem 3.4.1)

Percentage Accurate: 51.4% → 99.8%

Time: 10.9s

Alternatives: 6

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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 56.8%

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

   1. lift-cos.f64 N/A

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

   2. flip-- N/A

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

   3. metadata-eval N/A

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

   4. lift-cos.f64 N/A

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

   5. lift-cos.f64 N/A

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

   6. 1-sub-cos N/A

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

   7. associate-/l* N/A

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

   8. lower-*.f64 N/A

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

   9. lower-sin.f64 N/A

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

   10. lift-cos.f64 N/A

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

   11. hang-0p-tan N/A

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

   12. lower-tan.f64 N/A

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

   13. lower-/.f64 78.2

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

4. Applied egg-rr 78.2%

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

   1. lift-sin.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. lift-tan.f64 N/A

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

   4. times-frac N/A

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

   5. associate-*r/ N/A

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

   6. lower-/.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-/.f64 99.8

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

   9. lift-/.f64 N/A

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

   10. div-inv N/A

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

   11. lower-*.f64 N/A

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

   12. metadata-eval 99.8

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

6. Applied egg-rr 99.8%

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

Alternative 2: 74.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0054:\\ \;\;\;\;\mathsf{fma}\left(-0.041666666666666664, x \cdot x, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.0054)
   (fma -0.041666666666666664 (* x x) 0.5)
   (/ (/ (- 1.0 (cos x)) x) x)))

C

double code(double x) {
	double tmp;
	if (x <= 0.0054) {
		tmp = fma(-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 <= 0.0054)
		tmp = fma(-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, 0.0054], N[(-0.041666666666666664 * 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 < 0.0054000000000000003

   1. Initial program 38.5%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 63.2

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

   5. Simplified 63.2%

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

   if 0.0054000000000000003 < x

   1. Initial program 99.3%

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

   4. Applied egg-rr 99.5%

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

      1. lift-cos.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. distribute-frac-neg2 N/A

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

      5. distribute-frac-neg N/A

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

      6. lower-/.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. distribute-neg-frac N/A

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

      9. lower-/.f64 N/A

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

      10. lift-+.f64 N/A

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. metadata-eval N/A

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

      14. unsub-neg N/A

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

      15. lower--.f64 99.5

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

   6. Applied egg-rr 99.5%

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

Alternative 3: 73.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0054:\\ \;\;\;\;\mathsf{fma}\left(-0.041666666666666664, x \cdot x, 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.0054)
   (fma -0.041666666666666664 (* x x) 0.5)
   (/ (- 1.0 (cos x)) (* x x))))

C

double code(double x) {
	double tmp;
	if (x <= 0.0054) {
		tmp = fma(-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 <= 0.0054)
		tmp = fma(-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, 0.0054], N[(-0.041666666666666664 * 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 < 0.0054000000000000003

   1. Initial program 38.5%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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. lower-fma.f64 N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 63.2

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

   5. Simplified 63.2%

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

   if 0.0054000000000000003 < 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 4: 63.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.4 \cdot 10^{+58}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{x \cdot x} - \frac{x \cdot x}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 0.5\right) \cdot \frac{x}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 1.4e+58)
   0.5
   (if (<= x 1.35e+154)
     (- (/ 1.0 (* x x)) (/ (* x x) (* x (* x (* x x)))))
     (* (* x 0.5) (/ x (* x x))))))

C

double code(double x) {
	double tmp;
	if (x <= 1.4e+58) {
		tmp = 0.5;
	} else if (x <= 1.35e+154) {
		tmp = (1.0 / (x * x)) - ((x * x) / (x * (x * (x * x))));
	} else {
		tmp = (x * 0.5) * (x / (x * x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.4d+58) then
        tmp = 0.5d0
    else if (x <= 1.35d+154) then
        tmp = (1.0d0 / (x * x)) - ((x * x) / (x * (x * (x * x))))
    else
        tmp = (x * 0.5d0) * (x / (x * x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 1.4e+58) {
		tmp = 0.5;
	} else if (x <= 1.35e+154) {
		tmp = (1.0 / (x * x)) - ((x * x) / (x * (x * (x * x))));
	} else {
		tmp = (x * 0.5) * (x / (x * x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 1.4e+58:
		tmp = 0.5
	elif x <= 1.35e+154:
		tmp = (1.0 / (x * x)) - ((x * x) / (x * (x * (x * x))))
	else:
		tmp = (x * 0.5) * (x / (x * x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 1.4e+58)
		tmp = 0.5;
	elseif (x <= 1.35e+154)
		tmp = Float64(Float64(1.0 / Float64(x * x)) - Float64(Float64(x * x) / Float64(x * Float64(x * Float64(x * x)))));
	else
		tmp = Float64(Float64(x * 0.5) * Float64(x / Float64(x * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.4e+58)
		tmp = 0.5;
	elseif (x <= 1.35e+154)
		tmp = (1.0 / (x * x)) - ((x * x) / (x * (x * (x * x))));
	else
		tmp = (x * 0.5) * (x / (x * x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 1.4e+58], 0.5, If[LessEqual[x, 1.35e+154], N[(N[(1.0 / N[(x * x), $MachinePrecision]), $MachinePrecision] - N[(N[(x * x), $MachinePrecision] / N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * 0.5), $MachinePrecision] * N[(x / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 1.3999999999999999e58

   1. Initial program 42.9%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 59.7%

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

   if 1.3999999999999999e58 < x < 1.35000000000000003e154

   1. Initial program 99.0%

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

      1. lift-cos.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. div-sub N/A

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

      4. frac-sub N/A

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

      5. div-sub N/A

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

      6. *-lft-identity N/A

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

      7. unpow1 N/A

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

      8. pow2 N/A

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

      9. pow-div N/A

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

      10. metadata-eval N/A

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

      11. inv-pow N/A

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

      12. lower--.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lift-*.f64 N/A

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

      16. associate-*l* N/A

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

      17. lower-*.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. lift-*.f64 N/A

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

   4. Applied egg-rr 22.5%

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

   5. Taylor expanded in x around 0

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

   7. Simplified 17.4%

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

   if 1.35000000000000003e154 < x

   1. Initial program 99.7%

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

   4. Applied egg-rr 100.0%

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

      1. lift-cos.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. distribute-frac-neg2 N/A

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

      5. lift-/.f64 N/A

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

      6. associate-/l/ N/A

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

      7. lift-*.f64 N/A

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

      8. distribute-frac-neg N/A

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

      9. div-inv N/A

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

      10. lift-+.f64 N/A

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

      11. distribute-neg-in N/A

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

      12. lift-neg.f64 N/A

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

      13. *-lft-identity N/A

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

      14. metadata-eval N/A

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

      15. lift-/.f64 N/A

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

      16. distribute-lft1-in N/A

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

      17. *-lft-identity N/A

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

      18. distribute-rgt-out N/A

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

   6. Applied egg-rr 99.7%

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

      1. lift-*.f64 N/A

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

      2. lift-cos.f64 N/A

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

      3. lift--.f64 N/A

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

      4. associate-*l/ N/A

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

      5. *-lft-identity N/A

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

      6. lift--.f64 N/A

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

      7. sub-div N/A

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

      8. clear-num N/A

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

      9. frac-sub N/A

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

   8. Applied egg-rr 99.7%

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

   9. Taylor expanded in x around 0

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

       1. lower-*.f64 99.7

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

   11. Simplified 99.7%

       \[\leadsto \color{blue}{\left(0.5 \cdot x\right)} \cdot \frac{x}{x \cdot x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 63.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.4 \cdot 10^{+58}:\\ \;\;\;\;0.5\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+154}:\\ \;\;\;\;\frac{1}{x \cdot x} - \frac{x \cdot x}{x \cdot \left(x \cdot \left(x \cdot x\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot 0.5\right) \cdot \frac{x}{x \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 63.2% accurate, 17.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 3.6 \cdot 10^{+75}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 3.6e+75) 0.5 0.0))

C

double code(double x) {
	double tmp;
	if (x <= 3.6e+75) {
		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 <= 3.6d+75) 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 <= 3.6e+75) {
		tmp = 0.5;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 3.6e+75:
		tmp = 0.5
	else:
		tmp = 0.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 3.6e+75)
		tmp = 0.5;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 3.6e+75)
		tmp = 0.5;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 3.6e+75], 0.5, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 3.6e75

   1. Initial program 43.2%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 59.4%

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

   if 3.6e75 < x

   1. Initial program 99.5%

      \[\frac{1 - \cos x}{x \cdot x} \]
   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. Simplified 72.6%

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

      1. metadata-eval N/A

         \[\leadsto \frac{\color{blue}{0}}{x \cdot x} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{0}{\color{blue}{x \cdot x}} \]

      3. div0 72.6

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

   7. Applied egg-rr 72.6%

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

Alternative 6: 27.9% 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 56.8%

   \[\frac{1 - \cos x}{x \cdot x} \]
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. Simplified 33.1%

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

   1. metadata-eval N/A

      \[\leadsto \frac{\color{blue}{0}}{x \cdot x} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{0}{\color{blue}{x \cdot x}} \]

   3. div0 33.7

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

7. Applied egg-rr 33.7%

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

Reproduce

herbie shell --seed 2024207 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))