cos2 (problem 3.4.1)

Percentage Accurate: 50.3% → 99.8%

Time: 9.2s

Alternatives: 8

Speedup: 107.0×

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: 50.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{1 - \cos x}{x \cdot x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (- 1.0 (cos x)) (* x x)))

C

double code(double x) {
	return (1.0 - cos(x)) / (x * x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (1.0d0 - cos(x)) / (x * x)
end function

Java

public static double code(double x) {
	return (1.0 - Math.cos(x)) / (x * x);
}

Python

def code(x):
	return (1.0 - math.cos(x)) / (x * x)

Julia

function code(x)
	return Float64(Float64(1.0 - cos(x)) / Float64(x * x))
end

MATLAB

function tmp = code(x)
	tmp = (1.0 - cos(x)) / (x * x);
end

Wolfram

code[x_] := N[(N[(1.0 - N[Cos[x], $MachinePrecision]), $MachinePrecision] / N[(x * x), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 47.8%

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

   1. flip-- N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{1 \cdot 1 - \cos x \cdot \cos x}{1 + \cos x}\right), \mathsf{*.f64}\left(\color{blue}{x}, x\right)\right) \]

   2. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{1 - \cos x \cdot \cos x}{1 + \cos x}\right), \mathsf{*.f64}\left(x, x\right)\right) \]

   3. 1-sub-cos N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{\sin x \cdot \sin x}{1 + \cos x}\right), \mathsf{*.f64}\left(x, x\right)\right) \]

   4. associate-/l* N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\sin x \cdot \frac{\sin x}{1 + \cos x}\right), \mathsf{*.f64}\left(\color{blue}{x}, x\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\sin x, \left(\frac{\sin x}{1 + \cos x}\right)\right), \mathsf{*.f64}\left(\color{blue}{x}, x\right)\right) \]

   6. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\frac{\sin x}{1 + \cos x}\right)\right), \mathsf{*.f64}\left(x, x\right)\right) \]

   7. hang-0p-tan N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \tan \left(\frac{x}{2}\right)\right), \mathsf{*.f64}\left(x, x\right)\right) \]

   8. tan-lowering-tan.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{tan.f64}\left(\left(\frac{x}{2}\right)\right)\right), \mathsf{*.f64}\left(x, x\right)\right) \]

   9. /-lowering-/.f64 75.1%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{tan.f64}\left(\mathsf{/.f64}\left(x, 2\right)\right)\right), \mathsf{*.f64}\left(x, x\right)\right) \]

4. Applied egg-rr 75.1%

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

   1. times-frac N/A

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

   2. clear-num N/A

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

   3. un-div-inv N/A

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

   4. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{\sin x}{x}\right), \color{blue}{\left(\frac{x}{\tan \left(\frac{x}{2}\right)}\right)}\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\sin x, x\right), \left(\frac{\color{blue}{x}}{\tan \left(\frac{x}{2}\right)}\right)\right) \]

   6. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(x\right), x\right), \left(\frac{x}{\tan \left(\frac{x}{2}\right)}\right)\right) \]

   7. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(x\right), x\right), \mathsf{/.f64}\left(x, \color{blue}{\tan \left(\frac{x}{2}\right)}\right)\right) \]

   8. tan-lowering-tan.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(x\right), x\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(\left(\frac{x}{2}\right)\right)\right)\right) \]

   9. /-lowering-/.f64 99.8%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(x\right), x\right), \mathsf{/.f64}\left(x, \mathsf{tan.f64}\left(\mathsf{/.f64}\left(x, 2\right)\right)\right)\right) \]

6. Applied egg-rr 99.8%

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

Alternative 2: 99.8% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 47.8%

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

   1. flip-- N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{1 \cdot 1 - \cos x \cdot \cos x}{1 + \cos x}\right), \mathsf{*.f64}\left(\color{blue}{x}, x\right)\right) \]

   2. metadata-eval N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{1 - \cos x \cdot \cos x}{1 + \cos x}\right), \mathsf{*.f64}\left(x, x\right)\right) \]

   3. 1-sub-cos N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{\sin x \cdot \sin x}{1 + \cos x}\right), \mathsf{*.f64}\left(x, x\right)\right) \]

   4. associate-/l* N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\sin x \cdot \frac{\sin x}{1 + \cos x}\right), \mathsf{*.f64}\left(\color{blue}{x}, x\right)\right) \]

   5. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\sin x, \left(\frac{\sin x}{1 + \cos x}\right)\right), \mathsf{*.f64}\left(\color{blue}{x}, x\right)\right) \]

   6. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \left(\frac{\sin x}{1 + \cos x}\right)\right), \mathsf{*.f64}\left(x, x\right)\right) \]

   7. hang-0p-tan N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \tan \left(\frac{x}{2}\right)\right), \mathsf{*.f64}\left(x, x\right)\right) \]

   8. tan-lowering-tan.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{tan.f64}\left(\left(\frac{x}{2}\right)\right)\right), \mathsf{*.f64}\left(x, x\right)\right) \]

   9. /-lowering-/.f64 75.1%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{sin.f64}\left(x\right), \mathsf{tan.f64}\left(\mathsf{/.f64}\left(x, 2\right)\right)\right), \mathsf{*.f64}\left(x, x\right)\right) \]

4. Applied egg-rr 75.1%

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

   1. times-frac N/A

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

   2. associate-*r/ N/A

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

   3. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\left(\frac{\sin x}{x} \cdot \tan \left(\frac{x}{2}\right)\right), \color{blue}{x}\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\left(\frac{\sin x}{x}\right), \tan \left(\frac{x}{2}\right)\right), x\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\sin x, x\right), \tan \left(\frac{x}{2}\right)\right), x\right) \]

   6. sin-lowering-sin.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(x\right), x\right), \tan \left(\frac{x}{2}\right)\right), x\right) \]

   7. tan-lowering-tan.f64 N/A

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(x\right), x\right), \mathsf{tan.f64}\left(\left(\frac{x}{2}\right)\right)\right), x\right) \]

   8. /-lowering-/.f64 99.8%

      \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{sin.f64}\left(x\right), x\right), \mathsf{tan.f64}\left(\mathsf{/.f64}\left(x, 2\right)\right)\right), x\right) \]

6. Applied egg-rr 99.8%

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

Alternative 3: 75.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.032:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot \left(-0.041666666666666664 + \left(x \cdot x\right) \cdot 0.001388888888888889\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.032)
   (+
    0.5
    (* (* x x) (+ -0.041666666666666664 (* (* x x) 0.001388888888888889))))
   (/ (/ (- 1.0 (cos x)) x) x)))

C

double code(double x) {
	double tmp;
	if (x <= 0.032) {
		tmp = 0.5 + ((x * x) * (-0.041666666666666664 + ((x * x) * 0.001388888888888889)));
	} else {
		tmp = ((1.0 - cos(x)) / x) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.032d0) then
        tmp = 0.5d0 + ((x * x) * ((-0.041666666666666664d0) + ((x * x) * 0.001388888888888889d0)))
    else
        tmp = ((1.0d0 - cos(x)) / x) / x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.032) {
		tmp = 0.5 + ((x * x) * (-0.041666666666666664 + ((x * x) * 0.001388888888888889)));
	} else {
		tmp = ((1.0 - Math.cos(x)) / x) / x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.032:
		tmp = 0.5 + ((x * x) * (-0.041666666666666664 + ((x * x) * 0.001388888888888889)))
	else:
		tmp = ((1.0 - math.cos(x)) / x) / x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.032)
		tmp = Float64(0.5 + Float64(Float64(x * x) * Float64(-0.041666666666666664 + Float64(Float64(x * x) * 0.001388888888888889))));
	else
		tmp = Float64(Float64(Float64(1.0 - cos(x)) / x) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.032)
		tmp = 0.5 + ((x * x) * (-0.041666666666666664 + ((x * x) * 0.001388888888888889)));
	else
		tmp = ((1.0 - cos(x)) / x) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.032], N[(0.5 + N[(N[(x * x), $MachinePrecision] * N[(-0.041666666666666664 + N[(N[(x * x), $MachinePrecision] * 0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 < 0.032000000000000001

   1. Initial program 34.1%

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{720} \cdot {x}^{2} - \frac{1}{24}\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{1}{720} \cdot {x}^{2} - \frac{1}{24}\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{1}{720} \cdot {x}^{2}} - \frac{1}{24}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{1}{720} \cdot {x}^{2}} - \frac{1}{24}\right)\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{720} \cdot {x}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{24}\right)\right)}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{720} \cdot {x}^{2} + \frac{-1}{24}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{-1}{24} + \color{blue}{\frac{1}{720} \cdot {x}^{2}}\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{24}, \color{blue}{\left(\frac{1}{720} \cdot {x}^{2}\right)}\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{24}, \left({x}^{2} \cdot \color{blue}{\frac{1}{720}}\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{24}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{720}}\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{24}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{720}\right)\right)\right)\right) \]

      12. *-lowering-*.f64 68.5%

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{720}\right)\right)\right)\right) \]

   5. Simplified 68.5%

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

   if 0.032000000000000001 < x

   1. Initial program 99.2%

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

   4. Applied egg-rr 99.2%

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

   5. Taylor expanded in x around inf

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

      1. associate-*r/ N/A

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

      2. mul-1-neg N/A

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

      3. neg-sub0 N/A

         \[\leadsto \frac{0 - \left(\cos x - 1\right)}{{\color{blue}{x}}^{2}} \]

      4. associate-+l- N/A

         \[\leadsto \frac{\left(0 - \cos x\right) + 1}{{\color{blue}{x}}^{2}} \]

      5. neg-sub0 N/A

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

      6. +-commutative N/A

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

      7. sub-neg N/A

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

      8. unpow2 N/A

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

      9. associate-/r* N/A

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

      10. /-lowering-/.f64 N/A

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

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\left(1 - \cos x\right), x\right), x\right) \]

      12. --lowering--.f64 N/A

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \cos x\right), x\right), x\right) \]

      13. cos-lowering-cos.f64 99.4%

          \[\leadsto \mathsf{/.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(1, \mathsf{cos.f64}\left(x\right)\right), x\right), x\right) \]

   7. Simplified 99.4%

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

Alternative 4: 75.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.032:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot \left(-0.041666666666666664 + \left(x \cdot x\right) \cdot 0.001388888888888889\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 0.032)
   (+
    0.5
    (* (* x x) (+ -0.041666666666666664 (* (* x x) 0.001388888888888889))))
   (/ (- 1.0 (cos x)) (* x x))))

C

double code(double x) {
	double tmp;
	if (x <= 0.032) {
		tmp = 0.5 + ((x * x) * (-0.041666666666666664 + ((x * x) * 0.001388888888888889)));
	} else {
		tmp = (1.0 - cos(x)) / (x * x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.032d0) then
        tmp = 0.5d0 + ((x * x) * ((-0.041666666666666664d0) + ((x * x) * 0.001388888888888889d0)))
    else
        tmp = (1.0d0 - cos(x)) / (x * x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 0.032) {
		tmp = 0.5 + ((x * x) * (-0.041666666666666664 + ((x * x) * 0.001388888888888889)));
	} else {
		tmp = (1.0 - Math.cos(x)) / (x * x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 0.032:
		tmp = 0.5 + ((x * x) * (-0.041666666666666664 + ((x * x) * 0.001388888888888889)))
	else:
		tmp = (1.0 - math.cos(x)) / (x * x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 0.032)
		tmp = Float64(0.5 + Float64(Float64(x * x) * Float64(-0.041666666666666664 + Float64(Float64(x * x) * 0.001388888888888889))));
	else
		tmp = Float64(Float64(1.0 - cos(x)) / Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.032)
		tmp = 0.5 + ((x * x) * (-0.041666666666666664 + ((x * x) * 0.001388888888888889)));
	else
		tmp = (1.0 - cos(x)) / (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 0.032], N[(0.5 + N[(N[(x * x), $MachinePrecision] * N[(-0.041666666666666664 + N[(N[(x * x), $MachinePrecision] * 0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $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 < 0.032000000000000001

   1. Initial program 34.1%

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{720} \cdot {x}^{2} - \frac{1}{24}\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{1}{720} \cdot {x}^{2} - \frac{1}{24}\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{1}{720} \cdot {x}^{2}} - \frac{1}{24}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{1}{720} \cdot {x}^{2}} - \frac{1}{24}\right)\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{720} \cdot {x}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{24}\right)\right)}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{720} \cdot {x}^{2} + \frac{-1}{24}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{-1}{24} + \color{blue}{\frac{1}{720} \cdot {x}^{2}}\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{24}, \color{blue}{\left(\frac{1}{720} \cdot {x}^{2}\right)}\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{24}, \left({x}^{2} \cdot \color{blue}{\frac{1}{720}}\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{24}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{720}}\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{24}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{720}\right)\right)\right)\right) \]

      12. *-lowering-*.f64 68.5%

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{720}\right)\right)\right)\right) \]

   5. Simplified 68.5%

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

   if 0.032000000000000001 < x

   1. Initial program 99.2%

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

Alternative 5: 66.0% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.7:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot \left(-0.041666666666666664 + \left(x \cdot x\right) \cdot 0.001388888888888889\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.7)
   (+
    0.5
    (* (* x x) (+ -0.041666666666666664 (* (* x x) 0.001388888888888889))))
   (/ 2.0 (* x x))))

C

double code(double x) {
	double tmp;
	if (x <= 2.7) {
		tmp = 0.5 + ((x * x) * (-0.041666666666666664 + ((x * x) * 0.001388888888888889)));
	} else {
		tmp = 2.0 / (x * x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 2.7d0) then
        tmp = 0.5d0 + ((x * x) * ((-0.041666666666666664d0) + ((x * x) * 0.001388888888888889d0)))
    else
        tmp = 2.0d0 / (x * x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 2.7) {
		tmp = 0.5 + ((x * x) * (-0.041666666666666664 + ((x * x) * 0.001388888888888889)));
	} else {
		tmp = 2.0 / (x * x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2.7:
		tmp = 0.5 + ((x * x) * (-0.041666666666666664 + ((x * x) * 0.001388888888888889)))
	else:
		tmp = 2.0 / (x * x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.7)
		tmp = Float64(0.5 + Float64(Float64(x * x) * Float64(-0.041666666666666664 + Float64(Float64(x * x) * 0.001388888888888889))));
	else
		tmp = Float64(2.0 / Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2.7)
		tmp = 0.5 + ((x * x) * (-0.041666666666666664 + ((x * x) * 0.001388888888888889)));
	else
		tmp = 2.0 / (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2.7], N[(0.5 + N[(N[(x * x), $MachinePrecision] * N[(-0.041666666666666664 + N[(N[(x * x), $MachinePrecision] * 0.001388888888888889), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.7000000000000002

   1. Initial program 34.1%

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{720} \cdot {x}^{2} - \frac{1}{24}\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\left(\frac{1}{720} \cdot {x}^{2} - \frac{1}{24}\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\left(x \cdot x\right), \left(\color{blue}{\frac{1}{720} \cdot {x}^{2}} - \frac{1}{24}\right)\right)\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\color{blue}{\frac{1}{720} \cdot {x}^{2}} - \frac{1}{24}\right)\right)\right) \]

      5. sub-neg N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{720} \cdot {x}^{2} + \color{blue}{\left(\mathsf{neg}\left(\frac{1}{24}\right)\right)}\right)\right)\right) \]

      6. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{1}{720} \cdot {x}^{2} + \frac{-1}{24}\right)\right)\right) \]

      7. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \left(\frac{-1}{24} + \color{blue}{\frac{1}{720} \cdot {x}^{2}}\right)\right)\right) \]

      8. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{24}, \color{blue}{\left(\frac{1}{720} \cdot {x}^{2}\right)}\right)\right)\right) \]

      9. *-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{24}, \left({x}^{2} \cdot \color{blue}{\frac{1}{720}}\right)\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{24}, \mathsf{*.f64}\left(\left({x}^{2}\right), \color{blue}{\frac{1}{720}}\right)\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{24}, \mathsf{*.f64}\left(\left(x \cdot x\right), \frac{1}{720}\right)\right)\right)\right) \]

      12. *-lowering-*.f64 68.5%

          \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \mathsf{+.f64}\left(\frac{-1}{24}, \mathsf{*.f64}\left(\mathsf{*.f64}\left(x, x\right), \frac{1}{720}\right)\right)\right)\right) \]

   5. Simplified 68.5%

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

   if 2.7000000000000002 < x

   1. Initial program 99.2%

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

   4. Applied egg-rr 55.0%

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

   5. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left({x}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(x \cdot \color{blue}{x}\right)\right) \]

      3. *-lowering-*.f64 57.5%

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right) \]

   7. Simplified 57.5%

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

Alternative 6: 65.7% accurate, 8.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.4:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.4) (+ 0.5 (* (* x x) -0.041666666666666664)) (/ 2.0 (* x x))))

C

double code(double x) {
	double tmp;
	if (x <= 2.4) {
		tmp = 0.5 + ((x * x) * -0.041666666666666664);
	} else {
		tmp = 2.0 / (x * x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 2.4d0) then
        tmp = 0.5d0 + ((x * x) * (-0.041666666666666664d0))
    else
        tmp = 2.0d0 / (x * x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 2.4) {
		tmp = 0.5 + ((x * x) * -0.041666666666666664);
	} else {
		tmp = 2.0 / (x * x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2.4:
		tmp = 0.5 + ((x * x) * -0.041666666666666664)
	else:
		tmp = 2.0 / (x * x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.4)
		tmp = Float64(0.5 + Float64(Float64(x * x) * -0.041666666666666664));
	else
		tmp = Float64(2.0 / Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2.4)
		tmp = 0.5 + ((x * x) * -0.041666666666666664);
	else
		tmp = 2.0 / (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2.4], N[(0.5 + N[(N[(x * x), $MachinePrecision] * -0.041666666666666664), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.39999999999999991

   1. Initial program 34.1%

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

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \color{blue}{\left(\frac{-1}{24} \cdot {x}^{2}\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{-1}{24}, \color{blue}{\left({x}^{2}\right)}\right)\right) \]

      3. unpow2 N/A

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{-1}{24}, \left(x \cdot \color{blue}{x}\right)\right)\right) \]

      4. *-lowering-*.f64 68.3%

         \[\leadsto \mathsf{+.f64}\left(\frac{1}{2}, \mathsf{*.f64}\left(\frac{-1}{24}, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right)\right) \]

   5. Simplified 68.3%

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

   if 2.39999999999999991 < x

   1. Initial program 99.2%

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

   4. Applied egg-rr 55.0%

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

   5. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left({x}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(x \cdot \color{blue}{x}\right)\right) \]

      3. *-lowering-*.f64 57.5%

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right) \]

   7. Simplified 57.5%

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

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.4:\\ \;\;\;\;0.5 + \left(x \cdot x\right) \cdot -0.041666666666666664\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{x \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 66.1% accurate, 10.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 2.0) 0.5 (/ 2.0 (* x x))))

C

double code(double x) {
	double tmp;
	if (x <= 2.0) {
		tmp = 0.5;
	} else {
		tmp = 2.0 / (x * x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 2.0d0) then
        tmp = 0.5d0
    else
        tmp = 2.0d0 / (x * x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 2.0) {
		tmp = 0.5;
	} else {
		tmp = 2.0 / (x * x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2.0:
		tmp = 0.5
	else:
		tmp = 2.0 / (x * x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.0)
		tmp = 0.5;
	else
		tmp = Float64(2.0 / Float64(x * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2.0)
		tmp = 0.5;
	else
		tmp = 2.0 / (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2.0], 0.5, N[(2.0 / N[(x * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2

   1. Initial program 34.1%

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

   3. Taylor expanded in x around 0

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

   5. Simplified 68.6%

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

   if 2 < x

   1. Initial program 99.2%

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

   4. Applied egg-rr 55.0%

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

   5. Taylor expanded in x around 0

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

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \color{blue}{\left({x}^{2}\right)}\right) \]

      2. unpow2 N/A

         \[\leadsto \mathsf{/.f64}\left(2, \left(x \cdot \color{blue}{x}\right)\right) \]

      3. *-lowering-*.f64 57.5%

         \[\leadsto \mathsf{/.f64}\left(2, \mathsf{*.f64}\left(x, \color{blue}{x}\right)\right) \]

   7. Simplified 57.5%

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

Alternative 8: 52.1% accurate, 107.0× 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 47.8%

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

3. Taylor expanded in x around 0

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

5. Simplified 55.1%

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

Reproduce

herbie shell --seed 2024155 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))