cos2 (problem 3.4.1)

Percentage Accurate: 50.5% → 99.7%

Time: 9.7s

Alternatives: 9

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: 50.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 0.105:\\ \;\;\;\;0.5 + \left(-0.041666666666666664 \cdot {x}^{2} + \left(-2.48015873015873 \cdot 10^{-5} \cdot {x}^{6} + 0.001388888888888889 \cdot {x}^{4}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{x}, \cos x, \frac{-1}{x}\right)}{-x}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (if (<= x 0.105)
   (+
    0.5
    (+
     (* -0.041666666666666664 (pow x 2.0))
     (+
      (* -2.48015873015873e-5 (pow x 6.0))
      (* 0.001388888888888889 (pow x 4.0)))))
   (/ (fma (/ 1.0 x) (cos x) (/ -1.0 x)) (- x))))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (x <= 0.105) {
		tmp = 0.5 + ((-0.041666666666666664 * pow(x, 2.0)) + ((-2.48015873015873e-5 * pow(x, 6.0)) + (0.001388888888888889 * pow(x, 4.0))));
	} else {
		tmp = fma((1.0 / x), cos(x), (-1.0 / x)) / -x;
	}
	return tmp;
}

Julia

x = abs(x)
function code(x)
	tmp = 0.0
	if (x <= 0.105)
		tmp = Float64(0.5 + Float64(Float64(-0.041666666666666664 * (x ^ 2.0)) + Float64(Float64(-2.48015873015873e-5 * (x ^ 6.0)) + Float64(0.001388888888888889 * (x ^ 4.0)))));
	else
		tmp = Float64(fma(Float64(1.0 / x), cos(x), Float64(-1.0 / x)) / Float64(-x));
	end
	return tmp
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := If[LessEqual[x, 0.105], N[(0.5 + N[(N[(-0.041666666666666664 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] + N[(N[(-2.48015873015873e-5 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision] + N[(0.001388888888888889 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / x), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision] / (-x)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.104999999999999996

   1. Initial program 33.9%

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

   2. Taylor expanded in x around 0 67.9%

      \[\leadsto \color{blue}{0.5 + \left(-0.041666666666666664 \cdot {x}^{2} + \left(-2.48015873015873 \cdot 10^{-5} \cdot {x}^{6} + 0.001388888888888889 \cdot {x}^{4}\right)\right)} \]

   if 0.104999999999999996 < x

   1. Initial program 96.2%

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

      1. associate-/r* 99.2%

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

      2. div-inv 99.2%

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

   3. Applied egg-rr 99.2%

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

      1. *-commutative 99.2%

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

      2. frac-2neg 99.2%

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

      3. associate-*r/ 99.2%

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

      4. sub-neg 99.2%

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

      5. +-commutative 99.2%

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

      6. distribute-neg-in 99.2%

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

      7. add-sqr-sqrt 52.3%

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

      8. distribute-rgt-neg-in 52.3%

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

      9. distribute-lft-neg-out 52.3%

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

      10. sqr-neg 52.3%

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

      11. add-sqr-sqrt 99.2%

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

      12. metadata-eval 99.2%

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

   5. Applied egg-rr 99.2%

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

      1. distribute-lft-in 99.2%

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

      2. fma-def 99.3%

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

   7. Applied egg-rr 99.3%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.105:\\ \;\;\;\;0.5 + \left(-0.041666666666666664 \cdot {x}^{2} + \left(-2.48015873015873 \cdot 10^{-5} \cdot {x}^{6} + 0.001388888888888889 \cdot {x}^{4}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{x}, \cos x, \frac{-1}{x}\right)}{-x}\\ \end{array} \]

Alternative 2: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 0.032:\\ \;\;\;\;0.5 + \left(-0.041666666666666664 \cdot {x}^{2} + 0.001388888888888889 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos x + -1}{x}}{-x}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (if (<= x 0.032)
   (+
    0.5
    (+
     (* -0.041666666666666664 (pow x 2.0))
     (* 0.001388888888888889 (pow x 4.0))))
   (/ (/ (+ (cos x) -1.0) x) (- x))))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (x <= 0.032) {
		tmp = 0.5 + ((-0.041666666666666664 * pow(x, 2.0)) + (0.001388888888888889 * pow(x, 4.0)));
	} else {
		tmp = ((cos(x) + -1.0) / x) / -x;
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.032d0) then
        tmp = 0.5d0 + (((-0.041666666666666664d0) * (x ** 2.0d0)) + (0.001388888888888889d0 * (x ** 4.0d0)))
    else
        tmp = ((cos(x) + (-1.0d0)) / x) / -x
    end if
    code = tmp
end function

Java

x = Math.abs(x);
public static double code(double x) {
	double tmp;
	if (x <= 0.032) {
		tmp = 0.5 + ((-0.041666666666666664 * Math.pow(x, 2.0)) + (0.001388888888888889 * Math.pow(x, 4.0)));
	} else {
		tmp = ((Math.cos(x) + -1.0) / x) / -x;
	}
	return tmp;
}

Python

x = abs(x)
def code(x):
	tmp = 0
	if x <= 0.032:
		tmp = 0.5 + ((-0.041666666666666664 * math.pow(x, 2.0)) + (0.001388888888888889 * math.pow(x, 4.0)))
	else:
		tmp = ((math.cos(x) + -1.0) / x) / -x
	return tmp

Julia

x = abs(x)
function code(x)
	tmp = 0.0
	if (x <= 0.032)
		tmp = Float64(0.5 + Float64(Float64(-0.041666666666666664 * (x ^ 2.0)) + Float64(0.001388888888888889 * (x ^ 4.0))));
	else
		tmp = Float64(Float64(Float64(cos(x) + -1.0) / x) / Float64(-x));
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.032)
		tmp = 0.5 + ((-0.041666666666666664 * (x ^ 2.0)) + (0.001388888888888889 * (x ^ 4.0)));
	else
		tmp = ((cos(x) + -1.0) / x) / -x;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := If[LessEqual[x, 0.032], N[(0.5 + N[(N[(-0.041666666666666664 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] + N[(0.001388888888888889 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision] / x), $MachinePrecision] / (-x)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.032000000000000001

   1. Initial program 33.9%

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

   2. Taylor expanded in x around 0 68.4%

      \[\leadsto \color{blue}{0.5 + \left(-0.041666666666666664 \cdot {x}^{2} + 0.001388888888888889 \cdot {x}^{4}\right)} \]

   if 0.032000000000000001 < x

   1. Initial program 96.2%

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

      1. associate-/r* 99.2%

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

      2. div-inv 99.2%

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

   3. Applied egg-rr 99.2%

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

      1. *-commutative 99.2%

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

      2. frac-2neg 99.2%

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

      3. associate-*r/ 99.2%

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

      4. sub-neg 99.2%

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

      5. +-commutative 99.2%

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

      6. distribute-neg-in 99.2%

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

      7. add-sqr-sqrt 52.3%

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

      8. distribute-rgt-neg-in 52.3%

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

      9. distribute-lft-neg-out 52.3%

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

      10. sqr-neg 52.3%

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

      11. add-sqr-sqrt 99.2%

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

      12. metadata-eval 99.2%

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

   5. Applied egg-rr 99.2%

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

   6. Taylor expanded in x around inf 99.2%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.032:\\ \;\;\;\;0.5 + \left(-0.041666666666666664 \cdot {x}^{2} + 0.001388888888888889 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos x + -1}{x}}{-x}\\ \end{array} \]

Alternative 3: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 0.032:\\ \;\;\;\;0.5 + \left(-0.041666666666666664 \cdot {x}^{2} + 0.001388888888888889 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{x}, \cos x, \frac{-1}{x}\right)}{-x}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (if (<= x 0.032)
   (+
    0.5
    (+
     (* -0.041666666666666664 (pow x 2.0))
     (* 0.001388888888888889 (pow x 4.0))))
   (/ (fma (/ 1.0 x) (cos x) (/ -1.0 x)) (- x))))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (x <= 0.032) {
		tmp = 0.5 + ((-0.041666666666666664 * pow(x, 2.0)) + (0.001388888888888889 * pow(x, 4.0)));
	} else {
		tmp = fma((1.0 / x), cos(x), (-1.0 / x)) / -x;
	}
	return tmp;
}

Julia

x = abs(x)
function code(x)
	tmp = 0.0
	if (x <= 0.032)
		tmp = Float64(0.5 + Float64(Float64(-0.041666666666666664 * (x ^ 2.0)) + Float64(0.001388888888888889 * (x ^ 4.0))));
	else
		tmp = Float64(fma(Float64(1.0 / x), cos(x), Float64(-1.0 / x)) / Float64(-x));
	end
	return tmp
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := If[LessEqual[x, 0.032], N[(0.5 + N[(N[(-0.041666666666666664 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] + N[(0.001388888888888889 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 / x), $MachinePrecision] * N[Cos[x], $MachinePrecision] + N[(-1.0 / x), $MachinePrecision]), $MachinePrecision] / (-x)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.032000000000000001

   1. Initial program 33.9%

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

   2. Taylor expanded in x around 0 68.4%

      \[\leadsto \color{blue}{0.5 + \left(-0.041666666666666664 \cdot {x}^{2} + 0.001388888888888889 \cdot {x}^{4}\right)} \]

   if 0.032000000000000001 < x

   1. Initial program 96.2%

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

      1. associate-/r* 99.2%

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

      2. div-inv 99.2%

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

   3. Applied egg-rr 99.2%

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

      1. *-commutative 99.2%

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

      2. frac-2neg 99.2%

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

      3. associate-*r/ 99.2%

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

      4. sub-neg 99.2%

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

      5. +-commutative 99.2%

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

      6. distribute-neg-in 99.2%

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

      7. add-sqr-sqrt 52.3%

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

      8. distribute-rgt-neg-in 52.3%

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

      9. distribute-lft-neg-out 52.3%

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

      10. sqr-neg 52.3%

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

      11. add-sqr-sqrt 99.2%

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

      12. metadata-eval 99.2%

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

   5. Applied egg-rr 99.2%

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

      1. distribute-lft-in 99.2%

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

      2. fma-def 99.3%

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

   7. Applied egg-rr 99.3%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.032:\\ \;\;\;\;0.5 + \left(-0.041666666666666664 \cdot {x}^{2} + 0.001388888888888889 \cdot {x}^{4}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{1}{x}, \cos x, \frac{-1}{x}\right)}{-x}\\ \end{array} \]

Alternative 4: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 0.005:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos x + -1}{x}}{-x}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (if (<= x 0.005)
   (+ 0.5 (* -0.041666666666666664 (pow x 2.0)))
   (/ (/ (+ (cos x) -1.0) x) (- x))))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (x <= 0.005) {
		tmp = 0.5 + (-0.041666666666666664 * pow(x, 2.0));
	} else {
		tmp = ((cos(x) + -1.0) / x) / -x;
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.005d0) then
        tmp = 0.5d0 + ((-0.041666666666666664d0) * (x ** 2.0d0))
    else
        tmp = ((cos(x) + (-1.0d0)) / x) / -x
    end if
    code = tmp
end function

Java

x = Math.abs(x);
public static double code(double x) {
	double tmp;
	if (x <= 0.005) {
		tmp = 0.5 + (-0.041666666666666664 * Math.pow(x, 2.0));
	} else {
		tmp = ((Math.cos(x) + -1.0) / x) / -x;
	}
	return tmp;
}

Python

x = abs(x)
def code(x):
	tmp = 0
	if x <= 0.005:
		tmp = 0.5 + (-0.041666666666666664 * math.pow(x, 2.0))
	else:
		tmp = ((math.cos(x) + -1.0) / x) / -x
	return tmp

Julia

x = abs(x)
function code(x)
	tmp = 0.0
	if (x <= 0.005)
		tmp = Float64(0.5 + Float64(-0.041666666666666664 * (x ^ 2.0)));
	else
		tmp = Float64(Float64(Float64(cos(x) + -1.0) / x) / Float64(-x));
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.005)
		tmp = 0.5 + (-0.041666666666666664 * (x ^ 2.0));
	else
		tmp = ((cos(x) + -1.0) / x) / -x;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := If[LessEqual[x, 0.005], N[(0.5 + N[(-0.041666666666666664 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Cos[x], $MachinePrecision] + -1.0), $MachinePrecision] / x), $MachinePrecision] / (-x)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0050000000000000001

   1. Initial program 33.9%

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

   2. Taylor expanded in x around 0 68.0%

      \[\leadsto \color{blue}{0.5 + -0.041666666666666664 \cdot {x}^{2}} \]

   if 0.0050000000000000001 < x

   1. Initial program 96.2%

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

      1. associate-/r* 99.2%

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

      2. div-inv 99.2%

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

   3. Applied egg-rr 99.2%

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

      1. *-commutative 99.2%

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

      2. frac-2neg 99.2%

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

      3. associate-*r/ 99.2%

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

      4. sub-neg 99.2%

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

      5. +-commutative 99.2%

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

      6. distribute-neg-in 99.2%

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

      7. add-sqr-sqrt 52.3%

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

      8. distribute-rgt-neg-in 52.3%

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

      9. distribute-lft-neg-out 52.3%

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

      10. sqr-neg 52.3%

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

      11. add-sqr-sqrt 99.2%

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

      12. metadata-eval 99.2%

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

   5. Applied egg-rr 99.2%

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

   6. Taylor expanded in x around inf 99.2%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.005:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\cos x + -1}{x}}{-x}\\ \end{array} \]

Alternative 5: 99.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 0.005:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (if (<= x 0.005)
   (+ 0.5 (* -0.041666666666666664 (pow x 2.0)))
   (/ (- 1.0 (cos x)) (* x x))))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (x <= 0.005) {
		tmp = 0.5 + (-0.041666666666666664 * pow(x, 2.0));
	} else {
		tmp = (1.0 - cos(x)) / (x * x);
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.005d0) then
        tmp = 0.5d0 + ((-0.041666666666666664d0) * (x ** 2.0d0))
    else
        tmp = (1.0d0 - cos(x)) / (x * x)
    end if
    code = tmp
end function

Java

x = Math.abs(x);
public static double code(double x) {
	double tmp;
	if (x <= 0.005) {
		tmp = 0.5 + (-0.041666666666666664 * Math.pow(x, 2.0));
	} else {
		tmp = (1.0 - Math.cos(x)) / (x * x);
	}
	return tmp;
}

Python

x = abs(x)
def code(x):
	tmp = 0
	if x <= 0.005:
		tmp = 0.5 + (-0.041666666666666664 * math.pow(x, 2.0))
	else:
		tmp = (1.0 - math.cos(x)) / (x * x)
	return tmp

Julia

x = abs(x)
function code(x)
	tmp = 0.0
	if (x <= 0.005)
		tmp = Float64(0.5 + Float64(-0.041666666666666664 * (x ^ 2.0)));
	else
		tmp = Float64(Float64(1.0 - cos(x)) / Float64(x * x));
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.005)
		tmp = 0.5 + (-0.041666666666666664 * (x ^ 2.0));
	else
		tmp = (1.0 - cos(x)) / (x * x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := If[LessEqual[x, 0.005], N[(0.5 + N[(-0.041666666666666664 * N[Power[x, 2.0], $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 < 0.0050000000000000001

   1. Initial program 33.9%

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

   2. Taylor expanded in x around 0 68.0%

      \[\leadsto \color{blue}{0.5 + -0.041666666666666664 \cdot {x}^{2}} \]

   if 0.0050000000000000001 < x

   1. Initial program 96.2%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.005:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \end{array} \]

Alternative 6: 79.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 3.05:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{1}{x}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (if (<= x 3.05)
   (+ 0.5 (* -0.041666666666666664 (pow x 2.0)))
   (* (/ 1.0 x) (/ 1.0 x))))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (x <= 3.05) {
		tmp = 0.5 + (-0.041666666666666664 * pow(x, 2.0));
	} else {
		tmp = (1.0 / x) * (1.0 / x);
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 3.05d0) then
        tmp = 0.5d0 + ((-0.041666666666666664d0) * (x ** 2.0d0))
    else
        tmp = (1.0d0 / x) * (1.0d0 / x)
    end if
    code = tmp
end function

Java

x = Math.abs(x);
public static double code(double x) {
	double tmp;
	if (x <= 3.05) {
		tmp = 0.5 + (-0.041666666666666664 * Math.pow(x, 2.0));
	} else {
		tmp = (1.0 / x) * (1.0 / x);
	}
	return tmp;
}

Python

x = abs(x)
def code(x):
	tmp = 0
	if x <= 3.05:
		tmp = 0.5 + (-0.041666666666666664 * math.pow(x, 2.0))
	else:
		tmp = (1.0 / x) * (1.0 / x)
	return tmp

Julia

x = abs(x)
function code(x)
	tmp = 0.0
	if (x <= 3.05)
		tmp = Float64(0.5 + Float64(-0.041666666666666664 * (x ^ 2.0)));
	else
		tmp = Float64(Float64(1.0 / x) * Float64(1.0 / x));
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 3.05)
		tmp = 0.5 + (-0.041666666666666664 * (x ^ 2.0));
	else
		tmp = (1.0 / x) * (1.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := If[LessEqual[x, 3.05], N[(0.5 + N[(-0.041666666666666664 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / x), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 3.0499999999999998

   1. Initial program 33.9%

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

   2. Taylor expanded in x around 0 68.0%

      \[\leadsto \color{blue}{0.5 + -0.041666666666666664 \cdot {x}^{2}} \]

   if 3.0499999999999998 < x

   1. Initial program 96.2%

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

      1. associate-/r* 99.2%

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

      2. div-inv 99.2%

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

   3. Applied egg-rr 99.2%

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

      1. div-sub 99.1%

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

      2. add-sqr-sqrt 98.9%

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

      3. fma-neg 99.0%

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

      4. inv-pow 99.0%

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

      5. sqrt-pow1 99.2%

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

      6. metadata-eval 99.2%

         \[\leadsto \mathsf{fma}\left({x}^{\color{blue}{-0.5}}, \sqrt{\frac{1}{x}}, -\frac{\cos x}{x}\right) \cdot \frac{1}{x} \]

      7. inv-pow 99.2%

         \[\leadsto \mathsf{fma}\left({x}^{-0.5}, \sqrt{\color{blue}{{x}^{-1}}}, -\frac{\cos x}{x}\right) \cdot \frac{1}{x} \]

      8. sqrt-pow1 99.0%

         \[\leadsto \mathsf{fma}\left({x}^{-0.5}, \color{blue}{{x}^{\left(\frac{-1}{2}\right)}}, -\frac{\cos x}{x}\right) \cdot \frac{1}{x} \]

      9. metadata-eval 99.0%

         \[\leadsto \mathsf{fma}\left({x}^{-0.5}, {x}^{\color{blue}{-0.5}}, -\frac{\cos x}{x}\right) \cdot \frac{1}{x} \]

   5. Applied egg-rr 99.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{-0.5}, {x}^{-0.5}, -\frac{\cos x}{x}\right)} \cdot \frac{1}{x} \]

   6. Taylor expanded in x around inf 58.8%

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

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.05:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{1}{x}\\ \end{array} \]

Alternative 7: 79.0% accurate, 11.8× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 1.42:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{1}{x}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x) :precision binary64 (if (<= x 1.42) 0.5 (* (/ 1.0 x) (/ 1.0 x))))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (x <= 1.42) {
		tmp = 0.5;
	} else {
		tmp = (1.0 / x) * (1.0 / x);
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.42d0) then
        tmp = 0.5d0
    else
        tmp = (1.0d0 / x) * (1.0d0 / x)
    end if
    code = tmp
end function

Java

x = Math.abs(x);
public static double code(double x) {
	double tmp;
	if (x <= 1.42) {
		tmp = 0.5;
	} else {
		tmp = (1.0 / x) * (1.0 / x);
	}
	return tmp;
}

Python

x = abs(x)
def code(x):
	tmp = 0
	if x <= 1.42:
		tmp = 0.5
	else:
		tmp = (1.0 / x) * (1.0 / x)
	return tmp

Julia

x = abs(x)
function code(x)
	tmp = 0.0
	if (x <= 1.42)
		tmp = 0.5;
	else
		tmp = Float64(Float64(1.0 / x) * Float64(1.0 / x));
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.42)
		tmp = 0.5;
	else
		tmp = (1.0 / x) * (1.0 / x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := If[LessEqual[x, 1.42], 0.5, N[(N[(1.0 / x), $MachinePrecision] * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.4199999999999999

   1. Initial program 33.9%

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

   2. Taylor expanded in x around 0 68.4%

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

   if 1.4199999999999999 < x

   1. Initial program 96.2%

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

      1. associate-/r* 99.2%

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

      2. div-inv 99.2%

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

   3. Applied egg-rr 99.2%

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

      1. div-sub 99.1%

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

      2. add-sqr-sqrt 98.9%

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

      3. fma-neg 99.0%

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

      4. inv-pow 99.0%

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

      5. sqrt-pow1 99.2%

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

      6. metadata-eval 99.2%

         \[\leadsto \mathsf{fma}\left({x}^{\color{blue}{-0.5}}, \sqrt{\frac{1}{x}}, -\frac{\cos x}{x}\right) \cdot \frac{1}{x} \]

      7. inv-pow 99.2%

         \[\leadsto \mathsf{fma}\left({x}^{-0.5}, \sqrt{\color{blue}{{x}^{-1}}}, -\frac{\cos x}{x}\right) \cdot \frac{1}{x} \]

      8. sqrt-pow1 99.0%

         \[\leadsto \mathsf{fma}\left({x}^{-0.5}, \color{blue}{{x}^{\left(\frac{-1}{2}\right)}}, -\frac{\cos x}{x}\right) \cdot \frac{1}{x} \]

      9. metadata-eval 99.0%

         \[\leadsto \mathsf{fma}\left({x}^{-0.5}, {x}^{\color{blue}{-0.5}}, -\frac{\cos x}{x}\right) \cdot \frac{1}{x} \]

   5. Applied egg-rr 99.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{-0.5}, {x}^{-0.5}, -\frac{\cos x}{x}\right)} \cdot \frac{1}{x} \]

   6. Taylor expanded in x around inf 58.8%

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

4. Final simplification 65.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.42:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} \cdot \frac{1}{x}\\ \end{array} \]

Alternative 8: 76.0% accurate, 15.2× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 1.4 \cdot 10^{+77}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} \cdot 0\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x) :precision binary64 (if (<= x 1.4e+77) 0.5 (* (/ 1.0 x) 0.0)))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (x <= 1.4e+77) {
		tmp = 0.5;
	} else {
		tmp = (1.0 / x) * 0.0;
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.4d+77) then
        tmp = 0.5d0
    else
        tmp = (1.0d0 / x) * 0.0d0
    end if
    code = tmp
end function

Java

x = Math.abs(x);
public static double code(double x) {
	double tmp;
	if (x <= 1.4e+77) {
		tmp = 0.5;
	} else {
		tmp = (1.0 / x) * 0.0;
	}
	return tmp;
}

Python

x = abs(x)
def code(x):
	tmp = 0
	if x <= 1.4e+77:
		tmp = 0.5
	else:
		tmp = (1.0 / x) * 0.0
	return tmp

Julia

x = abs(x)
function code(x)
	tmp = 0.0
	if (x <= 1.4e+77)
		tmp = 0.5;
	else
		tmp = Float64(Float64(1.0 / x) * 0.0);
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.4e+77)
		tmp = 0.5;
	else
		tmp = (1.0 / x) * 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := If[LessEqual[x, 1.4e+77], 0.5, N[(N[(1.0 / x), $MachinePrecision] * 0.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.4e77

   1. Initial program 41.3%

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

   2. Taylor expanded in x around 0 61.4%

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

   if 1.4e77 < x

   1. Initial program 95.3%

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

      1. associate-/r* 99.6%

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

      2. div-inv 99.6%

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

   3. Applied egg-rr 99.6%

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

      1. div-sub 99.5%

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

      2. add-sqr-sqrt 99.4%

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

      3. fma-neg 99.5%

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

      4. inv-pow 99.5%

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

      5. sqrt-pow1 99.6%

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

      6. metadata-eval 99.6%

         \[\leadsto \mathsf{fma}\left({x}^{\color{blue}{-0.5}}, \sqrt{\frac{1}{x}}, -\frac{\cos x}{x}\right) \cdot \frac{1}{x} \]

      7. inv-pow 99.6%

         \[\leadsto \mathsf{fma}\left({x}^{-0.5}, \sqrt{\color{blue}{{x}^{-1}}}, -\frac{\cos x}{x}\right) \cdot \frac{1}{x} \]

      8. sqrt-pow1 99.5%

         \[\leadsto \mathsf{fma}\left({x}^{-0.5}, \color{blue}{{x}^{\left(\frac{-1}{2}\right)}}, -\frac{\cos x}{x}\right) \cdot \frac{1}{x} \]

      9. metadata-eval 99.5%

         \[\leadsto \mathsf{fma}\left({x}^{-0.5}, {x}^{\color{blue}{-0.5}}, -\frac{\cos x}{x}\right) \cdot \frac{1}{x} \]

   5. Applied egg-rr 99.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left({x}^{-0.5}, {x}^{-0.5}, -\frac{\cos x}{x}\right)} \cdot \frac{1}{x} \]

   6. Taylor expanded in x around 0 72.8%

      \[\leadsto \color{blue}{0} \cdot \frac{1}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 63.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.4 \cdot 10^{+77}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{x} \cdot 0\\ \end{array} \]

Alternative 9: 51.9% accurate, 107.0× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ 0.5 \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x) :precision binary64 0.5)

C

x = abs(x);
double code(double x) {
	return 0.5;
}

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    code = 0.5d0
end function

Java

x = Math.abs(x);
public static double code(double x) {
	return 0.5;
}

Python

x = abs(x)
def code(x):
	return 0.5

Julia

x = abs(x)
function code(x)
	return 0.5
end

MATLAB

x = abs(x)
function tmp = code(x)
	tmp = 0.5;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := 0.5

Derivation

1. Initial program 53.1%

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

2. Taylor expanded in x around 0 48.7%

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

3. Final simplification 48.7%

   \[\leadsto 0.5 \]

Reproduce

herbie shell --seed 2023320 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))