cos2 (problem 3.4.1)

Percentage Accurate: 50.1% → 99.8%

Time: 12.5s

Alternatives: 5

Speedup: 17.8×

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: 50.1% 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

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x_m \leq 5 \cdot 10^{-8}:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot {x_m}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sin x_m \cdot \left(-\tan \left(\frac{x_m}{2}\right)\right)}{x_m}}{-x_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 5e-8)
   (+ 0.5 (* -0.041666666666666664 (pow x_m 2.0)))
   (/ (/ (* (sin x_m) (- (tan (/ x_m 2.0)))) x_m) (- x_m))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 5e-8) {
		tmp = 0.5 + (-0.041666666666666664 * pow(x_m, 2.0));
	} else {
		tmp = ((sin(x_m) * -tan((x_m / 2.0))) / x_m) / -x_m;
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 5d-8) then
        tmp = 0.5d0 + ((-0.041666666666666664d0) * (x_m ** 2.0d0))
    else
        tmp = ((sin(x_m) * -tan((x_m / 2.0d0))) / x_m) / -x_m
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 5e-8) {
		tmp = 0.5 + (-0.041666666666666664 * Math.pow(x_m, 2.0));
	} else {
		tmp = ((Math.sin(x_m) * -Math.tan((x_m / 2.0))) / x_m) / -x_m;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 5e-8:
		tmp = 0.5 + (-0.041666666666666664 * math.pow(x_m, 2.0))
	else:
		tmp = ((math.sin(x_m) * -math.tan((x_m / 2.0))) / x_m) / -x_m
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 5e-8)
		tmp = Float64(0.5 + Float64(-0.041666666666666664 * (x_m ^ 2.0)));
	else
		tmp = Float64(Float64(Float64(sin(x_m) * Float64(-tan(Float64(x_m / 2.0)))) / x_m) / Float64(-x_m));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 5e-8)
		tmp = 0.5 + (-0.041666666666666664 * (x_m ^ 2.0));
	else
		tmp = ((sin(x_m) * -tan((x_m / 2.0))) / x_m) / -x_m;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 5e-8], N[(0.5 + N[(-0.041666666666666664 * N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Sin[x$95$m], $MachinePrecision] * (-N[Tan[N[(x$95$m / 2.0), $MachinePrecision]], $MachinePrecision])), $MachinePrecision] / x$95$m), $MachinePrecision] / (-x$95$m)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4.9999999999999998e-8

   1. Initial program 37.2%

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

   3. Taylor expanded in x around 0 64.0%

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

   if 4.9999999999999998e-8 < x

   1. Initial program 97.5%

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

      1. associate-/r* 99.4%

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

      2. div-inv 99.3%

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

   4. Applied egg-rr 99.3%

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

      1. *-commutative 99.3%

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

      2. frac-2neg 99.3%

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

      3. associate-*r/ 99.3%

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

      4. sub-neg 99.3%

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

      5. +-commutative 99.3%

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

      6. distribute-neg-in 99.3%

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

      7. add-sqr-sqrt 39.4%

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

      8. distribute-rgt-neg-in 39.4%

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

      9. distribute-lft-neg-out 39.4%

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

      10. sqr-neg 39.4%

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

      11. add-sqr-sqrt 99.3%

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

      12. metadata-eval 99.3%

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

   6. Applied egg-rr 99.3%

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

      1. metadata-eval 99.3%

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

      2. sub-neg 99.3%

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

      3. flip-- 98.5%

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

      4. frac-times 98.7%

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

      5. metadata-eval 98.7%

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

      6. sub-1-cos 98.8%

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

      7. *-un-lft-identity 98.8%

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

      8. pow2 98.8%

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

   8. Applied egg-rr 98.8%

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

      1. *-commutative 98.8%

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

      2. associate-/r* 98.7%

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

      3. distribute-frac-neg 98.7%

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

      4. unpow2 98.7%

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

      5. associate-*r/ 98.7%

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

      6. distribute-rgt-neg-in 98.7%

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

      7. +-commutative 98.7%

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

      8. hang-0p-tan 99.5%

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

   10. Simplified 99.5%

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

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{-8}:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\sin x \cdot \left(-\tan \left(\frac{x}{2}\right)\right)}{x}}{-x}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x_m \leq 0.0052:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot {x_m}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x_m}{x_m \cdot x_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.0052)
   (+ 0.5 (* -0.041666666666666664 (pow x_m 2.0)))
   (/ (- 1.0 (cos x_m)) (* x_m x_m))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.0052) {
		tmp = 0.5 + (-0.041666666666666664 * pow(x_m, 2.0));
	} else {
		tmp = (1.0 - cos(x_m)) / (x_m * x_m);
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 0.0052d0) then
        tmp = 0.5d0 + ((-0.041666666666666664d0) * (x_m ** 2.0d0))
    else
        tmp = (1.0d0 - cos(x_m)) / (x_m * x_m)
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 0.0052) {
		tmp = 0.5 + (-0.041666666666666664 * Math.pow(x_m, 2.0));
	} else {
		tmp = (1.0 - Math.cos(x_m)) / (x_m * x_m);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 0.0052:
		tmp = 0.5 + (-0.041666666666666664 * math.pow(x_m, 2.0))
	else:
		tmp = (1.0 - math.cos(x_m)) / (x_m * x_m)
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.0052)
		tmp = Float64(0.5 + Float64(-0.041666666666666664 * (x_m ^ 2.0)));
	else
		tmp = Float64(Float64(1.0 - cos(x_m)) / Float64(x_m * x_m));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 0.0052)
		tmp = 0.5 + (-0.041666666666666664 * (x_m ^ 2.0));
	else
		tmp = (1.0 - cos(x_m)) / (x_m * x_m);
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.0052], N[(0.5 + N[(-0.041666666666666664 * N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 - N[Cos[x$95$m], $MachinePrecision]), $MachinePrecision] / N[(x$95$m * x$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0051999999999999998

   1. Initial program 37.2%

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

   3. Taylor expanded in x around 0 64.0%

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

   if 0.0051999999999999998 < x

   1. Initial program 97.5%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0052:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{1 - \cos x}{x \cdot x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x_m \leq 0.0052:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot {x_m}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x_m}{x_m}}{x_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.0052)
   (+ 0.5 (* -0.041666666666666664 (pow x_m 2.0)))
   (/ (/ (- 1.0 (cos x_m)) x_m) x_m)))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.0052) {
		tmp = 0.5 + (-0.041666666666666664 * pow(x_m, 2.0));
	} else {
		tmp = ((1.0 - cos(x_m)) / x_m) / x_m;
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 0.0052d0) then
        tmp = 0.5d0 + ((-0.041666666666666664d0) * (x_m ** 2.0d0))
    else
        tmp = ((1.0d0 - cos(x_m)) / x_m) / x_m
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 0.0052) {
		tmp = 0.5 + (-0.041666666666666664 * Math.pow(x_m, 2.0));
	} else {
		tmp = ((1.0 - Math.cos(x_m)) / x_m) / x_m;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 0.0052:
		tmp = 0.5 + (-0.041666666666666664 * math.pow(x_m, 2.0))
	else:
		tmp = ((1.0 - math.cos(x_m)) / x_m) / x_m
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.0052)
		tmp = Float64(0.5 + Float64(-0.041666666666666664 * (x_m ^ 2.0)));
	else
		tmp = Float64(Float64(Float64(1.0 - cos(x_m)) / x_m) / x_m);
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 0.0052)
		tmp = 0.5 + (-0.041666666666666664 * (x_m ^ 2.0));
	else
		tmp = ((1.0 - cos(x_m)) / x_m) / x_m;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.0052], N[(0.5 + N[(-0.041666666666666664 * N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(1.0 - N[Cos[x$95$m], $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision] / x$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0051999999999999998

   1. Initial program 37.2%

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

   3. Taylor expanded in x around 0 64.0%

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

   if 0.0051999999999999998 < x

   1. Initial program 97.5%

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

      1. div-sub 97.4%

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

      2. pow2 97.4%

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

      3. pow-flip 97.3%

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

      4. metadata-eval 97.3%

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

      5. div-inv 97.2%

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

      6. pow2 97.2%

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

      7. pow-flip 99.3%

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

      8. metadata-eval 99.3%

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

   4. Applied egg-rr 99.3%

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

      1. *-un-lft-identity 99.3%

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

      2. distribute-rgt-out-- 99.5%

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

      3. sqr-pow 99.2%

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

      4. pow-prod-down 97.4%

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

      5. metadata-eval 97.4%

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

      6. inv-pow 97.4%

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

      7. associate-/r/ 97.4%

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

      8. associate-/l* 97.4%

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

      9. clear-num 99.4%

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

   6. Applied egg-rr 99.4%

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

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0052:\\ \;\;\;\;0.5 + -0.041666666666666664 \cdot {x}^{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1 - \cos x}{x}}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.6% accurate, 17.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x_m \leq 1.6 \cdot 10^{+77}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (if (<= x_m 1.6e+77) 0.5 0.0))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.6e+77) {
		tmp = 0.5;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 1.6d+77) then
        tmp = 0.5d0
    else
        tmp = 0.0d0
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 1.6e+77) {
		tmp = 0.5;
	} else {
		tmp = 0.0;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 1.6e+77:
		tmp = 0.5
	else:
		tmp = 0.0
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1.6e+77)
		tmp = 0.5;
	else
		tmp = 0.0;
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 1.6e+77)
		tmp = 0.5;
	else
		tmp = 0.0;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1.6e+77], 0.5, 0.0]

Derivation

1. Split input into 2 regimes

2. if x < 1.6000000000000001e77

   1. Initial program 42.4%

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

   3. Taylor expanded in x around 0 60.0%

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

   if 1.6000000000000001e77 < x

   1. Initial program 97.0%

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

      1. add-sqr-sqrt 96.8%

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

      2. pow2 96.8%

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

   4. Applied egg-rr 96.8%

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

      1. unpow2 96.8%

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

      2. add-sqr-sqrt 97.0%

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

      3. flip-- 95.9%

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

      4. metadata-eval 95.9%

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

      5. pow2 95.9%

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

      6. +-commutative 95.9%

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

   6. Applied egg-rr 95.9%

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

      1. unpow2 95.9%

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

      2. 1-sub-cos 96.1%

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

   8. Applied egg-rr 96.1%

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

      1. associate-/l/ 96.2%

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

      2. sqr-sin-a 96.0%

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

      3. div-sub 95.9%

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

      4. pow2 95.9%

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

      5. cos-2 96.1%

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

      6. cos-sum 95.9%

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

      7. add-sqr-sqrt 88.2%

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

      8. sqrt-prod 36.2%

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

      9. sqr-neg 36.2%

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

      10. sqrt-unprod 0.0%

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

      11. add-sqr-sqrt 62.1%

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

      12. sub-neg 62.1%

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

      13. +-inverses 62.1%

          \[\leadsto \frac{0.5}{{x}^{2} \cdot \left(\cos x + 1\right)} - \frac{0.5 \cdot \cos \color{blue}{0}}{\left(x \cdot x\right) \cdot \left(\cos x + 1\right)} \]

      14. pow2 62.1%

          \[\leadsto \frac{0.5}{{x}^{2} \cdot \left(\cos x + 1\right)} - \frac{0.5 \cdot \cos 0}{\color{blue}{{x}^{2}} \cdot \left(\cos x + 1\right)} \]

   10. Applied egg-rr 62.1%

       \[\leadsto \color{blue}{\frac{0.5}{{x}^{2} \cdot \left(\cos x + 1\right)} - \frac{0.5 \cdot \cos 0}{{x}^{2} \cdot \left(\cos x + 1\right)}} \]
   11. Step-by-step derivation

       1. cos-0 62.1%

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

       2. metadata-eval 62.1%

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

       3. +-inverses 62.1%

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

   12. Simplified 62.1%

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

4. Final simplification 60.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.6 \cdot 10^{+77}:\\ \;\;\;\;0.5\\ \mathbf{else}:\\ \;\;\;\;0\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 27.5% accurate, 107.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ 0 \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 0.0)

C

x_m = fabs(x);
double code(double x_m) {
	return 0.0;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    code = 0.0d0
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return 0.0;
}

Python

x_m = math.fabs(x)
def code(x_m):
	return 0.0

Julia

x_m = abs(x)
function code(x_m)
	return 0.0
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = 0.0;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := 0.0

Derivation

1. Initial program 53.7%

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

   1. add-sqr-sqrt 53.6%

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

   2. pow2 53.6%

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

4. Applied egg-rr 53.6%

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

   1. unpow2 53.6%

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

   2. add-sqr-sqrt 53.7%

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

   3. flip-- 53.4%

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

   4. metadata-eval 53.4%

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

   5. pow2 53.4%

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

   6. +-commutative 53.4%

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

6. Applied egg-rr 53.4%

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

   1. unpow2 53.4%

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

   2. 1-sub-cos 76.3%

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

8. Applied egg-rr 76.3%

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

   1. associate-/l/ 76.4%

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

   2. sqr-sin-a 53.5%

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

   3. div-sub 54.3%

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

   4. pow2 54.3%

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

   5. cos-2 54.2%

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

   6. cos-sum 54.3%

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

   7. add-sqr-sqrt 24.7%

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

   8. sqrt-prod 16.3%

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

   9. sqr-neg 16.3%

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

   10. sqrt-unprod 14.3%

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

   11. add-sqr-sqrt 27.2%

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

   12. sub-neg 27.2%

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

   13. +-inverses 27.2%

       \[\leadsto \frac{0.5}{{x}^{2} \cdot \left(\cos x + 1\right)} - \frac{0.5 \cdot \cos \color{blue}{0}}{\left(x \cdot x\right) \cdot \left(\cos x + 1\right)} \]

   14. pow2 27.2%

       \[\leadsto \frac{0.5}{{x}^{2} \cdot \left(\cos x + 1\right)} - \frac{0.5 \cdot \cos 0}{\color{blue}{{x}^{2}} \cdot \left(\cos x + 1\right)} \]

10. Applied egg-rr 27.2%

    \[\leadsto \color{blue}{\frac{0.5}{{x}^{2} \cdot \left(\cos x + 1\right)} - \frac{0.5 \cdot \cos 0}{{x}^{2} \cdot \left(\cos x + 1\right)}} \]
11. Step-by-step derivation

    1. cos-0 27.2%

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

    2. metadata-eval 27.2%

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

    3. +-inverses 27.9%

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

12. Simplified 27.9%

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

13. Final simplification 27.9%

    \[\leadsto 0 \]
14. Add Preprocessing

Reproduce

herbie shell --seed 2024013 
(FPCore (x)
  :name "cos2 (problem 3.4.1)"
  :precision binary64
  (/ (- 1.0 (cos x)) (* x x)))