Diagrams.TwoD.Apollonian:initialConfig from diagrams-contrib-1.3.0.5, A

Percentage Accurate: 68.9% → 99.9%

Time: 12.4s

Alternatives: 4

Speedup: 1.2×

• 40.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (/ (- (+ (* x x) (* y y)) (* z z)) (* y 2.0)))

C

double code(double x, double y, double z) {
	return (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (((x * x) + (y * y)) - (z * z)) / (y * 2.0d0)
end function

Java

public static double code(double x, double y, double z) {
	return (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
}

Python

def code(x, y, z):
	return (((x * x) + (y * y)) - (z * z)) / (y * 2.0)

Julia

function code(x, y, z)
	return Float64(Float64(Float64(Float64(x * x) + Float64(y * y)) - Float64(z * z)) / Float64(y * 2.0))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]

Initial Program: 68.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (/ (- (+ (* x x) (* y y)) (* z z)) (* y 2.0)))

C

double code(double x, double y, double z) {
	return (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (((x * x) + (y * y)) - (z * z)) / (y * 2.0d0)
end function

Java

public static double code(double x, double y, double z) {
	return (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
}

Python

def code(x, y, z):
	return (((x * x) + (y * y)) - (z * z)) / (y * 2.0)

Julia

function code(x, y, z)
	return Float64(Float64(Float64(Float64(x * x) + Float64(y * y)) - Float64(z * z)) / Float64(y * 2.0))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (((x * x) + (y * y)) - (z * z)) / (y * 2.0);
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(N[(x * x), $MachinePrecision] + N[(y * y), $MachinePrecision]), $MachinePrecision] - N[(z * z), $MachinePrecision]), $MachinePrecision] / N[(y * 2.0), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ 0.5 \cdot \left(y + \left(x\_m + z\right) \cdot \frac{x\_m - z}{y}\right) \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y z)
 :precision binary64
 (* 0.5 (+ y (* (+ x_m z) (/ (- x_m z) y)))))

C

x_m = fabs(x);
double code(double x_m, double y, double z) {
	return 0.5 * (y + ((x_m + z) * ((x_m - z) / y)));
}

Fortran

x_m = abs(x)
real(8) function code(x_m, y, z)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 0.5d0 * (y + ((x_m + z) * ((x_m - z) / y)))
end function

Java

x_m = Math.abs(x);
public static double code(double x_m, double y, double z) {
	return 0.5 * (y + ((x_m + z) * ((x_m - z) / y)));
}

Python

x_m = math.fabs(x)
def code(x_m, y, z):
	return 0.5 * (y + ((x_m + z) * ((x_m - z) / y)))

Julia

x_m = abs(x)
function code(x_m, y, z)
	return Float64(0.5 * Float64(y + Float64(Float64(x_m + z) * Float64(Float64(x_m - z) / y))))
end

MATLAB

x_m = abs(x);
function tmp = code(x_m, y, z)
	tmp = 0.5 * (y + ((x_m + z) * ((x_m - z) / y)));
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_, z_] := N[(0.5 * N[(y + N[(N[(x$95$m + z), $MachinePrecision] * N[(N[(x$95$m - z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 68.8%

   \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
2. Step-by-step derivation

   1. remove-double-neg 68.8%

      \[\leadsto \frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{\color{blue}{-\left(-y \cdot 2\right)}} \]

   2. distribute-lft-neg-out 68.8%

      \[\leadsto \frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{-\color{blue}{\left(-y\right) \cdot 2}} \]

   3. distribute-frac-neg2 68.8%

      \[\leadsto \color{blue}{-\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{\left(-y\right) \cdot 2}} \]

   4. distribute-frac-neg 68.8%

      \[\leadsto \color{blue}{\frac{-\left(\left(x \cdot x + y \cdot y\right) - z \cdot z\right)}{\left(-y\right) \cdot 2}} \]

   5. neg-mul-1 68.8%

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

   6. distribute-lft-neg-out 68.8%

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

   7. *-commutative 68.8%

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

   8. distribute-lft-neg-in 68.8%

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

   9. times-frac 68.8%

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

   10. metadata-eval 68.8%

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

   11. metadata-eval 68.8%

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

   12. associate--l+ 68.8%

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

   13. fma-define 71.6%

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

3. Simplified 71.6%

   \[\leadsto \color{blue}{0.5 \cdot \frac{\mathsf{fma}\left(x, x, y \cdot y - z \cdot z\right)}{y}} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 82.3%

   \[\leadsto 0.5 \cdot \color{blue}{\left(\left(y + \frac{{x}^{2}}{y}\right) - \frac{{z}^{2}}{y}\right)} \]
6. Step-by-step derivation

   1. associate--l+ 82.3%

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

   2. div-sub 85.8%

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

7. Simplified 85.8%

   \[\leadsto 0.5 \cdot \color{blue}{\left(y + \frac{{x}^{2} - {z}^{2}}{y}\right)} \]
8. Step-by-step derivation

   1. unpow2 85.8%

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

   2. unpow2 85.8%

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

   3. difference-of-squares 90.2%

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

9. Applied egg-rr 90.2%

   \[\leadsto 0.5 \cdot \left(y + \frac{\color{blue}{\left(x + z\right) \cdot \left(x - z\right)}}{y}\right) \]
10. Step-by-step derivation

    1. associate-/l* 99.9%

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

11. Applied egg-rr 99.9%

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

12. Final simplification 99.9%

    \[\leadsto 0.5 \cdot \left(y + \left(x + z\right) \cdot \frac{x - z}{y}\right) \]
13. Add Preprocessing

Alternative 2: 74.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;y \leq 1.8 \cdot 10^{+72} \lor \neg \left(y \leq 7.5 \cdot 10^{+87}\right) \land y \leq 2.9 \cdot 10^{+109}:\\ \;\;\;\;\left(x\_m + z\right) \cdot \left(0.5 \cdot \frac{x\_m - z}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y z)
 :precision binary64
 (if (or (<= y 1.8e+72) (and (not (<= y 7.5e+87)) (<= y 2.9e+109)))
   (* (+ x_m z) (* 0.5 (/ (- x_m z) y)))
   (* 0.5 y)))

C

x_m = fabs(x);
double code(double x_m, double y, double z) {
	double tmp;
	if ((y <= 1.8e+72) || (!(y <= 7.5e+87) && (y <= 2.9e+109))) {
		tmp = (x_m + z) * (0.5 * ((x_m - z) / y));
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m, y, z)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if ((y <= 1.8d+72) .or. (.not. (y <= 7.5d+87)) .and. (y <= 2.9d+109)) then
        tmp = (x_m + z) * (0.5d0 * ((x_m - z) / y))
    else
        tmp = 0.5d0 * y
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m, double y, double z) {
	double tmp;
	if ((y <= 1.8e+72) || (!(y <= 7.5e+87) && (y <= 2.9e+109))) {
		tmp = (x_m + z) * (0.5 * ((x_m - z) / y));
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m, y, z):
	tmp = 0
	if (y <= 1.8e+72) or (not (y <= 7.5e+87) and (y <= 2.9e+109)):
		tmp = (x_m + z) * (0.5 * ((x_m - z) / y))
	else:
		tmp = 0.5 * y
	return tmp

Julia

x_m = abs(x)
function code(x_m, y, z)
	tmp = 0.0
	if ((y <= 1.8e+72) || (!(y <= 7.5e+87) && (y <= 2.9e+109)))
		tmp = Float64(Float64(x_m + z) * Float64(0.5 * Float64(Float64(x_m - z) / y)));
	else
		tmp = Float64(0.5 * y);
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m, y, z)
	tmp = 0.0;
	if ((y <= 1.8e+72) || (~((y <= 7.5e+87)) && (y <= 2.9e+109)))
		tmp = (x_m + z) * (0.5 * ((x_m - z) / y));
	else
		tmp = 0.5 * y;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_, z_] := If[Or[LessEqual[y, 1.8e+72], And[N[Not[LessEqual[y, 7.5e+87]], $MachinePrecision], LessEqual[y, 2.9e+109]]], N[(N[(x$95$m + z), $MachinePrecision] * N[(0.5 * N[(N[(x$95$m - z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.80000000000000017e72 or 7.50000000000000014e87 < y < 2.9e109

   1. Initial program 76.7%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 79.7%

      \[\leadsto \color{blue}{y \cdot \left(0.5 + 0.5 \cdot \frac{{x}^{2} - {z}^{2}}{{y}^{2}}\right)} \]
   4. Step-by-step derivation

      1. *-commutative 79.7%

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

   5. Simplified 79.7%

      \[\leadsto \color{blue}{y \cdot \left(0.5 + \frac{{x}^{2} - {z}^{2}}{{y}^{2}} \cdot 0.5\right)} \]
   6. Step-by-step derivation

      1. unpow2 89.2%

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

      2. unpow2 89.2%

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

      3. difference-of-squares 93.6%

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

   7. Applied egg-rr 83.1%

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

   8. Taylor expanded in y around 0 70.4%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\left(x + z\right) \cdot \left(x - z\right)}{y}} \]
   9. Step-by-step derivation

      1. associate-*r/ 70.4%

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

      2. +-commutative 70.4%

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

   10. Simplified 70.4%

       \[\leadsto \color{blue}{\frac{0.5 \cdot \left(\left(z + x\right) \cdot \left(x - z\right)\right)}{y}} \]
   11. Step-by-step derivation

       1. associate-/l* 70.4%

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

       2. *-commutative 70.4%

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

       3. associate-*l/ 73.4%

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

       4. associate-*r* 73.4%

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

       5. +-commutative 73.4%

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

   12. Applied egg-rr 73.4%

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

   if 1.80000000000000017e72 < y < 7.50000000000000014e87 or 2.9e109 < y

   1. Initial program 37.3%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 73.4%

      \[\leadsto \color{blue}{0.5 \cdot y} \]
   4. Step-by-step derivation

      1. *-commutative 73.4%

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

   5. Simplified 73.4%

      \[\leadsto \color{blue}{y \cdot 0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.8 \cdot 10^{+72} \lor \neg \left(y \leq 7.5 \cdot 10^{+87}\right) \land y \leq 2.9 \cdot 10^{+109}:\\ \;\;\;\;\left(x + z\right) \cdot \left(0.5 \cdot \frac{x - z}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 43.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;y \leq 3.2 \cdot 10^{+23}:\\ \;\;\;\;0.5 \cdot \left(x\_m \cdot \frac{x\_m}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y z)
 :precision binary64
 (if (<= y 3.2e+23) (* 0.5 (* x_m (/ x_m y))) (* 0.5 y)))

C

x_m = fabs(x);
double code(double x_m, double y, double z) {
	double tmp;
	if (y <= 3.2e+23) {
		tmp = 0.5 * (x_m * (x_m / y));
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m, y, z)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= 3.2d+23) then
        tmp = 0.5d0 * (x_m * (x_m / y))
    else
        tmp = 0.5d0 * y
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m, double y, double z) {
	double tmp;
	if (y <= 3.2e+23) {
		tmp = 0.5 * (x_m * (x_m / y));
	} else {
		tmp = 0.5 * y;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m, y, z):
	tmp = 0
	if y <= 3.2e+23:
		tmp = 0.5 * (x_m * (x_m / y))
	else:
		tmp = 0.5 * y
	return tmp

Julia

x_m = abs(x)
function code(x_m, y, z)
	tmp = 0.0
	if (y <= 3.2e+23)
		tmp = Float64(0.5 * Float64(x_m * Float64(x_m / y)));
	else
		tmp = Float64(0.5 * y);
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m, y, z)
	tmp = 0.0;
	if (y <= 3.2e+23)
		tmp = 0.5 * (x_m * (x_m / y));
	else
		tmp = 0.5 * y;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_, z_] := If[LessEqual[y, 3.2e+23], N[(0.5 * N[(x$95$m * N[(x$95$m / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(0.5 * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.2e23

   1. Initial program 75.5%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   2. Step-by-step derivation

      1. remove-double-neg 75.5%

         \[\leadsto \frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{\color{blue}{-\left(-y \cdot 2\right)}} \]

      2. distribute-lft-neg-out 75.5%

         \[\leadsto \frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{-\color{blue}{\left(-y\right) \cdot 2}} \]

      3. distribute-frac-neg2 75.5%

         \[\leadsto \color{blue}{-\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{\left(-y\right) \cdot 2}} \]

      4. distribute-frac-neg 75.5%

         \[\leadsto \color{blue}{\frac{-\left(\left(x \cdot x + y \cdot y\right) - z \cdot z\right)}{\left(-y\right) \cdot 2}} \]

      5. neg-mul-1 75.5%

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

      6. distribute-lft-neg-out 75.5%

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

      7. *-commutative 75.5%

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

      8. distribute-lft-neg-in 75.5%

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

      9. times-frac 75.5%

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

      10. metadata-eval 75.5%

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

      11. metadata-eval 75.5%

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

      12. associate--l+ 75.5%

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

      13. fma-define 78.6%

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

   3. Simplified 78.6%

      \[\leadsto \color{blue}{0.5 \cdot \frac{\mathsf{fma}\left(x, x, y \cdot y - z \cdot z\right)}{y}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around inf 56.1%

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

      1. add-sqr-sqrt 30.9%

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

      2. pow2 30.9%

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

   7. Applied egg-rr 11.5%

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

   8. Taylor expanded in x around inf 14.7%

      \[\leadsto 0.5 \cdot {\color{blue}{\left(x \cdot \sqrt{\frac{1}{y}}\right)}}^{2} \]
   9. Step-by-step derivation

      1. unpow2 14.7%

         \[\leadsto 0.5 \cdot \color{blue}{\left(\left(x \cdot \sqrt{\frac{1}{y}}\right) \cdot \left(x \cdot \sqrt{\frac{1}{y}}\right)\right)} \]

      2. associate-*l* 14.7%

         \[\leadsto 0.5 \cdot \color{blue}{\left(x \cdot \left(\sqrt{\frac{1}{y}} \cdot \left(x \cdot \sqrt{\frac{1}{y}}\right)\right)\right)} \]

      3. sqrt-div 14.7%

         \[\leadsto 0.5 \cdot \left(x \cdot \left(\color{blue}{\frac{\sqrt{1}}{\sqrt{y}}} \cdot \left(x \cdot \sqrt{\frac{1}{y}}\right)\right)\right) \]

      4. metadata-eval 14.7%

         \[\leadsto 0.5 \cdot \left(x \cdot \left(\frac{\color{blue}{1}}{\sqrt{y}} \cdot \left(x \cdot \sqrt{\frac{1}{y}}\right)\right)\right) \]

      5. sqrt-div 14.7%

         \[\leadsto 0.5 \cdot \left(x \cdot \left(\frac{1}{\sqrt{y}} \cdot \left(x \cdot \color{blue}{\frac{\sqrt{1}}{\sqrt{y}}}\right)\right)\right) \]

      6. metadata-eval 14.7%

         \[\leadsto 0.5 \cdot \left(x \cdot \left(\frac{1}{\sqrt{y}} \cdot \left(x \cdot \frac{\color{blue}{1}}{\sqrt{y}}\right)\right)\right) \]

      7. un-div-inv 14.7%

         \[\leadsto 0.5 \cdot \left(x \cdot \left(\frac{1}{\sqrt{y}} \cdot \color{blue}{\frac{x}{\sqrt{y}}}\right)\right) \]

      8. times-frac 14.7%

         \[\leadsto 0.5 \cdot \left(x \cdot \color{blue}{\frac{1 \cdot x}{\sqrt{y} \cdot \sqrt{y}}}\right) \]

      9. *-un-lft-identity 14.7%

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

      10. add-sqr-sqrt 34.4%

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

   10. Applied egg-rr 34.4%

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

   if 3.2e23 < y

   1. Initial program 47.6%

      \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 65.0%

      \[\leadsto \color{blue}{0.5 \cdot y} \]
   4. Step-by-step derivation

      1. *-commutative 65.0%

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

   5. Simplified 65.0%

      \[\leadsto \color{blue}{y \cdot 0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 41.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.2 \cdot 10^{+23}:\\ \;\;\;\;0.5 \cdot \left(x \cdot \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 34.3% accurate, 5.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ 0.5 \cdot y \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m y z) :precision binary64 (* 0.5 y))

C

x_m = fabs(x);
double code(double x_m, double y, double z) {
	return 0.5 * y;
}

Fortran

x_m = abs(x)
real(8) function code(x_m, y, z)
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = 0.5d0 * y
end function

Java

x_m = Math.abs(x);
public static double code(double x_m, double y, double z) {
	return 0.5 * y;
}

Python

x_m = math.fabs(x)
def code(x_m, y, z):
	return 0.5 * y

Julia

x_m = abs(x)
function code(x_m, y, z)
	return Float64(0.5 * y)
end

MATLAB

x_m = abs(x);
function tmp = code(x_m, y, z)
	tmp = 0.5 * y;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_, y_, z_] := N[(0.5 * y), $MachinePrecision]

Derivation

1. Initial program 68.8%

   \[\frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
2. Add Preprocessing

3. Taylor expanded in y around inf 37.1%

   \[\leadsto \color{blue}{0.5 \cdot y} \]
4. Step-by-step derivation

   1. *-commutative 37.1%

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

5. Simplified 37.1%

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

6. Final simplification 37.1%

   \[\leadsto 0.5 \cdot y \]
7. Add Preprocessing

Developer target: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ y \cdot 0.5 - \left(\frac{0.5}{y} \cdot \left(z + x\right)\right) \cdot \left(z - x\right) \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (- (* y 0.5) (* (* (/ 0.5 y) (+ z x)) (- z x))))

C

double code(double x, double y, double z) {
	return (y * 0.5) - (((0.5 / y) * (z + x)) * (z - x));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (y * 0.5d0) - (((0.5d0 / y) * (z + x)) * (z - x))
end function

Java

public static double code(double x, double y, double z) {
	return (y * 0.5) - (((0.5 / y) * (z + x)) * (z - x));
}

Python

def code(x, y, z):
	return (y * 0.5) - (((0.5 / y) * (z + x)) * (z - x))

Julia

function code(x, y, z)
	return Float64(Float64(y * 0.5) - Float64(Float64(Float64(0.5 / y) * Float64(z + x)) * Float64(z - x)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (y * 0.5) - (((0.5 / y) * (z + x)) * (z - x));
end

Wolfram

code[x_, y_, z_] := N[(N[(y * 0.5), $MachinePrecision] - N[(N[(N[(0.5 / y), $MachinePrecision] * N[(z + x), $MachinePrecision]), $MachinePrecision] * N[(z - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024073 
(FPCore (x y z)
  :name "Diagrams.TwoD.Apollonian:initialConfig from diagrams-contrib-1.3.0.5, A"
  :precision binary64

  :alt
  (- (* y 0.5) (* (* (/ 0.5 y) (+ z x)) (- z x)))

  (/ (- (+ (* x x) (* y y)) (* z z)) (* y 2.0)))