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

Average Error: 27.7 → 0.1

Time: 11.9s

Precision: binary64

Cost: 832

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

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

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

↓

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

Java

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

↓

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

Python

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

↓

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

Julia

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

↓

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

MATLAB

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

↓

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

↓

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

Target

Original	27.7
Target	0.1
Herbie	0.1

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

Derivation

1. Initial program 27.7

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

2. Simplified 0.1

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

   [Start] 27.7	\[ \frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   sub-neg [=>] 27.7	\[ \frac{\color{blue}{\left(x \cdot x + y \cdot y\right) + \left(-z \cdot z\right)}}{y \cdot 2} \]
   +-commutative [=>] 27.7	\[ \frac{\color{blue}{\left(-z \cdot z\right) + \left(x \cdot x + y \cdot y\right)}}{y \cdot 2} \]
   neg-sub0 [=>] 27.7	\[ \frac{\color{blue}{\left(0 - z \cdot z\right)} + \left(x \cdot x + y \cdot y\right)}{y \cdot 2} \]
   associate-+l- [=>] 27.7	\[ \frac{\color{blue}{0 - \left(z \cdot z - \left(x \cdot x + y \cdot y\right)\right)}}{y \cdot 2} \]
   sub0-neg [=>] 27.7	\[ \frac{\color{blue}{-\left(z \cdot z - \left(x \cdot x + y \cdot y\right)\right)}}{y \cdot 2} \]
   neg-mul-1 [=>] 27.7	\[ \frac{\color{blue}{-1 \cdot \left(z \cdot z - \left(x \cdot x + y \cdot y\right)\right)}}{y \cdot 2} \]
   *-commutative [=>] 27.7	\[ \frac{\color{blue}{\left(z \cdot z - \left(x \cdot x + y \cdot y\right)\right) \cdot -1}}{y \cdot 2} \]
   times-frac [=>] 27.7	\[ \color{blue}{\frac{z \cdot z - \left(x \cdot x + y \cdot y\right)}{y} \cdot \frac{-1}{2}} \]

3. Final simplification 0.1

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

Alternatives

Alternative 1
Error	23.5
Cost	1108

\[\begin{array}{l} t_0 := 0.5 \cdot \frac{x}{\frac{y}{x}}\\ t_1 := z \cdot \left(z \cdot \frac{-0.5}{y}\right)\\ \mathbf{if}\;y \leq -1.6 \cdot 10^{+39}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;y \leq -2.3 \cdot 10^{-106}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{-120}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{-73}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 5.9 \cdot 10^{-58}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]

Alternative 2
Error	23.4
Cost	1108

\[\begin{array}{l} t_0 := 0.5 \cdot \frac{x}{\frac{y}{x}}\\ t_1 := -0.5 \cdot \left(z \cdot \frac{z}{y}\right)\\ \mathbf{if}\;y \leq -1.6 \cdot 10^{+39}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;y \leq -3.5 \cdot 10^{-100}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{-119}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 6.1 \cdot 10^{-74}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{-57}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]

Alternative 3
Error	23.5
Cost	1108

\[\begin{array}{l} t_0 := 0.5 \cdot \frac{x}{\frac{y}{x}}\\ t_1 := -0.5 \cdot \frac{z}{\frac{y}{z}}\\ \mathbf{if}\;y \leq -2.2 \cdot 10^{+39}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;y \leq -1.15 \cdot 10^{-102}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{-119}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 3.4 \cdot 10^{-76}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{-58}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]

Alternative 4
Error	23.5
Cost	1108

\[\begin{array}{l} t_0 := -0.5 \cdot \frac{z}{\frac{y}{z}}\\ \mathbf{if}\;y \leq -1.6 \cdot 10^{+39}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;y \leq -3.75 \cdot 10^{-100}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 2.55 \cdot 10^{-117}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{y}{x}}\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{-74}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{-59}:\\ \;\;\;\;\frac{x \cdot x}{y \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]

Alternative 5
Error	14.9
Cost	1106

\[\begin{array}{l} \mathbf{if}\;z \leq -1.4 \cdot 10^{-134} \lor \neg \left(z \leq -7.8 \cdot 10^{-173}\right) \land \left(z \leq -2.4 \cdot 10^{-201} \lor \neg \left(z \leq -8 \cdot 10^{-218}\right)\right):\\ \;\;\;\;-0.5 \cdot \left(\frac{z}{\frac{y}{z}} - y\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{y}{x}}\\ \end{array} \]

Alternative 6
Error	6.9
Cost	841

\[\begin{array}{l} \mathbf{if}\;z \leq -6.2 \cdot 10^{-116} \lor \neg \left(z \leq 4.4 \cdot 10^{-133}\right):\\ \;\;\;\;-0.5 \cdot \left(\frac{z}{\frac{y}{z}} - y\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y + x \cdot \frac{x}{y}\right)\\ \end{array} \]

Alternative 7
Error	22.9
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq -1.2 \cdot 10^{-32}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{-58}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]

Alternative 8
Error	27.2
Cost	192

\[y \cdot 0.5 \]

Reproduce

herbie shell --seed 2023018 
(FPCore (x y z)
  :name "Diagrams.TwoD.Apollonian:initialConfig from diagrams-contrib-1.3.0.5, A"
  :precision binary64

  :herbie-target
  (- (* y 0.5) (* (* (/ 0.5 y) (+ z x)) (- z x)))

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