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

Average Accuracy: 55.5% → 99.8%

Time: 13.9s

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	55.5%
Target	99.7%
Herbie	99.8%

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

Derivation

1. Initial program 55.5%

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

2. Simplified 99.8%

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

   [Start] 55.5	\[ \frac{\left(x \cdot x + y \cdot y\right) - z \cdot z}{y \cdot 2} \]
   sub-neg [=>] 55.5	\[ \frac{\color{blue}{\left(x \cdot x + y \cdot y\right) + \left(-z \cdot z\right)}}{y \cdot 2} \]
   +-commutative [=>] 55.5	\[ \frac{\color{blue}{\left(-z \cdot z\right) + \left(x \cdot x + y \cdot y\right)}}{y \cdot 2} \]
   neg-sub0 [=>] 55.5	\[ \frac{\color{blue}{\left(0 - z \cdot z\right)} + \left(x \cdot x + y \cdot y\right)}{y \cdot 2} \]
   associate-+l- [=>] 55.5	\[ \frac{\color{blue}{0 - \left(z \cdot z - \left(x \cdot x + y \cdot y\right)\right)}}{y \cdot 2} \]
   sub0-neg [=>] 55.5	\[ \frac{\color{blue}{-\left(z \cdot z - \left(x \cdot x + y \cdot y\right)\right)}}{y \cdot 2} \]
   neg-mul-1 [=>] 55.5	\[ \frac{\color{blue}{-1 \cdot \left(z \cdot z - \left(x \cdot x + y \cdot y\right)\right)}}{y \cdot 2} \]
   *-commutative [=>] 55.5	\[ \frac{\color{blue}{\left(z \cdot z - \left(x \cdot x + y \cdot y\right)\right) \cdot -1}}{y \cdot 2} \]
   times-frac [=>] 55.5	\[ \color{blue}{\frac{z \cdot z - \left(x \cdot x + y \cdot y\right)}{y} \cdot \frac{-1}{2}} \]

3. Final simplification 99.8%

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

Alternatives

Alternative 1
Accuracy	75.3%
Cost	2137

\[\begin{array}{l} t_0 := \left(x \cdot x\right) \cdot \frac{0.5}{y}\\ t_1 := -0.5 \cdot \left(z \cdot \frac{z}{y} - y\right)\\ \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{+16}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \cdot x \leq 5 \cdot 10^{+36}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \cdot x \leq 10^{+89}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{+136}:\\ \;\;\;\;x \cdot \frac{x}{y \cdot 2}\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{+203} \lor \neg \left(x \cdot x \leq 4 \cdot 10^{+263}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Accuracy	75.3%
Cost	2136

\[\begin{array}{l} t_0 := \left(x \cdot x\right) \cdot \frac{0.5}{y}\\ t_1 := -0.5 \cdot \left(\frac{z}{\frac{y}{z}} - y\right)\\ \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{+16}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \cdot x \leq 5 \cdot 10^{+36}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \cdot x \leq 10^{+89}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{+136}:\\ \;\;\;\;x \cdot \frac{x}{y \cdot 2}\\ \mathbf{elif}\;x \cdot x \leq 2 \cdot 10^{+203}:\\ \;\;\;\;-0.5 \cdot \left(z \cdot \frac{z}{y} - y\right)\\ \mathbf{elif}\;x \cdot x \leq 4 \cdot 10^{+263}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 3
Accuracy	87.4%
Cost	1357

\[\begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{-251}:\\ \;\;\;\;-0.5 \cdot \left(\frac{z}{\frac{y}{z}} - y\right)\\ \mathbf{elif}\;x \cdot x \leq 5 \cdot 10^{-90} \lor \neg \left(x \cdot x \leq 2 \cdot 10^{+16}\right):\\ \;\;\;\;0.5 \cdot \left(y + x \cdot \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(z \cdot \frac{z}{y} - y\right)\\ \end{array} \]

Alternative 4
Accuracy	63.3%
Cost	976

\[\begin{array}{l} t_0 := x \cdot \frac{x}{y \cdot 2}\\ \mathbf{if}\;y \leq -5.7 \cdot 10^{-7}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;y \leq -1.5 \cdot 10^{-138}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -5.2 \cdot 10^{-186}:\\ \;\;\;\;z \cdot \frac{-0.5}{\frac{y}{z}}\\ \mathbf{elif}\;y \leq 7 \cdot 10^{-106}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]

Alternative 5
Accuracy	63.4%
Cost	976

\[\begin{array}{l} \mathbf{if}\;y \leq -1.1 \cdot 10^{-7}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;y \leq -6.2 \cdot 10^{-139}:\\ \;\;\;\;x \cdot \frac{x}{y \cdot 2}\\ \mathbf{elif}\;y \leq -9.2 \cdot 10^{-186}:\\ \;\;\;\;z \cdot \frac{-0.5}{\frac{y}{z}}\\ \mathbf{elif}\;y \leq 7.8 \cdot 10^{-106}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]

Alternative 6
Accuracy	63.2%
Cost	976

\[\begin{array}{l} \mathbf{if}\;y \leq -0.055:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;y \leq -7 \cdot 10^{-138}:\\ \;\;\;\;x \cdot \frac{x}{y \cdot 2}\\ \mathbf{elif}\;y \leq -2.15 \cdot 10^{-182}:\\ \;\;\;\;-0.5 \cdot \frac{z \cdot z}{y}\\ \mathbf{elif}\;y \leq 5.4 \cdot 10^{-106}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]

Alternative 7
Accuracy	63.0%
Cost	844

\[\begin{array}{l} \mathbf{if}\;y \leq -1.1 \cdot 10^{-7}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;y \leq -1.65 \cdot 10^{-138}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \frac{0.5}{y}\\ \mathbf{elif}\;y \leq 8.1 \cdot 10^{-106}:\\ \;\;\;\;-0.5 \cdot \left(z \cdot \frac{z}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]

Alternative 8
Accuracy	99.8%
Cost	832

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

Alternative 9
Accuracy	63.3%
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -7.6 \cdot 10^{-78} \lor \neg \left(y \leq 6.7 \cdot 10^{-106}\right):\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(z \cdot \frac{-0.5}{y}\right)\\ \end{array} \]

Alternative 10
Accuracy	63.3%
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -8 \cdot 10^{-77} \lor \neg \left(y \leq 7.7 \cdot 10^{-106}\right):\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{-0.5}{\frac{y}{z}}\\ \end{array} \]

Alternative 11
Accuracy	63.4%
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -1.75 \cdot 10^{-78} \lor \neg \left(y \leq 8.6 \cdot 10^{-106}\right):\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(z \cdot \frac{z}{y}\right)\\ \end{array} \]

Alternative 12
Accuracy	56.9%
Cost	192

\[y \cdot 0.5 \]

Reproduce

herbie shell --seed 2023130 
(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)))