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

Average Error: 28.2 → 0.1

Time: 27.8s

Precision: binary64

Cost: 832

Math

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

↓

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

FPCore

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

↓

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

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 0.5 * (y + (((x - z) / y) * (x + z)));
}

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 = 0.5d0 * (y + (((x - z) / y) * (x + z)))
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 0.5 * (y + (((x - z) / y) * (x + z)));
}

Python

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

↓

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

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(0.5 * Float64(y + Float64(Float64(Float64(x - z) / y) * Float64(x + z))))
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 = 0.5 * (y + (((x - z) / y) * (x + z)));
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[(0.5 * N[(y + N[(N[(N[(x - z), $MachinePrecision] / y), $MachinePrecision] * N[(x + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	28.2
Target	0.2
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 28.2

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

2. Taylor expanded in y around 0 12.6

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

3. Simplified 12.6

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

4. Applied egg-rr 0.1

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

Alternatives

Alternative 1
Error	30.7
Cost	2032

\[\begin{array}{l} t_0 := \left(\frac{-0.5}{y} \cdot z\right) \cdot z\\ t_1 := \left(\frac{0.5}{y} \cdot x\right) \cdot x\\ \mathbf{if}\;z \leq -8.2 \cdot 10^{+34}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -3.8 \cdot 10^{-7}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;z \leq -5 \cdot 10^{-44}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -1.9 \cdot 10^{-207}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;z \leq -1.55 \cdot 10^{-223}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{-207}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;z \leq 9.2 \cdot 10^{-178}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.35 \cdot 10^{-150}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;z \leq 9.4 \cdot 10^{-132}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 0.0076:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;z \leq 8.2 \cdot 10^{+86}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 3.25 \cdot 10^{+209}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	30.8
Cost	2032

\[\begin{array}{l} t_0 := \left(\frac{-0.5}{y} \cdot z\right) \cdot z\\ t_1 := \left(\frac{0.5}{y} \cdot x\right) \cdot x\\ \mathbf{if}\;z \leq -3.4 \cdot 10^{+26}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -7.8 \cdot 10^{-6}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;z \leq -5 \cdot 10^{-44}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -6.8 \cdot 10^{-214}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;z \leq -1.55 \cdot 10^{-223}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.18 \cdot 10^{-208}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;z \leq 1.15 \cdot 10^{-177}:\\ \;\;\;\;\frac{x \cdot 0.5}{y} \cdot x\\ \mathbf{elif}\;z \leq 2.35 \cdot 10^{-150}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{-131}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 0.058:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;z \leq 2.8 \cdot 10^{+88}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 1.08 \cdot 10^{+209}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	30.6
Cost	2032

\[\begin{array}{l} t_0 := \frac{-0.5 \cdot \left(z \cdot z\right)}{y}\\ t_1 := \left(\frac{-0.5}{y} \cdot z\right) \cdot z\\ t_2 := \left(\frac{0.5}{y} \cdot x\right) \cdot x\\ \mathbf{if}\;z \leq -2.05 \cdot 10^{+34}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -3.8 \cdot 10^{-7}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;z \leq -5 \cdot 10^{-44}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -6.8 \cdot 10^{-214}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;z \leq -1.55 \cdot 10^{-223}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{-207}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{-177}:\\ \;\;\;\;\frac{x \cdot 0.5}{y} \cdot x\\ \mathbf{elif}\;z \leq 2.35 \cdot 10^{-150}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;z \leq 9.4 \cdot 10^{-132}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 0.068:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;z \leq 2.65 \cdot 10^{+88}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{+210}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 4
Error	30.6
Cost	2032

\[\begin{array}{l} t_0 := \frac{-0.5 \cdot \left(z \cdot z\right)}{y}\\ t_1 := \left(\frac{0.5}{y} \cdot x\right) \cdot x\\ \mathbf{if}\;z \leq -4.1 \cdot 10^{+34}:\\ \;\;\;\;\frac{z \cdot 0.5}{y} \cdot \left(-z\right)\\ \mathbf{elif}\;z \leq -3.8 \cdot 10^{-7}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;z \leq -5 \cdot 10^{-44}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -6.8 \cdot 10^{-214}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;z \leq -7.5 \cdot 10^{-225}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{-207}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;z \leq 9.2 \cdot 10^{-178}:\\ \;\;\;\;\frac{x \cdot 0.5}{y} \cdot x\\ \mathbf{elif}\;z \leq 2.15 \cdot 10^{-150}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;z \leq 9.4 \cdot 10^{-132}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 0.017:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;z \leq 8.2 \cdot 10^{+86}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 5.3 \cdot 10^{+208}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(\frac{-0.5}{y} \cdot z\right) \cdot z\\ \end{array} \]

Alternative 5
Error	17.3
Cost	1616

\[\begin{array}{l} t_0 := 0.5 \cdot \left(y + \frac{x}{y} \cdot x\right)\\ \mathbf{if}\;z \cdot z \leq 5 \cdot 10^{-39}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \cdot z \leq 4 \cdot 10^{-21}:\\ \;\;\;\;\frac{-0.5 \cdot \left(z \cdot z\right)}{y}\\ \mathbf{elif}\;z \cdot z \leq 10^{+51}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \cdot z \leq 4 \cdot 10^{+284}:\\ \;\;\;\;\frac{z \cdot 0.5}{y} \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	30.2
Cost	1504

\[\begin{array}{l} t_0 := \left(\frac{-0.5}{y} \cdot z\right) \cdot z\\ \mathbf{if}\;z \leq -8.2 \cdot 10^{+34}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq -3.8 \cdot 10^{-7}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;z \leq -5 \cdot 10^{-44}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{-207}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;z \leq 4.9 \cdot 10^{-178}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \frac{0.5}{y}\\ \mathbf{elif}\;z \leq 0.068:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{+87}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 4.7 \cdot 10^{+209}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 7
Error	6.8
Cost	904

\[\begin{array}{l} t_0 := 0.5 \cdot \left(y + \frac{z}{y} \cdot \left(-z\right)\right)\\ \mathbf{if}\;z \leq -7.5 \cdot 10^{-48}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{-31}:\\ \;\;\;\;0.5 \cdot \left(y + \frac{x}{y} \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 8
Error	23.3
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{-103}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;y \leq 2.9 \cdot 10^{-53}:\\ \;\;\;\;\left(x \cdot x\right) \cdot \frac{0.5}{y}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \]

Alternative 9
Error	27.6
Cost	192

\[0.5 \cdot y \]

Reproduce

herbie shell --seed 2023010 
(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)))