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

Average Error: 29.0 → 0.2

Time: 1.7min

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	29.0
Target	0.2
Herbie	0.2

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

Derivation

1. Initial program 29.0

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

2. Simplified 29.1

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

3. Taylor expanded in y around 0 12.6

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

4. Simplified 12.6

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

5. Applied egg-rr 0.2

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

Alternatives

Alternative 1
Error	22.9
Cost	1108

\[\begin{array}{l} t_0 := \frac{0.5 \cdot x}{y} \cdot x\\ t_1 := \frac{-0.5 \cdot z}{y} \cdot z\\ \mathbf{if}\;y \leq -290000:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;y \leq -1.3 \cdot 10^{-64}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -7.5 \cdot 10^{-203}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.65 \cdot 10^{-192}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{-87}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]

Alternative 2
Error	22.9
Cost	1108

\[\begin{array}{l} t_0 := \frac{0.5 \cdot x}{y} \cdot x\\ \mathbf{if}\;y \leq -290000:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;y \leq -2 \cdot 10^{-68}:\\ \;\;\;\;\frac{0.5}{y} \cdot \left(-z \cdot z\right)\\ \mathbf{elif}\;y \leq -2.45 \cdot 10^{-183}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{-190}:\\ \;\;\;\;\frac{-0.5 \cdot z}{y} \cdot z\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{-86}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]

Alternative 3
Error	15.3
Cost	1104

\[\begin{array}{l} t_0 := 0.5 \cdot \left(y + \frac{x}{y} \cdot x\right)\\ \mathbf{if}\;y \leq -3300000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -2.4 \cdot 10^{-63}:\\ \;\;\;\;\frac{0.5}{y} \cdot \left(-z \cdot z\right)\\ \mathbf{elif}\;y \leq -2.75 \cdot 10^{-218}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.95 \cdot 10^{-280}:\\ \;\;\;\;\frac{-0.5 \cdot z}{y} \cdot z\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Error	22.9
Cost	976

\[\begin{array}{l} t_0 := \frac{-0.5 \cdot z}{y} \cdot z\\ \mathbf{if}\;y \leq -290000:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;y \leq -1.45 \cdot 10^{-64}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -5.8 \cdot 10^{-193}:\\ \;\;\;\;\frac{0.5}{y} \cdot \left(x \cdot x\right)\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{-98}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]

Alternative 5
Error	6.9
Cost	900

\[\begin{array}{l} \mathbf{if}\;x \cdot x \leq 2 \cdot 10^{-177}:\\ \;\;\;\;0.5 \cdot \left(y + \frac{-z}{y} \cdot z\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \left(y + \frac{x}{y} \cdot x\right)\\ \end{array} \]

Alternative 6
Error	23.4
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq -2.4 \cdot 10^{-70}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;y \leq 1.95 \cdot 10^{-97}:\\ \;\;\;\;\frac{0.5}{y} \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]

Alternative 7
Error	27.2
Cost	192

\[y \cdot 0.5 \]

Reproduce

herbie shell --seed 2023033 
(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)))