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

Average Error: 28.2 → 0.1

Time: 8.9s

Precision: binary64

Cost: 832

Math

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

↓

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

FPCore

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

↓

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

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

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

Python

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

↓

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

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(Float64(Float64(z - x) / Float64(y / Float64(z + x))) - y))
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 * (((z - x) / (y / (z + x))) - y);
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[(N[(N[(z - x), $MachinePrecision] / N[(y / N[(z + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - y), $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. Simplified 0.1

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

3. Taylor expanded in x around 0 12.0

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

4. Simplified 0.1

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

5. Applied egg-rr 0.1

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

6. Final simplification 0.1

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

Alternatives

Alternative 1
Error	24.3
Cost	1108

\[\begin{array}{l} t_0 := x \cdot \frac{x}{y \cdot 2}\\ t_1 := -0.5 \cdot \frac{z}{\frac{y}{z}}\\ \mathbf{if}\;y \leq -2 \cdot 10^{+85}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;y \leq -4.3 \cdot 10^{-253}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{-149}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{-62}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 1.22 \cdot 10^{-54}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]

Alternative 2
Error	24.3
Cost	1108

\[\begin{array}{l} t_0 := x \cdot \frac{x}{y \cdot 2}\\ \mathbf{if}\;y \leq -2 \cdot 10^{+85}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;y \leq -4.3 \cdot 10^{-253}:\\ \;\;\;\;-0.5 \cdot \left(z \cdot \frac{z}{y}\right)\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{-149}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{-62}:\\ \;\;\;\;-0.5 \cdot \frac{z}{\frac{y}{z}}\\ \mathbf{elif}\;y \leq 1.22 \cdot 10^{-54}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]

Alternative 3
Error	6.6
Cost	904

\[\begin{array}{l} \mathbf{if}\;z \leq -5.176083897378314 \cdot 10^{-101}:\\ \;\;\;\;-0.5 \cdot \left(z \cdot \frac{z}{y} - y\right)\\ \mathbf{elif}\;z \leq 2.649889080294774 \cdot 10^{-20}:\\ \;\;\;\;-0.5 \cdot \left(\frac{x}{\frac{y}{-x}} - y\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\frac{z}{\frac{y}{z}} - y\right)\\ \end{array} \]

Alternative 4
Error	13.4
Cost	840

\[\begin{array}{l} t_0 := -0.5 \cdot \left(z \cdot \frac{z}{y} - y\right)\\ \mathbf{if}\;y \leq -4.3 \cdot 10^{-253}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-155}:\\ \;\;\;\;x \cdot \frac{x}{y \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	13.4
Cost	840

\[\begin{array}{l} \mathbf{if}\;y \leq -4.3 \cdot 10^{-253}:\\ \;\;\;\;-0.5 \cdot \left(z \cdot \frac{z}{y} - y\right)\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-155}:\\ \;\;\;\;x \cdot \frac{x}{y \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\frac{z}{\frac{y}{z}} - y\right)\\ \end{array} \]

Alternative 6
Error	24.3
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{+85}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;y \leq 5.8 \cdot 10^{-62}:\\ \;\;\;\;-0.5 \cdot \frac{z}{\frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]

Alternative 7
Error	26.8
Cost	192

\[y \cdot 0.5 \]

Reproduce

herbie shell --seed 2022284 
(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)))