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

Average Error: 27.8 → 0.1

Time: 3.7s

Precision: binary64

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]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Target

Original	27.8
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 27.8

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

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 12.2

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

4. Simplified 0.1

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

5. Final simplification 0.1

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

Reproduce

herbie shell --seed 2022162 
(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)))