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

Average Error: 28.1 → 0.2

Time: 5.8s

Precision: binary64

Math

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

↓

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

FPCore

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

↓

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

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

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

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

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

Target

Original	28.1
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 28.1

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

2. Simplified 0.2

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

3. Applied egg-rr 0.2

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

4. Applied egg-rr 0.2

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

5. Taylor expanded in x around 0 12.2

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

6. Simplified 0.2

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

7. Final simplification 0.2

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

Reproduce

herbie shell --seed 2022210 
(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)))