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

Average Error: 28.7 → 0.2

Time: 9.0s

Precision: binary64

Cost: 832

Math

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

↓

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

FPCore

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

↓

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

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

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

Python

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

↓

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

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

Target

Original	28.7
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 28.7

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

2. Simplified 0.2

   \[\leadsto \color{blue}{\left(\frac{z - x}{\frac{y}{x + z}} - y\right) \cdot -0.5} \]
   Proof
   (*.f64 (-.f64 (/.f64 (-.f64 z x) (/.f64 y (+.f64 x z))) y) -1/2): 0 points increase in error, 0 points decrease in error
   (*.f64 (-.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (-.f64 z x) (+.f64 x z)) y)) y) -1/2): 7 points increase in error, 0 points decrease in error
   (*.f64 (-.f64 (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 (+.f64 x z) (-.f64 z x))) y) y) -1/2): 0 points increase in error, 19 points decrease in error
   (*.f64 (-.f64 (/.f64 (*.f64 (Rewrite=> +-commutative_binary64 (+.f64 z x)) (-.f64 z x)) y) y) -1/2): 18 points increase in error, 0 points decrease in error
   (*.f64 (-.f64 (/.f64 (Rewrite<= difference-of-squares_binary64 (-.f64 (*.f64 z z) (*.f64 x x))) y) y) -1/2): 0 points increase in error, 18 points decrease in error
   (*.f64 (-.f64 (/.f64 (-.f64 (*.f64 z z) (*.f64 x x)) y) (Rewrite<= /-rgt-identity_binary64 (/.f64 y 1))) -1/2): 12 points increase in error, 0 points decrease in error
   (*.f64 (-.f64 (/.f64 (-.f64 (*.f64 z z) (*.f64 x x)) y) (/.f64 y (Rewrite<= *-inverses_binary64 (/.f64 y y)))) -1/2): 19 points increase in error, 0 points decrease in error
   (*.f64 (-.f64 (/.f64 (-.f64 (*.f64 z z) (*.f64 x x)) y) (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 y y) y))) -1/2): 0 points increase in error, 19 points decrease in error
   (*.f64 (Rewrite<= div-sub_binary64 (/.f64 (-.f64 (-.f64 (*.f64 z z) (*.f64 x x)) (*.f64 y y)) y)) -1/2): 12 points increase in error, 0 points decrease in error
   (*.f64 (/.f64 (Rewrite<= associate--r+_binary64 (-.f64 (*.f64 z z) (+.f64 (*.f64 x x) (*.f64 y y)))) y) -1/2): 1 points increase in error, 12 points decrease in error
   (*.f64 (/.f64 (-.f64 (*.f64 z z) (+.f64 (*.f64 x x) (*.f64 y y))) y) (Rewrite<= metadata-eval (/.f64 -1 2))): 12 points increase in error, 1 points decrease in error
   (Rewrite<= times-frac_binary64 (/.f64 (*.f64 (-.f64 (*.f64 z z) (+.f64 (*.f64 x x) (*.f64 y y))) -1) (*.f64 y 2))): 1 points increase in error, 12 points decrease in error
   (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 -1 (-.f64 (*.f64 z z) (+.f64 (*.f64 x x) (*.f64 y y))))) (*.f64 y 2)): 0 points increase in error, 1 points decrease in error
   (/.f64 (Rewrite<= neg-mul-1_binary64 (neg.f64 (-.f64 (*.f64 z z) (+.f64 (*.f64 x x) (*.f64 y y))))) (*.f64 y 2)): 12 points increase in error, 0 points decrease in error
   (/.f64 (Rewrite<= sub0-neg_binary64 (-.f64 0 (-.f64 (*.f64 z z) (+.f64 (*.f64 x x) (*.f64 y y))))) (*.f64 y 2)): 0 points increase in error, 12 points decrease in error
   (/.f64 (Rewrite<= associate-+l-_binary64 (+.f64 (-.f64 0 (*.f64 z z)) (+.f64 (*.f64 x x) (*.f64 y y)))) (*.f64 y 2)): 0 points increase in error, 0 points decrease in error
   (/.f64 (+.f64 (Rewrite<= neg-sub0_binary64 (neg.f64 (*.f64 z z))) (+.f64 (*.f64 x x) (*.f64 y y))) (*.f64 y 2)): 0 points increase in error, 0 points decrease in error
   (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (neg.f64 (*.f64 z z)))) (*.f64 y 2)): 12 points increase in error, 0 points decrease in error
   (/.f64 (Rewrite<= sub-neg_binary64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z))) (*.f64 y 2)): 0 points increase in error, 12 points decrease in error

3. Applied egg-rr 0.2

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

4. Final simplification 0.2

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

Alternatives

Alternative 1
Error	23.2
Cost	976

\[\begin{array}{l} t_0 := z \cdot \left(z \cdot \frac{-0.5}{y}\right)\\ \mathbf{if}\;y \leq -2.25 \cdot 10^{-93}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;y \leq -1.9 \cdot 10^{-129}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{-308}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \mathbf{elif}\;y \leq 0.11:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]

Alternative 2
Error	23.2
Cost	976

\[\begin{array}{l} t_0 := -0.5 \cdot \left(z \cdot \frac{z}{y}\right)\\ \mathbf{if}\;y \leq -2.5 \cdot 10^{-92}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;y \leq -2.3 \cdot 10^{-126}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -2.7 \cdot 10^{-308}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \mathbf{elif}\;y \leq 0.118:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]

Alternative 3
Error	23.2
Cost	976

\[\begin{array}{l} t_0 := -0.5 \cdot \left(z \cdot \frac{z}{y}\right)\\ \mathbf{if}\;y \leq -7.8 \cdot 10^{-94}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;y \leq -9.5 \cdot 10^{-130}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -6.5 \cdot 10^{-308}:\\ \;\;\;\;-0.5 \cdot \left(\left(z - x\right) \cdot \frac{x}{y}\right)\\ \mathbf{elif}\;y \leq 0.11:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]

Alternative 4
Error	13.8
Cost	841

\[\begin{array}{l} \mathbf{if}\;y \leq -3.7 \cdot 10^{-128} \lor \neg \left(y \leq 3.5 \cdot 10^{-308}\right):\\ \;\;\;\;-0.5 \cdot \left(\frac{z}{\frac{y}{z}} - y\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\left(z - x\right) \cdot \frac{x}{y}\right)\\ \end{array} \]

Alternative 5
Error	7.6
Cost	841

\[\begin{array}{l} \mathbf{if}\;x \leq -3.1 \cdot 10^{-79} \lor \neg \left(x \leq 7.5 \cdot 10^{+62}\right):\\ \;\;\;\;0.5 \cdot \left(y + x \cdot \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\frac{z}{\frac{y}{z}} - y\right)\\ \end{array} \]

Alternative 6
Error	23.2
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -2.8 \cdot 10^{-92} \lor \neg \left(y \leq 3.8 \cdot 10^{-114}\right):\\ \;\;\;\;y \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(x \cdot \frac{0.5}{y}\right)\\ \end{array} \]

Alternative 7
Error	27.3
Cost	192

\[y \cdot 0.5 \]

Reproduce

herbie shell --seed 2022343 
(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)))