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

Average Error: 28.2 → 0.2

Time: 10.7s

Precision: binary64

Cost: 832

Math

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

↓

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

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

Target

Original	28.2
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 28.2

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

2. Simplified 12.7

   \[\leadsto \color{blue}{-0.5 \cdot \left(\frac{\mathsf{fma}\left(z, z, x \cdot \left(-x\right)\right)}{y} - y\right)} \]
   Proof
   (*.f64 -1/2 (-.f64 (/.f64 (fma.f64 z z (*.f64 x (neg.f64 x))) y) y)): 0 points increase in error, 0 points decrease in error
   (*.f64 (Rewrite<= metadata-eval (/.f64 -1 2)) (-.f64 (/.f64 (fma.f64 z z (*.f64 x (neg.f64 x))) y) y)): 0 points increase in error, 0 points decrease in error
   (*.f64 (/.f64 -1 2) (-.f64 (/.f64 (fma.f64 z z (Rewrite<= distribute-rgt-neg-in_binary64 (neg.f64 (*.f64 x x)))) y) y)): 0 points increase in error, 0 points decrease in error
   (*.f64 (/.f64 -1 2) (-.f64 (/.f64 (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 z z) (*.f64 x x))) y) y)): 0 points increase in error, 0 points decrease in error
   (*.f64 (/.f64 -1 2) (-.f64 (/.f64 (-.f64 (*.f64 z z) (*.f64 x x)) y) (Rewrite<= /-rgt-identity_binary64 (/.f64 y 1)))): 0 points increase in error, 0 points decrease in error
   (*.f64 (/.f64 -1 2) (-.f64 (/.f64 (-.f64 (*.f64 z z) (*.f64 x x)) y) (/.f64 y (Rewrite<= *-inverses_binary64 (/.f64 y y))))): 0 points increase in error, 0 points decrease in error
   (*.f64 (/.f64 -1 2) (-.f64 (/.f64 (-.f64 (*.f64 z z) (*.f64 x x)) y) (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 y y) y)))): 70 points increase in error, 2 points decrease in error
   (*.f64 (/.f64 -1 2) (Rewrite<= div-sub_binary64 (/.f64 (-.f64 (-.f64 (*.f64 z z) (*.f64 x x)) (*.f64 y y)) y))): 0 points increase in error, 1 points decrease in error
   (*.f64 (/.f64 -1 2) (/.f64 (Rewrite<= associate--r+_binary64 (-.f64 (*.f64 z z) (+.f64 (*.f64 x x) (*.f64 y y)))) y)): 0 points increase in error, 0 points decrease in error
   (Rewrite<= times-frac_binary64 (/.f64 (*.f64 -1 (-.f64 (*.f64 z z) (+.f64 (*.f64 x x) (*.f64 y y)))) (*.f64 2 y))): 0 points increase in error, 0 points decrease in error
   (/.f64 (Rewrite<= neg-mul-1_binary64 (neg.f64 (-.f64 (*.f64 z z) (+.f64 (*.f64 x x) (*.f64 y y))))) (*.f64 2 y)): 0 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 2 y)): 0 points increase in error, 0 points decrease in error
   (/.f64 (Rewrite<= associate-+l-_binary64 (+.f64 (-.f64 0 (*.f64 z z)) (+.f64 (*.f64 x x) (*.f64 y y)))) (*.f64 2 y)): 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 2 y)): 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 2 y)): 0 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 2 y)): 0 points increase in error, 0 points decrease in error
   (/.f64 (-.f64 (+.f64 (*.f64 x x) (*.f64 y y)) (*.f64 z z)) (Rewrite<= *-commutative_binary64 (*.f64 y 2))): 0 points increase in error, 0 points decrease in error

3. Taylor expanded in z around 0 12.7

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

4. Simplified 0.2

   \[\leadsto -0.5 \cdot \left(\color{blue}{\frac{z - x}{y} \cdot \left(x + z\right)} - y\right) \]
   Proof
   (*.f64 (/.f64 (-.f64 z x) y) (+.f64 x z)): 0 points increase in error, 0 points decrease in error
   (*.f64 (/.f64 (-.f64 z x) y) (Rewrite<= +-commutative_binary64 (+.f64 z x))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= associate-/r/_binary64 (/.f64 (-.f64 z x) (/.f64 y (+.f64 z x)))): 35 points increase in error, 34 points decrease in error
   (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (-.f64 z x) (+.f64 z x)) y)): 76 points increase in error, 35 points decrease in error
   (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 (+.f64 z x) (-.f64 z x))) y): 0 points increase in error, 0 points decrease in error
   (/.f64 (Rewrite<= difference-of-squares_binary64 (-.f64 (*.f64 z z) (*.f64 x x))) y): 3 points increase in error, 0 points decrease in error
   (/.f64 (-.f64 (Rewrite<= unpow2_binary64 (pow.f64 z 2)) (*.f64 x x)) y): 0 points increase in error, 0 points decrease in error
   (/.f64 (-.f64 (pow.f64 z 2) (Rewrite<= unpow2_binary64 (pow.f64 x 2))) y): 0 points increase in error, 0 points decrease in error
   (Rewrite=> div-sub_binary64 (-.f64 (/.f64 (pow.f64 z 2) y) (/.f64 (pow.f64 x 2) y))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= unsub-neg_binary64 (+.f64 (/.f64 (pow.f64 z 2) y) (neg.f64 (/.f64 (pow.f64 x 2) y)))): 0 points increase in error, 0 points decrease in error
   (+.f64 (/.f64 (pow.f64 z 2) y) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (/.f64 (pow.f64 x 2) y)))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 -1 (/.f64 (pow.f64 x 2) y)) (/.f64 (pow.f64 z 2) y))): 0 points increase in error, 0 points decrease in error

5. Final simplification 0.2

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

Alternatives

Alternative 1
Error	7.5
Cost	1356

\[\begin{array}{l} t_0 := 0.5 \cdot \left(y + x \cdot \frac{x}{y}\right)\\ \mathbf{if}\;z \cdot z \leq 2 \cdot 10^{-145}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \cdot z \leq 5 \cdot 10^{-60}:\\ \;\;\;\;-0.5 \cdot \left(\frac{z}{\frac{y}{z}} - y\right)\\ \mathbf{elif}\;z \cdot z \leq 20000000:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(z \cdot \frac{z}{y} - y\right)\\ \end{array} \]

Alternative 2
Error	24.2
Cost	976

\[\begin{array}{l} t_0 := 0.5 \cdot \frac{x \cdot x}{y}\\ \mathbf{if}\;y \leq -1.45 \cdot 10^{-19}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;y \leq -9.2 \cdot 10^{-207}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 9 \cdot 10^{-170}:\\ \;\;\;\;-0.5 \cdot \left(z \cdot \frac{z}{y}\right)\\ \mathbf{elif}\;y \leq 2.95 \cdot 10^{+35}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]

Alternative 3
Error	24.2
Cost	976

\[\begin{array}{l} \mathbf{if}\;y \leq -1.45 \cdot 10^{-19}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;y \leq -9.2 \cdot 10^{-207}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{y}{x}}\\ \mathbf{elif}\;y \leq 9 \cdot 10^{-170}:\\ \;\;\;\;-0.5 \cdot \left(z \cdot \frac{z}{y}\right)\\ \mathbf{elif}\;y \leq 2.95 \cdot 10^{+35}:\\ \;\;\;\;0.5 \cdot \frac{x \cdot x}{y}\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]

Alternative 4
Error	14.7
Cost	840

\[\begin{array}{l} t_0 := -0.5 \cdot \left(\frac{z}{\frac{y}{z}} - y\right)\\ \mathbf{if}\;y \leq -6.7 \cdot 10^{-108}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -6.8 \cdot 10^{-186}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	14.7
Cost	840

\[\begin{array}{l} t_0 := -0.5 \cdot \left(z \cdot \frac{z}{y} - y\right)\\ \mathbf{if}\;y \leq -6.7 \cdot 10^{-108}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -6.8 \cdot 10^{-186}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	23.7
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq -1.85 \cdot 10^{-87}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;y \leq 2.35 \cdot 10^{-101}:\\ \;\;\;\;-0.5 \cdot \left(z \cdot \frac{z}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]

Alternative 7
Error	27.7
Cost	192

\[y \cdot 0.5 \]

Reproduce

herbie shell --seed 2022297 
(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)))