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

Average Error: 28.3 → 0.1

Time: 9.0s

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	28.3
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 28.3

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

2. Simplified 12.4

   \[\leadsto \color{blue}{-0.5 \cdot \left(\frac{\mathsf{fma}\left(z, z, x \cdot \left(-x\right)\right)}{y} - y\right)} \]
   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)))): 65 points increase in error, 1 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))): 2 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 12.4

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

4. Simplified 0.1

   \[\leadsto -0.5 \cdot \color{blue}{\left(\frac{z - x}{y} \cdot \left(x + z\right) - y\right)} \]
   Proof
   (-.f64 (*.f64 (/.f64 (-.f64 z x) y) (+.f64 x z)) y): 0 points increase in error, 0 points decrease in error
   (-.f64 (*.f64 (/.f64 (-.f64 z x) y) (Rewrite<= +-commutative_binary64 (+.f64 z x))) y): 0 points increase in error, 0 points decrease in error
   (-.f64 (Rewrite<= associate-/r/_binary64 (/.f64 (-.f64 z x) (/.f64 y (+.f64 z x)))) y): 25 points increase in error, 21 points decrease in error
   (-.f64 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (-.f64 z x) (+.f64 z x)) y)) y): 67 points increase in error, 20 points decrease in error
   (-.f64 (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 (+.f64 z x) (-.f64 z x))) y) y): 0 points increase in error, 0 points decrease in error
   (-.f64 (/.f64 (Rewrite<= difference-of-squares_binary64 (-.f64 (*.f64 z z) (*.f64 x x))) y) y): 0 points increase in error, 0 points decrease in error
   (-.f64 (/.f64 (-.f64 (Rewrite<= unpow2_binary64 (pow.f64 z 2)) (*.f64 x x)) y) y): 0 points increase in error, 0 points decrease in error
   (-.f64 (/.f64 (-.f64 (pow.f64 z 2) (Rewrite<= unpow2_binary64 (pow.f64 x 2))) y) y): 0 points increase in error, 0 points decrease in error
   (-.f64 (Rewrite=> div-sub_binary64 (-.f64 (/.f64 (pow.f64 z 2) y) (/.f64 (pow.f64 x 2) y))) y): 0 points increase in error, 1 points decrease in error
   (-.f64 (Rewrite<= unsub-neg_binary64 (+.f64 (/.f64 (pow.f64 z 2) y) (neg.f64 (/.f64 (pow.f64 x 2) y)))) y): 0 points increase in error, 0 points decrease in error
   (-.f64 (+.f64 (/.f64 (pow.f64 z 2) y) (Rewrite<= mul-1-neg_binary64 (*.f64 -1 (/.f64 (pow.f64 x 2) y)))) y): 0 points increase in error, 0 points decrease in error
   (-.f64 (Rewrite<= +-commutative_binary64 (+.f64 (*.f64 -1 (/.f64 (pow.f64 x 2) y)) (/.f64 (pow.f64 z 2) y))) y): 0 points increase in error, 0 points decrease in error

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	23.0
Cost	976

\[\begin{array}{l} t_0 := 0.5 \cdot \frac{x}{\frac{y}{x}}\\ \mathbf{if}\;y \leq -3.6 \cdot 10^{-65}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{-260}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{-138}:\\ \;\;\;\;-0.5 \cdot \left(z \cdot \frac{z}{y}\right)\\ \mathbf{elif}\;y \leq 10^{-47}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]

Alternative 2
Error	6.5
Cost	968

\[\begin{array}{l} \mathbf{if}\;z \leq -3.5152194781512994 \cdot 10^{-20}:\\ \;\;\;\;-0.5 \cdot \left(\frac{z}{\frac{y}{z}} - y\right)\\ \mathbf{elif}\;z \leq 3.5847376780375335 \cdot 10^{-89}:\\ \;\;\;\;-0.5 \cdot \left(\frac{z - x}{\frac{y}{x}} - y\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(z \cdot \frac{z}{y} - y\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022291 
(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)))