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

Average Error: 28.7 → 0.2

Time: 6.9s

Precision: binary64

Cost: 832

Math

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

↓

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

FPCore

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

↓

(FPCore (x y z) :precision binary64 (* -0.5 (- (* (+ x z) (/ (- z x) y)) 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 * (((x + z) * ((z - x) / y)) - 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) * (((x + z) * ((z - x) / y)) - 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 * (((x + z) * ((z - x) / y)) - 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 * (((x + z) * ((z - x) / y)) - 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(x + z) * Float64(Float64(z - x) / y)) - 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 * (((x + z) * ((z - x) / y)) - 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[(x + z), $MachinePrecision] * N[(N[(z - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] - y), $MachinePrecision]), $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

   \[\leadsto \color{blue}{-0.5 \cdot \left(\left(x + z\right) \cdot \frac{z - x}{y} - y\right)} \]
   Proof
   (*.f64 -1/2 (-.f64 (*.f64 (+.f64 x z) (/.f64 (-.f64 z x) y)) y)): 0 points increase in error, 0 points decrease in error
   (*.f64 (Rewrite<= metadata-eval (/.f64 -1 2)) (-.f64 (*.f64 (+.f64 x z) (/.f64 (-.f64 z x) y)) y)): 0 points increase in error, 0 points decrease in error
   (*.f64 (/.f64 -1 2) (-.f64 (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 (+.f64 x z) (-.f64 z x)) y)) y)): 63 points increase in error, 12 points decrease in error
   (*.f64 (/.f64 -1 2) (-.f64 (/.f64 (*.f64 (Rewrite=> +-commutative_binary64 (+.f64 z x)) (-.f64 z x)) y) y)): 0 points increase in error, 0 points decrease in error
   (*.f64 (/.f64 -1 2) (-.f64 (/.f64 (Rewrite<= difference-of-squares_binary64 (-.f64 (*.f64 z z) (*.f64 x x))) y) y)): 0 points increase in error, 1 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)))): 67 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. Final simplification 0.2

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

Alternatives

Alternative 1
Error	7.3
Cost	968

\[\begin{array}{l} \mathbf{if}\;z \leq -5.7 \cdot 10^{+20}:\\ \;\;\;\;-0.5 \cdot \left(\frac{z}{\frac{y}{z}} - y\right)\\ \mathbf{elif}\;z \leq 6.6 \cdot 10^{+16}:\\ \;\;\;\;0.5 \cdot \left(y + x \cdot \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\left(x + z\right) \cdot \frac{z}{y} - y\right)\\ \end{array} \]

Alternative 2
Error	14.9
Cost	840

\[\begin{array}{l} t_0 := -0.5 \cdot \left(\frac{z}{\frac{y}{z}} - y\right)\\ \mathbf{if}\;y \leq -2.25 \cdot 10^{-68}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 7.7 \cdot 10^{-90}:\\ \;\;\;\;x \cdot \frac{x}{y \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	7.3
Cost	836

\[\begin{array}{l} \mathbf{if}\;z \cdot z \leq 10^{+33}:\\ \;\;\;\;0.5 \cdot \left(y + x \cdot \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(\frac{z}{\frac{y}{z}} - y\right)\\ \end{array} \]

Alternative 4
Error	22.8
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq -1.52 \cdot 10^{-33}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{-79}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]

Alternative 5
Error	22.7
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq -2.7 \cdot 10^{-38}:\\ \;\;\;\;y \cdot 0.5\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{-79}:\\ \;\;\;\;x \cdot \frac{x}{y \cdot 2}\\ \mathbf{else}:\\ \;\;\;\;y \cdot 0.5\\ \end{array} \]

Alternative 6
Error	27.2
Cost	192

\[y \cdot 0.5 \]

Reproduce

herbie shell --seed 2022330 
(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)))