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

Average Error: 27.8 → 0.2

Time: 9.9s

Precision: binary64

Cost: 832

Math

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

↓

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

FPCore

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

↓

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

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

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

Python

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

↓

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

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

Target

Original	27.8
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 27.8

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

2. Taylor expanded in y around 0 12.1

   \[\leadsto \color{blue}{0.5 \cdot y + 0.5 \cdot \frac{{x}^{2} - {z}^{2}}{y}} \]

3. Simplified 0.2

   \[\leadsto \color{blue}{0.5 \cdot \left(y + \frac{z + x}{\frac{y}{x - z}}\right)} \]
   Proof
   (*.f64 1/2 (+.f64 y (/.f64 (+.f64 z x) (/.f64 y (-.f64 x z))))): 0 points increase in error, 0 points decrease in error
   (*.f64 1/2 (+.f64 y (/.f64 (Rewrite<= +-commutative_binary64 (+.f64 x z)) (/.f64 y (-.f64 x z))))): 0 points increase in error, 0 points decrease in error
   (*.f64 1/2 (+.f64 y (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (+.f64 x z) (-.f64 x z)) y)))): 63 points increase in error, 8 points decrease in error
   (*.f64 1/2 (+.f64 y (/.f64 (Rewrite<= difference-of-squares_binary64 (-.f64 (*.f64 x x) (*.f64 z z))) y))): 0 points increase in error, 0 points decrease in error
   (*.f64 1/2 (+.f64 y (/.f64 (-.f64 (Rewrite<= unpow2_binary64 (pow.f64 x 2)) (*.f64 z z)) y))): 0 points increase in error, 0 points decrease in error
   (*.f64 1/2 (+.f64 y (/.f64 (-.f64 (pow.f64 x 2) (Rewrite<= unpow2_binary64 (pow.f64 z 2))) y))): 0 points increase in error, 0 points decrease in error
   (Rewrite<= distribute-lft-out_binary64 (+.f64 (*.f64 1/2 y) (*.f64 1/2 (/.f64 (-.f64 (pow.f64 x 2) (pow.f64 z 2)) y)))): 0 points increase in error, 0 points decrease in error

4. Final simplification 0.2

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

Alternatives

Alternative 1
Error	22.9
Cost	1240

\[\begin{array}{l} t_0 := -0.5 \cdot \left(z \cdot \frac{z}{y}\right)\\ \mathbf{if}\;y \leq -3.5 \cdot 10^{-43}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;y \leq -1 \cdot 10^{-63}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -2.3 \cdot 10^{-164}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{-76}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 4 \cdot 10^{+26}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;y \leq 7.8 \cdot 10^{+39}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \]

Alternative 2
Error	23.1
Cost	1240

\[\begin{array}{l} t_0 := -0.5 \cdot \left(z \cdot \frac{z}{y}\right)\\ \mathbf{if}\;y \leq -1.85 \cdot 10^{-45}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;y \leq -1.2 \cdot 10^{-62}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -9 \cdot 10^{-152}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{-56}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{y}{x}}\\ \mathbf{elif}\;y \leq 4 \cdot 10^{+26}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;y \leq 7.8 \cdot 10^{+39}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \]

Alternative 3
Error	23.1
Cost	1240

\[\begin{array}{l} \mathbf{if}\;y \leq -1.7 \cdot 10^{-43}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;y \leq -9.5 \cdot 10^{-63}:\\ \;\;\;\;\frac{-0.5}{y} \cdot \left(z \cdot z\right)\\ \mathbf{elif}\;y \leq -9 \cdot 10^{-152}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{-53}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{y}{x}}\\ \mathbf{elif}\;y \leq 4 \cdot 10^{+26}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;y \leq 7.8 \cdot 10^{+39}:\\ \;\;\;\;-0.5 \cdot \left(z \cdot \frac{z}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \]

Alternative 4
Error	23.1
Cost	1240

\[\begin{array}{l} \mathbf{if}\;y \leq -1.18 \cdot 10^{-44}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;y \leq -9.5 \cdot 10^{-63}:\\ \;\;\;\;\frac{-0.5}{y} \cdot \left(z \cdot z\right)\\ \mathbf{elif}\;y \leq -7.5 \cdot 10^{-152}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{-56}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{y}{x}}\\ \mathbf{elif}\;y \leq 4 \cdot 10^{+26}:\\ \;\;\;\;0.5 \cdot y\\ \mathbf{elif}\;y \leq 7.8 \cdot 10^{+39}:\\ \;\;\;\;\frac{z \cdot -0.5}{\frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot y\\ \end{array} \]

Alternative 5
Error	6.6
Cost	840

\[\begin{array}{l} \mathbf{if}\;z \leq -6.3 \cdot 10^{-111}:\\ \;\;\;\;-0.5 \cdot \left(\frac{z}{\frac{y}{z}} - y\right)\\ \mathbf{elif}\;z \leq 1.45 \cdot 10^{-71}:\\ \;\;\;\;0.5 \cdot \left(y + x \cdot \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(z \cdot \frac{z}{y} - y\right)\\ \end{array} \]

Alternative 6
Error	6.7
Cost	840

\[\begin{array}{l} \mathbf{if}\;z \leq -5.8 \cdot 10^{-111}:\\ \;\;\;\;-0.5 \cdot \left(\frac{z}{\frac{y}{z}} - y\right)\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{-71}:\\ \;\;\;\;0.5 \cdot \left(y + \frac{x}{\frac{y}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(z \cdot \frac{z}{y} - y\right)\\ \end{array} \]

Alternative 7
Error	6.6
Cost	840

\[\begin{array}{l} \mathbf{if}\;z \leq -5.5 \cdot 10^{-108}:\\ \;\;\;\;-0.5 \cdot \left(\left(z + x\right) \cdot \frac{z}{y} - y\right)\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{-71}:\\ \;\;\;\;0.5 \cdot \left(y + \frac{x}{\frac{y}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(z \cdot \frac{z}{y} - y\right)\\ \end{array} \]

Alternative 8
Error	15.1
Cost	836

\[\begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{+182}:\\ \;\;\;\;-0.5 \cdot \left(z \cdot \frac{z}{y} - y\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{y}{x}}\\ \end{array} \]

Alternative 9
Error	15.1
Cost	836

\[\begin{array}{l} \mathbf{if}\;x \cdot x \leq 10^{+182}:\\ \;\;\;\;-0.5 \cdot \left(\frac{z}{\frac{y}{z}} - y\right)\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{x}{\frac{y}{x}}\\ \end{array} \]

Alternative 10
Error	0.2
Cost	832

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

Alternative 11
Error	26.7
Cost	192

\[0.5 \cdot y \]

Reproduce

herbie shell --seed 2022329 
(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)))