Diagrams.Backend.Cairo.Internal:setTexture from diagrams-cairo-1.3.0.3

Average Error: 12.7 → 3.2

Time: 18.4s

Precision: binary64

Cost: 448

Math

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

↓

\[\frac{x}{\frac{y}{y - z}} \]

FPCore

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

↓

(FPCore (x y z) :precision binary64 (/ x (/ y (- y z))))

C

double code(double x, double y, double z) {
	return (x * (y - z)) / y;
}

↓

double code(double x, double y, double z) {
	return x / (y / (y - 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 * (y - z)) / y
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 = x / (y / (y - z))
end function

Java

public static double code(double x, double y, double z) {
	return (x * (y - z)) / y;
}

↓

public static double code(double x, double y, double z) {
	return x / (y / (y - z));
}

Python

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

↓

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

Julia

function code(x, y, z)
	return Float64(Float64(x * Float64(y - z)) / y)
end

↓

function code(x, y, z)
	return Float64(x / Float64(y / Float64(y - z)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x * (y - z)) / y;
end

↓

function tmp = code(x, y, z)
	tmp = x / (y / (y - z));
end

Wolfram

code[x_, y_, z_] := N[(N[(x * N[(y - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]

↓

code[x_, y_, z_] := N[(x / N[(y / N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	12.7
Target	3.2
Herbie	3.2

\[\begin{array}{l} \mathbf{if}\;z < -2.060202331921739 \cdot 10^{+104}:\\ \;\;\;\;x - \frac{z \cdot x}{y}\\ \mathbf{elif}\;z < 1.6939766013828526 \cdot 10^{+213}:\\ \;\;\;\;\frac{x}{\frac{y}{y - z}}\\ \mathbf{else}:\\ \;\;\;\;\left(y - z\right) \cdot \frac{x}{y}\\ \end{array} \]

Derivation

1. Initial program 12.7

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

2. Simplified 3.4

   \[\leadsto \color{blue}{x \cdot \frac{y - z}{y}} \]
   Proof

   [Start] 12.7	\[ \frac{x \cdot \left(y - z\right)}{y} \]
   rational.json-simplify-2 [=>] 12.7	\[ \frac{\color{blue}{\left(y - z\right) \cdot x}}{y} \]
   rational.json-simplify-49 [=>] 3.4	\[ \color{blue}{x \cdot \frac{y - z}{y}} \]

3. Applied egg-rr 3.2

   \[\leadsto \color{blue}{\frac{x}{\frac{y}{y - z}}} \]

4. Final simplification 3.2

   \[\leadsto \frac{x}{\frac{y}{y - z}} \]

Alternatives

Alternative 1
Error	20.2
Cost	912

\[\begin{array}{l} t_0 := x \cdot \left(-\frac{z}{y}\right)\\ \mathbf{if}\;y \leq -7 \cdot 10^{-97}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{-169}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{-141}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{-39}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 2
Error	19.0
Cost	912

\[\begin{array}{l} t_0 := z \cdot \left(-\frac{x}{y}\right)\\ \mathbf{if}\;y \leq -1.15 \cdot 10^{-97}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.18 \cdot 10^{-179}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 3 \cdot 10^{-141}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.22 \cdot 10^{-39}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 3
Error	18.9
Cost	912

\[\begin{array}{l} t_0 := \frac{-z}{\frac{y}{x}}\\ \mathbf{if}\;y \leq -7.6 \cdot 10^{-98}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{-182}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.85 \cdot 10^{-144}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 9 \cdot 10^{-39}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4
Error	3.4
Cost	448

\[x \cdot \left(1 - \frac{z}{y}\right) \]

Alternative 5
Error	25.2
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023064 
(FPCore (x y z)
  :name "Diagrams.Backend.Cairo.Internal:setTexture from diagrams-cairo-1.3.0.3"
  :precision binary64

  :herbie-target
  (if (< z -2.060202331921739e+104) (- x (/ (* z x) y)) (if (< z 1.6939766013828526e+213) (/ x (/ y (- y z))) (* (- y z) (/ x y))))

  (/ (* x (- y z)) y))