Diagrams.Backend.Rasterific:rasterificRadialGradient from diagrams-rasterific-1.3.1.3

Average Error: 16.29% → 0.22%

Time: 6.7s

Precision: binary64

Cost: 576

Math

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

↓

\[y - \frac{y + -1}{\frac{z}{x}} \]

FPCore

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

↓

(FPCore (x y z) :precision binary64 (- y (/ (+ y -1.0) (/ z x))))

C

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

↓

double code(double x, double y, double z) {
	return y - ((y + -1.0) / (z / x));
}

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 - x))) / z
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 = y - ((y + (-1.0d0)) / (z / x))
end function

Java

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

↓

public static double code(double x, double y, double z) {
	return y - ((y + -1.0) / (z / x));
}

Python

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

↓

def code(x, y, z):
	return y - ((y + -1.0) / (z / x))

Julia

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

↓

function code(x, y, z)
	return Float64(y - Float64(Float64(y + -1.0) / Float64(z / x)))
end

MATLAB

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

↓

function tmp = code(x, y, z)
	tmp = y - ((y + -1.0) / (z / x));
end

Wolfram

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

↓

code[x_, y_, z_] := N[(y - N[(N[(y + -1.0), $MachinePrecision] / N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	16.29%
Target	0.07%
Herbie	0.22%

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

Derivation

1. Initial program 16.29

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

2. Taylor expanded in x around inf 5.25

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

3. Simplified 0.22

   \[\leadsto \color{blue}{y - \frac{y + -1}{\frac{z}{x}}} \]
   Proof

   [Start] 5.25	\[ y + \frac{\left(1 + -1 \cdot y\right) \cdot x}{z} \]
   associate-/l* [=>] 0.22	\[ y + \color{blue}{\frac{1 + -1 \cdot y}{\frac{z}{x}}} \]
   +-commutative [=>] 0.22	\[ y + \frac{\color{blue}{-1 \cdot y + 1}}{\frac{z}{x}} \]
   metadata-eval [<=] 0.22	\[ y + \frac{-1 \cdot y + \color{blue}{-1 \cdot -1}}{\frac{z}{x}} \]
   distribute-lft-in [<=] 0.22	\[ y + \frac{\color{blue}{-1 \cdot \left(y + -1\right)}}{\frac{z}{x}} \]
   metadata-eval [<=] 0.22	\[ y + \frac{-1 \cdot \left(y + \color{blue}{\left(-1\right)}\right)}{\frac{z}{x}} \]
   sub-neg [<=] 0.22	\[ y + \frac{-1 \cdot \color{blue}{\left(y - 1\right)}}{\frac{z}{x}} \]
   mul-1-neg [=>] 0.22	\[ y + \frac{\color{blue}{-\left(y - 1\right)}}{\frac{z}{x}} \]
   distribute-neg-frac [<=] 0.22	\[ y + \color{blue}{\left(-\frac{y - 1}{\frac{z}{x}}\right)} \]
   associate-/l* [<=] 5.25	\[ y + \left(-\color{blue}{\frac{\left(y - 1\right) \cdot x}{z}}\right) \]
   unsub-neg [=>] 5.25	\[ \color{blue}{y - \frac{\left(y - 1\right) \cdot x}{z}} \]
   associate-/l* [=>] 0.22	\[ y - \color{blue}{\frac{y - 1}{\frac{z}{x}}} \]
   sub-neg [=>] 0.22	\[ y - \frac{\color{blue}{y + \left(-1\right)}}{\frac{z}{x}} \]
   metadata-eval [=>] 0.22	\[ y - \frac{y + \color{blue}{-1}}{\frac{z}{x}} \]

4. Final simplification 0.22

   \[\leadsto y - \frac{y + -1}{\frac{z}{x}} \]

Alternatives

Alternative 1
Error	2.14%
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -2.3 \cdot 10^{+37} \lor \neg \left(y \leq 1\right):\\ \;\;\;\;y \cdot \frac{z - x}{z}\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \]

Alternative 2
Error	2.13%
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -2.3 \cdot 10^{+37} \lor \neg \left(y \leq 1\right):\\ \;\;\;\;y - \frac{y}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \]

Alternative 3
Error	2.14%
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq -2.3 \cdot 10^{+37}:\\ \;\;\;\;y - \frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{z - x}}\\ \end{array} \]

Alternative 4
Error	31.69%
Cost	456

\[\begin{array}{l} \mathbf{if}\;y \leq -4.7 \cdot 10^{-94}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{-26}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 5
Error	14.11%
Cost	320

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

Alternative 6
Error	49.28%
Cost	64

\[y \]

Reproduce

herbie shell --seed 2023090 
(FPCore (x y z)
  :name "Diagrams.Backend.Rasterific:rasterificRadialGradient from diagrams-rasterific-1.3.1.3"
  :precision binary64

  :herbie-target
  (- (+ y (/ x z)) (/ y (/ z x)))

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