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

Average Error: 15.99% → 0.08%

Time: 7.1s

Precision: binary64

Cost: 576

Math

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

↓

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

FPCore

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

↓

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

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 + ((x / z) * (1.0 - 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 + (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 + ((x / z) * (1.0d0 - y))
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 + ((x / z) * (1.0 - y));
}

Python

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

↓

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

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(x / z) * Float64(1.0 - y)))
end

MATLAB

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

↓

function tmp = code(x, y, z)
	tmp = y + ((x / z) * (1.0 - y));
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[(x / z), $MachinePrecision] * N[(1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Target

Original	15.99%
Target	0.07%
Herbie	0.08%

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

Derivation

1. Initial program 15.99

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

2. Simplified 15.98

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y, z - x, x\right)}{z}} \]
   Proof

   [Start] 15.99	\[ \frac{x + y \cdot \left(z - x\right)}{z} \]
   +-commutative [=>] 15.99	\[ \frac{\color{blue}{y \cdot \left(z - x\right) + x}}{z} \]
   fma-def [=>] 15.98	\[ \frac{\color{blue}{\mathsf{fma}\left(y, z - x, x\right)}}{z} \]

3. Taylor expanded in z around 0 5.41

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

4. Simplified 0.08

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

   [Start] 5.41	\[ -1 \cdot \frac{y \cdot x}{z} + \left(y + \frac{x}{z}\right) \]
   +-commutative [=>] 5.41	\[ -1 \cdot \frac{y \cdot x}{z} + \color{blue}{\left(\frac{x}{z} + y\right)} \]
   associate-+r+ [=>] 5.41	\[ \color{blue}{\left(-1 \cdot \frac{y \cdot x}{z} + \frac{x}{z}\right) + y} \]
   +-commutative [=>] 5.41	\[ \color{blue}{y + \left(-1 \cdot \frac{y \cdot x}{z} + \frac{x}{z}\right)} \]
   mul-1-neg [=>] 5.41	\[ y + \left(\color{blue}{\left(-\frac{y \cdot x}{z}\right)} + \frac{x}{z}\right) \]
   neg-sub0 [=>] 5.41	\[ y + \left(\color{blue}{\left(0 - \frac{y \cdot x}{z}\right)} + \frac{x}{z}\right) \]
   remove-double-neg [<=] 5.41	\[ y + \left(\left(0 - \frac{y \cdot x}{z}\right) + \frac{\color{blue}{-\left(-x\right)}}{z}\right) \]
   neg-mul-1 [=>] 5.41	\[ y + \left(\left(0 - \frac{y \cdot x}{z}\right) + \frac{-\color{blue}{-1 \cdot x}}{z}\right) \]
   distribute-lft-neg-in [=>] 5.41	\[ y + \left(\left(0 - \frac{y \cdot x}{z}\right) + \frac{\color{blue}{\left(--1\right) \cdot x}}{z}\right) \]
   metadata-eval [=>] 5.41	\[ y + \left(\left(0 - \frac{y \cdot x}{z}\right) + \frac{\color{blue}{1} \cdot x}{z}\right) \]
   associate-*l/ [<=] 5.56	\[ y + \left(\left(0 - \frac{y \cdot x}{z}\right) + \color{blue}{\frac{1}{z} \cdot x}\right) \]
   associate-/r/ [<=] 5.56	\[ y + \left(\left(0 - \frac{y \cdot x}{z}\right) + \color{blue}{\frac{1}{\frac{z}{x}}}\right) \]
   associate--r- [<=] 5.56	\[ y + \color{blue}{\left(0 - \left(\frac{y \cdot x}{z} - \frac{1}{\frac{z}{x}}\right)\right)} \]
   associate-/l* [=>] 0.22	\[ y + \left(0 - \left(\color{blue}{\frac{y}{\frac{z}{x}}} - \frac{1}{\frac{z}{x}}\right)\right) \]
   div-sub [<=] 0.22	\[ y + \left(0 - \color{blue}{\frac{y - 1}{\frac{z}{x}}}\right) \]
   associate-/l* [<=] 5.41	\[ y + \left(0 - \color{blue}{\frac{\left(y - 1\right) \cdot x}{z}}\right) \]
   neg-sub0 [<=] 5.41	\[ y + \color{blue}{\left(-\frac{\left(y - 1\right) \cdot x}{z}\right)} \]
   associate-/l* [=>] 0.22	\[ y + \left(-\color{blue}{\frac{y - 1}{\frac{z}{x}}}\right) \]
   distribute-neg-frac [=>] 0.22	\[ y + \color{blue}{\frac{-\left(y - 1\right)}{\frac{z}{x}}} \]

5. Final simplification 0.08

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

Alternatives

Alternative 1
Error	1.22%
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 0.245\right):\\ \;\;\;\;y \cdot \left(1 - \frac{x}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \]

Alternative 2
Error	1.22%
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq -1:\\ \;\;\;\;y \cdot \left(1 - \frac{x}{z}\right)\\ \mathbf{elif}\;y \leq 0.245:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{z - x}}\\ \end{array} \]

Alternative 3
Error	31.59%
Cost	456

\[\begin{array}{l} \mathbf{if}\;y \leq -1.4 \cdot 10^{-142}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{-28}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 4
Error	13.63%
Cost	320

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

Alternative 5
Error	49.82%
Cost	64

\[y \]

Reproduce

herbie shell --seed 2023115 
(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))