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

Average Error: 10.0 → 0.0

Time: 6.3s

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	10.0
Target	0.0
Herbie	0.0

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

Derivation

1. Initial program 10.0

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

2. Simplified 10.0

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

   [Start] 10.0	\[ \frac{x + y \cdot \left(z - x\right)}{z} \]
   +-commutative [=>] 10.0	\[ \frac{\color{blue}{y \cdot \left(z - x\right) + x}}{z} \]
   fma-def [=>] 10.0	\[ \frac{\color{blue}{\mathsf{fma}\left(y, z - x, x\right)}}{z} \]

3. Taylor expanded in z around 0 3.5

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

4. Simplified 0.0

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

   [Start] 3.5	\[ -1 \cdot \frac{y \cdot x}{z} + \left(y + \frac{x}{z}\right) \]
   +-commutative [=>] 3.5	\[ -1 \cdot \frac{y \cdot x}{z} + \color{blue}{\left(\frac{x}{z} + y\right)} \]
   associate-+r+ [=>] 3.5	\[ \color{blue}{\left(-1 \cdot \frac{y \cdot x}{z} + \frac{x}{z}\right) + y} \]
   +-commutative [=>] 3.5	\[ \color{blue}{y + \left(-1 \cdot \frac{y \cdot x}{z} + \frac{x}{z}\right)} \]
   mul-1-neg [=>] 3.5	\[ y + \left(\color{blue}{\left(-\frac{y \cdot x}{z}\right)} + \frac{x}{z}\right) \]
   neg-sub0 [=>] 3.5	\[ y + \left(\color{blue}{\left(0 - \frac{y \cdot x}{z}\right)} + \frac{x}{z}\right) \]
   remove-double-neg [<=] 3.5	\[ y + \left(\left(0 - \frac{y \cdot x}{z}\right) + \frac{\color{blue}{-\left(-x\right)}}{z}\right) \]
   neg-mul-1 [=>] 3.5	\[ y + \left(\left(0 - \frac{y \cdot x}{z}\right) + \frac{-\color{blue}{-1 \cdot x}}{z}\right) \]
   distribute-lft-neg-in [=>] 3.5	\[ y + \left(\left(0 - \frac{y \cdot x}{z}\right) + \frac{\color{blue}{\left(--1\right) \cdot x}}{z}\right) \]
   metadata-eval [=>] 3.5	\[ y + \left(\left(0 - \frac{y \cdot x}{z}\right) + \frac{\color{blue}{1} \cdot x}{z}\right) \]
   associate-*l/ [<=] 3.6	\[ y + \left(\left(0 - \frac{y \cdot x}{z}\right) + \color{blue}{\frac{1}{z} \cdot x}\right) \]
   associate-/r/ [<=] 3.6	\[ y + \left(\left(0 - \frac{y \cdot x}{z}\right) + \color{blue}{\frac{1}{\frac{z}{x}}}\right) \]
   associate--r- [<=] 3.6	\[ y + \color{blue}{\left(0 - \left(\frac{y \cdot x}{z} - \frac{1}{\frac{z}{x}}\right)\right)} \]
   associate-/l* [=>] 0.1	\[ y + \left(0 - \left(\color{blue}{\frac{y}{\frac{z}{x}}} - \frac{1}{\frac{z}{x}}\right)\right) \]
   div-sub [<=] 0.1	\[ y + \left(0 - \color{blue}{\frac{y - 1}{\frac{z}{x}}}\right) \]
   associate-/l* [<=] 3.5	\[ y + \left(0 - \color{blue}{\frac{\left(y - 1\right) \cdot x}{z}}\right) \]
   neg-sub0 [<=] 3.5	\[ y + \color{blue}{\left(-\frac{\left(y - 1\right) \cdot x}{z}\right)} \]
   associate-/l* [=>] 0.1	\[ y + \left(-\color{blue}{\frac{y - 1}{\frac{z}{x}}}\right) \]
   distribute-neg-frac [=>] 0.1	\[ y + \color{blue}{\frac{-\left(y - 1\right)}{\frac{z}{x}}} \]
   mul-1-neg [<=] 0.1	\[ y + \frac{\color{blue}{-1 \cdot \left(y - 1\right)}}{\frac{z}{x}} \]
   sub-neg [=>] 0.1	\[ y + \frac{-1 \cdot \color{blue}{\left(y + \left(-1\right)\right)}}{\frac{z}{x}} \]
   metadata-eval [=>] 0.1	\[ y + \frac{-1 \cdot \left(y + \color{blue}{-1}\right)}{\frac{z}{x}} \]
   distribute-lft-in [=>] 0.1	\[ y + \frac{\color{blue}{-1 \cdot y + -1 \cdot -1}}{\frac{z}{x}} \]
   metadata-eval [=>] 0.1	\[ y + \frac{-1 \cdot y + \color{blue}{1}}{\frac{z}{x}} \]
   +-commutative [<=] 0.1	\[ y + \frac{\color{blue}{1 + -1 \cdot y}}{\frac{z}{x}} \]
   associate-/l* [<=] 3.5	\[ y + \color{blue}{\frac{\left(1 + -1 \cdot y\right) \cdot x}{z}} \]
   *-commutative [=>] 3.5	\[ y + \frac{\color{blue}{x \cdot \left(1 + -1 \cdot y\right)}}{z} \]
   associate-/l* [=>] 3.1	\[ y + \color{blue}{\frac{x}{\frac{z}{1 + -1 \cdot y}}} \]
   associate-/r/ [=>] 0.0	\[ y + \color{blue}{\frac{x}{z} \cdot \left(1 + -1 \cdot y\right)} \]
   mul-1-neg [=>] 0.0	\[ y + \frac{x}{z} \cdot \left(1 + \color{blue}{\left(-y\right)}\right) \]
   unsub-neg [=>] 0.0	\[ y + \frac{x}{z} \cdot \color{blue}{\left(1 - y\right)} \]

5. Final simplification 0.0

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

Alternatives

Alternative 1
Error	7.9
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -4.1 \cdot 10^{-153} \lor \neg \left(z \leq 2.2 \cdot 10^{-112}\right):\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \left(1 - y\right)\\ \end{array} \]

Alternative 2
Error	4.3
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -2.7 \cdot 10^{+14} \lor \neg \left(y \leq 6.1 \cdot 10^{-6}\right):\\ \;\;\;\;\left(z - x\right) \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \]

Alternative 3
Error	4.3
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -2.7 \cdot 10^{+14} \lor \neg \left(y \leq 6.1 \cdot 10^{-6}\right):\\ \;\;\;\;y - x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \]

Alternative 4
Error	0.9
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -2.7 \cdot 10^{+14} \lor \neg \left(y \leq 6.1 \cdot 10^{-6}\right):\\ \;\;\;\;\frac{y}{\frac{z}{z - x}}\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \]

Alternative 5
Error	20.4
Cost	456

\[\begin{array}{l} \mathbf{if}\;y \leq -8 \cdot 10^{-14}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{-31}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 6
Error	9.2
Cost	320

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

Alternative 7
Error	31.4
Cost	64

\[y \]

Reproduce

herbie shell --seed 2022356 
(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))