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

Average Error: 10.1 → 0.0

Time: 11.1s

Precision: binary64

Cost: 576

Math

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

↓

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

FPCore

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

↓

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

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

Python

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

↓

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

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

MATLAB

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

↓

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

Target

Original	10.1
Target	0.0
Herbie	0.0

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

Derivation

1. Initial program 10.1

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

2. Taylor expanded in x around inf 3.4

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

3. Simplified 0.0

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

   [Start] 3.4	\[ y + \frac{\left(1 + -1 \cdot y\right) \cdot x}{z} \]
   rational.json-simplify-2 [=>] 3.4	\[ y + \frac{\color{blue}{x \cdot \left(1 + -1 \cdot y\right)}}{z} \]
   rational.json-simplify-49 [=>] 0.0	\[ y + \color{blue}{\left(1 + -1 \cdot y\right) \cdot \frac{x}{z}} \]
   rational.json-simplify-17 [=>] 0.0	\[ y + \color{blue}{\left(-1 \cdot y - -1\right)} \cdot \frac{x}{z} \]
   rational.json-simplify-2 [=>] 0.0	\[ y + \left(\color{blue}{y \cdot -1} - -1\right) \cdot \frac{x}{z} \]
   rational.json-simplify-9 [=>] 0.0	\[ y + \left(\color{blue}{\left(-y\right)} - -1\right) \cdot \frac{x}{z} \]
   rational.json-simplify-12 [=>] 0.0	\[ y + \left(\color{blue}{\left(0 - y\right)} - -1\right) \cdot \frac{x}{z} \]
   rational.json-simplify-42 [=>] 0.0	\[ y + \color{blue}{\left(\left(0 - -1\right) - y\right)} \cdot \frac{x}{z} \]
   metadata-eval [=>] 0.0	\[ y + \left(\color{blue}{1} - y\right) \cdot \frac{x}{z} \]

4. Final simplification 0.0

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

Alternatives

Alternative 1
Error	20.5
Cost	720

\[\begin{array}{l} \mathbf{if}\;y \leq -4.5 \cdot 10^{-34}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{-147}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{-101}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 9.6 \cdot 10^{-33}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 2
Error	0.7
Cost	712

\[\begin{array}{l} t_0 := y \cdot \left(1 - \frac{x}{z}\right)\\ \mathbf{if}\;y \leq -1:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	0.7
Cost	712

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

Alternative 4
Error	9.6
Cost	648

\[\begin{array}{l} \mathbf{if}\;y \leq -5.8 \cdot 10^{+211}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq -2.7 \cdot 10^{+174}:\\ \;\;\;\;y \cdot \frac{x}{-z}\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \]

Alternative 5
Error	9.6
Cost	648

\[\begin{array}{l} \mathbf{if}\;y \leq -5.8 \cdot 10^{+211}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq -2.3 \cdot 10^{+174}:\\ \;\;\;\;\frac{y}{-\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \]

Alternative 6
Error	8.8
Cost	320

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

Alternative 7
Error	31.2
Cost	64

\[y \]

Reproduce

herbie shell --seed 2023068 
(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))