Graphics.Rasterific.Shading:$sgradientColorAt from Rasterific-0.6.1

Average Error: 0.04% → 0.03%

Time: 5.7s

Precision: binary64

Cost: 704

Math

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

↓

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

FPCore

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

↓

(FPCore (x y z) :precision binary64 (- (/ y (- y z)) (/ x (- 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 (y / (y - z)) - (x / (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 = (y / (y - z)) - (x / (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 (y / (y - z)) - (x / (y - z));
}

Python

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

↓

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

Julia

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

↓

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

MATLAB

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

↓

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

Wolfram

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

↓

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

Target

Original	0.04%
Target	0.03%
Herbie	0.03%

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

Derivation

1. Initial program 0.04

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

2. Simplified 0.04

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

   [Start] 0.04	\[ \frac{x - y}{z - y} \]
   sub-neg [=>] 0.04	\[ \frac{\color{blue}{x + \left(-y\right)}}{z - y} \]
   +-commutative [=>] 0.04	\[ \frac{\color{blue}{\left(-y\right) + x}}{z - y} \]
   neg-sub0 [=>] 0.04	\[ \frac{\color{blue}{\left(0 - y\right)} + x}{z - y} \]
   associate-+l- [=>] 0.04	\[ \frac{\color{blue}{0 - \left(y - x\right)}}{z - y} \]
   sub0-neg [=>] 0.04	\[ \frac{\color{blue}{-\left(y - x\right)}}{z - y} \]
   neg-mul-1 [=>] 0.04	\[ \frac{\color{blue}{-1 \cdot \left(y - x\right)}}{z - y} \]
   sub-neg [=>] 0.04	\[ \frac{-1 \cdot \left(y - x\right)}{\color{blue}{z + \left(-y\right)}} \]
   +-commutative [=>] 0.04	\[ \frac{-1 \cdot \left(y - x\right)}{\color{blue}{\left(-y\right) + z}} \]
   neg-sub0 [=>] 0.04	\[ \frac{-1 \cdot \left(y - x\right)}{\color{blue}{\left(0 - y\right)} + z} \]
   associate-+l- [=>] 0.04	\[ \frac{-1 \cdot \left(y - x\right)}{\color{blue}{0 - \left(y - z\right)}} \]
   sub0-neg [=>] 0.04	\[ \frac{-1 \cdot \left(y - x\right)}{\color{blue}{-\left(y - z\right)}} \]
   neg-mul-1 [=>] 0.04	\[ \frac{-1 \cdot \left(y - x\right)}{\color{blue}{-1 \cdot \left(y - z\right)}} \]
   times-frac [=>] 0.04	\[ \color{blue}{\frac{-1}{-1} \cdot \frac{y - x}{y - z}} \]
   metadata-eval [=>] 0.04	\[ \color{blue}{1} \cdot \frac{y - x}{y - z} \]
   *-lft-identity [=>] 0.04	\[ \color{blue}{\frac{y - x}{y - z}} \]

3. Applied egg-rr 0.03

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

4. Final simplification 0.03

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

Alternatives

Alternative 1
Error	16.37%
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -1.95 \cdot 10^{+41} \lor \neg \left(z \leq 3600\right):\\ \;\;\;\;\frac{x - y}{z}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{x}{y - z}\\ \end{array} \]

Alternative 2
Error	16.36%
Cost	712

\[\begin{array}{l} \mathbf{if}\;z \leq -4.6 \cdot 10^{+41}:\\ \;\;\;\;\frac{x - y}{z}\\ \mathbf{elif}\;z \leq 9500:\\ \;\;\;\;1 - \frac{x}{y - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} - \frac{y}{z}\\ \end{array} \]

Alternative 3
Error	39.97%
Cost	652

\[\begin{array}{l} \mathbf{if}\;y \leq -8.4 \cdot 10^{+61}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.02 \cdot 10^{-73}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;y \leq 4.8 \cdot 10^{+40}:\\ \;\;\;\;\frac{-x}{y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 4
Error	39.73%
Cost	652

\[\begin{array}{l} \mathbf{if}\;y \leq -1.76 \cdot 10^{+61}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{-73}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;y \leq 2.9 \cdot 10^{+38}:\\ \;\;\;\;\frac{-y}{z}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 5
Error	23.68%
Cost	585

\[\begin{array}{l} \mathbf{if}\;y \leq -4.4 \cdot 10^{+66} \lor \neg \left(y \leq 1.5 \cdot 10^{+14}\right):\\ \;\;\;\;\frac{y}{y - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z - y}\\ \end{array} \]

Alternative 6
Error	24.43%
Cost	585

\[\begin{array}{l} \mathbf{if}\;y \leq -430000000000 \lor \neg \left(y \leq 7.2 \cdot 10^{+35}\right):\\ \;\;\;\;\frac{y - x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{z}\\ \end{array} \]

Alternative 7
Error	30.72%
Cost	584

\[\begin{array}{l} \mathbf{if}\;y \leq -5.5 \cdot 10^{+86}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 8.8 \cdot 10^{+40}:\\ \;\;\;\;\frac{x}{z - y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 8
Error	39.17%
Cost	456

\[\begin{array}{l} \mathbf{if}\;y \leq -1.85 \cdot 10^{+61}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 9.2 \cdot 10^{+39}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 9
Error	0.04%
Cost	448

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

Alternative 10
Error	63.48%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023090 
(FPCore (x y z)
  :name "Graphics.Rasterific.Shading:$sgradientColorAt from Rasterific-0.6.1"
  :precision binary64

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

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