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

Average Accuracy: 100.0% → 99.9%

Time: 9.7s

Precision: binary64

Cost: 960

Math

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

↓

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

FPCore

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

↓

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

Python

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

↓

def code(x, y, z):
	return ((y - x) / (y + z)) / ((y - z) / (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(Float64(y - x) / Float64(y + z)) / Float64(Float64(y - z) / 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 - x) / (y + z)) / ((y - z) / (y + z));
end

Wolfram

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

↓

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

Target

Original	100.0%
Target	100.0%
Herbie	99.9%

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

Derivation

1. Initial program 100.0%

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

2. Simplified 100.0%

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

   [Start] 100.0	\[ \frac{x - y}{z - y} \]
   sub-neg [=>] 100.0	\[ \frac{\color{blue}{x + \left(-y\right)}}{z - y} \]
   +-commutative [=>] 100.0	\[ \frac{\color{blue}{\left(-y\right) + x}}{z - y} \]
   neg-sub0 [=>] 100.0	\[ \frac{\color{blue}{\left(0 - y\right)} + x}{z - y} \]
   associate-+l- [=>] 100.0	\[ \frac{\color{blue}{0 - \left(y - x\right)}}{z - y} \]
   sub0-neg [=>] 100.0	\[ \frac{\color{blue}{-\left(y - x\right)}}{z - y} \]
   neg-mul-1 [=>] 100.0	\[ \frac{\color{blue}{-1 \cdot \left(y - x\right)}}{z - y} \]
   sub-neg [=>] 100.0	\[ \frac{-1 \cdot \left(y - x\right)}{\color{blue}{z + \left(-y\right)}} \]
   +-commutative [=>] 100.0	\[ \frac{-1 \cdot \left(y - x\right)}{\color{blue}{\left(-y\right) + z}} \]
   neg-sub0 [=>] 100.0	\[ \frac{-1 \cdot \left(y - x\right)}{\color{blue}{\left(0 - y\right)} + z} \]
   associate-+l- [=>] 100.0	\[ \frac{-1 \cdot \left(y - x\right)}{\color{blue}{0 - \left(y - z\right)}} \]
   sub0-neg [=>] 100.0	\[ \frac{-1 \cdot \left(y - x\right)}{\color{blue}{-\left(y - z\right)}} \]
   neg-mul-1 [=>] 100.0	\[ \frac{-1 \cdot \left(y - x\right)}{\color{blue}{-1 \cdot \left(y - z\right)}} \]
   times-frac [=>] 100.0	\[ \color{blue}{\frac{-1}{-1} \cdot \frac{y - x}{y - z}} \]
   metadata-eval [=>] 100.0	\[ \color{blue}{1} \cdot \frac{y - x}{y - z} \]
   *-lft-identity [=>] 100.0	\[ \color{blue}{\frac{y - x}{y - z}} \]

3. Applied egg-rr 35.3%

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

   [Start] 100.0	\[ \frac{y - x}{y - z} \]
   div-sub [=>] 100.0	\[ \color{blue}{\frac{y}{y - z} - \frac{x}{y - z}} \]
   frac-sub [=>] 49.5	\[ \color{blue}{\frac{y \cdot \left(y - z\right) - \left(y - z\right) \cdot x}{\left(y - z\right) \cdot \left(y - z\right)}} \]
   flip-- [=>] 49.5	\[ \frac{y \cdot \left(y - z\right) - \left(y - z\right) \cdot x}{\left(y - z\right) \cdot \color{blue}{\frac{y \cdot y - z \cdot z}{y + z}}} \]
   associate-*r/ [=>] 37.0	\[ \frac{y \cdot \left(y - z\right) - \left(y - z\right) \cdot x}{\color{blue}{\frac{\left(y - z\right) \cdot \left(y \cdot y - z \cdot z\right)}{y + z}}} \]
   associate-/r/ [=>] 35.3	\[ \color{blue}{\frac{y \cdot \left(y - z\right) - \left(y - z\right) \cdot x}{\left(y - z\right) \cdot \left(y \cdot y - z \cdot z\right)} \cdot \left(y + z\right)} \]
   *-commutative [=>] 35.3	\[ \frac{y \cdot \left(y - z\right) - \color{blue}{x \cdot \left(y - z\right)}}{\left(y - z\right) \cdot \left(y \cdot y - z \cdot z\right)} \cdot \left(y + z\right) \]

4. Simplified 99.9%

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

   [Start] 35.3	\[ \frac{y \cdot \left(y - z\right) - x \cdot \left(y - z\right)}{\left(y - z\right) \cdot \left(y \cdot y - z \cdot z\right)} \cdot \left(y + z\right) \]
   associate-*l/ [=>] 32.0	\[ \color{blue}{\frac{\left(y \cdot \left(y - z\right) - x \cdot \left(y - z\right)\right) \cdot \left(y + z\right)}{\left(y - z\right) \cdot \left(y \cdot y - z \cdot z\right)}} \]
   *-commutative [=>] 32.0	\[ \frac{\left(y \cdot \left(y - z\right) - x \cdot \left(y - z\right)\right) \cdot \left(y + z\right)}{\color{blue}{\left(y \cdot y - z \cdot z\right) \cdot \left(y - z\right)}} \]
   times-frac [=>] 49.5	\[ \color{blue}{\frac{y \cdot \left(y - z\right) - x \cdot \left(y - z\right)}{y \cdot y - z \cdot z} \cdot \frac{y + z}{y - z}} \]
   distribute-rgt-out-- [=>] 49.5	\[ \frac{\color{blue}{\left(y - z\right) \cdot \left(y - x\right)}}{y \cdot y - z \cdot z} \cdot \frac{y + z}{y - z} \]
   sqr-neg [<=] 49.5	\[ \frac{\left(y - z\right) \cdot \left(y - x\right)}{y \cdot y - \color{blue}{\left(-z\right) \cdot \left(-z\right)}} \cdot \frac{y + z}{y - z} \]
   difference-of-squares [=>] 49.5	\[ \frac{\left(y - z\right) \cdot \left(y - x\right)}{\color{blue}{\left(y + \left(-z\right)\right) \cdot \left(y - \left(-z\right)\right)}} \cdot \frac{y + z}{y - z} \]
   sub-neg [<=] 49.5	\[ \frac{\left(y - z\right) \cdot \left(y - x\right)}{\color{blue}{\left(y - z\right)} \cdot \left(y - \left(-z\right)\right)} \cdot \frac{y + z}{y - z} \]
   times-frac [=>] 99.9	\[ \color{blue}{\left(\frac{y - z}{y - z} \cdot \frac{y - x}{y - \left(-z\right)}\right)} \cdot \frac{y + z}{y - z} \]
   sub-neg [=>] 99.9	\[ \left(\frac{y - z}{y - z} \cdot \frac{y - x}{\color{blue}{y + \left(-\left(-z\right)\right)}}\right) \cdot \frac{y + z}{y - z} \]
   +-commutative [=>] 99.9	\[ \left(\frac{y - z}{y - z} \cdot \frac{y - x}{\color{blue}{\left(-\left(-z\right)\right) + y}}\right) \cdot \frac{y + z}{y - z} \]
   remove-double-neg [=>] 99.9	\[ \left(\frac{y - z}{y - z} \cdot \frac{y - x}{\color{blue}{z} + y}\right) \cdot \frac{y + z}{y - z} \]
   +-commutative [=>] 99.9	\[ \left(\frac{y - z}{y - z} \cdot \frac{y - x}{z + y}\right) \cdot \frac{\color{blue}{z + y}}{y - z} \]

5. Applied egg-rr 99.9%

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

   [Start] 99.9	\[ \left(\frac{y - z}{y - z} \cdot \frac{y - x}{z + y}\right) \cdot \frac{z + y}{y - z} \]
   associate-*r/ [=>] 100.0	\[ \color{blue}{\frac{\left(\frac{y - z}{y - z} \cdot \frac{y - x}{z + y}\right) \cdot \left(z + y\right)}{y - z}} \]
   *-inverses [=>] 100.0	\[ \frac{\left(\color{blue}{1} \cdot \frac{y - x}{z + y}\right) \cdot \left(z + y\right)}{y - z} \]
   *-un-lft-identity [<=] 100.0	\[ \frac{\color{blue}{\frac{y - x}{z + y}} \cdot \left(z + y\right)}{y - z} \]
   associate-/l* [=>] 99.9	\[ \color{blue}{\frac{\frac{y - x}{z + y}}{\frac{y - z}{z + y}}} \]
   +-commutative [=>] 99.9	\[ \frac{\frac{y - x}{\color{blue}{y + z}}}{\frac{y - z}{z + y}} \]
   +-commutative [=>] 99.9	\[ \frac{\frac{y - x}{y + z}}{\frac{y - z}{\color{blue}{y + z}}} \]

6. Final simplification 99.9%

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

Alternatives

Alternative 1
Accuracy	73.7%
Cost	848

\[\begin{array}{l} \mathbf{if}\;y \leq -9.6 \cdot 10^{+30}:\\ \;\;\;\;\frac{y - x}{y}\\ \mathbf{elif}\;y \leq -3.3 \cdot 10^{-73}:\\ \;\;\;\;\frac{x}{z - y}\\ \mathbf{elif}\;y \leq -6.4 \cdot 10^{-78}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{-129}:\\ \;\;\;\;\frac{x - y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{y - z}\\ \end{array} \]

Alternative 2
Accuracy	74.9%
Cost	585

\[\begin{array}{l} \mathbf{if}\;y \leq -3.2 \cdot 10^{+61} \lor \neg \left(y \leq 5 \cdot 10^{-22}\right):\\ \;\;\;\;\frac{y}{y - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z - y}\\ \end{array} \]

Alternative 3
Accuracy	73.7%
Cost	585

\[\begin{array}{l} \mathbf{if}\;y \leq -9.5 \cdot 10^{+30} \lor \neg \left(y \leq 2.8 \cdot 10^{-129}\right):\\ \;\;\;\;\frac{y}{y - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{z}\\ \end{array} \]

Alternative 4
Accuracy	58.0%
Cost	584

\[\begin{array}{l} \mathbf{if}\;y \leq -9.2 \cdot 10^{+30}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 3.2 \cdot 10^{-131}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{z}{y}\\ \end{array} \]

Alternative 5
Accuracy	68.9%
Cost	584

\[\begin{array}{l} \mathbf{if}\;y \leq -1.95 \cdot 10^{+62}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 3 \cdot 10^{+62}:\\ \;\;\;\;\frac{x}{z - y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 6
Accuracy	58.3%
Cost	456

\[\begin{array}{l} \mathbf{if}\;y \leq -6 \cdot 10^{+31}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{-129}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 7
Accuracy	100.0%
Cost	448

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

Alternative 8
Accuracy	36.4%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023136 
(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)))