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

Average Error: 0.04% → 0.04%

Time: 3.7s

Precision: binary64

Cost: 448

Math

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

↓

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

FPCore

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

↓

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

C

double code(double x, double y, double z) {
	return (x - y) / (z - y);
}

↓

double code(double x, double y, double z) {
	return (x - y) / (z - 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 - 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 = (x - y) / (z - y)
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 (x - y) / (z - y);
}

Python

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

↓

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

Julia

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

↓

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

MATLAB

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

↓

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

Wolfram

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

↓

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

Target

Original	0.04%
Target	0.03%
Herbie	0.04%

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

Derivation

1. Initial program 0.04

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

2. Final simplification 0.04

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

Alternatives

Alternative 1
Error	25.37%
Cost	1176

\[\begin{array}{l} t_0 := \frac{x - y}{z}\\ t_1 := 1 - \frac{x}{y}\\ \mathbf{if}\;y \leq -3.5 \cdot 10^{+131}:\\ \;\;\;\;\frac{y}{y - z}\\ \mathbf{elif}\;y \leq -9.2 \cdot 10^{+76}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -2.6 \cdot 10^{+34}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -3400:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -3.8 \cdot 10^{-81}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{+51}:\\ \;\;\;\;\frac{-x}{y - z}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	44.23%
Cost	1048

\[\begin{array}{l} t_0 := \frac{-x}{y}\\ \mathbf{if}\;y \leq -4.6 \cdot 10^{+177}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -3.6 \cdot 10^{+131}:\\ \;\;\;\;\frac{-y}{z}\\ \mathbf{elif}\;y \leq -0.00115:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 2.9 \cdot 10^{+49}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;y \leq 4.1 \cdot 10^{+114}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.38 \cdot 10^{+166}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 3
Error	25.32%
Cost	982

\[\begin{array}{l} \mathbf{if}\;y \leq -1.35 \cdot 10^{+131}:\\ \;\;\;\;\frac{y}{y - z}\\ \mathbf{elif}\;y \leq -3.1 \cdot 10^{+80} \lor \neg \left(y \leq -2.9 \cdot 10^{+34}\right) \land \left(y \leq -52 \lor \neg \left(y \leq 1.4 \cdot 10^{+49}\right)\right):\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - y}{z}\\ \end{array} \]

Alternative 4
Error	42.25%
Cost	916

\[\begin{array}{l} t_0 := \frac{-x}{y}\\ \mathbf{if}\;y \leq -1.4 \cdot 10^{+108}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq -0.00165:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 6.8 \cdot 10^{+52}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{+114}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.38 \cdot 10^{+166}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 5
Error	30%
Cost	848

\[\begin{array}{l} t_0 := \frac{y}{y - z}\\ t_1 := 1 - \frac{x}{y}\\ \mathbf{if}\;y \leq -4.8 \cdot 10^{+132}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq -320000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -1.4 \cdot 10^{-73}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 3.35 \cdot 10^{-71}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	30.36%
Cost	585

\[\begin{array}{l} \mathbf{if}\;y \leq -0.0005 \lor \neg \left(y \leq 3.5 \cdot 10^{-71}\right):\\ \;\;\;\;1 - \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]

Alternative 7
Error	40.88%
Cost	456

\[\begin{array}{l} \mathbf{if}\;y \leq -1.8 \cdot 10^{+79}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 9 \cdot 10^{+48}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 8
Error	63.94%
Cost	64

\[1 \]

Reproduce

herbie shell --seed 2023121 
(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)))