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

Average Error: 0.0 → 0.0

Time: 4.4s

Precision: binary64

Cost: 768

Math

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

↓

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

FPCore

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

↓

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

Python

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

↓

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

MATLAB

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

↓

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

Wolfram

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

↓

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

Target

Original	0.0
Target	0.0
Herbie	0.0

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

Derivation

1. Initial program 0.0

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

2. Taylor expanded in x around 0 0.0

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

3. Simplified 0.0

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

   [Start] 0.0	\[ -1 \cdot \frac{y}{z - y} + \frac{x}{z - y} \]
   rational_best_oopsla_all_46_json_45_simplify-74 [=>] 0.0	\[ \color{blue}{\frac{y}{z - y} \cdot -1} + \frac{x}{z - y} \]
   rational_best_oopsla_all_46_json_45_simplify-92 [=>] 0.0	\[ \color{blue}{\left(-\frac{y}{z - y}\right)} + \frac{x}{z - y} \]

4. Final simplification 0.0

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

Alternatives

Alternative 1
Error	15.6
Cost	712

\[\begin{array}{l} t_0 := -\frac{x - y}{y}\\ \mathbf{if}\;y \leq -900000000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{+37}:\\ \;\;\;\;\frac{x}{z} - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	25.7
Cost	652

\[\begin{array}{l} \mathbf{if}\;y \leq -8.8 \cdot 10^{+63}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 9 \cdot 10^{-78}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;y \leq 3 \cdot 10^{+39}:\\ \;\;\;\;-\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 3
Error	25.4
Cost	652

\[\begin{array}{l} \mathbf{if}\;y \leq -8.4 \cdot 10^{+61}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{-73}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{+37}:\\ \;\;\;\;-\frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 4
Error	15.2
Cost	648

\[\begin{array}{l} t_0 := -\frac{y}{z - y}\\ \mathbf{if}\;y \leq -4.4 \cdot 10^{+66}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{+14}:\\ \;\;\;\;\frac{x}{z - y}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	15.7
Cost	648

\[\begin{array}{l} t_0 := -\frac{x - y}{y}\\ \mathbf{if}\;y \leq -2700000000000:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{+34}:\\ \;\;\;\;\frac{x - y}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	19.7
Cost	584

\[\begin{array}{l} \mathbf{if}\;y \leq -8.5 \cdot 10^{+85}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{+40}:\\ \;\;\;\;\frac{x}{z - y}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 7
Error	25.1
Cost	456

\[\begin{array}{l} \mathbf{if}\;y \leq -1.76 \cdot 10^{+61}:\\ \;\;\;\;1\\ \mathbf{elif}\;y \leq 2.3 \cdot 10^{+39}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 8
Error	0.0
Cost	448

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

Alternative 9
Error	40.6
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)))