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

Average Accuracy: 83.7% → 99.9%

Time: 8.8s

Precision: binary64

Cost: 704

Math

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

↓

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

FPCore

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

↓

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

C

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

↓

double code(double x, double y, double z) {
	return (x / z) + ((1.0 - (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 - 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 = (x / z) + ((1.0d0 - (x / z)) * y)
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 (x / z) + ((1.0 - (x / z)) * y);
}

Python

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

↓

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

Julia

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

↓

function code(x, y, z)
	return Float64(Float64(x / z) + Float64(Float64(1.0 - Float64(x / z)) * y))
end

MATLAB

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

↓

function tmp = code(x, y, z)
	tmp = (x / z) + ((1.0 - (x / z)) * y);
end

Wolfram

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

↓

code[x_, y_, z_] := N[(N[(x / z), $MachinePrecision] + N[(N[(1.0 - N[(x / z), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]

Target

Original	83.7%
Target	99.9%
Herbie	99.9%

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

Derivation

1. Initial program 83.7%

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

2. Taylor expanded in y around 0 99.9%

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

3. Final simplification 99.9%

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

Alternatives

Alternative 1
Accuracy	81.3%
Cost	912

\[\begin{array}{l} t_0 := y \cdot \frac{-x}{z}\\ \mathbf{if}\;y \leq -1.85 \cdot 10^{+161}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq -7.5 \cdot 10^{+129}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{+44}:\\ \;\;\;\;\frac{x}{z} + y\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{+198}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 2
Accuracy	81.3%
Cost	912

\[\begin{array}{l} \mathbf{if}\;y \leq -1.65 \cdot 10^{+161}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq -6 \cdot 10^{+129}:\\ \;\;\;\;y \cdot \frac{-x}{z}\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{+44}:\\ \;\;\;\;\frac{x}{z} + y\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{+198}:\\ \;\;\;\;\frac{y}{\frac{-z}{x}}\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 3
Accuracy	99.4%
Cost	840

\[\begin{array}{l} \mathbf{if}\;y \leq -8 \cdot 10^{+55}:\\ \;\;\;\;\left(1 - \frac{x}{z}\right) \cdot y\\ \mathbf{elif}\;y \leq 54000000000000:\\ \;\;\;\;\frac{x + y \cdot \left(z - x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{z - x}}\\ \end{array} \]

Alternative 4
Accuracy	67.7%
Cost	720

\[\begin{array}{l} \mathbf{if}\;y \leq -2.5 \cdot 10^{-18}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq -1.1 \cdot 10^{-170}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{elif}\;y \leq -1.02 \cdot 10^{-178}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 2.7 \cdot 10^{-70}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 5
Accuracy	99.0%
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -0.99 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;\left(1 - \frac{x}{z}\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} + y\\ \end{array} \]

Alternative 6
Accuracy	99.0%
Cost	712

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

Alternative 7
Accuracy	85.2%
Cost	320

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

Alternative 8
Accuracy	51.1%
Cost	64

\[y \]

Reproduce

herbie shell --seed 2023151 
(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))