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

Percentage Accurate: 87.6% → 99.6%

Time: 6.4s

Precision: binary64

Cost: 7113

• 20.7% of points produce a very large (infinite) output. You may want to add a precondition.

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq -6.6 \cdot 10^{+47} \lor \neg \left(y \leq 5.5 \cdot 10^{+14}\right):\\ \;\;\;\;y - y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, z - x, x\right)}{z}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -6.6e+47) (not (<= y 5.5e+14)))
   (- y (* y (/ x z)))
   (/ (fma y (- z x) x) z)))

C

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

↓

double code(double x, double y, double z) {
	double tmp;
	if ((y <= -6.6e+47) || !(y <= 5.5e+14)) {
		tmp = y - (y * (x / z));
	} else {
		tmp = fma(y, (z - x), x) / z;
	}
	return tmp;
}

Julia

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

↓

function code(x, y, z)
	tmp = 0.0
	if ((y <= -6.6e+47) || !(y <= 5.5e+14))
		tmp = Float64(y - Float64(y * Float64(x / z)));
	else
		tmp = Float64(fma(y, Float64(z - x), x) / z);
	end
	return tmp
end

Wolfram

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

↓

code[x_, y_, z_] := If[Or[LessEqual[y, -6.6e+47], N[Not[LessEqual[y, 5.5e+14]], $MachinePrecision]], N[(y - N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * N[(z - x), $MachinePrecision] + x), $MachinePrecision] / z), $MachinePrecision]]

Target

Original	87.6%
Target	93.8%
Herbie	99.6%

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

Derivation

1. Split input into 2 regimes

2. if y < -6.5999999999999998e47 or 5.5e14 < y

   1. Initial program 76.8%

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

   2. Taylor expanded in y around inf 76.8%

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

   3. Taylor expanded in z around 0 88.9%

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

   4. Simplified 100.0%

      \[\leadsto \color{blue}{y - y \cdot \frac{x}{z}} \]
      Step-by-step derivation

      [Start] 88.9%	\[ -1 \cdot \frac{y \cdot x}{z} + y \]
      +-commutative [=>] 88.9%	\[ \color{blue}{y + -1 \cdot \frac{y \cdot x}{z}} \]
      mul-1-neg [=>] 88.9%	\[ y + \color{blue}{\left(-\frac{y \cdot x}{z}\right)} \]
      unsub-neg [=>] 88.9%	\[ \color{blue}{y - \frac{y \cdot x}{z}} \]
      associate-*r/ [<=] 100.0%	\[ y - \color{blue}{y \cdot \frac{x}{z}} \]

   if -6.5999999999999998e47 < y < 5.5e14

   1. Initial program 99.9%

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

   2. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(y, z - x, x\right)}{z}} \]
      Step-by-step derivation

      [Start] 99.9%	\[ \frac{x + y \cdot \left(z - x\right)}{z} \]
      +-commutative [=>] 99.9%	\[ \frac{\color{blue}{y \cdot \left(z - x\right) + x}}{z} \]
      fma-def [=>] 100.0%	\[ \frac{\color{blue}{\mathsf{fma}\left(y, z - x, x\right)}}{z} \]

3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.6 \cdot 10^{+47} \lor \neg \left(y \leq 5.5 \cdot 10^{+14}\right):\\ \;\;\;\;y - y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, z - x, x\right)}{z}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.6%
Cost	7113

\[\begin{array}{l} \mathbf{if}\;y \leq -6.6 \cdot 10^{+47} \lor \neg \left(y \leq 5.5 \cdot 10^{+14}\right):\\ \;\;\;\;y - y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(y, z - x, x\right)}{z}\\ \end{array} \]

Alternative 2
Accuracy	77.7%
Cost	912

\[\begin{array}{l} t_0 := y + \frac{x}{z}\\ t_1 := y \cdot \frac{-x}{z}\\ \mathbf{if}\;y \leq 1.5 \cdot 10^{+35}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{+140}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 2.25 \cdot 10^{+229}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 2 \cdot 10^{+272}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Accuracy	77.6%
Cost	912

\[\begin{array}{l} t_0 := y + \frac{x}{z}\\ \mathbf{if}\;y \leq 3.3 \cdot 10^{+35}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 5.5 \cdot 10^{+140}:\\ \;\;\;\;y \cdot \frac{-x}{z}\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{+229}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{+272}:\\ \;\;\;\;\frac{y}{\frac{-z}{x}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 4
Accuracy	99.6%
Cost	841

\[\begin{array}{l} \mathbf{if}\;y \leq -6.6 \cdot 10^{+47} \lor \neg \left(y \leq 210000000000\right):\\ \;\;\;\;y - y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y \cdot \left(z - x\right)}{z}\\ \end{array} \]

Alternative 5
Accuracy	83.7%
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -1.26 \cdot 10^{-79} \lor \neg \left(z \leq 4.1 \cdot 10^{-50}\right):\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1 - y}{z}\\ \end{array} \]

Alternative 6
Accuracy	98.2%
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -2.4 \cdot 10^{+40} \lor \neg \left(y \leq 1\right):\\ \;\;\;\;y - y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \]

Alternative 7
Accuracy	60.8%
Cost	456

\[\begin{array}{l} \mathbf{if}\;y \leq -1.65 \cdot 10^{-30}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{-50}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 8
Accuracy	77.7%
Cost	320

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

Alternative 9
Accuracy	40.5%
Cost	64

\[y \]

Reproduce

herbie shell --seed 2023165 
(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))