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

Average Error: 10.3 → 0.2

Time: 5.7s

Precision: binary64

Cost: 1864

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x + y \cdot \left(z - x\right)}{z}\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;\frac{z - x}{\frac{z}{y}}\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+302}:\\ \;\;\;\;y + \frac{x - x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(z - x\right) \cdot \frac{y}{z}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (+ x (* y (- z x))) z)))
   (if (<= t_0 (- INFINITY))
     (/ (- z x) (/ z y))
     (if (<= t_0 5e+302) (+ y (/ (- x (* x y)) z)) (* (- z x) (/ y 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 t_0 = (x + (y * (z - x))) / z;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = (z - x) / (z / y);
	} else if (t_0 <= 5e+302) {
		tmp = y + ((x - (x * y)) / z);
	} else {
		tmp = (z - x) * (y / z);
	}
	return tmp;
}

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) {
	double t_0 = (x + (y * (z - x))) / z;
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = (z - x) / (z / y);
	} else if (t_0 <= 5e+302) {
		tmp = y + ((x - (x * y)) / z);
	} else {
		tmp = (z - x) * (y / z);
	}
	return tmp;
}

Python

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

↓

def code(x, y, z):
	t_0 = (x + (y * (z - x))) / z
	tmp = 0
	if t_0 <= -math.inf:
		tmp = (z - x) / (z / y)
	elif t_0 <= 5e+302:
		tmp = y + ((x - (x * y)) / z)
	else:
		tmp = (z - x) * (y / 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)
	t_0 = Float64(Float64(x + Float64(y * Float64(z - x))) / z)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = Float64(Float64(z - x) / Float64(z / y));
	elseif (t_0 <= 5e+302)
		tmp = Float64(y + Float64(Float64(x - Float64(x * y)) / z));
	else
		tmp = Float64(Float64(z - x) * Float64(y / z));
	end
	return tmp
end

MATLAB

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

↓

function tmp_2 = code(x, y, z)
	t_0 = (x + (y * (z - x))) / z;
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = (z - x) / (z / y);
	elseif (t_0 <= 5e+302)
		tmp = y + ((x - (x * y)) / z);
	else
		tmp = (z - x) * (y / z);
	end
	tmp_2 = tmp;
end

Wolfram

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

↓

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(x + N[(y * N[(z - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], N[(N[(z - x), $MachinePrecision] / N[(z / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 5e+302], N[(y + N[(N[(x - N[(x * y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(N[(z - x), $MachinePrecision] * N[(y / z), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	10.3
Target	0.0
Herbie	0.2

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

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x (*.f64 y (-.f64 z x))) z) < -inf.0

   1. Initial program 64.0

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

   2. Taylor expanded in y around inf 64.0

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

   3. Simplified 0.1

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

      [Start] 64.0	\[ \frac{y \cdot \left(z - x\right)}{z} \]
      *-commutative [=>] 64.0	\[ \frac{\color{blue}{\left(z - x\right) \cdot y}}{z} \]
      associate-/l* [=>] 0.1	\[ \color{blue}{\frac{z - x}{\frac{z}{y}}} \]

   if -inf.0 < (/.f64 (+.f64 x (*.f64 y (-.f64 z x))) z) < 5e302

   1. Initial program 0.1

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

   2. Taylor expanded in x around 0 4.3

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

   3. Simplified 0.0

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

      [Start] 4.3	\[ \left(-1 \cdot \frac{y}{z} + \frac{1}{z}\right) \cdot x + y \]
      +-commutative [=>] 4.3	\[ \color{blue}{y + \left(-1 \cdot \frac{y}{z} + \frac{1}{z}\right) \cdot x} \]
      *-commutative [=>] 4.3	\[ y + \color{blue}{x \cdot \left(-1 \cdot \frac{y}{z} + \frac{1}{z}\right)} \]
      distribute-rgt-in [=>] 4.3	\[ y + \color{blue}{\left(\left(-1 \cdot \frac{y}{z}\right) \cdot x + \frac{1}{z} \cdot x\right)} \]
      associate-*r/ [=>] 4.3	\[ y + \left(\color{blue}{\frac{-1 \cdot y}{z}} \cdot x + \frac{1}{z} \cdot x\right) \]
      associate-/r/ [<=] 0.2	\[ y + \left(\color{blue}{\frac{-1 \cdot y}{\frac{z}{x}}} + \frac{1}{z} \cdot x\right) \]
      associate-*r/ [<=] 0.2	\[ y + \left(\color{blue}{-1 \cdot \frac{y}{\frac{z}{x}}} + \frac{1}{z} \cdot x\right) \]
      associate-/l* [<=] 0.2	\[ y + \left(-1 \cdot \color{blue}{\frac{y \cdot x}{z}} + \frac{1}{z} \cdot x\right) \]
      associate-*l/ [=>] 0.0	\[ y + \left(-1 \cdot \frac{y \cdot x}{z} + \color{blue}{\frac{1 \cdot x}{z}}\right) \]
      *-lft-identity [=>] 0.0	\[ y + \left(-1 \cdot \frac{y \cdot x}{z} + \frac{\color{blue}{x}}{z}\right) \]
      mul-1-neg [=>] 0.0	\[ y + \left(\color{blue}{\left(-\frac{y \cdot x}{z}\right)} + \frac{x}{z}\right) \]
      neg-sub0 [=>] 0.0	\[ y + \left(\color{blue}{\left(0 - \frac{y \cdot x}{z}\right)} + \frac{x}{z}\right) \]
      associate-+l- [=>] 0.0	\[ y + \color{blue}{\left(0 - \left(\frac{y \cdot x}{z} - \frac{x}{z}\right)\right)} \]
      associate-/l* [=>] 0.0	\[ y + \left(0 - \left(\color{blue}{\frac{y}{\frac{z}{x}}} - \frac{x}{z}\right)\right) \]
      *-lft-identity [<=] 0.0	\[ y + \left(0 - \left(\frac{y}{\frac{z}{x}} - \frac{\color{blue}{1 \cdot x}}{z}\right)\right) \]
      associate-/l* [=>] 0.2	\[ y + \left(0 - \left(\frac{y}{\frac{z}{x}} - \color{blue}{\frac{1}{\frac{z}{x}}}\right)\right) \]
      div-sub [<=] 0.2	\[ y + \left(0 - \color{blue}{\frac{y - 1}{\frac{z}{x}}}\right) \]
      associate-/l* [<=] 0.0	\[ y + \left(0 - \color{blue}{\frac{\left(y - 1\right) \cdot x}{z}}\right) \]
      neg-sub0 [<=] 0.0	\[ y + \color{blue}{\left(-\frac{\left(y - 1\right) \cdot x}{z}\right)} \]
      mul-1-neg [<=] 0.0	\[ y + \color{blue}{-1 \cdot \frac{\left(y - 1\right) \cdot x}{z}} \]

   if 5e302 < (/.f64 (+.f64 x (*.f64 y (-.f64 z x))) z)

   1. Initial program 61.6

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

   2. Taylor expanded in y around inf 62.5

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

   3. Simplified 1.8

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

      [Start] 62.5	\[ \frac{y \cdot \left(z - x\right)}{z} \]
      associate-/l* [=>] 1.0	\[ \color{blue}{\frac{y}{\frac{z}{z - x}}} \]
      associate-/r/ [=>] 1.8	\[ \color{blue}{\frac{y}{z} \cdot \left(z - x\right)} \]

3. Recombined 3 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + y \cdot \left(z - x\right)}{z} \leq -\infty:\\ \;\;\;\;\frac{z - x}{\frac{z}{y}}\\ \mathbf{elif}\;\frac{x + y \cdot \left(z - x\right)}{z} \leq 5 \cdot 10^{+302}:\\ \;\;\;\;y + \frac{x - x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\left(z - x\right) \cdot \frac{y}{z}\\ \end{array} \]

Alternatives

Alternative 1
Error	10.4
Cost	912

\[\begin{array}{l} t_0 := \frac{-y}{\frac{z}{x}}\\ \mathbf{if}\;y \leq -1.75 \cdot 10^{+140}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq -2.3 \cdot 10^{+112}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{+19}:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{elif}\;y \leq 6.6 \cdot 10^{+98}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 2
Error	10.4
Cost	912

\[\begin{array}{l} \mathbf{if}\;y \leq -1.75 \cdot 10^{+140}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq -1.3 \cdot 10^{+112}:\\ \;\;\;\;\frac{-y}{\frac{z}{x}}\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{+24}:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{+98}:\\ \;\;\;\;y \cdot \frac{-x}{z}\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 3
Error	4.3
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -3.6 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;\left(z - x\right) \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \]

Alternative 4
Error	3.9
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -3.6 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;\frac{z - x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \]

Alternative 5
Error	20.4
Cost	456

\[\begin{array}{l} \mathbf{if}\;y \leq -1.1 \cdot 10^{-67}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 10^{-89}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 6
Error	8.9
Cost	320

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

Alternative 7
Error	31.4
Cost	64

\[y \]

Reproduce

herbie shell --seed 2023057 
(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))