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

Average Error: 10.3 → 0.2

Time: 8.9s

Precision: binary64

Cost: 840

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq -1000:\\ \;\;\;\;y \cdot \frac{z - x}{z}\\ \mathbf{elif}\;y \leq 720000:\\ \;\;\;\;\frac{x + y \cdot \left(z - x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{z - x}}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= y -1000.0)
   (* y (/ (- z x) z))
   (if (<= y 720000.0) (/ (+ x (* y (- z x))) z) (/ y (/ z (- z x))))))

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 <= -1000.0) {
		tmp = y * ((z - x) / z);
	} else if (y <= 720000.0) {
		tmp = (x + (y * (z - x))) / z;
	} else {
		tmp = y / (z / (z - x));
	}
	return tmp;
}

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
    real(8) :: tmp
    if (y <= (-1000.0d0)) then
        tmp = y * ((z - x) / z)
    else if (y <= 720000.0d0) then
        tmp = (x + (y * (z - x))) / z
    else
        tmp = y / (z / (z - x))
    end if
    code = tmp
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) {
	double tmp;
	if (y <= -1000.0) {
		tmp = y * ((z - x) / z);
	} else if (y <= 720000.0) {
		tmp = (x + (y * (z - x))) / z;
	} else {
		tmp = y / (z / (z - x));
	}
	return tmp;
}

Python

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

↓

def code(x, y, z):
	tmp = 0
	if y <= -1000.0:
		tmp = y * ((z - x) / z)
	elif y <= 720000.0:
		tmp = (x + (y * (z - x))) / z
	else:
		tmp = y / (z / (z - x))
	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 <= -1000.0)
		tmp = Float64(y * Float64(Float64(z - x) / z));
	elseif (y <= 720000.0)
		tmp = Float64(Float64(x + Float64(y * Float64(z - x))) / z);
	else
		tmp = Float64(y / Float64(z / Float64(z - x)));
	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)
	tmp = 0.0;
	if (y <= -1000.0)
		tmp = y * ((z - x) / z);
	elseif (y <= 720000.0)
		tmp = (x + (y * (z - x))) / z;
	else
		tmp = y / (z / (z - x));
	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_] := If[LessEqual[y, -1000.0], N[(y * N[(N[(z - x), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 720000.0], N[(N[(x + N[(y * N[(z - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(y / N[(z / N[(z - x), $MachinePrecision]), $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 y < -1e3

   1. Initial program 23.8

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

   2. Taylor expanded in y around inf 24.2

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

   3. Simplified 0.6

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

      [Start] 24.2	\[ \frac{y \cdot \left(z - x\right)}{z} \]
      rational.json-simplify-2 [=>] 24.2	\[ \frac{\color{blue}{\left(z - x\right) \cdot y}}{z} \]
      rational.json-simplify-49 [=>] 0.6	\[ \color{blue}{y \cdot \frac{z - x}{z}} \]

   if -1e3 < y < 7.2e5

   1. Initial program 0.1

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

   if 7.2e5 < y

   1. Initial program 23.3

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

   2. Taylor expanded in y around inf 23.5

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

   3. Simplified 0.3

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

      [Start] 23.5	\[ \frac{y \cdot \left(z - x\right)}{z} \]
      rational.json-simplify-2 [=>] 23.5	\[ \frac{\color{blue}{\left(z - x\right) \cdot y}}{z} \]
      rational.json-simplify-49 [=>] 0.3	\[ \color{blue}{y \cdot \frac{z - x}{z}} \]

   4. Applied egg-rr 0.3

      \[\leadsto \color{blue}{\frac{y}{\frac{z}{z - x}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1000:\\ \;\;\;\;y \cdot \frac{z - x}{z}\\ \mathbf{elif}\;y \leq 720000:\\ \;\;\;\;\frac{x + y \cdot \left(z - x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{z - x}}\\ \end{array} \]

Alternatives

Alternative 1
Error	1.0
Cost	712

\[\begin{array}{l} t_0 := y \cdot \frac{z - x}{z}\\ \mathbf{if}\;y \leq -5.5 \cdot 10^{+18}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	1.0
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq -5.5 \cdot 10^{+18}:\\ \;\;\;\;y \cdot \frac{z - x}{z}\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{z - x}}\\ \end{array} \]

Alternative 3
Error	20.7
Cost	456

\[\begin{array}{l} \mathbf{if}\;y \leq -1.18 \cdot 10^{-91}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 1.05 \cdot 10^{-116}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 4
Error	8.9
Cost	320

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

Alternative 5
Error	31.5
Cost	64

\[y \]

Reproduce

herbie shell --seed 2023074 
(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))