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

Percentage Accurate: 88.1% → 99.9%

Time: 5.8s

Precision: binary64

Cost: 840

• 21.3% of points produce a very large (infinite) output. You may want to add a precondition.

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq -1.3 \cdot 10^{+15}:\\ \;\;\;\;y - \frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;y \leq 2000000000000:\\ \;\;\;\;\frac{x + y \cdot \left(z - x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \frac{x}{z}\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.3e+15)
   (- y (/ y (/ z x)))
   (if (<= y 2000000000000.0)
     (/ (+ x (* y (- z x))) z)
     (* y (- 1.0 (/ 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 <= -1.3e+15) {
		tmp = y - (y / (z / x));
	} else if (y <= 2000000000000.0) {
		tmp = (x + (y * (z - x))) / z;
	} else {
		tmp = y * (1.0 - (x / z));
	}
	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 <= (-1.3d+15)) then
        tmp = y - (y / (z / x))
    else if (y <= 2000000000000.0d0) then
        tmp = (x + (y * (z - x))) / z
    else
        tmp = y * (1.0d0 - (x / z))
    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 <= -1.3e+15) {
		tmp = y - (y / (z / x));
	} else if (y <= 2000000000000.0) {
		tmp = (x + (y * (z - x))) / z;
	} else {
		tmp = y * (1.0 - (x / z));
	}
	return tmp;
}

Python

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

↓

def code(x, y, z):
	tmp = 0
	if y <= -1.3e+15:
		tmp = y - (y / (z / x))
	elif y <= 2000000000000.0:
		tmp = (x + (y * (z - x))) / z
	else:
		tmp = y * (1.0 - (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 <= -1.3e+15)
		tmp = Float64(y - Float64(y / Float64(z / x)));
	elseif (y <= 2000000000000.0)
		tmp = Float64(Float64(x + Float64(y * Float64(z - x))) / z);
	else
		tmp = Float64(y * Float64(1.0 - Float64(x / 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)
	tmp = 0.0;
	if (y <= -1.3e+15)
		tmp = y - (y / (z / x));
	elseif (y <= 2000000000000.0)
		tmp = (x + (y * (z - x))) / z;
	else
		tmp = y * (1.0 - (x / 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_] := If[LessEqual[y, -1.3e+15], N[(y - N[(y / N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2000000000000.0], N[(N[(x + N[(y * N[(z - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(y * N[(1.0 - N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Target

Original	88.1%
Target	93.5%
Herbie	99.9%

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

Derivation

1. Split input into 3 regimes

2. if y < -1.3e15

   1. Initial program 79.7%

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

   2. Taylor expanded in y around 0 99.9%

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

   3. Taylor expanded in y around inf 99.9%

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

   4. Simplified 100.0%

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

      [Start] 99.9	\[ \left(1 - \frac{x}{z}\right) \cdot y \]
      sub-neg [=>] 99.9	\[ \color{blue}{\left(1 + \left(-\frac{x}{z}\right)\right)} \cdot y \]
      +-commutative [=>] 99.9	\[ \color{blue}{\left(\left(-\frac{x}{z}\right) + 1\right)} \cdot y \]
      distribute-rgt1-in [<=] 99.9	\[ \color{blue}{y + \left(-\frac{x}{z}\right) \cdot y} \]
      *-commutative [<=] 99.9	\[ y + \color{blue}{y \cdot \left(-\frac{x}{z}\right)} \]
      distribute-rgt-neg-out [=>] 99.9	\[ y + \color{blue}{\left(-y \cdot \frac{x}{z}\right)} \]
      unsub-neg [=>] 99.9	\[ \color{blue}{y - y \cdot \frac{x}{z}} \]
      associate-*r/ [=>] 94.1	\[ y - \color{blue}{\frac{y \cdot x}{z}} \]
      associate-/l* [=>] 100.0	\[ y - \color{blue}{\frac{y}{\frac{z}{x}}} \]

   if -1.3e15 < y < 2e12

   1. Initial program 99.9%

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

   if 2e12 < y

   1. Initial program 69.1%

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

   2. Taylor expanded in y around 0 89.5%

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

   3. Taylor expanded in y around inf 100.0%

      \[\leadsto \color{blue}{\left(1 - \frac{x}{z}\right) \cdot y} \]
3. Recombined 3 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.3 \cdot 10^{+15}:\\ \;\;\;\;y - \frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;y \leq 2000000000000:\\ \;\;\;\;\frac{x + y \cdot \left(z - x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \frac{x}{z}\right)\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	99.1%
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -4.2 \lor \neg \left(y \leq 1\right):\\ \;\;\;\;y \cdot \left(1 - \frac{x}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \]

Alternative 2
Accuracy	99.1%
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq -4.2:\\ \;\;\;\;y - \frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;y \leq 1:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(1 - \frac{x}{z}\right)\\ \end{array} \]

Alternative 3
Accuracy	63.2%
Cost	585

\[\begin{array}{l} \mathbf{if}\;y \leq -6.4 \cdot 10^{-45} \lor \neg \left(y \leq 0.0005\right):\\ \;\;\;\;z \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z}\\ \end{array} \]

Alternative 4
Accuracy	76.8%
Cost	516

\[\begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{+183}:\\ \;\;\;\;y \cdot \frac{x}{-z}\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \]

Alternative 5
Accuracy	76.7%
Cost	516

\[\begin{array}{l} \mathbf{if}\;x \leq -2.6 \cdot 10^{+183}:\\ \;\;\;\;\frac{y}{\frac{-z}{x}}\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \]

Alternative 6
Accuracy	57.1%
Cost	456

\[\begin{array}{l} \mathbf{if}\;z \leq -2.5 \cdot 10^{-93}:\\ \;\;\;\;y\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{-29}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 7
Accuracy	77.9%
Cost	320

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

Alternative 8
Accuracy	40.3%
Cost	64

\[y \]

Reproduce

herbie shell --seed 2023161 
(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))