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

Average Accuracy: 84.2% → 99.9%

Time: 7.9s

Precision: binary64

Cost: 969

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq -1.2 \cdot 10^{+26} \lor \neg \left(y \leq 5 \cdot 10^{+26}\right):\\ \;\;\;\;\frac{y}{\frac{z}{z - x}}\\ \mathbf{else}:\\ \;\;\;\;\left(y + \frac{x}{z}\right) - \frac{x}{\frac{z}{y}}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (or (<= y -1.2e+26) (not (<= y 5e+26)))
   (/ y (/ z (- z x)))
   (- (+ y (/ x z)) (/ 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) {
	double tmp;
	if ((y <= -1.2e+26) || !(y <= 5e+26)) {
		tmp = y / (z / (z - x));
	} else {
		tmp = (y + (x / z)) - (x / (z / y));
	}
	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 <= (-1.2d+26)) .or. (.not. (y <= 5d+26))) then
        tmp = y / (z / (z - x))
    else
        tmp = (y + (x / z)) - (x / (z / y))
    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 <= -1.2e+26) || !(y <= 5e+26)) {
		tmp = y / (z / (z - x));
	} else {
		tmp = (y + (x / z)) - (x / (z / y));
	}
	return tmp;
}

Python

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

↓

def code(x, y, z):
	tmp = 0
	if (y <= -1.2e+26) or not (y <= 5e+26):
		tmp = y / (z / (z - x))
	else:
		tmp = (y + (x / z)) - (x / (z / y))
	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 <= -1.2e+26) || !(y <= 5e+26))
		tmp = Float64(y / Float64(z / Float64(z - x)));
	else
		tmp = Float64(Float64(y + Float64(x / z)) - Float64(x / Float64(z / y)));
	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 <= -1.2e+26) || ~((y <= 5e+26)))
		tmp = y / (z / (z - x));
	else
		tmp = (y + (x / z)) - (x / (z / y));
	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[Or[LessEqual[y, -1.2e+26], N[Not[LessEqual[y, 5e+26]], $MachinePrecision]], N[(y / N[(z / N[(z - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y + N[(x / z), $MachinePrecision]), $MachinePrecision] - N[(x / N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Target

Original	84.2%
Target	99.9%
Herbie	99.9%

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

Derivation

1. Split input into 2 regimes

2. if y < -1.20000000000000002e26 or 5.0000000000000001e26 < y

   1. Initial program 61.3%

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

   2. Taylor expanded in y around inf 61.3%

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

   3. Simplified 99.9%

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

      [Start] 61.3	\[ \frac{y \cdot \left(z - x\right)}{z} \]
      associate-/l* [=>] 99.9	\[ \color{blue}{\frac{y}{\frac{z}{z - x}}} \]

   if -1.20000000000000002e26 < y < 5.0000000000000001e26

   1. Initial program 99.8%

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

   2. Taylor expanded in z around 0 99.9%

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

   3. Simplified 99.9%

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

      [Start] 99.9	\[ -1 \cdot \frac{y \cdot x}{z} + \left(y + \frac{x}{z}\right) \]
      +-commutative [=>] 99.9	\[ \color{blue}{\left(y + \frac{x}{z}\right) + -1 \cdot \frac{y \cdot x}{z}} \]
      mul-1-neg [=>] 99.9	\[ \left(y + \frac{x}{z}\right) + \color{blue}{\left(-\frac{y \cdot x}{z}\right)} \]
      unsub-neg [=>] 99.9	\[ \color{blue}{\left(y + \frac{x}{z}\right) - \frac{y \cdot x}{z}} \]
      *-commutative [=>] 99.9	\[ \left(y + \frac{x}{z}\right) - \frac{\color{blue}{x \cdot y}}{z} \]
      associate-/l* [=>] 99.9	\[ \left(y + \frac{x}{z}\right) - \color{blue}{\frac{x}{\frac{z}{y}}} \]

3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.2 \cdot 10^{+26} \lor \neg \left(y \leq 5 \cdot 10^{+26}\right):\\ \;\;\;\;\frac{y}{\frac{z}{z - x}}\\ \mathbf{else}:\\ \;\;\;\;\left(y + \frac{x}{z}\right) - \frac{x}{\frac{z}{y}}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	94.0%
Cost	844

\[\begin{array}{l} t_0 := \left(z - x\right) \cdot \frac{y}{z}\\ \mathbf{if}\;y \leq -1:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{-16}:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{+222}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 2
Accuracy	99.8%
Cost	841

\[\begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{+17} \lor \neg \left(y \leq 380000000\right):\\ \;\;\;\;\frac{y}{\frac{z}{z - x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + y \cdot \left(z - x\right)}{z}\\ \end{array} \]

Alternative 3
Accuracy	97.8%
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -1 \lor \neg \left(y \leq 1.42 \cdot 10^{-36}\right):\\ \;\;\;\;\frac{y}{\frac{z}{z - x}}\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \]

Alternative 4
Accuracy	69.2%
Cost	456

\[\begin{array}{l} \mathbf{if}\;y \leq -3.05 \cdot 10^{-43}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 1.4 \cdot 10^{-72}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 5
Accuracy	86.0%
Cost	320

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

Alternative 6
Accuracy	50.6%
Cost	64

\[y \]

Reproduce

herbie shell --seed 2023133 
(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))