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

Average Accuracy: 83.3% → 99.5%

Time: 7.4s

Precision: binary64

Cost: 840

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq -5.4 \cdot 10^{+69}:\\ \;\;\;\;\frac{y}{\frac{z}{z - x}}\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{+105}:\\ \;\;\;\;y + \frac{x}{\frac{z}{1 - y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z - x}{z}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= y -5.4e+69)
   (/ y (/ z (- z x)))
   (if (<= y 6.5e+105) (+ y (/ x (/ z (- 1.0 y)))) (* y (/ (- z 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 <= -5.4e+69) {
		tmp = y / (z / (z - x));
	} else if (y <= 6.5e+105) {
		tmp = y + (x / (z / (1.0 - y)));
	} else {
		tmp = y * ((z - 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 <= (-5.4d+69)) then
        tmp = y / (z / (z - x))
    else if (y <= 6.5d+105) then
        tmp = y + (x / (z / (1.0d0 - y)))
    else
        tmp = y * ((z - 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 <= -5.4e+69) {
		tmp = y / (z / (z - x));
	} else if (y <= 6.5e+105) {
		tmp = y + (x / (z / (1.0 - y)));
	} else {
		tmp = y * ((z - 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 <= -5.4e+69:
		tmp = y / (z / (z - x))
	elif y <= 6.5e+105:
		tmp = y + (x / (z / (1.0 - y)))
	else:
		tmp = y * ((z - 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 <= -5.4e+69)
		tmp = Float64(y / Float64(z / Float64(z - x)));
	elseif (y <= 6.5e+105)
		tmp = Float64(y + Float64(x / Float64(z / Float64(1.0 - y))));
	else
		tmp = Float64(y * Float64(Float64(z - 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 <= -5.4e+69)
		tmp = y / (z / (z - x));
	elseif (y <= 6.5e+105)
		tmp = y + (x / (z / (1.0 - y)));
	else
		tmp = y * ((z - 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, -5.4e+69], N[(y / N[(z / N[(z - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 6.5e+105], N[(y + N[(x / N[(z / N[(1.0 - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(N[(z - x), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]]

Target

Original	83.3%
Target	99.9%
Herbie	99.5%

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

Derivation

1. Split input into 3 regimes

2. if y < -5.3999999999999996e69

   1. Initial program 53.2%

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

   2. Taylor expanded in y around inf 53.2%

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

   3. Simplified 99.9%

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

      [Start] 53.2	\[ \frac{y \cdot \left(z - x\right)}{z} \]
      associate-/l* [=>] 99.9	\[ \color{blue}{\frac{y}{\frac{z}{z - x}}} \]

   if -5.3999999999999996e69 < y < 6.50000000000000049e105

   1. Initial program 97.5%

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

   2. Taylor expanded in x around inf 99.1%

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

   3. Simplified 99.3%

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

      [Start] 99.1	\[ y + \frac{\left(1 + -1 \cdot y\right) \cdot x}{z} \]
      *-commutative [=>] 99.1	\[ y + \frac{\color{blue}{x \cdot \left(1 + -1 \cdot y\right)}}{z} \]
      associate-/l* [=>] 99.3	\[ y + \color{blue}{\frac{x}{\frac{z}{1 + -1 \cdot y}}} \]
      mul-1-neg [=>] 99.3	\[ y + \frac{x}{\frac{z}{1 + \color{blue}{\left(-y\right)}}} \]
      unsub-neg [=>] 99.3	\[ y + \frac{x}{\frac{z}{\color{blue}{1 - y}}} \]

   if 6.50000000000000049e105 < y

   1. Initial program 47.9%

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

   2. Simplified 47.9%

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

      [Start] 47.9	\[ \frac{x + y \cdot \left(z - x\right)}{z} \]
      +-commutative [=>] 47.9	\[ \frac{\color{blue}{y \cdot \left(z - x\right) + x}}{z} \]
      fma-def [=>] 47.9	\[ \frac{\color{blue}{\mathsf{fma}\left(y, z - x, x\right)}}{z} \]

   3. Taylor expanded in y around inf 47.9%

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

   4. Simplified 99.9%

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

      [Start] 47.9	\[ \frac{y \cdot \left(z - x\right)}{z} \]
      *-commutative [=>] 47.9	\[ \frac{\color{blue}{\left(z - x\right) \cdot y}}{z} \]
      associate-/l* [=>] 85.4	\[ \color{blue}{\frac{z - x}{\frac{z}{y}}} \]
      associate-/r/ [=>] 99.9	\[ \color{blue}{\frac{z - x}{z} \cdot y} \]
      remove-double-neg [<=] 99.9	\[ \frac{\color{blue}{-\left(-\left(z - x\right)\right)}}{z} \cdot y \]
      neg-mul-1 [=>] 99.9	\[ \frac{-\color{blue}{-1 \cdot \left(z - x\right)}}{z} \cdot y \]
      *-commutative [=>] 99.9	\[ \frac{-\color{blue}{\left(z - x\right) \cdot -1}}{z} \cdot y \]
      distribute-rgt-neg-in [=>] 99.9	\[ \frac{\color{blue}{\left(z - x\right) \cdot \left(--1\right)}}{z} \cdot y \]
      metadata-eval [=>] 99.9	\[ \frac{\left(z - x\right) \cdot \color{blue}{1}}{z} \cdot y \]
      associate-*r/ [<=] 99.7	\[ \color{blue}{\left(\left(z - x\right) \cdot \frac{1}{z}\right)} \cdot y \]
      unpow-1 [<=] 99.7	\[ \left(\left(z - x\right) \cdot \color{blue}{{z}^{-1}}\right) \cdot y \]
      *-commutative [<=] 99.7	\[ \color{blue}{y \cdot \left(\left(z - x\right) \cdot {z}^{-1}\right)} \]
      unpow-1 [=>] 99.7	\[ y \cdot \left(\left(z - x\right) \cdot \color{blue}{\frac{1}{z}}\right) \]
      associate-*r/ [=>] 99.9	\[ y \cdot \color{blue}{\frac{\left(z - x\right) \cdot 1}{z}} \]
      metadata-eval [<=] 99.9	\[ y \cdot \frac{\left(z - x\right) \cdot \color{blue}{\left(--1\right)}}{z} \]
      distribute-rgt-neg-in [<=] 99.9	\[ y \cdot \frac{\color{blue}{-\left(z - x\right) \cdot -1}}{z} \]
      *-commutative [<=] 99.9	\[ y \cdot \frac{-\color{blue}{-1 \cdot \left(z - x\right)}}{z} \]
      neg-mul-1 [<=] 99.9	\[ y \cdot \frac{-\color{blue}{\left(-\left(z - x\right)\right)}}{z} \]
      remove-double-neg [=>] 99.9	\[ y \cdot \frac{\color{blue}{z - x}}{z} \]

3. Recombined 3 regimes into one program.

4. Final simplification 99.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.4 \cdot 10^{+69}:\\ \;\;\;\;\frac{y}{\frac{z}{z - x}}\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{+105}:\\ \;\;\;\;y + \frac{x}{\frac{z}{1 - y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z - x}{z}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	98.9%
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -116000000 \lor \neg \left(y \leq 0.07\right):\\ \;\;\;\;y \cdot \frac{z - x}{z}\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \]

Alternative 2
Accuracy	98.9%
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq -116000000:\\ \;\;\;\;\frac{y}{\frac{z}{z - x}}\\ \mathbf{elif}\;y \leq 0.07:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z - x}{z}\\ \end{array} \]

Alternative 3
Accuracy	84.9%
Cost	648

\[\begin{array}{l} \mathbf{if}\;y \leq 1.35 \cdot 10^{+72}:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{+140}:\\ \;\;\;\;y \cdot \frac{-x}{z}\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 4
Accuracy	84.8%
Cost	648

\[\begin{array}{l} \mathbf{if}\;y \leq 7 \cdot 10^{+70}:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{elif}\;y \leq 1.9 \cdot 10^{+140}:\\ \;\;\;\;\frac{-y}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 5
Accuracy	69.1%
Cost	456

\[\begin{array}{l} \mathbf{if}\;y \leq -1.76 \cdot 10^{-90}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 3.25 \cdot 10^{-18}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 6
Accuracy	86.5%
Cost	320

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

Alternative 7
Accuracy	50.0%
Cost	64

\[y \]

Reproduce

herbie shell --seed 2023135 
(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))