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

Average Error: 10.2 → 0.1

Time: 8.5s

Precision: binary64

Cost: 840

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq -2.05 \cdot 10^{+16}:\\ \;\;\;\;y \cdot \left(1 - \frac{x}{z}\right)\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{+14}:\\ \;\;\;\;\frac{x + y \cdot \left(z - x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{z - x}}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= y -2.05e+16)
   (* y (- 1.0 (/ x z)))
   (if (<= y 9.5e+14) (/ (+ 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 <= -2.05e+16) {
		tmp = y * (1.0 - (x / z));
	} else if (y <= 9.5e+14) {
		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 <= (-2.05d+16)) then
        tmp = y * (1.0d0 - (x / z))
    else if (y <= 9.5d+14) 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 <= -2.05e+16) {
		tmp = y * (1.0 - (x / z));
	} else if (y <= 9.5e+14) {
		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 <= -2.05e+16:
		tmp = y * (1.0 - (x / z))
	elif y <= 9.5e+14:
		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 <= -2.05e+16)
		tmp = Float64(y * Float64(1.0 - Float64(x / z)));
	elseif (y <= 9.5e+14)
		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 <= -2.05e+16)
		tmp = y * (1.0 - (x / z));
	elseif (y <= 9.5e+14)
		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, -2.05e+16], N[(y * N[(1.0 - N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 9.5e+14], 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	10.2
Target	0.0
Herbie	0.1

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

Derivation

1. Split input into 3 regimes

2. if y < -2.05e16

   1. Initial program 24.6

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

   2. Simplified 24.6

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

      [Start] 24.6	\[ \frac{x + y \cdot \left(z - x\right)}{z} \]
      +-commutative [=>] 24.6	\[ \frac{\color{blue}{y \cdot \left(z - x\right) + x}}{z} \]
      fma-def [=>] 24.6	\[ \frac{\color{blue}{\mathsf{fma}\left(y, z - x, x\right)}}{z} \]

   3. Applied egg-rr 50.1

      \[\leadsto \frac{\color{blue}{\frac{{x}^{3} + {\left(y \cdot \left(z - x\right)\right)}^{3}}{\left(y \cdot \left(z - x\right)\right) \cdot \left(y \cdot \left(z - x\right)\right) + x \cdot \left(x - y \cdot \left(z - x\right)\right)}}}{z} \]

   4. Taylor expanded in y around inf 33.0

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

   5. Taylor expanded in y around inf 0.1

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

   if -2.05e16 < y < 9.5e14

   1. Initial program 0.1

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

   if 9.5e14 < y

   1. Initial program 24.3

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

   2. Taylor expanded in y around inf 24.3

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

   3. Simplified 0.1

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

      [Start] 24.3	\[ \frac{y \cdot \left(z - x\right)}{z} \]
      associate-/l* [=>] 0.1	\[ \color{blue}{\frac{y}{\frac{z}{z - x}}} \]

3. Recombined 3 regimes into one program.

4. Final simplification 0.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.05 \cdot 10^{+16}:\\ \;\;\;\;y \cdot \left(1 - \frac{x}{z}\right)\\ \mathbf{elif}\;y \leq 9.5 \cdot 10^{+14}:\\ \;\;\;\;\frac{x + y \cdot \left(z - x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{z - x}}\\ \end{array} \]

Alternatives

Alternative 1
Error	1.3
Cost	840

\[\begin{array}{l} \mathbf{if}\;y \leq -3.8 \cdot 10^{+103}:\\ \;\;\;\;y \cdot \left(1 - \frac{x}{z}\right)\\ \mathbf{elif}\;y \leq 1.42 \cdot 10^{-37}:\\ \;\;\;\;y + \frac{x}{\frac{z}{1 - y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{z - x}}\\ \end{array} \]

Alternative 2
Error	1.6
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -45000000 \lor \neg \left(y \leq 1.42 \cdot 10^{-37}\right):\\ \;\;\;\;y \cdot \left(1 - \frac{x}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;y + \frac{x}{z}\\ \end{array} \]

Alternative 3
Error	1.6
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq -45000000:\\ \;\;\;\;y \cdot \left(1 - \frac{x}{z}\right)\\ \mathbf{elif}\;y \leq 1.42 \cdot 10^{-37}:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{z - x}}\\ \end{array} \]

Alternative 4
Error	10.0
Cost	648

\[\begin{array}{l} \mathbf{if}\;y \leq 7 \cdot 10^{+34}:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{elif}\;y \leq 2.55 \cdot 10^{+122}:\\ \;\;\;\;\frac{-x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 5
Error	9.9
Cost	648

\[\begin{array}{l} \mathbf{if}\;y \leq 1.25 \cdot 10^{+34}:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{elif}\;y \leq 9.4 \cdot 10^{+123}:\\ \;\;\;\;\frac{-y}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 6
Error	19.4
Cost	456

\[\begin{array}{l} \mathbf{if}\;y \leq -3.2 \cdot 10^{-44}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 3.7 \cdot 10^{-38}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 7
Error	8.6
Cost	320

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

Alternative 8
Error	31.7
Cost	64

\[y \]

Reproduce

herbie shell --seed 2023060 
(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))