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

Average Error: 10.5 → 0.2

Time: 7.3s

Precision: binary64

Cost: 840

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq -1.7691363646300464 \cdot 10^{+23}:\\ \;\;\;\;y \cdot \left(1 - \frac{x}{z}\right)\\ \mathbf{elif}\;y \leq 6321005.695493938:\\ \;\;\;\;\frac{x + y \cdot \left(z - x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;y - \frac{y}{\frac{z}{x}}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.7691363646300464e+23)
   (* y (- 1.0 (/ x z)))
   (if (<= y 6321005.695493938)
     (/ (+ x (* y (- z x))) z)
     (- y (/ y (/ 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 <= -1.7691363646300464e+23) {
		tmp = y * (1.0 - (x / z));
	} else if (y <= 6321005.695493938) {
		tmp = (x + (y * (z - x))) / z;
	} else {
		tmp = y - (y / (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 <= (-1.7691363646300464d+23)) then
        tmp = y * (1.0d0 - (x / z))
    else if (y <= 6321005.695493938d0) then
        tmp = (x + (y * (z - x))) / z
    else
        tmp = y - (y / (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 <= -1.7691363646300464e+23) {
		tmp = y * (1.0 - (x / z));
	} else if (y <= 6321005.695493938) {
		tmp = (x + (y * (z - x))) / z;
	} else {
		tmp = y - (y / (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 <= -1.7691363646300464e+23:
		tmp = y * (1.0 - (x / z))
	elif y <= 6321005.695493938:
		tmp = (x + (y * (z - x))) / z
	else:
		tmp = y - (y / (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 <= -1.7691363646300464e+23)
		tmp = Float64(y * Float64(1.0 - Float64(x / z)));
	elseif (y <= 6321005.695493938)
		tmp = Float64(Float64(x + Float64(y * Float64(z - x))) / z);
	else
		tmp = Float64(y - Float64(y / 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 <= -1.7691363646300464e+23)
		tmp = y * (1.0 - (x / z));
	elseif (y <= 6321005.695493938)
		tmp = (x + (y * (z - x))) / z;
	else
		tmp = y - (y / (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, -1.7691363646300464e+23], N[(y * N[(1.0 - N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 6321005.695493938], N[(N[(x + N[(y * N[(z - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], N[(y - N[(y / N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Target

Original	10.5
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 < -1.76913636463004642e23

   1. Initial program 23.3

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

   2. Taylor expanded in y around inf 23.3

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

   3. Simplified 0.1

      \[\leadsto \color{blue}{y \cdot \left(1 - \frac{x}{z}\right)} \]
      Proof
      (*.f64 y (-.f64 1 (/.f64 x z))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= distribute-rgt-out--_binary64 (-.f64 (*.f64 1 y) (*.f64 (/.f64 x z) y))): 1 points increase in error, 1 points decrease in error
      (-.f64 (*.f64 1 y) (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 x y) z))): 40 points increase in error, 17 points decrease in error
      (-.f64 (*.f64 1 y) (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 y x)) z)): 0 points increase in error, 0 points decrease in error
      (Rewrite=> fma-neg_binary64 (fma.f64 1 y (neg.f64 (/.f64 (*.f64 y x) z)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (Rewrite<= *-inverses_binary64 (/.f64 z z)) y (neg.f64 (/.f64 (*.f64 y x) z))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 (/.f64 z z) y) (/.f64 (*.f64 y x) z))): 0 points increase in error, 0 points decrease in error
      (-.f64 (Rewrite<= associate-/r/_binary64 (/.f64 z (/.f64 z y))) (/.f64 (*.f64 y x) z)): 65 points increase in error, 9 points decrease in error
      (-.f64 (/.f64 z (/.f64 z y)) (/.f64 (Rewrite=> *-commutative_binary64 (*.f64 x y)) z)): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 z (/.f64 z y)) (Rewrite=> associate-/l*_binary64 (/.f64 x (/.f64 z y)))): 26 points increase in error, 38 points decrease in error
      (Rewrite<= div-sub_binary64 (/.f64 (-.f64 z x) (/.f64 z y))): 4 points increase in error, 1 points decrease in error
      (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (-.f64 z x) y) z)): 86 points increase in error, 79 points decrease in error
      (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 y (-.f64 z x))) z): 0 points increase in error, 0 points decrease in error

   if -1.76913636463004642e23 < y < 6321005.6954939384

   1. Initial program 0.1

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

   if 6321005.6954939384 < y

   1. Initial program 25.0

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

   2. Taylor expanded in y around inf 25.3

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

   3. Simplified 0.3

      \[\leadsto \color{blue}{y \cdot \left(1 - \frac{x}{z}\right)} \]
      Proof
      (*.f64 y (-.f64 1 (/.f64 x z))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= distribute-rgt-out--_binary64 (-.f64 (*.f64 1 y) (*.f64 (/.f64 x z) y))): 1 points increase in error, 1 points decrease in error
      (-.f64 (*.f64 1 y) (Rewrite=> associate-*l/_binary64 (/.f64 (*.f64 x y) z))): 40 points increase in error, 17 points decrease in error
      (-.f64 (*.f64 1 y) (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 y x)) z)): 0 points increase in error, 0 points decrease in error
      (Rewrite=> fma-neg_binary64 (fma.f64 1 y (neg.f64 (/.f64 (*.f64 y x) z)))): 0 points increase in error, 0 points decrease in error
      (fma.f64 (Rewrite<= *-inverses_binary64 (/.f64 z z)) y (neg.f64 (/.f64 (*.f64 y x) z))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= fma-neg_binary64 (-.f64 (*.f64 (/.f64 z z) y) (/.f64 (*.f64 y x) z))): 0 points increase in error, 0 points decrease in error
      (-.f64 (Rewrite<= associate-/r/_binary64 (/.f64 z (/.f64 z y))) (/.f64 (*.f64 y x) z)): 65 points increase in error, 9 points decrease in error
      (-.f64 (/.f64 z (/.f64 z y)) (/.f64 (Rewrite=> *-commutative_binary64 (*.f64 x y)) z)): 0 points increase in error, 0 points decrease in error
      (-.f64 (/.f64 z (/.f64 z y)) (Rewrite=> associate-/l*_binary64 (/.f64 x (/.f64 z y)))): 26 points increase in error, 38 points decrease in error
      (Rewrite<= div-sub_binary64 (/.f64 (-.f64 z x) (/.f64 z y))): 4 points increase in error, 1 points decrease in error
      (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (-.f64 z x) y) z)): 86 points increase in error, 79 points decrease in error
      (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 y (-.f64 z x))) z): 0 points increase in error, 0 points decrease in error

   4. Taylor expanded in y around 0 0.3

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

   5. Simplified 0.3

      \[\leadsto \color{blue}{y - \frac{y}{\frac{z}{x}}} \]
      Proof
      (-.f64 y (/.f64 y (/.f64 z x))): 0 points increase in error, 0 points decrease in error
      (-.f64 (Rewrite<= *-rgt-identity_binary64 (*.f64 y 1)) (/.f64 y (/.f64 z x))): 0 points increase in error, 0 points decrease in error
      (-.f64 (*.f64 y 1) (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 y x) z))): 37 points increase in error, 18 points decrease in error
      (-.f64 (*.f64 y 1) (Rewrite<= associate-*r/_binary64 (*.f64 y (/.f64 x z)))): 17 points increase in error, 40 points decrease in error
      (Rewrite=> distribute-lft-out--_binary64 (*.f64 y (-.f64 1 (/.f64 x z)))): 1 points increase in error, 1 points decrease in error
      (Rewrite<= *-commutative_binary64 (*.f64 (-.f64 1 (/.f64 x z)) y)): 0 points increase in error, 0 points decrease in error
3. Recombined 3 regimes into one program.

4. Final simplification 0.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.7691363646300464 \cdot 10^{+23}:\\ \;\;\;\;y \cdot \left(1 - \frac{x}{z}\right)\\ \mathbf{elif}\;y \leq 6321005.695493938:\\ \;\;\;\;\frac{x + y \cdot \left(z - x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;y - \frac{y}{\frac{z}{x}}\\ \end{array} \]

Alternatives

Alternative 1
Error	8.3
Cost	712

\[\begin{array}{l} t_0 := y + \frac{x}{z}\\ \mathbf{if}\;z \leq -1.32 \cdot 10^{-278}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{-168}:\\ \;\;\;\;\frac{1 - y}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	0.8
Cost	712

\[\begin{array}{l} t_0 := y - \frac{y}{\frac{z}{x}}\\ \mathbf{if}\;y \leq -47976.82319527205:\\ \;\;\;\;t_0\\ \mathbf{elif}\;y \leq 1.9963099193206504 \cdot 10^{-9}:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	0.8
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq -47976.82319527205:\\ \;\;\;\;y \cdot \left(1 - \frac{x}{z}\right)\\ \mathbf{elif}\;y \leq 1.9963099193206504 \cdot 10^{-9}:\\ \;\;\;\;y + \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y - \frac{y}{\frac{z}{x}}\\ \end{array} \]

Alternative 4
Error	10.3
Cost	648

\[\begin{array}{l} t_0 := y + \frac{x}{z}\\ \mathbf{if}\;z \leq -3.5 \cdot 10^{-280}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 5.7 \cdot 10^{-235}:\\ \;\;\;\;\frac{y \cdot x}{-z}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 5
Error	10.2
Cost	648

\[\begin{array}{l} t_0 := y + \frac{x}{z}\\ \mathbf{if}\;z \leq -8 \cdot 10^{-287}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 8 \cdot 10^{-237}:\\ \;\;\;\;y \cdot \left(-\frac{x}{z}\right)\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 6
Error	20.3
Cost	456

\[\begin{array}{l} \mathbf{if}\;y \leq -2.441066437601481 \cdot 10^{-28}:\\ \;\;\;\;y\\ \mathbf{elif}\;y \leq 1.9963099193206504 \cdot 10^{-9}:\\ \;\;\;\;\frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]

Alternative 7
Error	9.1
Cost	320

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

Alternative 8
Error	31.1
Cost	64

\[y \]

Reproduce

herbie shell --seed 2022297 
(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))