Diagrams.Backend.Cairo.Internal:setTexture from diagrams-cairo-1.3.0.3

Average Error: 12.1 → 1.8

Time: 4.4s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_0 := \mathsf{fma}\left(\frac{-z}{y}, x, x\right)\\ \mathbf{if}\;x \leq -1 \cdot 10^{+92}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;x \leq 8.829242912010182 \cdot 10^{-304}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{-x}{y}, x\right)\\ \mathbf{elif}\;x \leq 4.142754075651002 \cdot 10^{-166}:\\ \;\;\;\;x - \frac{x \cdot z}{y}\\ \mathbf{elif}\;x \leq 1.9913091815903685 \cdot 10^{-11}:\\ \;\;\;\;x - \frac{z}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (fma (/ (- z) y) x x)))
   (if (<= x -1e+92)
     t_0
     (if (<= x 8.829242912010182e-304)
       (fma z (/ (- x) y) x)
       (if (<= x 4.142754075651002e-166)
         (- x (/ (* x z) y))
         (if (<= x 1.9913091815903685e-11) (- x (/ z (/ y x))) t_0))))))

C

double code(double x, double y, double z) {
	return (x * (y - z)) / y;
}

↓

double code(double x, double y, double z) {
	double t_0 = fma((-z / y), x, x);
	double tmp;
	if (x <= -1e+92) {
		tmp = t_0;
	} else if (x <= 8.829242912010182e-304) {
		tmp = fma(z, (-x / y), x);
	} else if (x <= 4.142754075651002e-166) {
		tmp = x - ((x * z) / y);
	} else if (x <= 1.9913091815903685e-11) {
		tmp = x - (z / (y / x));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	return Float64(Float64(x * Float64(y - z)) / y)
end

↓

function code(x, y, z)
	t_0 = fma(Float64(Float64(-z) / y), x, x)
	tmp = 0.0
	if (x <= -1e+92)
		tmp = t_0;
	elseif (x <= 8.829242912010182e-304)
		tmp = fma(z, Float64(Float64(-x) / y), x);
	elseif (x <= 4.142754075651002e-166)
		tmp = Float64(x - Float64(Float64(x * z) / y));
	elseif (x <= 1.9913091815903685e-11)
		tmp = Float64(x - Float64(z / Float64(y / x)));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := N[(N[(x * N[(y - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]

↓

code[x_, y_, z_] := Block[{t$95$0 = N[(N[((-z) / y), $MachinePrecision] * x + x), $MachinePrecision]}, If[LessEqual[x, -1e+92], t$95$0, If[LessEqual[x, 8.829242912010182e-304], N[(z * N[((-x) / y), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[x, 4.142754075651002e-166], N[(x - N[(N[(x * z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.9913091815903685e-11], N[(x - N[(z / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]]]

Target

Original	12.1
Target	3.1
Herbie	1.8

\[\begin{array}{l} \mathbf{if}\;z < -2.060202331921739 \cdot 10^{+104}:\\ \;\;\;\;x - \frac{z \cdot x}{y}\\ \mathbf{elif}\;z < 1.6939766013828526 \cdot 10^{+213}:\\ \;\;\;\;\frac{x}{\frac{y}{y - z}}\\ \mathbf{else}:\\ \;\;\;\;\left(y - z\right) \cdot \frac{x}{y}\\ \end{array} \]

Derivation

1. Split input into 4 regimes

2. if x < -1e92 or 1.9913091815903685e-11 < x

   1. Initial program 24.2

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

   2. Simplified 8.1

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

   3. Applied egg-rr 0.1

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

   if -1e92 < x < 8.8292429120101816e-304

   1. Initial program 5.4

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

   2. Simplified 3.0

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

   3. Applied egg-rr 3.0

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

   if 8.8292429120101816e-304 < x < 4.1427540756510021e-166

   1. Initial program 8.2

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

   2. Simplified 4.0

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

   3. Taylor expanded in z around 0 4.0

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

   if 4.1427540756510021e-166 < x < 1.9913091815903685e-11

   1. Initial program 1.9

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

   2. Simplified 1.1

      \[\leadsto \color{blue}{x - \frac{z}{\frac{y}{x}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 1.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+92}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-z}{y}, x, x\right)\\ \mathbf{elif}\;x \leq 8.829242912010182 \cdot 10^{-304}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{-x}{y}, x\right)\\ \mathbf{elif}\;x \leq 4.142754075651002 \cdot 10^{-166}:\\ \;\;\;\;x - \frac{x \cdot z}{y}\\ \mathbf{elif}\;x \leq 1.9913091815903685 \cdot 10^{-11}:\\ \;\;\;\;x - \frac{z}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-z}{y}, x, x\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022211 
(FPCore (x y z)
  :name "Diagrams.Backend.Cairo.Internal:setTexture from diagrams-cairo-1.3.0.3"
  :precision binary64

  :herbie-target
  (if (< z -2.060202331921739e+104) (- x (/ (* z x) y)) (if (< z 1.6939766013828526e+213) (/ x (/ y (- y z))) (* (- y z) (/ x y))))

  (/ (* x (- y z)) y))