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

Average Error: 19.35% → 3.29%

Time: 5.0s

Precision: binary64

Cost: 8332

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x \cdot \left(y - z\right)}{y}\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{+35}:\\ \;\;\;\;\frac{y - z}{\frac{y}{x}}\\ \mathbf{elif}\;t_0 \leq 50000000:\\ \;\;\;\;x - \frac{x}{\frac{y}{z}}\\ \mathbf{elif}\;t_0 \leq 2 \cdot 10^{+251}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{-z}{y}, x\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* x (- y z)) y)))
   (if (<= t_0 -1e+35)
     (/ (- y z) (/ y x))
     (if (<= t_0 50000000.0)
       (- x (/ x (/ y z)))
       (if (<= t_0 2e+251) t_0 (fma x (/ (- z) y) x))))))

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 = (x * (y - z)) / y;
	double tmp;
	if (t_0 <= -1e+35) {
		tmp = (y - z) / (y / x);
	} else if (t_0 <= 50000000.0) {
		tmp = x - (x / (y / z));
	} else if (t_0 <= 2e+251) {
		tmp = t_0;
	} else {
		tmp = fma(x, (-z / y), x);
	}
	return tmp;
}

Julia

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

↓

function code(x, y, z)
	t_0 = Float64(Float64(x * Float64(y - z)) / y)
	tmp = 0.0
	if (t_0 <= -1e+35)
		tmp = Float64(Float64(y - z) / Float64(y / x));
	elseif (t_0 <= 50000000.0)
		tmp = Float64(x - Float64(x / Float64(y / z)));
	elseif (t_0 <= 2e+251)
		tmp = t_0;
	else
		tmp = fma(x, Float64(Float64(-z) / y), x);
	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[(x * N[(y - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$0, -1e+35], N[(N[(y - z), $MachinePrecision] / N[(y / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 50000000.0], N[(x - N[(x / N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 2e+251], t$95$0, N[(x * N[((-z) / y), $MachinePrecision] + x), $MachinePrecision]]]]]

Target

Original	19.35%
Target	5.07%
Herbie	3.29%

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

Derivation

1. Split input into 4 regimes

2. if (/.f64 (*.f64 x (-.f64 y z)) y) < -9.9999999999999997e34

   1. Initial program 28.61

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

   2. Simplified 9.68

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

      [Start] 28.61	\[ \frac{x \cdot \left(y - z\right)}{y} \]
      *-commutative [=>] 28.61	\[ \frac{\color{blue}{\left(y - z\right) \cdot x}}{y} \]
      associate-/l* [=>] 9.68	\[ \color{blue}{\frac{y - z}{\frac{y}{x}}} \]

   if -9.9999999999999997e34 < (/.f64 (*.f64 x (-.f64 y z)) y) < 5e7

   1. Initial program 9.04

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

   2. Simplified 0.2

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

      [Start] 9.04	\[ \frac{x \cdot \left(y - z\right)}{y} \]
      associate-*r/ [<=] 0.32	\[ \color{blue}{x \cdot \frac{y - z}{y}} \]
      div-sub [=>] 0.31	\[ x \cdot \color{blue}{\left(\frac{y}{y} - \frac{z}{y}\right)} \]
      distribute-rgt-out-- [<=] 0.3	\[ \color{blue}{\frac{y}{y} \cdot x - \frac{z}{y} \cdot x} \]
      *-inverses [=>] 0.3	\[ \color{blue}{1} \cdot x - \frac{z}{y} \cdot x \]
      *-lft-identity [=>] 0.3	\[ \color{blue}{x} - \frac{z}{y} \cdot x \]
      associate-*l/ [=>] 4.99	\[ x - \color{blue}{\frac{z \cdot x}{y}} \]
      *-commutative [<=] 4.99	\[ x - \frac{\color{blue}{x \cdot z}}{y} \]
      associate-/l* [=>] 0.2	\[ x - \color{blue}{\frac{x}{\frac{y}{z}}} \]

   if 5e7 < (/.f64 (*.f64 x (-.f64 y z)) y) < 2.0000000000000001e251

   1. Initial program 0.34

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

   if 2.0000000000000001e251 < (/.f64 (*.f64 x (-.f64 y z)) y)

   1. Initial program 72.17

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

   2. Simplified 6.08

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

      [Start] 72.17	\[ \frac{x \cdot \left(y - z\right)}{y} \]
      *-commutative [=>] 72.17	\[ \frac{\color{blue}{\left(y - z\right) \cdot x}}{y} \]
      associate-*l/ [<=] 6.1	\[ \color{blue}{\frac{y - z}{y} \cdot x} \]
      div-sub [=>] 6.09	\[ \color{blue}{\left(\frac{y}{y} - \frac{z}{y}\right)} \cdot x \]
      sub-neg [=>] 6.09	\[ \color{blue}{\left(\frac{y}{y} + \left(-\frac{z}{y}\right)\right)} \cdot x \]
      +-commutative [=>] 6.09	\[ \color{blue}{\left(\left(-\frac{z}{y}\right) + \frac{y}{y}\right)} \cdot x \]
      *-inverses [=>] 6.09	\[ \left(\left(-\frac{z}{y}\right) + \color{blue}{1}\right) \cdot x \]
      distribute-lft1-in [<=] 6.09	\[ \color{blue}{\left(-\frac{z}{y}\right) \cdot x + x} \]
      *-commutative [=>] 6.09	\[ \color{blue}{x \cdot \left(-\frac{z}{y}\right)} + x \]
      fma-def [=>] 6.08	\[ \color{blue}{\mathsf{fma}\left(x, -\frac{z}{y}, x\right)} \]
      distribute-neg-frac [=>] 6.08	\[ \mathsf{fma}\left(x, \color{blue}{\frac{-z}{y}}, x\right) \]

3. Recombined 4 regimes into one program.

4. Final simplification 3.29

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(y - z\right)}{y} \leq -1 \cdot 10^{+35}:\\ \;\;\;\;\frac{y - z}{\frac{y}{x}}\\ \mathbf{elif}\;\frac{x \cdot \left(y - z\right)}{y} \leq 50000000:\\ \;\;\;\;x - \frac{x}{\frac{y}{z}}\\ \mathbf{elif}\;\frac{x \cdot \left(y - z\right)}{y} \leq 2 \cdot 10^{+251}:\\ \;\;\;\;\frac{x \cdot \left(y - z\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{-z}{y}, x\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	3.44%
Cost	1997

\[\begin{array}{l} t_0 := \frac{x \cdot \left(y - z\right)}{y}\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{+35}:\\ \;\;\;\;\left(y - z\right) \cdot \frac{x}{y}\\ \mathbf{elif}\;t_0 \leq 50000000 \lor \neg \left(t_0 \leq 5 \cdot 10^{+262}\right):\\ \;\;\;\;x - \frac{x}{\frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 2
Error	3.14%
Cost	1997

\[\begin{array}{l} t_0 := \frac{x \cdot \left(y - z\right)}{y}\\ \mathbf{if}\;t_0 \leq -1 \cdot 10^{+35}:\\ \;\;\;\;\frac{y - z}{\frac{y}{x}}\\ \mathbf{elif}\;t_0 \leq 50000000 \lor \neg \left(t_0 \leq 5 \cdot 10^{+262}\right):\\ \;\;\;\;x - \frac{x}{\frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	32.57%
Cost	913

\[\begin{array}{l} \mathbf{if}\;y \leq -1.38 \cdot 10^{+44}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq -5.6 \cdot 10^{-150} \lor \neg \left(y \leq -3.4 \cdot 10^{-209}\right) \land y \leq 1.45 \cdot 10^{-43}:\\ \;\;\;\;z \cdot \frac{-x}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4
Error	15.66%
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -1.35 \cdot 10^{-119} \lor \neg \left(z \leq 5.4 \cdot 10^{-168}\right):\\ \;\;\;\;\left(y - z\right) \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5
Error	6.09%
Cost	713

\[\begin{array}{l} \mathbf{if}\;z \leq -820000000000 \lor \neg \left(z \leq 3.6 \cdot 10^{+47}\right):\\ \;\;\;\;\left(y - z\right) \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{x}{\frac{y}{z}}\\ \end{array} \]

Alternative 6
Error	40.21%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023102 
(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))