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

Average Error: 12.3 → 0.6

Time: 5.3s

Precision: binary64

Cost: 2512

Math

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

↓

\[\begin{array}{l} t_0 := \frac{x \cdot \left(y - z\right)}{y}\\ t_1 := x - \frac{x}{\frac{y}{z}}\\ \mathbf{if}\;t_0 \leq -\infty:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq -4 \cdot 10^{+142}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;t_0 \leq 2 \cdot 10^{-117}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t_0 \leq 5 \cdot 10^{+293}:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{y}{y - z}}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (* x (- y z)) y)) (t_1 (- x (/ x (/ y z)))))
   (if (<= t_0 (- INFINITY))
     t_1
     (if (<= t_0 -4e+142)
       t_0
       (if (<= t_0 2e-117)
         t_1
         (if (<= t_0 5e+293) t_0 (/ x (/ y (- y z)))))))))

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 t_1 = x - (x / (y / z));
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_0 <= -4e+142) {
		tmp = t_0;
	} else if (t_0 <= 2e-117) {
		tmp = t_1;
	} else if (t_0 <= 5e+293) {
		tmp = t_0;
	} else {
		tmp = x / (y / (y - z));
	}
	return tmp;
}

Java

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

↓

public static double code(double x, double y, double z) {
	double t_0 = (x * (y - z)) / y;
	double t_1 = x - (x / (y / z));
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else if (t_0 <= -4e+142) {
		tmp = t_0;
	} else if (t_0 <= 2e-117) {
		tmp = t_1;
	} else if (t_0 <= 5e+293) {
		tmp = t_0;
	} else {
		tmp = x / (y / (y - z));
	}
	return tmp;
}

Python

def code(x, y, z):
	return (x * (y - z)) / y

↓

def code(x, y, z):
	t_0 = (x * (y - z)) / y
	t_1 = x - (x / (y / z))
	tmp = 0
	if t_0 <= -math.inf:
		tmp = t_1
	elif t_0 <= -4e+142:
		tmp = t_0
	elif t_0 <= 2e-117:
		tmp = t_1
	elif t_0 <= 5e+293:
		tmp = t_0
	else:
		tmp = x / (y / (y - z))
	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)
	t_1 = Float64(x - Float64(x / Float64(y / z)))
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_0 <= -4e+142)
		tmp = t_0;
	elseif (t_0 <= 2e-117)
		tmp = t_1;
	elseif (t_0 <= 5e+293)
		tmp = t_0;
	else
		tmp = Float64(x / Float64(y / Float64(y - z)));
	end
	return tmp
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x * (y - z)) / y;
end

↓

function tmp_2 = code(x, y, z)
	t_0 = (x * (y - z)) / y;
	t_1 = x - (x / (y / z));
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = t_1;
	elseif (t_0 <= -4e+142)
		tmp = t_0;
	elseif (t_0 <= 2e-117)
		tmp = t_1;
	elseif (t_0 <= 5e+293)
		tmp = t_0;
	else
		tmp = x / (y / (y - z));
	end
	tmp_2 = 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]}, Block[{t$95$1 = N[(x - N[(x / N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], t$95$1, If[LessEqual[t$95$0, -4e+142], t$95$0, If[LessEqual[t$95$0, 2e-117], t$95$1, If[LessEqual[t$95$0, 5e+293], t$95$0, N[(x / N[(y / N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Target

Original	12.3
Target	2.8
Herbie	0.6

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

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 x (-.f64 y z)) y) < -inf.0 or -4.0000000000000002e142 < (/.f64 (*.f64 x (-.f64 y z)) y) < 2.00000000000000006e-117

   1. Initial program 14.4

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

   2. Simplified 0.7

      \[\leadsto \color{blue}{x - \frac{x}{\frac{y}{z}}} \]
      Proof
      (-.f64 x (/.f64 x (/.f64 y z))): 0 points increase in error, 0 points decrease in error
      (-.f64 x (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 x z) y))): 1 points increase in error, 0 points decrease in error
      (-.f64 x (/.f64 (Rewrite=> *-commutative_binary64 (*.f64 z x)) y)): 0 points increase in error, 1 points decrease in error
      (-.f64 x (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 z y) x))): 0 points increase in error, 0 points decrease in error
      (-.f64 (Rewrite<= *-lft-identity_binary64 (*.f64 1 x)) (*.f64 (/.f64 z y) x)): 1 points increase in error, 0 points decrease in error
      (-.f64 (*.f64 (Rewrite<= *-inverses_binary64 (/.f64 y y)) x) (*.f64 (/.f64 z y) x)): 0 points increase in error, 1 points decrease in error
      (Rewrite=> distribute-rgt-out--_binary64 (*.f64 x (-.f64 (/.f64 y y) (/.f64 z y)))): 4 points increase in error, 0 points decrease in error
      (*.f64 x (Rewrite<= div-sub_binary64 (/.f64 (-.f64 y z) y))): 0 points increase in error, 4 points decrease in error
      (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 x (-.f64 y z)) y)): 0 points increase in error, 0 points decrease in error

   if -inf.0 < (/.f64 (*.f64 x (-.f64 y z)) y) < -4.0000000000000002e142 or 2.00000000000000006e-117 < (/.f64 (*.f64 x (-.f64 y z)) y) < 5.00000000000000033e293

   1. Initial program 0.3

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

   if 5.00000000000000033e293 < (/.f64 (*.f64 x (-.f64 y z)) y)

   1. Initial program 58.5

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

   2. Simplified 1.5

      \[\leadsto \color{blue}{\frac{x}{\frac{y}{y - z}}} \]
      Proof
      (-.f64 x (/.f64 x (/.f64 y z))): 0 points increase in error, 0 points decrease in error
      (-.f64 x (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 x z) y))): 1 points increase in error, 0 points decrease in error
      (-.f64 x (/.f64 (Rewrite=> *-commutative_binary64 (*.f64 z x)) y)): 0 points increase in error, 1 points decrease in error
      (-.f64 x (Rewrite<= associate-*l/_binary64 (*.f64 (/.f64 z y) x))): 0 points increase in error, 0 points decrease in error
      (-.f64 (Rewrite<= *-lft-identity_binary64 (*.f64 1 x)) (*.f64 (/.f64 z y) x)): 1 points increase in error, 0 points decrease in error
      (-.f64 (*.f64 (Rewrite<= *-inverses_binary64 (/.f64 y y)) x) (*.f64 (/.f64 z y) x)): 0 points increase in error, 1 points decrease in error
      (Rewrite=> distribute-rgt-out--_binary64 (*.f64 x (-.f64 (/.f64 y y) (/.f64 z y)))): 4 points increase in error, 0 points decrease in error
      (*.f64 x (Rewrite<= div-sub_binary64 (/.f64 (-.f64 y z) y))): 0 points increase in error, 4 points decrease in error
      (Rewrite=> associate-*r/_binary64 (/.f64 (*.f64 x (-.f64 y z)) y)): 0 points increase in error, 0 points decrease in error
3. Recombined 3 regimes into one program.

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x \cdot \left(y - z\right)}{y} \leq -\infty:\\ \;\;\;\;x - \frac{x}{\frac{y}{z}}\\ \mathbf{elif}\;\frac{x \cdot \left(y - z\right)}{y} \leq -4 \cdot 10^{+142}:\\ \;\;\;\;\frac{x \cdot \left(y - z\right)}{y}\\ \mathbf{elif}\;\frac{x \cdot \left(y - z\right)}{y} \leq 2 \cdot 10^{-117}:\\ \;\;\;\;x - \frac{x}{\frac{y}{z}}\\ \mathbf{elif}\;\frac{x \cdot \left(y - z\right)}{y} \leq 5 \cdot 10^{+293}:\\ \;\;\;\;\frac{x \cdot \left(y - z\right)}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{y}{y - z}}\\ \end{array} \]

Alternatives

Alternative 1
Error	8.8
Cost	977

\[\begin{array}{l} \mathbf{if}\;y \leq -3.65 \cdot 10^{+224}:\\ \;\;\;\;x\\ \mathbf{elif}\;y \leq 9 \cdot 10^{+80} \lor \neg \left(y \leq 3.3 \cdot 10^{+182}\right) \land y \leq 8.5 \cdot 10^{+219}:\\ \;\;\;\;\left(y - z\right) \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 2
Error	19.1
Cost	912

\[\begin{array}{l} t_0 := \frac{x}{y} \cdot \left(-z\right)\\ \mathbf{if}\;z \leq -4.5 \cdot 10^{-24}:\\ \;\;\;\;t_0\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{-148}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 3.1 \cdot 10^{-135}:\\ \;\;\;\;\frac{x}{-\frac{y}{z}}\\ \mathbf{elif}\;z \leq 1.22 \cdot 10^{-17}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t_0\\ \end{array} \]

Alternative 3
Error	3.5
Cost	713

\[\begin{array}{l} \mathbf{if}\;y \leq -7.8 \cdot 10^{-237} \lor \neg \left(y \leq 3.6 \cdot 10^{-199}\right):\\ \;\;\;\;x - \frac{x}{\frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;-\frac{x \cdot z}{y}\\ \end{array} \]

Alternative 4
Error	3.5
Cost	712

\[\begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{-237}:\\ \;\;\;\;\frac{x}{\frac{y}{y - z}}\\ \mathbf{elif}\;y \leq 2 \cdot 10^{-198}:\\ \;\;\;\;-\frac{x \cdot z}{y}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{x}{\frac{y}{z}}\\ \end{array} \]

Alternative 5
Error	19.0
Cost	649

\[\begin{array}{l} \mathbf{if}\;z \leq -3.8 \cdot 10^{-28} \lor \neg \left(z \leq 5.5 \cdot 10^{-20}\right):\\ \;\;\;\;\frac{-z}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6
Error	18.8
Cost	649

\[\begin{array}{l} \mathbf{if}\;z \leq -9.2 \cdot 10^{-25} \lor \neg \left(z \leq 1.22 \cdot 10^{-17}\right):\\ \;\;\;\;\frac{x}{y} \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 7
Error	24.9
Cost	64

\[x \]

Reproduce

herbie shell --seed 2022340 
(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))