Optimisation.CirclePacking:place from circle-packing-0.1.0.4, E

Average Error: 6.2 → 0.7

Time: 12.8s

Precision: binary64

Cost: 7113

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;y \leq -7.8 \cdot 10^{-23} \lor \neg \left(y \leq 2 \cdot 10^{-17}\right):\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= y -7.8e-23) (not (<= y 2e-17)))
   (fma y (/ (- z t) a) x)
   (+ x (/ (* y (- z t)) a))))

C

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

↓

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((y <= -7.8e-23) || !(y <= 2e-17)) {
		tmp = fma(y, ((z - t) / a), x);
	} else {
		tmp = x + ((y * (z - t)) / a);
	}
	return tmp;
}

Julia

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

↓

function code(x, y, z, t, a)
	tmp = 0.0
	if ((y <= -7.8e-23) || !(y <= 2e-17))
		tmp = fma(y, Float64(Float64(z - t) / a), x);
	else
		tmp = Float64(x + Float64(Float64(y * Float64(z - t)) / a));
	end
	return tmp
end

Wolfram

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

↓

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[y, -7.8e-23], N[Not[LessEqual[y, 2e-17]], $MachinePrecision]], N[(y * N[(N[(z - t), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], N[(x + N[(N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]]

Target

Original	6.2
Target	0.7
Herbie	0.7

\[\begin{array}{l} \mathbf{if}\;y < -1.0761266216389975 \cdot 10^{-10}:\\ \;\;\;\;x + \frac{1}{\frac{\frac{a}{z - t}}{y}}\\ \mathbf{elif}\;y < 2.894426862792089 \cdot 10^{-49}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \end{array} \]

Derivation

1. Split input into 2 regimes

2. if y < -7.8e-23 or 2.00000000000000014e-17 < y

   1. Initial program 13.9

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

   2. Simplified 0.9

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

      [Start] 13.9	\[ x + \frac{y \cdot \left(z - t\right)}{a} \]
      +-commutative [=>] 13.9	\[ \color{blue}{\frac{y \cdot \left(z - t\right)}{a} + x} \]
      associate-*r/ [<=] 0.9	\[ \color{blue}{y \cdot \frac{z - t}{a}} + x \]
      fma-def [=>] 0.9	\[ \color{blue}{\mathsf{fma}\left(y, \frac{z - t}{a}, x\right)} \]

   if -7.8e-23 < y < 2.00000000000000014e-17

   1. Initial program 0.5

      \[x + \frac{y \cdot \left(z - t\right)}{a} \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.8 \cdot 10^{-23} \lor \neg \left(y \leq 2 \cdot 10^{-17}\right):\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.4
Cost	1352

\[\begin{array}{l} t_1 := y \cdot \left(z - t\right)\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{+271}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+248}:\\ \;\;\;\;x + \frac{t_1}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \left(z - t\right) \cdot \frac{y}{a}\\ \end{array} \]

Alternative 2
Error	0.5
Cost	1352

\[\begin{array}{l} t_1 := y \cdot \left(z - t\right)\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{+271}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+176}:\\ \;\;\;\;x + \frac{t_1}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{z - t}{\frac{a}{y}}\\ \end{array} \]

Alternative 3
Error	19.8
Cost	1108

\[\begin{array}{l} t_1 := \left(z - t\right) \cdot \frac{y}{a}\\ \mathbf{if}\;x \leq -1.8 \cdot 10^{+74}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -3.7 \cdot 10^{+24}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -1.2 \cdot 10^{-61}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{-221}:\\ \;\;\;\;y \cdot \frac{z - t}{a}\\ \mathbf{elif}\;x \leq 6.5 \cdot 10^{-17}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4
Error	20.7
Cost	977

\[\begin{array}{l} \mathbf{if}\;x \leq -3.8 \cdot 10^{+78}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -3.5 \cdot 10^{+24} \lor \neg \left(x \leq -9 \cdot 10^{-62}\right) \land x \leq 8.5 \cdot 10^{-86}:\\ \;\;\;\;y \cdot \frac{z - t}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5
Error	27.6
Cost	912

\[\begin{array}{l} t_1 := \frac{-y}{\frac{a}{t}}\\ \mathbf{if}\;x \leq -5.5 \cdot 10^{-64}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -4.6 \cdot 10^{-291}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{-155}:\\ \;\;\;\;y \cdot \frac{z}{a}\\ \mathbf{elif}\;x \leq 1.32 \cdot 10^{-112}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6
Error	27.6
Cost	912

\[\begin{array}{l} \mathbf{if}\;x \leq -5.5 \cdot 10^{-64}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -2.6 \cdot 10^{-292}:\\ \;\;\;\;y \cdot \frac{-t}{a}\\ \mathbf{elif}\;x \leq 1.1 \cdot 10^{-148}:\\ \;\;\;\;y \cdot \frac{z}{a}\\ \mathbf{elif}\;x \leq 1.82 \cdot 10^{-113}:\\ \;\;\;\;\frac{-y}{\frac{a}{t}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 7
Error	27.2
Cost	912

\[\begin{array}{l} t_1 := t \cdot \left(-\frac{y}{a}\right)\\ \mathbf{if}\;x \leq -5.5 \cdot 10^{-64}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -3.1 \cdot 10^{-291}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.82 \cdot 10^{-149}:\\ \;\;\;\;y \cdot \frac{z}{a}\\ \mathbf{elif}\;x \leq 3.9 \cdot 10^{-89}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 8
Error	27.2
Cost	912

\[\begin{array}{l} \mathbf{if}\;x \leq -4.4 \cdot 10^{-63}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -3.6 \cdot 10^{-292}:\\ \;\;\;\;t \cdot \left(-\frac{y}{a}\right)\\ \mathbf{elif}\;x \leq 3.4 \cdot 10^{-150}:\\ \;\;\;\;y \cdot \frac{z}{a}\\ \mathbf{elif}\;x \leq 7 \cdot 10^{-90}:\\ \;\;\;\;\frac{y \cdot \left(-t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 9
Error	17.1
Cost	844

\[\begin{array}{l} \mathbf{if}\;t \leq -6.6 \cdot 10^{+105}:\\ \;\;\;\;\left(z - t\right) \cdot \frac{y}{a}\\ \mathbf{elif}\;t \leq 5 \cdot 10^{-305}:\\ \;\;\;\;x + \frac{y \cdot z}{a}\\ \mathbf{elif}\;t \leq 1.1 \cdot 10^{+214}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-t}{\frac{a}{y}}\\ \end{array} \]

Alternative 10
Error	10.1
Cost	844

\[\begin{array}{l} t_1 := x - t \cdot \frac{y}{a}\\ \mathbf{if}\;t \leq -1.7 \cdot 10^{-91}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 2.65 \cdot 10^{-306}:\\ \;\;\;\;x + \frac{y \cdot z}{a}\\ \mathbf{elif}\;t \leq 6.6 \cdot 10^{-30}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Error	16.8
Cost	712

\[\begin{array}{l} \mathbf{if}\;t \leq -4.4 \cdot 10^{+106}:\\ \;\;\;\;\left(z - t\right) \cdot \frac{y}{a}\\ \mathbf{elif}\;t \leq 1.1 \cdot 10^{+214}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z}}\\ \mathbf{else}:\\ \;\;\;\;\frac{-t}{\frac{a}{y}}\\ \end{array} \]

Alternative 12
Error	2.2
Cost	708

\[\begin{array}{l} \mathbf{if}\;y \leq -2.6 \cdot 10^{-91}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \mathbf{else}:\\ \;\;\;\;x + \left(z - t\right) \cdot \frac{y}{a}\\ \end{array} \]

Alternative 13
Error	27.8
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -2.15 \cdot 10^{-165}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{-129}:\\ \;\;\;\;y \cdot \frac{z}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 14
Error	27.8
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -1.55 \cdot 10^{-167}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{-113}:\\ \;\;\;\;\frac{y}{\frac{a}{z}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 15
Error	2.5
Cost	576

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

Alternative 16
Error	30.5
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023060 
(FPCore (x y z t a)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, E"
  :precision binary64

  :herbie-target
  (if (< y -1.0761266216389975e-10) (+ x (/ 1.0 (/ (/ a (- z t)) y))) (if (< y 2.894426862792089e-49) (+ x (/ (* y (- z t)) a)) (+ x (/ y (/ a (- z t))))))

  (+ x (/ (* y (- z t)) a)))