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

Average Error: 9.5% → 0.45%

Time: 14.3s

Precision: binary64

Cost: 7236

Math

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

↓

\[\begin{array}{l} t_1 := y \cdot \left(z - t\right)\\ t_2 := \frac{z - t}{a}\\ \mathbf{if}\;t_1 \leq -2 \cdot 10^{+261}:\\ \;\;\;\;\mathsf{fma}\left(y, t_2, x\right)\\ \mathbf{elif}\;t_1 \leq 10^{+278}:\\ \;\;\;\;x + \frac{t_1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{x + y \cdot t_2}}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* y (- z t))) (t_2 (/ (- z t) a)))
   (if (<= t_1 -2e+261)
     (fma y t_2 x)
     (if (<= t_1 1e+278) (+ x (/ t_1 a)) (/ 1.0 (/ 1.0 (+ x (* y t_2))))))))

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 t_1 = y * (z - t);
	double t_2 = (z - t) / a;
	double tmp;
	if (t_1 <= -2e+261) {
		tmp = fma(y, t_2, x);
	} else if (t_1 <= 1e+278) {
		tmp = x + (t_1 / a);
	} else {
		tmp = 1.0 / (1.0 / (x + (y * t_2)));
	}
	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)
	t_1 = Float64(y * Float64(z - t))
	t_2 = Float64(Float64(z - t) / a)
	tmp = 0.0
	if (t_1 <= -2e+261)
		tmp = fma(y, t_2, x);
	elseif (t_1 <= 1e+278)
		tmp = Float64(x + Float64(t_1 / a));
	else
		tmp = Float64(1.0 / Float64(1.0 / Float64(x + Float64(y * t_2))));
	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_] := Block[{t$95$1 = N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(z - t), $MachinePrecision] / a), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+261], N[(y * t$95$2 + x), $MachinePrecision], If[LessEqual[t$95$1, 1e+278], N[(x + N[(t$95$1 / a), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(1.0 / N[(x + N[(y * t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Target

Original	9.5%
Target	0.97%
Herbie	0.45%

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

Derivation

1. Split input into 3 regimes

2. if (*.f64 y (-.f64 z t)) < -1.9999999999999999e261

   1. Initial program 67.8

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

   2. Simplified 0.64

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

      [Start] 67.8	\[ x + \frac{y \cdot \left(z - t\right)}{a} \]
      +-commutative [=>] 67.8	\[ \color{blue}{\frac{y \cdot \left(z - t\right)}{a} + x} \]
      associate-*r/ [<=] 0.66	\[ \color{blue}{y \cdot \frac{z - t}{a}} + x \]
      fma-def [=>] 0.64	\[ \color{blue}{\mathsf{fma}\left(y, \frac{z - t}{a}, x\right)} \]

   if -1.9999999999999999e261 < (*.f64 y (-.f64 z t)) < 9.99999999999999964e277

   1. Initial program 0.41

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

   if 9.99999999999999964e277 < (*.f64 y (-.f64 z t))

   1. Initial program 77.01

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

   2. Applied egg-rr 0.77

      \[\leadsto \color{blue}{\frac{1}{\frac{1}{y \cdot \frac{z - t}{a} + x}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 0.45

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \leq -2 \cdot 10^{+261}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{a}, x\right)\\ \mathbf{elif}\;y \cdot \left(z - t\right) \leq 10^{+278}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{x + y \cdot \frac{z - t}{a}}}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.44%
Cost	1608

\[\begin{array}{l} t_1 := y \cdot \left(z - t\right)\\ \mathbf{if}\;t_1 \leq -2.5 \cdot 10^{+261}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \mathbf{elif}\;t_1 \leq 10^{+278}:\\ \;\;\;\;x + \frac{t_1}{a}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{x + y \cdot \frac{z - t}{a}}}\\ \end{array} \]

Alternative 2
Error	0.42%
Cost	1352

\[\begin{array}{l} t_1 := y \cdot \left(z - t\right)\\ \mathbf{if}\;t_1 \leq -2.5 \cdot 10^{+261}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \mathbf{elif}\;t_1 \leq 10^{+264}:\\ \;\;\;\;x + \frac{t_1}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \left(z - t\right) \cdot \frac{y}{a}\\ \end{array} \]

Alternative 3
Error	2.71%
Cost	1097

\[\begin{array}{l} \mathbf{if}\;z - t \leq -2 \cdot 10^{+92} \lor \neg \left(z - t \leq 10^{+42}\right):\\ \;\;\;\;x + \left(z - t\right) \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \end{array} \]

Alternative 4
Error	47.2%
Cost	912

\[\begin{array}{l} t_1 := \frac{y}{a} \cdot \left(-t\right)\\ \mathbf{if}\;t \leq -7.5 \cdot 10^{+194}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -1.85 \cdot 10^{-201}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq -2.8 \cdot 10^{-229}:\\ \;\;\;\;\frac{y}{\frac{a}{z}}\\ \mathbf{elif}\;t \leq 2.5 \cdot 10^{+94}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	4.99%
Cost	841

\[\begin{array}{l} \mathbf{if}\;t \leq -1.2 \cdot 10^{-130} \lor \neg \left(t \leq 5.4 \cdot 10^{-230}\right):\\ \;\;\;\;x + \left(z - t\right) \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z}{a}\\ \end{array} \]

Alternative 6
Error	44.09%
Cost	716

\[\begin{array}{l} \mathbf{if}\;x \leq -1.12 \cdot 10^{-127}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 10^{-235}:\\ \;\;\;\;\frac{-y}{\frac{a}{t}}\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{-120}:\\ \;\;\;\;\frac{y \cdot z}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 7
Error	28.85%
Cost	713

\[\begin{array}{l} \mathbf{if}\;t \leq -1.06 \cdot 10^{+195} \lor \neg \left(t \leq 2.4 \cdot 10^{+94}\right):\\ \;\;\;\;\frac{y}{a} \cdot \left(-t\right)\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z}{a}\\ \end{array} \]

Alternative 8
Error	28.97%
Cost	713

\[\begin{array}{l} \mathbf{if}\;t \leq -1 \cdot 10^{+195} \lor \neg \left(t \leq 1.6 \cdot 10^{+92}\right):\\ \;\;\;\;\frac{y}{a} \cdot \left(-t\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot z}{a}\\ \end{array} \]

Alternative 9
Error	15.45%
Cost	713

\[\begin{array}{l} \mathbf{if}\;t \leq -5.2 \cdot 10^{-9} \lor \neg \left(t \leq 4 \cdot 10^{-32}\right):\\ \;\;\;\;x - t \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z}{a}\\ \end{array} \]

Alternative 10
Error	33.44%
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -3 \cdot 10^{+58}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.12 \cdot 10^{-119}:\\ \;\;\;\;y \cdot \frac{z - t}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 11
Error	44.43%
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -7.6 \cdot 10^{-107}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.5 \cdot 10^{-124}:\\ \;\;\;\;y \cdot \frac{z}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 12
Error	44.09%
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{-37}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{-133}:\\ \;\;\;\;z \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 13
Error	44.34%
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -2.1 \cdot 10^{-107}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.3 \cdot 10^{-132}:\\ \;\;\;\;\frac{y}{\frac{a}{z}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 14
Error	44.7%
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{-37}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 4.4 \cdot 10^{-124}:\\ \;\;\;\;\frac{y \cdot z}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 15
Error	48.33%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023090 
(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)))