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

Average Error: 9.47% → 0.62%

Time: 11.8s

Precision: binary64

Cost: 7236

Math

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

↓

\[\begin{array}{l} t_1 := y \cdot \left(z - t\right)\\ \mathbf{if}\;t_1 \leq -2 \cdot 10^{+283}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{a}, x\right)\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+198}:\\ \;\;\;\;x + \frac{t_1}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \left(z - t\right) \cdot \frac{y}{a}\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* y (- z t))))
   (if (<= t_1 -2e+283)
     (fma y (/ (- z t) a) x)
     (if (<= t_1 2e+198) (+ x (/ t_1 a)) (+ x (* (- z t) (/ y 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 t_1 = y * (z - t);
	double tmp;
	if (t_1 <= -2e+283) {
		tmp = fma(y, ((z - t) / a), x);
	} else if (t_1 <= 2e+198) {
		tmp = x + (t_1 / a);
	} else {
		tmp = x + ((z - t) * (y / 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)
	t_1 = Float64(y * Float64(z - t))
	tmp = 0.0
	if (t_1 <= -2e+283)
		tmp = fma(y, Float64(Float64(z - t) / a), x);
	elseif (t_1 <= 2e+198)
		tmp = Float64(x + Float64(t_1 / a));
	else
		tmp = Float64(x + Float64(Float64(z - t) * Float64(y / 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_] := Block[{t$95$1 = N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+283], N[(y * N[(N[(z - t), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 2e+198], N[(x + N[(t$95$1 / a), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(z - t), $MachinePrecision] * N[(y / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Target

Original	9.47%
Target	1.06%
Herbie	0.62%

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

Derivation

1. Split input into 3 regimes

2. if (*.f64 y (-.f64 z t)) < -1.99999999999999991e283

   1. Initial program 77.55

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

   2. Simplified 0.31

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

      [Start] 77.55	\[ x + \frac{y \cdot \left(z - t\right)}{a} \]
      +-commutative [=>] 77.55	\[ \color{blue}{\frac{y \cdot \left(z - t\right)}{a} + x} \]
      associate-*r/ [<=] 0.31	\[ \color{blue}{y \cdot \frac{z - t}{a}} + x \]
      fma-def [=>] 0.31	\[ \color{blue}{\mathsf{fma}\left(y, \frac{z - t}{a}, x\right)} \]

   if -1.99999999999999991e283 < (*.f64 y (-.f64 z t)) < 2.00000000000000004e198

   1. Initial program 0.58

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

   if 2.00000000000000004e198 < (*.f64 y (-.f64 z t))

   1. Initial program 44.52

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

   2. Simplified 1.15

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

      [Start] 44.52	\[ x + \frac{y \cdot \left(z - t\right)}{a} \]
      associate-*l/ [<=] 1.15	\[ x + \color{blue}{\frac{y}{a} \cdot \left(z - t\right)} \]

3. Recombined 3 regimes into one program.

4. Final simplification 0.62

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \leq -2 \cdot 10^{+283}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{a}, x\right)\\ \mathbf{elif}\;y \cdot \left(z - t\right) \leq 2 \cdot 10^{+198}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \left(z - t\right) \cdot \frac{y}{a}\\ \end{array} \]

Alternatives

Alternative 1
Error	0.87%
Cost	1353

\[\begin{array}{l} t_1 := y \cdot \left(z - t\right)\\ \mathbf{if}\;t_1 \leq -5 \cdot 10^{+147} \lor \neg \left(t_1 \leq 2 \cdot 10^{+198}\right):\\ \;\;\;\;x + \left(z - t\right) \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{t_1}{a}\\ \end{array} \]

Alternative 2
Error	43.08%
Cost	1176

\[\begin{array}{l} t_1 := \frac{y}{a} \cdot \left(-t\right)\\ \mathbf{if}\;x \leq -2.05 \cdot 10^{-139}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.7 \cdot 10^{-305}:\\ \;\;\;\;z \cdot \frac{y}{a}\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{-287}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{-156}:\\ \;\;\;\;\frac{z}{\frac{a}{y}}\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-115}:\\ \;\;\;\;\left(-y\right) \cdot \frac{t}{a}\\ \mathbf{elif}\;x \leq 7.2 \cdot 10^{-98}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 3
Error	43.16%
Cost	1112

\[\begin{array}{l} t_1 := \frac{-y}{\frac{a}{t}}\\ \mathbf{if}\;x \leq -2.1 \cdot 10^{-139}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{-303}:\\ \;\;\;\;z \cdot \frac{y}{a}\\ \mathbf{elif}\;x \leq 1.9 \cdot 10^{-284}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{-151}:\\ \;\;\;\;\frac{z}{\frac{a}{y}}\\ \mathbf{elif}\;x \leq 1.4 \cdot 10^{-115}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{-109}:\\ \;\;\;\;\frac{y}{\frac{a}{z}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4
Error	43.17%
Cost	1112

\[\begin{array}{l} \mathbf{if}\;x \leq -2.1 \cdot 10^{-139}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.1 \cdot 10^{-307}:\\ \;\;\;\;z \cdot \frac{y}{a}\\ \mathbf{elif}\;x \leq 2.05 \cdot 10^{-287}:\\ \;\;\;\;\frac{-y}{\frac{a}{t}}\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{-147}:\\ \;\;\;\;\frac{z}{\frac{a}{y}}\\ \mathbf{elif}\;x \leq 2 \cdot 10^{-115}:\\ \;\;\;\;\left(-y\right) \cdot \frac{t}{a}\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{-110}:\\ \;\;\;\;\frac{y}{\frac{a}{z}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5
Error	17.23%
Cost	976

\[\begin{array}{l} t_1 := x - t \cdot \frac{y}{a}\\ t_2 := x + \frac{z}{\frac{a}{y}}\\ \mathbf{if}\;z \leq -3.2 \cdot 10^{-39}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{-181}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 4.5 \cdot 10^{-30}:\\ \;\;\;\;x + y \cdot \frac{z}{a}\\ \mathbf{elif}\;z \leq 9.7 \cdot 10^{+61}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 6
Error	47.13%
Cost	848

\[\begin{array}{l} t_1 := z \cdot \frac{y}{a}\\ \mathbf{if}\;z \leq -4.2 \cdot 10^{+273}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 3 \cdot 10^{-78}:\\ \;\;\;\;x\\ \mathbf{elif}\;z \leq 5.2 \cdot 10^{-31}:\\ \;\;\;\;y \cdot \frac{z}{a}\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{+159}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 7
Error	4.67%
Cost	841

\[\begin{array}{l} \mathbf{if}\;x \leq -1.35 \cdot 10^{-197} \lor \neg \left(x \leq 2 \cdot 10^{-120}\right):\\ \;\;\;\;x + \left(z - t\right) \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \end{array} \]

Alternative 8
Error	24.29%
Cost	713

\[\begin{array}{l} \mathbf{if}\;x \leq -3.35 \cdot 10^{-187} \lor \neg \left(x \leq 5.5 \cdot 10^{-106}\right):\\ \;\;\;\;x + y \cdot \frac{z}{a}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z - t}{a}\\ \end{array} \]

Alternative 9
Error	22.05%
Cost	713

\[\begin{array}{l} \mathbf{if}\;x \leq -3.8 \cdot 10^{-190} \lor \neg \left(x \leq 1.9 \cdot 10^{-107}\right):\\ \;\;\;\;x + \frac{z}{\frac{a}{y}}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z - t}{a}\\ \end{array} \]

Alternative 10
Error	32.2%
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{-66}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 2.7 \cdot 10^{-104}:\\ \;\;\;\;y \cdot \frac{z - t}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 11
Error	43.49%
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -1.3 \cdot 10^{-143}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.12 \cdot 10^{-110}:\\ \;\;\;\;y \cdot \frac{z}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 12
Error	4%
Cost	576

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

Alternative 13
Error	47.91%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023121 
(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)))