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

Average Accuracy: 90.6% → 98.8%

Time: 12.3s

Precision: binary64

Cost: 7625

Math

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

↓

\[\begin{array}{l} t_1 := y \cdot \left(z - t\right)\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{+142} \lor \neg \left(t_1 \leq 5 \cdot 10^{+211}\right):\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{t_1}{a}\\ \end{array} \]

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 (or (<= t_1 -1e+142) (not (<= t_1 5e+211)))
     (fma y (/ (- z t) a) x)
     (+ x (/ t_1 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 <= -1e+142) || !(t_1 <= 5e+211)) {
		tmp = fma(y, ((z - t) / a), x);
	} else {
		tmp = x + (t_1 / 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 <= -1e+142) || !(t_1 <= 5e+211))
		tmp = fma(y, Float64(Float64(z - t) / a), x);
	else
		tmp = Float64(x + Float64(t_1 / 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[Or[LessEqual[t$95$1, -1e+142], N[Not[LessEqual[t$95$1, 5e+211]], $MachinePrecision]], N[(y * N[(N[(z - t), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], N[(x + N[(t$95$1 / a), $MachinePrecision]), $MachinePrecision]]]

Target

Original	90.6%
Target	98.9%
Herbie	98.8%

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (-.f64 z t)) < -1.00000000000000005e142 or 4.9999999999999995e211 < (*.f64 y (-.f64 z t))

   1. Initial program 62.1%

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

   2. Simplified 97.1%

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

      [Start] 62.1	\[ x + \frac{y \cdot \left(z - t\right)}{a} \]
      +-commutative [=>] 62.1	\[ \color{blue}{\frac{y \cdot \left(z - t\right)}{a} + x} \]
      associate-*r/ [<=] 97.1	\[ \color{blue}{y \cdot \frac{z - t}{a}} + x \]
      fma-def [=>] 97.1	\[ \color{blue}{\mathsf{fma}\left(y, \frac{z - t}{a}, x\right)} \]

   if -1.00000000000000005e142 < (*.f64 y (-.f64 z t)) < 4.9999999999999995e211

   1. Initial program 99.3%

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

4. Final simplification 98.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \leq -1 \cdot 10^{+142} \lor \neg \left(y \cdot \left(z - t\right) \leq 5 \cdot 10^{+211}\right):\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	98.6%
Cost	1352

\[\begin{array}{l} t_1 := y \cdot \left(z - t\right)\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{+142}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \mathbf{elif}\;t_1 \leq 5 \cdot 10^{+128}:\\ \;\;\;\;x + \frac{t_1}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \left(z - t\right) \cdot \frac{y}{a}\\ \end{array} \]

Alternative 2
Accuracy	97.6%
Cost	1097

\[\begin{array}{l} \mathbf{if}\;z - t \leq -1 \cdot 10^{+27} \lor \neg \left(z - t \leq 5 \cdot 10^{+17}\right):\\ \;\;\;\;x + \left(z - t\right) \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{\frac{a}{z - t}}\\ \end{array} \]

Alternative 3
Accuracy	70.0%
Cost	976

\[\begin{array}{l} t_1 := \left(z - t\right) \cdot \frac{y}{a}\\ \mathbf{if}\;x \leq -1.06 \cdot 10^{-5}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -1.05 \cdot 10^{-233}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 3.5 \cdot 10^{-155}:\\ \;\;\;\;y \cdot \frac{z - t}{a}\\ \mathbf{elif}\;x \leq 2.6 \cdot 10^{-47}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4
Accuracy	53.6%
Cost	849

\[\begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-143}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -3.1 \cdot 10^{-292} \lor \neg \left(x \leq 1.65 \cdot 10^{-152}\right) \land x \leq 1.45 \cdot 10^{-80}:\\ \;\;\;\;z \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 5
Accuracy	56.1%
Cost	780

\[\begin{array}{l} \mathbf{if}\;x \leq -6.5 \cdot 10^{-143}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -2.2 \cdot 10^{-251}:\\ \;\;\;\;z \cdot \frac{y}{a}\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{-97}:\\ \;\;\;\;\frac{-y}{\frac{a}{t}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6
Accuracy	77.3%
Cost	713

\[\begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{-213} \lor \neg \left(x \leq 1.4 \cdot 10^{-153}\right):\\ \;\;\;\;x + z \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z - t}{a}\\ \end{array} \]

Alternative 7
Accuracy	85.9%
Cost	713

\[\begin{array}{l} \mathbf{if}\;t \leq -4.6 \cdot 10^{-13} \lor \neg \left(t \leq 1.05 \cdot 10^{+14}\right):\\ \;\;\;\;x - t \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;x + z \cdot \frac{y}{a}\\ \end{array} \]

Alternative 8
Accuracy	68.4%
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -3.4 \cdot 10^{-5}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 5.5 \cdot 10^{-48}:\\ \;\;\;\;y \cdot \frac{z - t}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 9
Accuracy	54.8%
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -9.5 \cdot 10^{-147}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-259}:\\ \;\;\;\;y \cdot \frac{z}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 10
Accuracy	95.9%
Cost	576

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

Alternative 11
Accuracy	51.8%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023137 
(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)))