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

Average Accuracy: 90.9% → 99.3%

Time: 17.3s

Precision: binary64

Cost: 7236

Math

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

↓

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

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 (- INFINITY))
     (fma y (/ (- t z) a) x)
     (if (<= t_1 1e+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 <= -((double) INFINITY)) {
		tmp = fma(y, ((t - z) / a), x);
	} else if (t_1 <= 1e+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 <= Float64(-Inf))
		tmp = fma(y, Float64(Float64(t - z) / a), x);
	elseif (t_1 <= 1e+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, (-Infinity)], N[(y * N[(N[(t - z), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 1e+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	90.9%
Target	98.9%
Herbie	99.3%

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

Derivation

1. Split input into 3 regimes

2. if (*.f64 y (-.f64 z t)) < -inf.0

   1. Initial program 0.0%

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

   2. Simplified 99.6%

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

      [Start] 0.0	\[ x - \frac{y \cdot \left(z - t\right)}{a} \]
      sub-neg [=>] 0.0	\[ \color{blue}{x + \left(-\frac{y \cdot \left(z - t\right)}{a}\right)} \]
      +-commutative [=>] 0.0	\[ \color{blue}{\left(-\frac{y \cdot \left(z - t\right)}{a}\right) + x} \]
      *-commutative [=>] 0.0	\[ \left(-\frac{\color{blue}{\left(z - t\right) \cdot y}}{a}\right) + x \]
      associate-/l* [=>] 99.7	\[ \left(-\color{blue}{\frac{z - t}{\frac{a}{y}}}\right) + x \]
      distribute-neg-frac [=>] 99.7	\[ \color{blue}{\frac{-\left(z - t\right)}{\frac{a}{y}}} + x \]
      associate-/r/ [=>] 99.6	\[ \color{blue}{\frac{-\left(z - t\right)}{a} \cdot y} + x \]
      *-commutative [=>] 99.6	\[ \color{blue}{y \cdot \frac{-\left(z - t\right)}{a}} + x \]
      fma-def [=>] 99.6	\[ \color{blue}{\mathsf{fma}\left(y, \frac{-\left(z - t\right)}{a}, x\right)} \]
      sub-neg [=>] 99.6	\[ \mathsf{fma}\left(y, \frac{-\color{blue}{\left(z + \left(-t\right)\right)}}{a}, x\right) \]
      distribute-neg-in [=>] 99.6	\[ \mathsf{fma}\left(y, \frac{\color{blue}{\left(-z\right) + \left(-\left(-t\right)\right)}}{a}, x\right) \]
      +-commutative [=>] 99.6	\[ \mathsf{fma}\left(y, \frac{\color{blue}{\left(-\left(-t\right)\right) + \left(-z\right)}}{a}, x\right) \]
      remove-double-neg [=>] 99.6	\[ \mathsf{fma}\left(y, \frac{\color{blue}{t} + \left(-z\right)}{a}, x\right) \]
      sub-neg [<=] 99.6	\[ \mathsf{fma}\left(y, \frac{\color{blue}{t - z}}{a}, x\right) \]

   if -inf.0 < (*.f64 y (-.f64 z t)) < 1.00000000000000002e198

   1. Initial program 99.3%

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

   if 1.00000000000000002e198 < (*.f64 y (-.f64 z t))

   1. Initial program 55.5%

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

   2. Simplified 98.9%

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

      [Start] 55.5	\[ x - \frac{y \cdot \left(z - t\right)}{a} \]
      associate-*l/ [<=] 98.9	\[ x - \color{blue}{\frac{y}{a} \cdot \left(z - t\right)} \]

3. Recombined 3 regimes into one program.

4. Final simplification 99.3%

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

Alternatives

Alternative 1
Accuracy	99.2%
Cost	1736

\[\begin{array}{l} t_1 := \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{if}\;t_1 \leq -2 \cdot 10^{+286}:\\ \;\;\;\;x - \left(z - t\right) \cdot \frac{y}{a}\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+306}:\\ \;\;\;\;x - t_1\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\frac{t - z}{a}}{\frac{1}{y}}\\ \end{array} \]

Alternative 2
Accuracy	99.3%
Cost	1353

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

Alternative 3
Accuracy	54.9%
Cost	1308

\[\begin{array}{l} \mathbf{if}\;x \leq -1.3:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -3.5 \cdot 10^{-45}:\\ \;\;\;\;t \cdot \frac{y}{a}\\ \mathbf{elif}\;x \leq -3.1 \cdot 10^{-102}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{-258}:\\ \;\;\;\;\frac{y \cdot t}{a}\\ \mathbf{elif}\;x \leq 1.6 \cdot 10^{-219}:\\ \;\;\;\;\frac{y \cdot \left(-z\right)}{a}\\ \mathbf{elif}\;x \leq 7 \cdot 10^{-156}:\\ \;\;\;\;\frac{y}{\frac{a}{t}}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-105}:\\ \;\;\;\;\frac{-y}{\frac{a}{z}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 4
Accuracy	83.3%
Cost	1108

\[\begin{array}{l} t_1 := x - z \cdot \frac{y}{a}\\ t_2 := x + \frac{t}{\frac{a}{y}}\\ \mathbf{if}\;t \leq -4 \cdot 10^{-16}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 4.8 \cdot 10^{-87}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 2.1 \cdot 10^{-24}:\\ \;\;\;\;x + y \cdot \frac{t}{a}\\ \mathbf{elif}\;t \leq 6.5 \cdot 10^{+94}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 2.1 \cdot 10^{+114}:\\ \;\;\;\;\frac{t - z}{\frac{a}{y}}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 5
Accuracy	55.1%
Cost	1044

\[\begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -5.3 \cdot 10^{-47}:\\ \;\;\;\;t \cdot \frac{y}{a}\\ \mathbf{elif}\;x \leq -1.7 \cdot 10^{-95}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 9.2 \cdot 10^{-156}:\\ \;\;\;\;\frac{t}{\frac{a}{y}}\\ \mathbf{elif}\;x \leq 5.4 \cdot 10^{-104}:\\ \;\;\;\;y \cdot \frac{-z}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 6
Accuracy	55.2%
Cost	1044

\[\begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -4.6 \cdot 10^{-46}:\\ \;\;\;\;t \cdot \frac{y}{a}\\ \mathbf{elif}\;x \leq -1.55 \cdot 10^{-95}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 9 \cdot 10^{-156}:\\ \;\;\;\;\frac{t}{\frac{a}{y}}\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{-104}:\\ \;\;\;\;\frac{y}{a} \cdot \left(-z\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 7
Accuracy	55.1%
Cost	1044

\[\begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -3 \cdot 10^{-45}:\\ \;\;\;\;t \cdot \frac{y}{a}\\ \mathbf{elif}\;x \leq -1.5 \cdot 10^{-95}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.02 \cdot 10^{-159}:\\ \;\;\;\;\frac{t}{\frac{a}{y}}\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{-102}:\\ \;\;\;\;\frac{-y}{\frac{a}{z}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 8
Accuracy	74.6%
Cost	978

\[\begin{array}{l} \mathbf{if}\;x \leq -1.22 \cdot 10^{-74} \lor \neg \left(x \leq 1.25 \cdot 10^{-228}\right) \land \left(x \leq 2.1 \cdot 10^{-132} \lor \neg \left(x \leq 1.5 \cdot 10^{-98}\right)\right):\\ \;\;\;\;x + y \cdot \frac{t}{a}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{t - z}{a}\\ \end{array} \]

Alternative 9
Accuracy	75.8%
Cost	977

\[\begin{array}{l} \mathbf{if}\;x \leq -1.65 \cdot 10^{-72}:\\ \;\;\;\;x + y \cdot \frac{t}{a}\\ \mathbf{elif}\;x \leq 6.4 \cdot 10^{-228} \lor \neg \left(x \leq 2.7 \cdot 10^{-136}\right) \land x \leq 1.6 \cdot 10^{-105}:\\ \;\;\;\;y \cdot \frac{t - z}{a}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{t}{\frac{a}{y}}\\ \end{array} \]

Alternative 10
Accuracy	55.4%
Cost	849

\[\begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -7.4 \cdot 10^{-47} \lor \neg \left(x \leq -1.8 \cdot 10^{-95}\right) \land x \leq 2.7 \cdot 10^{-99}:\\ \;\;\;\;t \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 11
Accuracy	55.4%
Cost	848

\[\begin{array}{l} \mathbf{if}\;x \leq -1.25:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq -4 \cdot 10^{-45}:\\ \;\;\;\;t \cdot \frac{y}{a}\\ \mathbf{elif}\;x \leq -1.65 \cdot 10^{-95}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 9.5 \cdot 10^{-101}:\\ \;\;\;\;\frac{t}{\frac{a}{y}}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 12
Accuracy	84.2%
Cost	713

\[\begin{array}{l} \mathbf{if}\;t \leq -3.2 \cdot 10^{-14} \lor \neg \left(t \leq 4.8 \cdot 10^{-87}\right):\\ \;\;\;\;x + \frac{t}{\frac{a}{y}}\\ \mathbf{else}:\\ \;\;\;\;x - z \cdot \frac{y}{a}\\ \end{array} \]

Alternative 13
Accuracy	69.4%
Cost	712

\[\begin{array}{l} \mathbf{if}\;x \leq -20:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{-38}:\\ \;\;\;\;y \cdot \frac{t - z}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 14
Accuracy	55.5%
Cost	584

\[\begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{-95}:\\ \;\;\;\;x\\ \mathbf{elif}\;x \leq 1.14 \cdot 10^{-104}:\\ \;\;\;\;y \cdot \frac{t}{a}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 15
Accuracy	95.8%
Cost	576

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

Alternative 16
Accuracy	50.8%
Cost	64

\[x \]

Reproduce

herbie shell --seed 2023133 
(FPCore (x y z t a)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, F"
  :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)))