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

Average Error: 5.9 → 1.8

Time: 4.3s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := \mathsf{fma}\left(\frac{y}{a}, t - z, x\right)\\ \mathbf{if}\;z - t \leq -1.8034715821452683 \cdot 10^{-59}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z - t \leq 4.204200830287601 \cdot 10^{-132}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t - z}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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 (fma (/ y a) (- t z) x)))
   (if (<= (- z t) -1.8034715821452683e-59)
     t_1
     (if (<= (- z t) 4.204200830287601e-132) (fma y (/ (- t z) a) x) t_1))))

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 = fma((y / a), (t - z), x);
	double tmp;
	if ((z - t) <= -1.8034715821452683e-59) {
		tmp = t_1;
	} else if ((z - t) <= 4.204200830287601e-132) {
		tmp = fma(y, ((t - z) / a), x);
	} else {
		tmp = t_1;
	}
	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 = fma(Float64(y / a), Float64(t - z), x)
	tmp = 0.0
	if (Float64(z - t) <= -1.8034715821452683e-59)
		tmp = t_1;
	elseif (Float64(z - t) <= 4.204200830287601e-132)
		tmp = fma(y, Float64(Float64(t - z) / a), x);
	else
		tmp = t_1;
	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[(N[(y / a), $MachinePrecision] * N[(t - z), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[N[(z - t), $MachinePrecision], -1.8034715821452683e-59], t$95$1, If[LessEqual[N[(z - t), $MachinePrecision], 4.204200830287601e-132], N[(y * N[(N[(t - z), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	5.9
Target	0.6
Herbie	1.8

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

Derivation

1. Split input into 2 regimes

2. if (-.f64 z t) < -1.8034715821452683e-59 or 4.204200830287601e-132 < (-.f64 z t)

   1. Initial program 6.9

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

   2. Simplified 6.9

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

   3. Taylor expanded in y around 0 6.9

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

   4. Simplified 1.9

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

   if -1.8034715821452683e-59 < (-.f64 z t) < 4.204200830287601e-132

   1. Initial program 0.9

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

   2. Simplified 1.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(y, \frac{t - z}{a}, x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 1.8

   \[\leadsto \begin{array}{l} \mathbf{if}\;z - t \leq -1.8034715821452683 \cdot 10^{-59}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - z, x\right)\\ \mathbf{elif}\;z - t \leq 4.204200830287601 \cdot 10^{-132}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t - z}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - z, x\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022131 
(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)))