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

Average Error: 6.3 → 0.9

Time: 6.4s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{if}\;t_1 \leq -2.960246524217243 \cdot 10^{+182}:\\ \;\;\;\;\mathsf{fma}\left(t - z, \frac{y}{a}, x\right)\\ \mathbf{elif}\;t_1 \leq 1.3593815966461325 \cdot 10^{+298}:\\ \;\;\;\;x + \frac{y \cdot \left(t - z\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t - z}{a}, x\right)\\ \end{array} \]

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)) a)))
   (if (<= t_1 -2.960246524217243e+182)
     (fma (- t z) (/ y a) x)
     (if (<= t_1 1.3593815966461325e+298)
       (+ x (/ (* y (- t z)) a))
       (fma y (/ (- t z) a) x)))))

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)) / a;
	double tmp;
	if (t_1 <= -2.960246524217243e+182) {
		tmp = fma((t - z), (y / a), x);
	} else if (t_1 <= 1.3593815966461325e+298) {
		tmp = x + ((y * (t - z)) / a);
	} else {
		tmp = fma(y, ((t - z) / a), x);
	}
	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(Float64(y * Float64(z - t)) / a)
	tmp = 0.0
	if (t_1 <= -2.960246524217243e+182)
		tmp = fma(Float64(t - z), Float64(y / a), x);
	elseif (t_1 <= 1.3593815966461325e+298)
		tmp = Float64(x + Float64(Float64(y * Float64(t - z)) / a));
	else
		tmp = fma(y, Float64(Float64(t - z) / a), x);
	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 * N[(z - t), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]}, If[LessEqual[t$95$1, -2.960246524217243e+182], N[(N[(t - z), $MachinePrecision] * N[(y / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 1.3593815966461325e+298], N[(x + N[(N[(y * N[(t - z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(y * N[(N[(t - z), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision]]]]

Error

Bits error versus x

Bits error versus y

Bits error versus z

Bits error versus t

Bits error versus a

Target

Original	6.3
Target	0.7
Herbie	0.9

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

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 y (-.f64 z t)) a) < -2.96024652421724321e182

   1. Initial program 23.7

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

   2. Simplified 13.6

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

   3. Taylor expanded in y around 0 23.7

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

   4. Simplified 4.3

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

   if -2.96024652421724321e182 < (/.f64 (*.f64 y (-.f64 z t)) a) < 1.35938159664613252e298

   1. Initial program 0.3

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

   if 1.35938159664613252e298 < (/.f64 (*.f64 y (-.f64 z t)) a)

   1. Initial program 57.4

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

   2. Simplified 3.1

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

   3. Applied egg-rr 3.2

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{{\left(\frac{a}{t - z}\right)}^{-1}}, x\right) \]

   4. Applied egg-rr 3.1

      \[\leadsto \mathsf{fma}\left(y, \color{blue}{1 \cdot \frac{t - z}{a}}, x\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 0.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y \cdot \left(z - t\right)}{a} \leq -2.960246524217243 \cdot 10^{+182}:\\ \;\;\;\;\mathsf{fma}\left(t - z, \frac{y}{a}, x\right)\\ \mathbf{elif}\;\frac{y \cdot \left(z - t\right)}{a} \leq 1.3593815966461325 \cdot 10^{+298}:\\ \;\;\;\;x + \frac{y \cdot \left(t - z\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t - z}{a}, x\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022148 
(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)))