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

Average Error: 6.3 → 0.4

Time: 4.4s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := y \cdot \left(z - t\right)\\ \mathbf{if}\;t_1 \leq -2.404461501901897 \cdot 10^{+254}:\\ \;\;\;\;\mathsf{fma}\left(z - t, y \cdot \frac{1}{a}, x\right)\\ \mathbf{elif}\;t_1 \leq 1.8220340511756164 \cdot 10^{+236}:\\ \;\;\;\;x + \frac{t_1}{a}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a}\\ \end{array} \]

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 -2.404461501901897e+254)
     (fma (- z t) (* y (/ 1.0 a)) x)
     (if (<= t_1 1.8220340511756164e+236)
       (+ x (/ t_1 a))
       (+ x (* y (/ (- z t) 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 <= -2.404461501901897e+254) {
		tmp = fma((z - t), (y * (1.0 / a)), x);
	} else if (t_1 <= 1.8220340511756164e+236) {
		tmp = x + (t_1 / a);
	} else {
		tmp = x + (y * ((z - t) / 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 <= -2.404461501901897e+254)
		tmp = fma(Float64(z - t), Float64(y * Float64(1.0 / a)), x);
	elseif (t_1 <= 1.8220340511756164e+236)
		tmp = Float64(x + Float64(t_1 / a));
	else
		tmp = Float64(x + Float64(y * Float64(Float64(z - t) / 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, -2.404461501901897e+254], N[(N[(z - t), $MachinePrecision] * N[(y * N[(1.0 / a), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 1.8220340511756164e+236], N[(x + N[(t$95$1 / a), $MachinePrecision]), $MachinePrecision], N[(x + N[(y * N[(N[(z - t), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $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	0.4

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

Derivation

1. Split input into 3 regimes

2. if (*.f64 y (-.f64 z t)) < -2.40446150190189726e254

   1. Initial program 40.8

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

   2. Applied egg-rr 0.3

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

   if -2.40446150190189726e254 < (*.f64 y (-.f64 z t)) < 1.8220340511756164e236

   1. Initial program 0.3

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

   if 1.8220340511756164e236 < (*.f64 y (-.f64 z t))

   1. Initial program 37.0

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

   2. Applied egg-rr 0.7

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

   3. Applied egg-rr 0.6

      \[\leadsto \color{blue}{y \cdot \frac{z - t}{a} + x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 0.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \leq -2.404461501901897 \cdot 10^{+254}:\\ \;\;\;\;\mathsf{fma}\left(z - t, y \cdot \frac{1}{a}, x\right)\\ \mathbf{elif}\;y \cdot \left(z - t\right) \leq 1.8220340511756164 \cdot 10^{+236}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{z - t}{a}\\ \end{array} \]

Reproduce

herbie shell --seed 2022148 
(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)))