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

Average Error: 6.2 → 0.5

Time: 8.4s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := \frac{-z}{a}\\ t_2 := y \cdot \left(z - t\right)\\ \mathbf{if}\;t_2 \leq -1 \cdot 10^{+159}:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(1, \frac{t}{a}, t_1\right), x\right)\\ \mathbf{elif}\;t_2 \leq 5 \cdot 10^{+289}:\\ \;\;\;\;x + \frac{y \cdot \left(t - z\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(t, {a}^{-1}, t_1\right), x\right)\\ \end{array} \]

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 (/ (- z) a)) (t_2 (* y (- z t))))
   (if (<= t_2 -1e+159)
     (fma y (fma 1.0 (/ t a) t_1) x)
     (if (<= t_2 5e+289)
       (+ x (/ (* y (- t z)) a))
       (fma y (fma t (pow a -1.0) t_1) 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 = -z / a;
	double t_2 = y * (z - t);
	double tmp;
	if (t_2 <= -1e+159) {
		tmp = fma(y, fma(1.0, (t / a), t_1), x);
	} else if (t_2 <= 5e+289) {
		tmp = x + ((y * (t - z)) / a);
	} else {
		tmp = fma(y, fma(t, pow(a, -1.0), t_1), 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(-z) / a)
	t_2 = Float64(y * Float64(z - t))
	tmp = 0.0
	if (t_2 <= -1e+159)
		tmp = fma(y, fma(1.0, Float64(t / a), t_1), x);
	elseif (t_2 <= 5e+289)
		tmp = Float64(x + Float64(Float64(y * Float64(t - z)) / a));
	else
		tmp = fma(y, fma(t, (a ^ -1.0), t_1), 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[((-z) / a), $MachinePrecision]}, Block[{t$95$2 = N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+159], N[(y * N[(1.0 * N[(t / a), $MachinePrecision] + t$95$1), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$2, 5e+289], N[(x + N[(N[(y * N[(t - z), $MachinePrecision]), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], N[(y * N[(t * N[Power[a, -1.0], $MachinePrecision] + t$95$1), $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	6.2
Target	0.7
Herbie	0.5

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

Derivation

1. Split input into 3 regimes

2. if (*.f64 y (-.f64 z t)) < -9.9999999999999993e158

   1. Initial program 22.2

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

   2. Simplified 1.5

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

   3. Applied egg-rr 1.5

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

   if -9.9999999999999993e158 < (*.f64 y (-.f64 z t)) < 5.00000000000000031e289

   1. Initial program 0.4

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

   if 5.00000000000000031e289 < (*.f64 y (-.f64 z t))

   1. Initial program 56.2

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

   2. Simplified 0.2

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

   3. Applied egg-rr 0.2

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

4. Final simplification 0.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \leq -1 \cdot 10^{+159}:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(1, \frac{t}{a}, \frac{-z}{a}\right), x\right)\\ \mathbf{elif}\;y \cdot \left(z - t\right) \leq 5 \cdot 10^{+289}:\\ \;\;\;\;x + \frac{y \cdot \left(t - z\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \mathsf{fma}\left(t, {a}^{-1}, \frac{-z}{a}\right), x\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022166 
(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)))