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

Average Error: 6.5 → 0.6

Time: 5.4s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := y \cdot \left(z - t\right)\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;x + {\left(\frac{\frac{a}{z - t}}{y}\right)}^{-1}\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+179}:\\ \;\;\;\;x + \frac{t_1}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{a}, x\right)\\ \end{array} \]

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))))
   (if (<= t_1 (- INFINITY))
     (+ x (pow (/ (/ a (- z t)) y) -1.0))
     (if (<= t_1 2e+179) (+ x (/ t_1 a)) (fma y (/ (- z t) 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);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = x + pow(((a / (z - t)) / y), -1.0);
	} else if (t_1 <= 2e+179) {
		tmp = x + (t_1 / a);
	} else {
		tmp = fma(y, ((z - t) / 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(y * Float64(z - t))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(x + (Float64(Float64(a / Float64(z - t)) / y) ^ -1.0));
	elseif (t_1 <= 2e+179)
		tmp = Float64(x + Float64(t_1 / a));
	else
		tmp = fma(y, Float64(Float64(z - t) / 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[(y * N[(z - t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(x + N[Power[N[(N[(a / N[(z - t), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], -1.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+179], N[(x + N[(t$95$1 / a), $MachinePrecision]), $MachinePrecision], N[(y * N[(N[(z - t), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision]]]]

Target

Original	6.5
Target	0.7
Herbie	0.6

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 64.0

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

   2. Applied egg-rr 0.3

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

   if -inf.0 < (*.f64 y (-.f64 z t)) < 1.99999999999999996e179

   1. Initial program 0.4

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

   if 1.99999999999999996e179 < (*.f64 y (-.f64 z t))

   1. Initial program 24.7

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

   2. Simplified 1.6

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

4. Final simplification 0.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \left(z - t\right) \leq -\infty:\\ \;\;\;\;x + {\left(\frac{\frac{a}{z - t}}{y}\right)}^{-1}\\ \mathbf{elif}\;y \cdot \left(z - t\right) \leq 2 \cdot 10^{+179}:\\ \;\;\;\;x + \frac{y \cdot \left(z - t\right)}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{a}, x\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022190 
(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)))