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

Average Error: 6.2 → 0.7

Time: 6.5s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := x + \frac{y \cdot \left(z - x\right)}{t}\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(x, 1, \left(z - x\right) \cdot \frac{y}{t}\right)\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+307}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - x}{t}, x\right)\\ \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (+ x (/ (* y (- z x)) t)))

↓

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ x (/ (* y (- z x)) t))))
   (if (<= t_1 (- INFINITY))
     (fma x 1.0 (* (- z x) (/ y t)))
     (if (<= t_1 2e+307) t_1 (fma y (/ (- z x) t) x)))))

C

double code(double x, double y, double z, double t) {
	return x + ((y * (z - x)) / t);
}

↓

double code(double x, double y, double z, double t) {
	double t_1 = x + ((y * (z - x)) / t);
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = fma(x, 1.0, ((z - x) * (y / t)));
	} else if (t_1 <= 2e+307) {
		tmp = t_1;
	} else {
		tmp = fma(y, ((z - x) / t), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	return Float64(x + Float64(Float64(y * Float64(z - x)) / t))
end

↓

function code(x, y, z, t)
	t_1 = Float64(x + Float64(Float64(y * Float64(z - x)) / t))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = fma(x, 1.0, Float64(Float64(z - x) * Float64(y / t)));
	elseif (t_1 <= 2e+307)
		tmp = t_1;
	else
		tmp = fma(y, Float64(Float64(z - x) / t), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := N[(x + N[(N[(y * N[(z - x), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]

↓

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x + N[(N[(y * N[(z - x), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(x * 1.0 + N[(N[(z - x), $MachinePrecision] * N[(y / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+307], t$95$1, N[(y * N[(N[(z - x), $MachinePrecision] / t), $MachinePrecision] + x), $MachinePrecision]]]]

Target

Original	6.2
Target	1.9
Herbie	0.7

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

Derivation

1. Split input into 3 regimes

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

   1. Initial program 64.0

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

   2. Simplified 0.2

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

   3. Applied egg-rr 1.3

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\mathsf{fma}\left(z - x, \frac{y}{t}, x\right)}\right)}^{3}} \]

   4. Applied egg-rr 2.2

      \[\leadsto {\color{blue}{\left({\left(\sqrt[3]{\sqrt[3]{\mathsf{fma}\left(z - x, \frac{y}{t}, x\right)}}\right)}^{3}\right)}}^{3} \]

   5. Applied egg-rr 0.2

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

   if -inf.0 < (+.f64 x (/.f64 (*.f64 y (-.f64 z x)) t)) < 1.99999999999999997e307

   1. Initial program 0.7

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

   if 1.99999999999999997e307 < (+.f64 x (/.f64 (*.f64 y (-.f64 z x)) t))

   1. Initial program 63.1

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

   2. Simplified 0.4

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

   3. Taylor expanded in z around 0 63.1

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

   4. Simplified 0.4

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

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{y \cdot \left(z - x\right)}{t} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(x, 1, \left(z - x\right) \cdot \frac{y}{t}\right)\\ \mathbf{elif}\;x + \frac{y \cdot \left(z - x\right)}{t} \leq 2 \cdot 10^{+307}:\\ \;\;\;\;x + \frac{y \cdot \left(z - x\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - x}{t}, x\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022190 
(FPCore (x y z t)
  :name "Optimisation.CirclePacking:place from circle-packing-0.1.0.4, D"
  :precision binary64

  :herbie-target
  (- x (+ (* x (/ y t)) (* (- z) (/ y t))))

  (+ x (/ (* y (- z x)) t)))