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

Average Error: 6.4 → 1.4

Time: 4.3s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := x + \frac{y \cdot \left(z - x\right)}{t}\\ \mathbf{if}\;t_1 \leq 10^{-241}:\\ \;\;\;\;\mathsf{fma}\left(z - x, \frac{y}{t}, x\right)\\ \mathbf{elif}\;t_1 \leq 10^{+299}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - x}{t}, x\right)\\ \end{array} \]

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 1e-241)
     (fma (- z x) (/ y t) x)
     (if (<= t_1 1e+299) 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 <= 1e-241) {
		tmp = fma((z - x), (y / t), x);
	} else if (t_1 <= 1e+299) {
		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 <= 1e-241)
		tmp = fma(Float64(z - x), Float64(y / t), x);
	elseif (t_1 <= 1e+299)
		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, 1e-241], N[(N[(z - x), $MachinePrecision] * N[(y / t), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 1e+299], t$95$1, N[(y * N[(N[(z - x), $MachinePrecision] / t), $MachinePrecision] + x), $MachinePrecision]]]]

Target

Original	6.4
Target	2.0
Herbie	1.4

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

Derivation

1. Split input into 3 regimes

2. if (+.f64 x (/.f64 (*.f64 y (-.f64 z x)) t)) < 9.9999999999999997e-242

   1. Initial program 6.4

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

   2. Simplified 2.2

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

   if 9.9999999999999997e-242 < (+.f64 x (/.f64 (*.f64 y (-.f64 z x)) t)) < 1.0000000000000001e299

   1. Initial program 0.4

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

   if 1.0000000000000001e299 < (+.f64 x (/.f64 (*.f64 y (-.f64 z x)) t))

   1. Initial program 56.2

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

   2. Simplified 1.1

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

   3. Taylor expanded in z around 0 56.2

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

   4. Simplified 2.3

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

4. Final simplification 1.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \frac{y \cdot \left(z - x\right)}{t} \leq 10^{-241}:\\ \;\;\;\;\mathsf{fma}\left(z - x, \frac{y}{t}, x\right)\\ \mathbf{elif}\;x + \frac{y \cdot \left(z - x\right)}{t} \leq 10^{+299}:\\ \;\;\;\;x + \frac{y \cdot \left(z - x\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - x}{t}, x\right)\\ \end{array} \]

Reproduce

herbie shell --seed 2022200 
(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)))