Numeric.Signal.Multichannel:$cget from hsignal-0.2.7.1

Average Error: 2.0 → 1.7

Time: 11.5s

Precision: binary64

Cost: 6980

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+42}:\\ \;\;\;\;\left(t + x \cdot \frac{z}{y}\right) - x \cdot \frac{t}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, z - t, t\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (x y z t)
 :precision binary64
 (if (<= x -1e+42)
   (- (+ t (* x (/ z y))) (* x (/ t y)))
   (fma (/ x y) (- z t) t)))

C

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

↓

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -1e+42) {
		tmp = (t + (x * (z / y))) - (x * (t / y));
	} else {
		tmp = fma((x / y), (z - t), t);
	}
	return tmp;
}

Julia

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

↓

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -1e+42)
		tmp = Float64(Float64(t + Float64(x * Float64(z / y))) - Float64(x * Float64(t / y)));
	else
		tmp = fma(Float64(x / y), Float64(z - t), t);
	end
	return tmp
end

Wolfram

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

↓

code[x_, y_, z_, t_] := If[LessEqual[x, -1e+42], N[(N[(t + N[(x * N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x * N[(t / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / y), $MachinePrecision] * N[(z - t), $MachinePrecision] + t), $MachinePrecision]]

Target

Original	2.0
Target	2.2
Herbie	1.7

\[\begin{array}{l} \mathbf{if}\;z < 2.759456554562692 \cdot 10^{-282}:\\ \;\;\;\;\frac{x}{y} \cdot \left(z - t\right) + t\\ \mathbf{elif}\;z < 2.326994450874436 \cdot 10^{-110}:\\ \;\;\;\;x \cdot \frac{z - t}{y} + t\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} \cdot \left(z - t\right) + t\\ \end{array} \]

Derivation

1. Split input into 2 regimes

2. if x < -1.00000000000000004e42

   1. Initial program 4.1

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

   2. Applied egg-rr 3.5

      \[\leadsto \color{blue}{\frac{1}{\frac{\frac{y}{x}}{z - t}}} + t \]

   3. Applied egg-rr 2.0

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

   if -1.00000000000000004e42 < x

   1. Initial program 1.6

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

   2. Simplified 1.6

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

      [Start] 1.6	\[ \frac{x}{y} \cdot \left(z - t\right) + t \]
      fma-def [=>] 1.6	\[ \color{blue}{\mathsf{fma}\left(\frac{x}{y}, z - t, t\right)} \]

3. Recombined 2 regimes into one program.

4. Final simplification 1.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{+42}:\\ \;\;\;\;\left(t + x \cdot \frac{z}{y}\right) - x \cdot \frac{t}{y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, z - t, t\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	22.9
Cost	1684

\[\begin{array}{l} t_1 := \frac{z}{\frac{y}{x}}\\ \mathbf{if}\;\frac{x}{y} \leq -1 \cdot 10^{+158}:\\ \;\;\;\;\frac{x \cdot z}{y}\\ \mathbf{elif}\;\frac{x}{y} \leq -50000000000000:\\ \;\;\;\;\frac{x}{y} \cdot \left(-t\right)\\ \mathbf{elif}\;\frac{x}{y} \leq -2 \cdot 10^{-107}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{x}{y} \leq 7 \cdot 10^{-46}:\\ \;\;\;\;t\\ \mathbf{elif}\;\frac{x}{y} \leq 5 \cdot 10^{+185}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{-t}{y}\\ \end{array} \]

Alternative 2
Error	22.9
Cost	1684

\[\begin{array}{l} t_1 := \frac{z}{\frac{y}{x}}\\ \mathbf{if}\;\frac{x}{y} \leq -1 \cdot 10^{+158}:\\ \;\;\;\;\frac{x \cdot z}{y}\\ \mathbf{elif}\;\frac{x}{y} \leq -50000000000000:\\ \;\;\;\;\frac{x}{y} \cdot \left(-t\right)\\ \mathbf{elif}\;\frac{x}{y} \leq -2 \cdot 10^{-107}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\frac{x}{y} \leq 7 \cdot 10^{-46}:\\ \;\;\;\;t\\ \mathbf{elif}\;\frac{x}{y} \leq 5 \cdot 10^{+185}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(-t\right)}{y}\\ \end{array} \]

Alternative 3
Error	15.8
Cost	1490

\[\begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -0.1 \lor \neg \left(\frac{x}{y} \leq -2 \cdot 10^{-67} \lor \neg \left(\frac{x}{y} \leq -5 \cdot 10^{-130}\right) \land \frac{x}{y} \leq 5 \cdot 10^{-14}\right):\\ \;\;\;\;x \cdot \frac{z - t}{y}\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \]

Alternative 4
Error	5.0
Cost	1488

\[\begin{array}{l} t_1 := \frac{z}{\frac{y}{x}}\\ t_2 := x \cdot \frac{z - t}{y}\\ \mathbf{if}\;\frac{x}{y} \leq -50000000000000:\\ \;\;\;\;t_2\\ \mathbf{elif}\;\frac{x}{y} \leq 0.01:\\ \;\;\;\;t + t_1\\ \mathbf{elif}\;\frac{x}{y} \leq 10^{+32}:\\ \;\;\;\;t - t \cdot \frac{x}{y}\\ \mathbf{elif}\;\frac{x}{y} \leq 10^{+44}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]

Alternative 5
Error	4.9
Cost	1488

\[\begin{array}{l} t_1 := \frac{z}{\frac{y}{x}}\\ \mathbf{if}\;\frac{x}{y} \leq -50000000000000:\\ \;\;\;\;x \cdot \frac{z - t}{y}\\ \mathbf{elif}\;\frac{x}{y} \leq 0.01:\\ \;\;\;\;t + t_1\\ \mathbf{elif}\;\frac{x}{y} \leq 10^{+32}:\\ \;\;\;\;t - t \cdot \frac{x}{y}\\ \mathbf{elif}\;\frac{x}{y} \leq 10^{+44}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{y}{z - t}}\\ \end{array} \]

Alternative 6
Error	23.6
Cost	1362

\[\begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -0.1 \lor \neg \left(\frac{x}{y} \leq -2 \cdot 10^{-67}\right) \land \left(\frac{x}{y} \leq -2 \cdot 10^{-107} \lor \neg \left(\frac{x}{y} \leq 7 \cdot 10^{-46}\right)\right):\\ \;\;\;\;z \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \]

Alternative 7
Error	23.5
Cost	1362

\[\begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -0.1 \lor \neg \left(\frac{x}{y} \leq -2 \cdot 10^{-67}\right) \land \left(\frac{x}{y} \leq -2 \cdot 10^{-107} \lor \neg \left(\frac{x}{y} \leq 7 \cdot 10^{-46}\right)\right):\\ \;\;\;\;\frac{z}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \]

Alternative 8
Error	23.3
Cost	1361

\[\begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -1 \cdot 10^{+158}:\\ \;\;\;\;\frac{x \cdot z}{y}\\ \mathbf{elif}\;\frac{x}{y} \leq -50000000000000:\\ \;\;\;\;\frac{-t}{\frac{y}{x}}\\ \mathbf{elif}\;\frac{x}{y} \leq -2 \cdot 10^{-107} \lor \neg \left(\frac{x}{y} \leq 7 \cdot 10^{-46}\right):\\ \;\;\;\;\frac{z}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \]

Alternative 9
Error	23.3
Cost	1361

\[\begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -1 \cdot 10^{+158}:\\ \;\;\;\;\frac{x \cdot z}{y}\\ \mathbf{elif}\;\frac{x}{y} \leq -50000000000000:\\ \;\;\;\;\frac{x}{y} \cdot \left(-t\right)\\ \mathbf{elif}\;\frac{x}{y} \leq -2 \cdot 10^{-107} \lor \neg \left(\frac{x}{y} \leq 7 \cdot 10^{-46}\right):\\ \;\;\;\;\frac{z}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \]

Alternative 10
Error	2.5
Cost	1229

\[\begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -\infty:\\ \;\;\;\;x \cdot \frac{z - t}{y}\\ \mathbf{elif}\;\frac{x}{y} \leq -50000000000000 \lor \neg \left(\frac{x}{y} \leq 0.5\right):\\ \;\;\;\;\frac{z - t}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;t + \frac{z}{\frac{y}{x}}\\ \end{array} \]

Alternative 11
Error	2.5
Cost	1229

\[\begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -\infty:\\ \;\;\;\;x \cdot \left(\left(z - t\right) \cdot \frac{1}{y}\right)\\ \mathbf{elif}\;\frac{x}{y} \leq -50000000000000 \lor \neg \left(\frac{x}{y} \leq 0.5\right):\\ \;\;\;\;\frac{z - t}{\frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;t + \frac{z}{\frac{y}{x}}\\ \end{array} \]

Alternative 12
Error	1.4
Cost	1220

\[\begin{array}{l} t_1 := t + \frac{x}{y} \cdot \left(z - t\right)\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;\frac{x}{\frac{y}{z - t}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 13
Error	4.5
Cost	969

\[\begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -50000000000000 \lor \neg \left(\frac{x}{y} \leq 5\right):\\ \;\;\;\;x \cdot \frac{z - t}{y}\\ \mathbf{else}:\\ \;\;\;\;t + \frac{z}{\frac{y}{x}}\\ \end{array} \]

Alternative 14
Error	31.9
Cost	64

\[t \]

Reproduce

herbie shell --seed 2023016 
(FPCore (x y z t)
  :name "Numeric.Signal.Multichannel:$cget from hsignal-0.2.7.1"
  :precision binary64

  :herbie-target
  (if (< z 2.759456554562692e-282) (+ (* (/ x y) (- z t)) t) (if (< z 2.326994450874436e-110) (+ (* x (/ (- z t) y)) t) (+ (* (/ x y) (- z t)) t)))

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